PA RIOK J. SINKO
Dr. Murtadha Alshareifi e-Library
i
MARTIN’S PHYSICAL PHARMACY
AND PHARMACEUTICAL SCIENCES
Physical Chemical and Biopharmaceutical Principles
in the Pharmaceutical Sciences
SIXTH EDITION
Editor
PATRICK J. SINKO, PhD, RPh
Professor II (Distinguished)
Parke-Davis Chair Professor in Pharmaceutics and Drug Delivery
Ernest Mario School of Pharmacy
Rutgers, The State University of New Jersey
Piscataway, New Jersey
Assistant Editor
YASHVEER SINGH, PhD
Assistant Research Professor
Department of Pharmaceutics
Ernest Mario School of Pharmacy
Rutgers, The State University of New Jersey
Piscataway, New Jersey
1
Dr. Murtadha Alshareifi e-Library
Editor: David B. Troy
Product Manager: Meredith L. Brittain
Vendor Manager: Kevin Johnson
Designer: Holly McLaughlin
Compositor: Aptara® , Inc.
Sixth Edition
c 2011, 2006 Lippincott Williams & Wilkins, a Wolters Kluwer business.
Copyright 351 West Camden Street
Baltimore, MD 21201
530 Walnut St.
Philadelphia, PA 19106
Printed in China
All rights reserved. This book is protected by copyright. No part of this book may be reproduced
or transmitted in any form or by any means, including photocopies or scanned-in or other electronic
copies, or utilized by any information storage and retrieval system without written permission from
the copyright owner, except for brief quotations embodied in critical articles and reviews. Materials
appearing in this book prepared by individuals as part of their official duties as U.S. government
employees are not covered by the above-mentioned copyright. To request permission, please contact
Lippincott Williams & Wilkins at 530 Walnut Street, Philadelphia, PA 19106, via e-mail at permissions@lww.com, or via Website at lww.com (products and services).
987654321
Library of Congress Cataloging-in-Publication Data
Martin’s physical pharmacy and pharmaceutical sciences : physical
chemical and biopharmaceutical principles in the pharmaceutical
sciences.—6th ed. / editor, Patrick J. Sinko ; assistant editor,
Yashveer Singh.
p. ; cm.
Includes bibliographical references and index.
ISBN 978-0-7817-9766-5
1. Pharmaceutical chemistry. 2. Chemistry, Physical and theoretical.
I. Martin, Alfred N. II. Sinko, Patrick J. III. Singh, Yashveer.
IV. Title: Physical pharmacy and pharmaceutical sciences.
[DNLM: 1. Chemistry, Pharmaceutical. 2. Chemistry, Physical. QV 744
M386 2011]
RS403.M34 2011
615 .19—dc22
2009046514
DISCLAIMER
Care has been taken to confirm the accuracy of the information present and to describe generally
accepted practices. However, the authors, editors, and publisher are not responsible for errors or
omissions or for any consequences from application of the information in this book and make no
warranty, expressed or implied, with respect to the currency, completeness, or accuracy of the contents
of the publication. Application of this information in a particular situation remains the professional
responsibility of the practitioner; the clinical treatments described and recommended may not be
considered absolute and universal recommendations.
The authors, editors, and publisher have exerted every effort to ensure that drug selection and
dosage set forth in this text are in accordance with the current recommendations and practice at the
time of publication. However, in view of ongoing research, changes in government regulations, and the
constant flow of information relating to drug therapy and drug reactions, the reader is urged to check
the package insert for each drug for any change in indications and dosage and for added warnings
and precautions. This is particularly important when the recommended agent is a new or infrequently
employed drug.
Some drugs and medical devices presented in this publication have Food and Drug Administration
(FDA) clearance for limited use in restricted research settings. It is the responsibility of the health
care providers to ascertain the FDA status of each drug or device planned for use in their clinical
practice.
To purchase additional copies of this book, call our customer service department at (800) 638-3030
or fax orders to (301) 223-2320. International customers should call (301) 223-2300.
Visit Lippincott Williams & Wilkins on the Internet: at http://www.lww.com. Lippincott Williams &
Wilkins customer service representatives are available from 8:30 am to 6:00 pm, EST.
2
Dr. Murtadha Alshareifi e-Library
Dedicated to my parents Patricia and Patrick Sinko,
my wife Renee, and my children Pat, Katie (and Maggie)
Dr.Murtadha Alshareifi
cn=Dr.Murtadha Alshareifi, o=College Of
Pharmacy/University Of Baghdad, ou=http://
facebook.com/ebooks.drmurtadha,
email=ebooks_alshareifi@yahoo.com, c=IQ
3
Dr. Murtadha Alshareifi e-Library
DEDICATION
ALFRED N. MARTIN (1919–2003)
This fiftieth anniversary edition of Martin’s Physical Pharmacy and Pharmaceutical Sciences is dedicated to the memory of Professor Alfred N. Martin, whose vision, creativity,
dedication, and untiring effort and attention to detail led to
the publication of the first edition in 1960. Because of his
national reputation as a leader and pioneer in the then emerging specialty of physical pharmacy, I made the decision to
join Professor Martin’s group of graduate students at Purdue University in 1960 and had the opportunity to witness
the excitement and the many accolades of colleagues from
far and near that accompanied the publication of the first
edition of Physical Pharmacy. The completion of that work
represented the culmination of countless hours of painstaking study, research, documentation, and revision on the part
of Dr. Martin, many of his graduate students, and his wife,
Mary, who typed the original manuscript. It also represented
the fruition of Professor Martin’s dream of a textbook that
would revolutionize pharmaceutical education and research.
Physical Pharmacy was for Professor Martin truly a labor of
love, and it remained so throughout his lifetime, as he worked
unceasingly and with steadfast dedication on the subsequent
revisions of the book.
The publication of the first edition of Physical Pharmacy
generated broad excitement throughout the national and international academic and industrial research communities in
pharmacy and the pharmaceutical sciences. It was the world’s
first textbook in the emerging discipline of physical pharmacy
and has remained the “gold standard” textbook on the application of physical chemical principles in pharmacy and the
pharmaceutical sciences. Physical Pharmacy, upon its publication in 1960, provided great clarity and definition to a discipline that had been widely discussed throughout the 1950s
but not fully understood or adopted. Alfred Martin’s Physical Pharmacy had a profound effect in shaping the direction
of research and education throughout the world of pharmaceutical education and research in the pharmaceutical industry and academia. The publication of this book transformed
pharmacy and pharmaceutical research from an essentially
empirical mix of art and descriptive science to a quantitative application of fundamental physical and chemical scientific principles to pharmaceutical systems and dosage forms.
Physical Pharmacy literally changed the direction, scope,
focus, and philosophy of pharmaceutical education during the
1960s and the 1970s and paved the way for the specialty disciplines of biopharmaceutics and pharmacokinetics which,
along with physical pharmacy, were necessary underpinnings
of a scientifically based clinical emphasis in the teaching of
pharmacy students, which is now pervasive throughout pharmaceutical education.
From the time of the initial publication of Physical Pharmacy to the present, this pivotal and classic book has been
widely used both as a teaching textbook and as an indispensible reference for academic and industrial researchers in
the pharmaceutical sciences throughout the world. This sixth
edition of Martin’s Physical Pharmacy and Pharmaceutical
Sciences serves as a most fitting tribute to the extraordinary,
heroic, and inspired vision and dedication of Professor Martin. That this book continues to be a valuable and widely
used textbook in schools and colleges of pharmacy throughout the world, and a valuable reference to pharmaceutical
scientists and researchers, is a most appropriate recognition
of the life’s work of Alfred Martin. All who have contributed
to the thorough revision that has resulted in the publication
of the current edition have retained the original format and
fundamental organization of basic principles and topics that
were the hallmarks of Professor Martin’s classic first edition
of this seminal book.
Professor Martin always demanded the best of himself, his
students, and his colleagues. The fact that the subsequent and
current editions of Martin’s Physical Pharmacy and Pharmaceutical Sciences have remained faithful to his vision of
scientific excellence as applied to understanding and applying the principles underlying the pharmaceutical sciences is
indeed a most appropriate tribute to Professor Martin’s memory. It is in that spirit that this fiftieth anniversary edition is
formally dedicated to the memory of that visionary and creative pioneer in the discipline of physical pharmacy, Alfred
N. Martin.
John L. Colaizzi, PhD
Rutgers, The State University of New Jersey
Piscataway, New Jersey
November 2009
iv
4
Dr. Murtadha Alshareifi e-Library
PREFACE
Ever since the First Edition of Martin’s Physical Pharmacy
was published in 1960, Dr. Alfred Martin’s vision was to provide a text that introduced pharmacy students to the application of physical chemical principles to the pharmaceutical sciences. This remains a primary objective of the Sixth Edition.
Martin’s Physical Pharmacy has been used by generations of
pharmacy and pharmaceutical science graduate students for
50 years and, while some topics change from time to time,
the basic principles remain constant, and it is my hope that
each edition reflects the pharmaceutical sciences at that point
in time.
moved to the Web (see the “Additional Resources” section
later in this preface).
SIGNIFICANT CHANGES FROM THE FIFTH EDITION
Important changes include new chapters on Pharmaceutical
Biotechnology and Oral Solid Dosage Forms. Three chapters were rewritten de novo on the basis of the valuable
feedback received since the publication of the Fifth Edition. These include Chapter 1 (“Introduction”), which is
now called Interpretive Tools; Chapter 20 (“Biomaterials”),
which is now called Pharmaceutical Polymers; and Chapter 23 (“Drug Delivery Systems”), which is now called
Drug Delivery and Targeting.
ORGANIZATION
As with prior editions, this edition represents an updating of
most chapters, a significant expansion of others, and the addition of new chapters in order to reflect the applications of the
physical chemical principles that are important to the Pharmaceutical Sciences today. As was true when Dr. Martin was
at the helm, this edition is a work in progress that reflects
the many suggestions made by students and colleagues in
academia and industry. There are 23 chapters in the Sixth
Edition, as compared with 22 in the Fifth Edition. All chapters have been reformatted and updated in order to make
the material more accessible to students. Efforts were made
to shorten chapters in order to focus on the most important
subjects taught in Pharmacy education today. Care has been
taken to present the information in “layers” from the basic
to more in-depth discussions of topics. This approach allows
the instructor to customize their course needs and focus their
course and the students’ attention on the appropriate topics
and subtopics.
With the publication of the Sixth Edition, a Web-based
resource is also available for students and faculty members
(see the “Additional Resources” section later in this preface).
ADDITIONAL RESOURCES
Martin’s Physical Pharmacy and Pharmaceutical Sciences,
Sixth Edition, includes additional resources for both instructors and students that are available on the book’s companion
Web site at thepoint.lww.com/Sinko6e.
Instructors
Approved adopting instructors will be given access to the
following additional resources:
■
Practice problems and answers to ascertain student understanding.
Students
Students who have purchased Martin’s Physical Pharmacy
and Pharmaceutical Sciences, Sixth Edition, have access to
the following additional resources:
■
FEATURES
A separate set of practice problems and answers to reinforce concepts learned in the text.
In addition, purchasers of the text can access the searchable
Full Text Online by going to the Martin’s Physical Pharmacy and Pharmaceutical Sciences, Sixth Edition, Web site
at thePoint.lww.com/Sinko6e. See the inside front cover of
this text for more details, including the passcode you will
need to gain access to the Web site.
Each chapter begins with a listing of Chapter Objectives that
introduce information to be learned in the chapter. Key Concept Boxes highlight important concepts, and each Chapter
Summary reinforces chapter content. In addition, illustrative Examples have been retained, updated, and expanded.
Recommended Readings point out instructive additional
sources for possible reference. Practice Problems have been
Patrick Sinko
Piscataway, New Jersey
v
5
Dr. Murtadha Alshareifi e-Library
CONTRIBUTORS
GREGORY E. AMIDON, PhD
TERUNA J. SIAHAAN, PhD
Research Professor
Department of Pharmaceutical Sciences
College of Pharmacy
University of Michigan
Ann Arbor, Michigan
Professor
Department of Pharmaceutical Chemistry
University of Kansas
Lawrence, Kansas
YASHVEER SINGH, PhD
CHARLES RUSSELL MIDDAUGH, PhD
Distinguished Professor
Department of Pharmaceutical Chemistry
University of Kansas
Lawrence, Kansas
Assistant Research Professor
Department of Pharmaceutics
Ernest Mario School of Pharmacy
Rutgers, The State University of New Jersey
Piscataway, New Jersey
HOSSEIN OMIDIAN, PhD
PATRICK J. SINKO, PhD, RPh
Assistant Professor
Department of Pharmaceutical Sciences
College of Pharmacy
Nova Southeastern University
Ft. Lauderdale, Florida
Professor II (Distinguished)
Parke-Davis Chair Professor in Pharmaceutics and Drug Delivery
Ernest Mario School of Pharmacy
Rutgers, The State University of New Jersey
Piscataway, New Jersey
KINAM PARK, PhD
HAIAN ZHENG, PhD
Showalter Distinguished Professor
Department of Biomedical Engineering
Professor of Pharmaceutics
Departments of Biomedical Engineering and Pharmaceutics
Purdue University
West Lafayette, Indiana
Assistant Professor
Department of Pharmaceutical Sciences
Albany College of Pharmacy and Health Sciences
Albany, New York
vi
6
Dr. Murtadha Alshareifi e-Library
ACKNOWLEDGMENTS
I would like to acknowledge Dr. Mayur Lodaya for his contributions to the continuous processing section of Chapter 22
on Oral Dosage forms.
Numerous graduate students contributed in many ways
to this edition, and I am always appreciative of their insights, criticisms, and suggestions. Thanks also to Mrs. Amy
Grabowski for her invaluable assistance with coordination
efforts and support interactions with all contributors.
To all of the people at LWW who kept the project moving forward with the highest level of professionalism, skill,
and patience. In particular, to David Troy for supporting our
vision for this project and Meredith Brittain for her exceptional eye for detail and her persistent efforts to keep us on
track.
And to my wonderful wife, Renee, who deserves enormous credit for juggling her hectic professional life as a
pharmacist and her expert skill as the family organizer while
maintaining a sense of calmness in what is an otherwise
chaotic life.
The Sixth Edition reflects the hard work and dedication of
many people. In particular, I acknowledge Drs. Gregory Amidon (Ch 22), Russell Middaugh (Ch 21), Hamid Omidian
(Chs 20 and 23), Kinam Park (Ch 20), Teruna Siahaan (Ch
21), and Yashveer Singh (Ch 23) for their hard work in spearheading the efforts to write new chapters or rewrite existing
chapters de novo. In addition, Dr. Singh went beyond the
call of duty and took on the responsibilities of Assistant
Editor during the proofing stages of the production of the
manuscripts. Through his efforts, I hope that we have caught
many of the minor errors from the fourth and fifth editions. I
also thank HaiAn Zheng, who edited the online practice problems for this edition, and Miss Xun Gong, who assisted him.
The figures and experimental data shown in Chapter 6
were produced by Chris Olsen, Yuhong Zeng, Weiqiang
Cheng, Mangala Roshan Liyanage, Jaya Bhattacharyya,
Jared Trefethen, Vidyashankara Iyer, Aaron Markham, Julian
Kissmann and Sangeeta Joshi of the Department of Pharmaceutical Chemistry at the University of Kansas. The section
on drying of biopharmaceuticals is based on a series of lectures and overheads presented by Dr. Pikal of the University
of Connecticut in April of 2009 at the University of Kansas.
Patrick Sinko
Piscataway, New Jersey
vii
7
Dr. Murtadha Alshareifi e-Library
CONTENTS
1
INTERPRETIVE TOOLS 1
13
DRUG RELEASE AND DISSOLUTION 300
2
STATES OF MATTER 17
14
CHEMICAL KINETICS AND STABILITY 318
3
THERMODYNAMICS 54
15
INTERFACIAL PHENOMENA 355
4
DETERMINATION OF THE PHYSICAL PROPERTIES
OF MOLECULES 77
16
COLLOIDAL DISPERSIONS 386
5
NONELECTROLYTES 109
17
COARSE DISPERSIONS 410
6
ELECTROLYTE SOLUTIONS 129
18
MICROMERITICS 442
7
IONIC EQUILIBRIA 146
19
RHEOLOGY 469
8
BUFFERED AND ISOTONIC SOLUTIONS 163
20
PHARMACEUTICAL POLYMERS 492
9
SOLUBILITY AND DISTRIBUTION PHENOMENA 182
21
PHARMACEUTICAL BIOTECHNOLOGY 516
10
COMPLEXATION AND PROTEIN BINDING 197
22
ORAL SOLID DOSAGE FORMS 563
11
DIFFUSION 223
23
DRUG DELIVERY AND TARGETING 594
12
BIOPHARMACEUTICS 258
Index 647
viii
8
Dr. Murtadha Alshareifi e-Library
1 Interpretive Tools
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Understand the basic tools required to analyze and interpret data sets from the clinic,
laboratory, or literature.
2. Describe the differences between classic dosage forms and modern drug delivery
systems.
3. Use dimensional analysis.
4. Understand and apply the concept of significant figures.
5. Define determinant and indeterminant errors, precision, and accuracy.
6. Calculate the mean, median, and mode of a data set.
7. Understand the concept of variability.
8. Calculate standard deviation and coefficient of variation and understand when it is
appropriate to use these parameters.
9. Use graphic methods to determine the slope of lines.
10. Interpret slopes of lines and how they relate to absorption and elimination from the
body.
Introduction
“One of the earmarks of evidence-based medicine is that the practitioner should not just accept the
conventional wisdom of his/her mentor. Evidence-based medicine uses the scientific method of using
observations and literature searches to form a hypothesis as a basis for appropriate medical therapy.
This process necessitates education in basic sciences and an understanding of basic scientific
principles.”1,2 Today more than ever before, the pharmacist and the pharmaceutical scientist are called
upon to demonstrate a sound knowledge of biopharmaceutics, biochemistry, chemistry, pharmacology,
physiology, and toxicology and an intimate understanding of the physical, chemical, and
biopharmaceutical properties of medicinal products. Whether engaged in research and development,
teaching, manufacturing, the practice of pharmacy, or any of the allied branches of the profession, the
pharmacist must recognize the need to rely heavily on the basic sciences. This stems from the fact that
pharmacy is an applied science, composed of principles and methods that have been culled from other
disciplines. The pharmacist engaged in advanced studies must work at the boundaries between the
various sciences and must keep abreast of advances in the physical, chemical, and biological fields in
order to understand and contribute to the rapid developments in his or her profession. You are also
expected to provide concise and practical interpretations of highly technical drug information to your
patients and colleagues. With the abundance of information and misinformation that is freely and
publicly available (e.g., on the Internet), having the tools and ability to provide meaningful interpretations
of results is critical.
Historically, physical pharmacy has been associated with the area of pharmacy that dealt with the
quantitative and theoretical principles of physicochemical science as they applied to the practice of
pharmacy. Physical pharmacy attempted to integrate the factual knowledge of pharmacy through the
development of broad principles of its own, and it aided the pharmacist and the pharmaceutical scientist
in their attempt to predict the solubility, stability, compatibility, and biologic action of drug products.
Although this remains true today, the field has become even more highly integrated into the biomedical
aspects of the practice of pharmacy. As such, the field is more broadly known today as
the pharmaceutical sciences and the chapters that follow reflect the high degree of integration of the
biological and physical–chemical aspects of the field.
Developing new drugs and delivery systems and improving upon the various modes of administration
are still the primary goals of the pharmaceutical scientist. A practicing pharmacist must also possess a
thorough understanding of modern drug delivery systems as he or she advises patients on the best use
of prescribed medicines. In the past, drug delivery focused nearly exclusively on pharmaceutical
9
Dr. Murtadha Alshareifi e-Library
techn
nology (in other words, the manufacture a
and testing of tablets, capsu
ules, creams, ointments,
soluttions, etc.). Th
his area of study is still very important tod
day. However, the pharmaciist needs to
unde
erstand how th
hese delivery systems
s
perfo
orm in and res
spond to the no
ormal and patthophysiologic
c
state
es of the patient. The integra
ation of physiccal–chemical and
a biological aspects is rellatively new in
n the
pharrmaceutical scciences. As the
e field progressses toward th
he complete in
ntegration of thhese
subd
disciplines, the
e impact of the
e biopharmace
eutical science
es and drug delivery will beccome enormo
ous.
The advent and co
ommercialization of molecu lar, nanoscale
e, and microsc
copic drug deliivery technolo
ogies
is a d
direct result off the integratio
on of the biolo
ogical and physical–chemica
al sciences. Inn the past, a
dosa
age (or dose) form
f
and a dru
ug delivery syystem were considered to be
e one and the same. A dosa
age
form
m is the entity that is adminis
stered to patie nts so that the
ey receive an effective dosee of a drug. Th
he
traditional understtanding of how
w an oral dosa
age form, such
h as a tablet, works
w
is that a patient takes
s it by
mouth with some fluid,
f
the table
et disintegratess, and the dru
ug dissolves in the stomach and is then
abso
orbed through the intestines
s into the blood
dstream. If the
e dose is
P.2
too h
high, a lower-d
dose tablet ma
ay be prescrib
bed. If a lower--dose tablet is
s not commerccially available
e, the
patie
ent may be insstructed to divide the tablet. However, a pharmacist
p
who dispenses a nifedipine
(Proccardia XL) exttended-release tablet or an oxybutynin (D
Ditropan XL) extended-releaase tablet to a
patie
ent would adviise the patientt not to bite, ch
hew, or divide
e the “tablet.” The
T reason forr this is that th
he
table
et dosage form
m is actually an
n elegant osm
motic pump dru
ug delivery sys
stem that lookks like a
convventional table
et (see Key Co
oncept Box on
n Dosage Form
ms and Drug Delivery
D
Systeems). This crea
ative
and elegant appro
oach solves nu
umerous challe
enges to the delivery
d
of pha
armaceutical ccare to patientts.
On the one hand, it provides a sustained-rele
s
ease drug deliv
very system to
o patients so tthat they take their
medication less fre
equently, there
eby enhancing
g patient compliance and po
ositively influeencing the suc
ccess
rate of therapeuticc regimens. On
n the other ha
and, patients see
s a familiar dosage
d
form thhat they can take
by a familiar route of administration. In essen ce, these osm
motic pumps are delivery sysstems packag
ged
into a dosage form
m that is familia
ar to the patie
ent. The subtle
e differences between
b
dose forms and de
elivery
syste
ems will becom
me even more
e profound in tthe years to co
ome as drug delivery
d
system
ms successfully
migrrate to the mollecular scale.
Key Co
oncept
Pha
armaceutica
al Sciences
s
Pharmacy, like many
m
other applied
a
scien
nces, has pas
ssed through
h a descriptivve and empirric
era. Over the pa
ast decade a firm scientifiic foundation
n has been developed, al lowing the “a
art”
of ph
harmacy to transform
t
itse
elf into a qua
antitative and
d mechanistic
c field of studdy. The
integ
gration of the
e biological, chemical,
c
an
nd physical sciences
s
remains critical tto the continuing
evollution of the pharmaceutical sciencess. The theore
etical links be
etween the d iverse scienttific
discciplines that serve
s
as the foundation fo
or pharmacy
y are reflected in this boook. The scientific
princciples of pha
armacy are not as comple
ex as some would
w
believe
e, and certaiinly they are not
beyo
ond the unde
erstanding off the well-edu
ucated pharm
macist of today.
Key Co
oncept
Dos
sage Forms
s and Drug Delivery Sy
ystems
A Prrocardia XL extended-rel
e
ease tablet iis similar in appearance
a
to
t a conventiional tablet. It
conssists, howeve
er, of a semipermeable m
membrane su
urrounding an
a osmoticallyy active drug
g
core
e. The core iss divided into
o two layers: an “active” layer contain
ning the drug and a “push
h”
laye
er containing pharmacologically inert b
but osmotica
ally active components. A
As fluid from the
gasttrointestinal (GI)
(
tract entters the table
et, pressure increases
i
in the osmotic layer and
“pusshes” against the drug lay
yer, releasing
g drug through the precis
sion laser-dririlled tablet orifice
in th
he active laye
er. Procardia
a XL is design
ned to provid
de nifedipine at an approx
oximately
consstant rate over 24 hr. This controlled rate of drug delivery into the GI lumeen is indepen
ndent
of pH
H or GI motillity. The nifed
depine relea
ase profile fro
om Procardia
a XL dependss on the
10
Dr. Murtadha Alshareifi e-Library
existence of an osmotic
o
grad
dient betwee n the conten
nts of the bila
ayer core andd the fluid in the
GI trract. Drug de
elivery is ess
sentially consstant as long as the osmo
otic gradient remains
consstant, and then gradually
y falls to zero
o. Upon being
g swallowed,, the biologiccally inert
com
mponents of the tablet rem
main intact du
uring GI tran
nsit and are eliminated
e
in the feces as
s an
inso
oluble shell. The
T information that the p
pharmacist provides
p
to th
he patient inccludes “Do not
crussh, chew, or break the ex
xtended-relea
ase form of Procardia
P
XL. These tableets are specially
form
mulated to rellease the me
edication slow
wly into the body.
b
Swallow the tabletss whole with a
glasss of water orr another liqu
uid. Occasion
nally, you ma
ay find a tablet form in thhe stool. Do not
n
be a
alarmed, this is the outer shell of the ttablet only, th
he medicatio
on has been absorbed by
y the
bodyy.” On examining the figu
ure, you will n
notice how th
he osmotic pump
p
tablet loooks identica
al to
a co
onventional ta
ablet.
Rem
member, mosst of the time
e when a patiient takes a tablet,
t
it is also the deliveery system. It
has been optimizzed so that itt can be masss-produced and can rele
ease the drugg in a
reprroducible and
d reliable ma
anner. Comp lete disintegration and de
eaggregationn occurs and
d
there is little, if any,
a
evidence
e of the table
et dose form that can be found
f
in the stool. Howev
ver,
with an osmotic pump delivery system, th
he “tablet” do
oes not disintegrate evenn though all of
o
the d
drug will be released.
r
Ev
ventually, the
e outer shell of
o the depleted “tablet” paasses out off the
bodyy in the stool.
This course should
d mark the turrning point in tthe study patte
ern of the stud
dent, for in thee latter part of the
pharrmacy curriculum, emphasis
s is placed upo
on the applica
ation of scientiffic principles tto practical
profe
essional problems. Although
h facts must b
be the foundation upon whic
ch any body off knowledge is
s
built,, the rote mem
morization of disjointed
d
“partticles” of know
wledge does not lead to logiccal and system
matic
thought. This chap
pter provides a foundation fo
or interpreting
g the observations and resullts that come from
f
carefful
P.3
ntific study. Att the conclusio
on of this chap
pter, you should have the ab
bility to integraate facts and ideas
scien
into a meaningful whole
w
and con
ncisely conveyy a sense of th
hat meaning to a third partyy. For example
e, if
you a
are a pharmaccy practitionerr, you should b
be able to tran
nslate a complex scientific pprinciple to a
simp
ple, practical, and
a useful rec
commendation
n for a patient.
Key Co
oncept
11
Dr. Murtadha Alshareifi e-Library
“
I HEAR AN
ND I FORGE
ET. I SEE A
AND I REM
MEMBER. I DO AND I
U
UNDERSTA
AND.”
The ancient Chin
nese proverb
b emphasize
es the value of
o active partticipation in tthe learning
proccess. Throug
gh the illustra
ative example
es and practice problems
s in this bookk and on the
onlin
ne companio
on Web site, the student iis encourage
ed to actively
y participate.
The comprehensio
on of course material
m
is prim
marily the resp
ponsibility of th
he student. Thhe teacher can
n
guide
e and direct, explain,
e
and clarify, but com
mpetence in so
olving problem
ms in the classrroom and the
laborratory depend
ds largely on th
he student's u
understanding of theory, recall of facts, abbility to integra
ate
know
wledge, and willingness
w
to devote
d
sufficie
ent time and efffort to the task. Each assignnment should be
read and outlined, and assigned
d problems sh
hould be solved outside the classroom. Thhe teacher's
comm
ments then wiill serve to clarify questiona ble points and
d aid the stude
ent to improvee his or her
judgm
ment and reassoning abilities
s.
Me
easureme
ents, Data
a, Propag
gation off Uncerta
ainty
The goal of this ch
hapter is to pro
ovide a founda
ation for the quantitative
q
rea
asoning skills tthat are
fundamental to the
e pharmacy prractitioner and
d pharmaceutiical scientist. “As
“ mathemattics is the lang
guage
of sccience, statistics is the logic of science.”3 Mathematics and statistics are fundamenntal tools of th
he
pharrmaceutical scciences. You need
n
to undersstand how and
d when to use
e these tools, aand how to
interrpret what theyy tell us. You must
m
also be ccareful not to overinterpret
o
results.
r
On thee one hand, yo
ou
may ask “do we re
eally need to know
k
how thesse equations and
a formulas were
w
derived iin order to use
e
them
m effectively?” Logically, the answer would
d seem to be no. By analog
gy, you do not need to know
w how
to bu
uild a compute
er in order to use
u one to sen
nd an e-mail message,
m
do you? On the otther hand,
grap
phically represented data convey a sense dynamics tha
at benefit from understandinng a bit more about
a
the ffundamental equations
e
behind the behaviior. These equ
uations are me
erely tools (thaat you should not
mem
morize!) that alllow for the tra
ansformation o
of a bunch of numbers
n
into a behavior thaat you can
interrpret.
The mathematics and statistics covered in thiis chapter and
d this book are
e presented in a format to
prom
mote understan
nding and pra
actical use. The
erefore, many
y of the basic mathematical
m
“tutorial” elem
ments
have
e been remove
ed from the six
xth edition, an
nd in particularr this chapter, because of thhe migration of
o
numerous college-level topics to
o secondary sschool courses
s over the years. However, if you believe that
you n
need a refresh
her in basic mathematical
m
cconcepts, this information is still available in the online
comp
panion to this text (at thePo
oint.lww.com/S
Sinko6e). Statistical formula
as and graphiccal method
expla
anations have
e also been dra
amatically red
duced in this edition.
e
Depending on your ppersonal goals
s and
the p
philosophy of your
y
program of study, you may well need an in-depth treatment of tthe subject ma
atter.
Additional detailed
d treatments can
c be found o
on the Web sitte and in the re
ecommended readings.
Data Analys
sis Tools
s
12
Dr. Murtadha Alshareifi e-Library
Read
dily available tools
t
such as programmable
e calculators, computer spreadsheet proggrams (e.g.,
Micro
osoft Excel, Apple
A
Numbers
s, or OpenOffiice.org Calc), and statisticall software pacckages (e.g.,
Minittab, SAS or SPSS) make th
he processing of data relativ
vely easy. Spreadsheet proggrams have tw
wo
distin
nct advantage
es: (1) data collection/entry iis simple and can often be automated,
a
reeducing the
posssibility of errorss in transcriptiion, and (2) si mple data ma
anipulations an
nd elementaryy statistical
calcu
ulations are allso easy to perform. In addittion, many spreadsheet pro
ograms seamleessly interface
e with
statisstical package
es when more robust statist ical analysis is
s required. With very little ef
effort, you can add
data sets and generate pages of
o analysis. Th
he student sho
ould appreciate
e that while it may be possible to
automate data enttry and have the computer p
perform calcullations, the final interpretatioon of the results
and statistical ana
alysis is your re
esponsibility! A
As you set out to analyze data keep in m ind the simple
e
acronym GIGO—G
Garbage In, Garbage
G
Out. I n other words
s, solid scientiffic results andd sound metho
ods of
analyysis will yield meaningful intterpretations a
and conclusions. However, if the scientificc foundation is
s
weakk, there are no
o known statis
stical tools tha
at can make ba
ad data significant.
Dim
mensiona
al Analys
sis
Dime
ensional analyysis (also calle
ed the factor-la
abel method or
o the unit factor method) is a problem-solving
meth
hod that uses the fact that any
a number orr expression can be multiplie
ed by 1 withouut changing its
s
value
e. For example, since 2.2 kg
g = 1 lb, you ccan divide both sides of the relationship bby “1 lb,” resulting
in 2.2
2 kg/1 lb = 1. This is known as a converssion factor,
P.4
whicch is a ratio of like-dimensioned quantitiess and is equal to the dimens
sionless unity (in other word
ds, is
equa
al to 1). On the
e face of it, the
e concept mayy seem a bit abstract
a
and no
ot very practiccal. However,
dime
ensional analyysis is very use
eful for any va
alue that has a “unit of meas
sure” associate
ted with it, which is
nearrly everything in the pharma
aceutical scien
nces. Simply put,
p this is a prractical methodd for convertin
ng
the u
units of one ite
em to the units
s of another ite
em.
Exa
ample 1-1
Solvving problems using dime
ensional ana
alysis is straig
ghtforward. You
Y do not neeed to worry
y
abou
ut the actual numbers un
ntil the very e
end. At first, simply
s
focus on the unitss. Plug in all of
o
the cconversion fa
actors that cancel out the
e units you do
d not want until
u you end up with the units
that you do wantt. Only then do
d you need
d to worry abo
out doing the
e calculation . If the units work
out, you will get the right ans
swer every tim
me. In this example, the goal is to illuustrate how to
use the method for convertin
ng one value to another.
Que
estion: How many
m
second
ds are there in 1 year?
Con
nversion Facttors:
365 days = 1 year 24 hr = 1 day 60 miin = 1 hr 60
0 sec = 1 min
n
Rea
arrange Convversion Facto
ors:
Solvve (arrange conversion
c
fa
actors so tha
at the units th
hat you do no
ot want canceel out):
as yyou see the units
u
become
e seconds.
Calcculate: Now, plug the num
mbers carefu
ully into your calculator an
nd the resultting answer is
31,5
536,000 sec//year.
Exa
ample 1-2
Thiss example will demonstra
ate the use off dimensiona
al analysis in performing a calculation
n.
How
w many calorries are there
e in 3.00 joule
es? One sho
ould first reca
all a relationsship or ratio that
t
conn
nects calorie
es and joules
s. The relatio n 1 cal = 4.184 joules comes to mindd. This is the key
convversion facto
or required to
o solve this p
problem. The
e question can then be assked keeping
g in
13
Dr. Murtadha Alshareifi e-Library
mind
d the converrsion factor: If 1 cal equalls 4.184 joule
es, how man
ny calories arre there in 3.00
joule
es? Write do
own the conversion factorr, being careful to express each quanntity in its pro
oper
unitss. For the un
nknown quan
ntity, use anX
X.
Table 1-1 Writin
ng or Interp
preting Sig
gnificant fig
gures in Nuumbers
R
Rule
Exxample
All nonzerro digits are
consideredd significantt
998.513 has five
f significcant figuress: 9, 8, 5, 1,
aand 3
Leading zeeros are not
significantt
00.00361 hass three signiificant figurres: 3, 6,
aand 1
Trailing zeeros in a num
mber
containingg a decimal point
p
are significcant
9998.100 hass six significcant figuress: 9, 9, 8,
11, 0, and 0
The signifi
ficance of traailing
zeros in a number
n
not
containingg a decimal point
p
can be ambbiguous
T
The numberr of significcant figures in
nnumbers lik
ke 11,000 is uncertain bbecause a
ddecimal poiint is missin
ng. If the nuumber was
w
written as 11,000, it wo
ould be cleaar that
tthere are fiv
ve significan
nt figures
Zeros appeearing anyw
where
between tw
wo nonzero
digits are significant
s
6607.132 hass six significcant figuress: 6, 0, 7,
11, 3, and 2
Exa
ample 1-3
How
w many gallons are equiv
valent to 2.0 lliters? It wou
uld be necess
sary to set upp successive
e
prop
portions to so
olve this prob
blem. In the m
method involving the identity of units on both side
es of
the e
equation, the
e quantity de
esired, X (galllons), is plac
ced on the le
eft and its equuivalent, 2.0
literss, is set down on the righ
ht side of the equation. Th
he right side must then bbe multiplied by
know
wn relations in ratio form, such as 1 p
pint per 473 mL, to give the units of ggallons. Carry
ying
out tthe indicated
d operations yields the re
esult with its proper
p
units:
One may be conce
erned about th
he apparent d isregard for th
he rules of significant figuress in the
equivvalents such as
a 1 pint = 473
3 mL. The qua
antity of pints can be measu
ured as accuraately as that of
o
14
Dr. Murtadha Alshareifi e-Library
milliliters, so that we
w assume 1.0
00 pint is mea
ant here. The quantities
q
1 ga
allon and 1 liteer are also exa
act
by de
efinition, and significant
s
figu
ures need not be considered
d in such case
es.
Sig
gnificant Figures
A sig
gnificant figure
e is any digit used
u
to represe
ent a magnitu
ude or a quantity in the placee in which it
stand
ds. The rules for interpreting
g significant fi gures and som
me examples are shown in Table 1-1.
Significant figures give a sense of the accuraccy of a numbe
er. They includ
de all digits exxcept leading and
a
trailin
ng zeros wherre they are used merely to l ocate the dec
cimal point. An
nother way to sstate this is, th
he
signiificant figures of a number include all certtain digits plus
s the first unce
ertain digit. Foor example, on
ne
may use a ruler, th
he smallest su
ubdivisions of which are cen
ntimeters, to measure
m
the leength of a piec
ce of
P.5
glasss tubing. If one
e finds that the
e tubing meassures slightly greater
g
than 27
2 cm in lengthh, it is proper to
t
estim
mate the doubtful fraction, say 0.4, and exxpress the num
mber as 27.4 cm. A replicatte measureme
ent
may yield the valu
ue 27.6 or 27.2
2 cm, so that tthe result is ex
xpressed as 27.4 ± 0.2 cm. When a value
e
such
h as 27.4 cm iss encountered
d in the literatu
ure without furrther qualification, the readeer should assu
ume
that the final figure
e is correct to within about ± 1 in the last decimal
d
place, which is meaant to signify the
t
mean deviation off a single meas
surement. How
wever, when a statement su
uch as “not lesss than 99” is
given
n in an official compendium, it means 99..0 and not 98.9.
Exa
ample 1-4
How
w Many Significant Fig
gures in the
e Number 0.00750?
The two zeros im
mmediately fo
ollowing the decimal poin
nt in the num
mber 0.007500 merely loca
ate
the d
decimal poin
nt and are no
ot significant. However, th
he zero follow
wing the 5 is significant
beca
ause it is nott needed to write
w
the num
mber; if it werre not signific
cant, it couldd be omitted.
Thus, the value contains thre
ee significantt figures.
How
w Many Significant Fig
gures in the
e Number 75
500?
The question of significant fig
gures in the number 7500 is ambiguo
ous. One doees not know
whe
ether any or all
a of the zero
os are mean t to be signifficant or whether they aree simply used
d to
indiccate the mag
gnitude of the
e number. Hiint: To expre
ess the signifficant figuress of such a va
alue
in an
n unambiguo
ous way, it is
s best to use exponential notation. Thus, the expreession 7.5 ×
3
10 signifies thatt the numberr contains tw
wo significant figures, and the zeros inn 7500 are no
ot to
be ta
aken as sign
nificant. In the
e value 7.500
0 × 103, both
h zeros are significant,
s
annd the number
conttains a total of
o four significant figures .
Sign
nificant figures are particularrly useful for in
ndicating the precision
p
of a result. The prooper interpreta
ation
of a value may be questioned specifically in ccases when pe
erforming calc
culations (e.g.,, when spuriou
us
digitss introduced by
b calculations
s carried out to
o greater accu
uracy than that of the originaal data) or whe
en
repo
orting measure
ements to a grreater precisio
on than the equipment supports. It is impoortant to remember
that the instrumen
nt used to mak
ke the measure
rement limits th
he precision of
o the resultingg value that is
repo
orted. For exam
mple, a measu
uring rule marrked off in centtimeter divisio
ons will not prooduce as great a
precision as one marked
m
off in 0.1
0 cm or mm.. One may obttain a length of
o 27.4 ± 0.2 ccm with the firs
st
rulerr and a value of,
o say, 27.46 ± 0.02 cm witth the second. The latter ruler, yielding a rresult with fou
ur
signiificant figures, is obviously the
t more preccise one. The number
n
27.46 implies a prec
ecision of abou
ut 2
partss in 3000, whe
ereas 27.4 imp
plies a precisio
on of only 2 pa
arts in 300.
The absolute mag
gnitude of a va
alue should no
ot be confused
d with its precis
sion. We conssider the numb
ber
0.00053 mole/literr as a relatively
y small quantiity because th
hree zeros imm
mediately follow
w the decimal
pointt. These zeross are not significant, howeve
er, and tell us nothing about the precisionn of the
measurement. Wh
hen such a res
sult is expresssed as 5.3 × 10-4 mole/liter, or better as 5 .3 (±0.1) × 104
mo
ole/liter, both itts precision an
nd its magnitud
de are readily
y apparent.
Key Co
oncept
“Wh
hen Signific
cant Figures do Not Ap
pply”
15
Dr. Murtadha Alshareifi e-Library
Since significant figure rules are based upon estimations derived from statistical rules for
handling probability distributions, they apply only to measuredvalues. The concept of
significant figures does not pertain to values that are known to be exact. For example, integer
counts (e.g., the number of tablets dispensed in a prescription bottle); legally defined
conversions such as 1 pint = 473 mL; constants that are defined arbitrarily (e.g., a centimeter
is 0.01 m); scalar operations such as “doubling” or “halving”; and mathematical constants,
such as π and e. However, physical constants such as Avogadro's number have a limited
number of significant figures since the values for these constants are derived from
measurements.
Example 1-5
The following example is used to illustrate excessive precision. If a faucet is turned on and
100 mL of water flows from the spigot in 31.47 sec, what is the average volumetric flow rate?
By dividing the volume by time using a calculator, we get a rate of 3.177629488401652
mL/sec. Directly stating the uncertainty is the simplest way to indicate the precision of any
result. Indicating the flow rate as 3.177 ± 0.061 mL/sec is one way to accomplish this. This is
particularly appropriate when the uncertainty itself is important and precisely known. If the
degree of precision in the answer is not important, it is acceptable to express trailing digits
that are not known exactly, for example, 3.1776 mL/sec. If the precision of the result is not
known you must be careful in how you report the value. Otherwise, you may overstate the
accuracy or diminish the precision of the result.
In dealing with experimental data, certain rules pertain to the figures that enter into the computations:
1.
2.
3.
In rejecting superfluous figures, increase by 1 the last figure retained if the following figure
rejected is 5 or greater. Do not alter the last figure if the rejected figure has a value of less than
5.
Thus, if the value 13.2764 is to be rounded off to four significant figures, it is written as 13.28.
The value 13.2744 is rounded off to 13.27.
In addition or subtraction include only as many figures to the right of the decimal point as there
are present in the number with the least such figures. Thus, in adding 442.78, 58.4, and 2.684,
obtain the sum and then round off the result so that it contains only one figure following the
decimal point:
This figure is rounded off to 503.9.
Rule 2 of course cannot apply to the weights and volumes of ingredients in the monograph of a
pharmaceutical preparation. The minimum weight or volume of each ingredient in a
pharmaceutical formula or a prescription
P.6
should be large enough that the error introduced is no greater than, say, 5 in 100 (5%), using
the weighing and measuring apparatus at hand. Accuracy and precision in prescription
compounding are discussed in some detail by Brecht.5
4.
In multiplication or division, the rule commonly used is to retain the same number of significant
figures in the result as appears in the value with the least number of significant figures. In
multiplying 2.67 and 3.2, the result is recorded as 8.5 rather than as 8.544. A better rule here is
to retain in the result the number of figures that produces a percentage error no greater than
that in the value with the largest percentage uncertainty.
16
Dr. Murtadha Alshareifi e-Library
5.
6.
In the use
e of logarithms for multiplica
ation and divis
sion, retain the
e same numbeer of significan
nt
figures in
n the mantissa
a as there are in the original numbers. The
e characteristiic signifies only the
magnitud
de of the numb
ber and accord
dingly is not significant. Bec
cause calculattions involved in
theoretica
al pharmacy usually
u
require
e no more than
n three signific
cant figures, a four-place
logarithm
m table yields sufficient
s
prec ision for our work.
w
Such a ta
able is found oon the inside back
b
cover of this
t
book. The
e calculator is more conveniient, however,, and tables off logarithms are
rarely use
ed today.
If the result is to be use
ed in further ccalculations, re
etain at least one
o digit more than suggestted in
the rules just given. Th
he final result iis then rounde
ed off to the last significant ffigure.
Rem
member, signifiicant figures are
a not meant to be a perfec
ct representation of uncertaiinty. Instead, they
t
are u
used to preven
nt the loss of precision
p
whe n rounding nu
umbers. They also help you avoid stating more
inforrmation than you
y actually kn
now. Error and
d uncertainty are
a not the sam
me. For exam ple, if you perrform
an experiment in triplicate
t
(in other words, yo
ou repeat the experiment
e
thrree times), youu will get a value
that looks somethiing like 4.351±
±0.076. This d
does not mean
n that you mad
de an error in the experimen
nt or
the ccollection of th
he data. It simp
ply means tha
at the outcome
e is naturally statistical.
s
Data Types
s
The scientist is continually attem
mpting to relatte phenomena
a and establish
h generalizatioons with which
h to
conssolidate and in
nterpret experimental data. T
The problem frequently
f
reso
olves into a seearch for the
relationship betwe
een two quantiities that are cchanging at a certain rate orr in a particulaar manner. The
e
depe
endence of on
ne property, the dependent vvariable, y, on
n the change or
o alteration off another
measurable quanttity, the indepe
endent variablle x, is expressed mathema
atically as
aries directly as
a x” or “y is d irectly proporttional to x.” A proportionalityy is changed to an
whicch is read “y va
equa
ation as follow
ws. If y is propo
ortional to x in general, then
n all pairs of sp
pecific values of y and x,
say y1 and x1, y2 and
a x2,…, are proportional. T
Thus,
Beca
ause the ratio of any y to its
s correspondin
ng x is equal to
o any other ratio of y and x, the ratios are
e
consstant, or, in general
Hencce, it is a simp
ple matter to change
c
a propo
ortionality to an
a equality by introducing a proportionalitty
consstant, k. To summarize, if
then
w the relationsship between x and y by the
e use of the moore general
It is ffrequently dessirable to show
notation
qual, for exam
mple, to 2x, to 227x2, or to 0.0
0051
whicch is read “y is some function of x.” That iss, y may be eq
+ log
g (a/x). The functional notatiion in equation
n (1-5) merely
y signifies that y and x are reelated in some
e
way without speciffying the actua
al equation byy which they are connected.
As w
we begin to layy the foundatio
on for the interrpretation of data
d
using descriptive statisttics, some
backkground inform
mation about th
he types of da
ata that you wiill encounter in
n the pharmacceutical scienc
ces is
need
ded. In 1946, Stevens
S
define
ed measurem
ment as “the as
ssignment of numbers
n
to obj
bjects or eventts
acco
ording to a rule
e.”4 He propos
sed a classificcation system that is widely used today too define data ty
ypes.
The first two, interrvals and ratios, are categorrized as contin
nuous variable
es. These wouuld include res
sults
of lab
boratory meassurements for nearly all of t he data that are
a normally co
ollected in thee laboratory (e.g.,
conccentrations, we
eights). Only ratio
r
or interva
al measurements can have units of measuurement, and
17
Dr. Murtadha Alshareifi e-Library
these variables are quantitative in nature. In other words, if you were given a set of “interval” data you
would be able to calculate the exact differences between the different values. This makes this type of
data “quantitative.” Since the interval between measurements can be very small, we can also say that
the data are “continuous.” Another laboratory example of interval data measures is temperature. Think
of the gradations on a common thermometer (in Celsius or Fahrenheit scale)—they are typically spaced
apart by 1 degree with minor gradations at the 1/10th degree. The intervals could become even smaller;
however, because of the physical limitations of common thermometers, smaller gradations are not
possible since they cannot be read accurately. Of course, with digital thermometers the gradations (or
intervals) could be much smaller but then the precision of the thermometer may become questionable.
Another temperature scale that will be used in various sections of this text is the Kelvin scale, a
thermodynamic temperature scale. By international agreement,
P.7
the Kelvin and Celsius scales are related through the definition of absolute zero (in other words, 0 K = 273.15°C). Since the thermodynamic temperature is measured relative to absolute zero, the Kelvin
scale is considered a ratio measurement. This also holds true for other physical quantities such as
length or mass. The third common data type in the pharmaceutical sciences is ordinal
scale measurements. Ordinal measurements represent the rank order of what is being measured.
“Ordinals” are more subjective than interval or ratio measurements.
The final type of measurement is called nominal data. In this type of measurement, there is no order or
sequence of the observations. They are merely assigned different groupings such as by name, make, or
some similar characteristic. For example, you may have three groups of tablets: white tablets, red
tablets, and yellow tablets. The only way to associate the various tablets is by their color. In clinical
research, variables measured at a nominal level include sex, marital status, or race. There are a variety
of ways to classify data types and the student is referred to texts devoted to statistics such as those
listed in the recommended readings at the end of this chapter.6,7
Error and Describing Variability
If one is to maintain a high degree of accuracy in the compounding of prescriptions, the manufacture of
products on a large scale, or the analysis of clinical or laboratory research results, one must know how
to locate and eliminate constant and accidental errors as far as possible. Pharmacists must recognize,
however, that just as they cannot hope to produce a perfect pharmaceutical product, neither can they
make an absolute measurement. In addition to the inescapable imperfections in mechanical apparatus
and the slight impurities that are always present in chemicals, perfect accuracy is impossible because of
the inability of the operator to make a measurement or estimate a quantity to a degree finer than the
smallest division of the instrument scale.
Error may be defined as a deviation from the absolute value or from the true average of a large number
of results. Two types of errors are recognized: determinate(constant) and indeterminate (random or
accidental).
Determinate Errors
Determinate or constant errors are those that, although sometimes unsuspected, can be avoided or
determined and corrected once they are uncovered. They are usually present in each measurement and
affect all observations of a series in the same way. Examples of determinate errors are those inherent in
the particular method used, errors in the calibration and the operation of the measuring instruments,
impurities in the reagents and drugs, and biased personal errors that, for example, might recur
consistently in the reading of a meniscus, in pouring and mixing, in weighing operations, in matching
colors, and in making calculations. The change of volume of solutions with temperature, although not
constant, is a systematic error that can also be determined and accounted for once the coefficient of
expansion is known.
Determinate errors can be reduced in analytic work by using a calibrated apparatus, using blanks and
controls, using several different analytic procedures and apparatus, eliminating impurities, and carrying
out the experiment under varying conditions. In pharmaceutical manufacturing, determinate errors can
18
Dr. Murtadha Alshareifi e-Library
be eliminated by calibrating
c
the weights and o
other apparatu
us and by checking calculattions and resu
ults
with other workerss. Adequate co
orrections for d
determinate errors
e
must be made before the estimation of
indetterminate erro
ors can have any
a significancce.
Ind
determina
ate Errorrs
Indeterminate erro
ors occur by accident or cha
ance, and they
y vary from on
ne measuremeent to the nextt.
Whe
en one fires a number
n
of bullets at a targe
et, some may hit
h the bull's eye, whereas oothers will be
scatttered around this
t
central po
oint. The greatter the skill of the
t marksman
n, the less scaattered will be the
patte
ern on the targ
get. Likewise, in a chemical analysis, the results of a se
eries of tests w
will yield a random
patte
ern around an average or ce
entral value, kknown as the mean.
m
Random
m errors will aalso occur in filling
a number of capsu
ules with a dru
ug, and the fin
nished productts will show a definite variattion in weight.
Indeterminate erro
ors cannot be allowed for orr corrected because of the natural
n
fluctuat
ations that occur in
all m
measurements.
Thosse errors that arise from ran
ndom fluctuatio
ons in temperature or other external factoors and from the
varia
ations involved
d in reading instruments are
e not to be con
nsidered accid
dental or randoom. Instead, they
belon
ng to the classs of determina
ate errors and are often called pseudoacc
cidental or varriable
determinate errorss. These errors may be redu
uced by contro
olling condition
ns through thee use of consttant
temp
perature bathss and ovens, the use of bufffers, and the maintenance
m
of
o constant hum
midity and
pressure where indicated. Care in reading fra
actions of units
s on graduates
s, balances, aand other
appa
aratus can also reduce pseu
udoaccidental errors. Variab
ble determinatte errors, althoough seeming
gly
indetterminate, can
n thus be dete
ermined and co
orrected by ca
areful analysis
s and refinemeent of techniqu
ue on
the p
part of the worrker. Only erro
ors that result from pure random fluctuatio
ons in nature aare considered
truly indeterminate
e.
Pre
ecision and
a Accu
uracy
Preccision is a mea
asure of the ag
greement amo
ong the values
s in a group off data, whereaas accuracy is
s the
agre
eement betwee
en the data an
nd the true val ue. Indetermin
nate or chance errors influeence the precis
sion
of the results, and the measurem
ment of the prrecision is acc
complished best by statisticaal means.
Dete
erminate or constant errors affect
a
the accu
uracy of data.
P.8
The techniques ussed in analyzin
ng the precisio
on of results, which
w
in turn supply
s
a meassure of the
ors, will be con
nsidered first, and the detec
ction and elimination of deteerminate errors
s or
indetterminate erro
inacccuracies will be
b discussed la
ater.
19
Dr. Murtadha Alshareifi e-Library
Fig.. 1-1. The normal
n
curvee for the disstribution off indetermin
nate errors.
Indeterminate or chance
c
errors obey the lawss of probability
y, both positive
e and negativee errors being
equa
ally probable, and larger errors being lesss probable tha
an smaller one
es. If one plotss a large numb
ber of
results having variious errors alo
ong the vertica
al axis againstt the magnitud
de of the errorss on the horizontal
axis,, one obtains a bell-shaped curve, known
n as a normal frequency
f
disttribution curvee, as shown
in Fig
gure 1-1. If the
e distribution of
o results follo
ows the norma
al probability la
aw, the deviatiions will be
repre
esented exacttly by the curve for an infinitte number of observations,
o
which
w
constituute
the u
universe or po
opulation. Whe
ereas the popu
ulation is the whole
w
of the ca
ategory underr consideration
n, the
samp
ple is that porttion of the pop
pulation used in the analysis
s.
Des
scriptive
e Statistic
cs
Since the typical pharmacy
p
stud
dent has sufficcient exposure
e to descriptive
e statistics in oother courses, this
sectiion will focus on
o introducing
g (or reintroduccing) some of the key conce
epts that will bbe used nume
erous
timess in later chap
pters. The stud
dent who requ
uires additiona
al background in statistics iss advised to se
eek
out o
one of the man
ny outstanding
g texts that ha
ave been publiished.6,7 Desc
criptive statisttics depict the
basicc features of a data set colle
ected from an experimental study. They give
g
summariees about the
samp
ple and the measures. How
wever, viewing the individual data and tables of results aalone is not
alwa
ays sufficient to
o understand the behavior o
of the data. Ty
ypically, a grap
phic analysis iis paired with a
tabular description
n to perform a quantitative a
analysis of the
e data set. The
e third componnent of descrip
ptive
statisstics is “summ
mary” statistics
s. These are s ingle numbers
s that summarrize the data. W
With interval data
d
(e.g., the dose stre
ength of individual tablets in
n a batch of 10
0,000 tablets),, summary staatistics focus on
o
how big the value is and the varriability among
g the values. The
T first of the
ese aspects reelates to meas
sures
of “central tendenccy” (e.g., whatt is the averag
ge?), while the
e second referrs to “dispersioon” (in other words,
w
the ““variation” amo
ong a group of
o values).
Central Ten
ndency: Mean,
M
Me
edian, Mo
ode
Centtral tendency can
c be describ
bed using a su
ummary statis
stic (the mean, median, or m
mode) that give
es an
indiccation of the avverage value in the data sett. The theoretical mean for a large numbeer of
measurements (th
he universe or population) iss known as the
e universe or population
p
meean and is give
en
the ssymbol µ (mu)).
The arithmetic me
ean [X with barr above] is obttained by adding together th
he results of thhe various
measurements an
nd dividing the total by the n umber N of th
he measureme
ents. In matheematical notation,
the a
arithmetic mea
an for a small group of value
es is expresse
ed as
in wh
hich Σ stands for “the sum of,”
o Xi is the ith
h individual me
easurement of the group, annd N is the
number of values. [X with bar ab
bove] is an esstimate of µ an
nd approaches
s it as the num
mber of
measurements N is increased. Remember,
R
th
he “equations”” used in all off the calculatioons are really a
shorrthand notation
n describing th
he various rela
ationships that define some parameter.
Exa
ample 1-6
A ne
ew student has just joined
d the lab and
d is being tra
ained to pipettte liquids coorrectly. She is
usin
ng a 1-mL pip
pettor and is asked to witthdraw 1 mL of water from
m a beaker aand weigh it on
o a
bala
ance in a weighing boat. To
T determine
e her pipettin
ng skill, she is asked to reepeat this 10
0
time
es and take the average. What is the average volu
ume of waterr that the stuudent withdra
aws
afterr 10 repeats?
? The density of water is 1 g/mL.
20
Dr. Murtadha Alshareifi e-Library
AttemptWeight (g)
1
1.05
2
0.98
3
0.95
4
1.00
5
1.02
6
1.00
7
1.10
8
1.03
9
0.96
10
0.98
If ΣXi = 9.99 and N = 10, so 9.99/10 = 0.999. Given the number of significant figures, the
average would be reported as 1.00 g, which equals 1 mL since the density of water is 1 g/mL.
The median is the middle value of a range of values when they are arranged in rank order (e.g., from
lowest to highest). So, the median value of the list [1, 2, 3, 4, 5] is the number 3. In this case, the mean
is also 3. So, which value is a better indicator of the central tendency of the data? The answer in this
case is neither—both indicate central tendency equally well. However, the value of the median as a
summary statistic
P.9
becomes more obvious when the data set is skewed (in other words, when there are outliers or data
points with values that are quite different from most of the others in the data set). For example, in the
data set [1, 2, 2, 3, 10] the mean would be 3.6 but the median would be 2. In this case, the median is a
better summary statistic than the mean because it gives a better representation of central tendency of
the data set. Sometimes the median is referred to as a more “robust” statistic since it gives a reasonable
outcome even with outlier results in the data set.
Example 1-7
As you have seen, calculating the median of a data set with an odd number of results is
straightforward. But, what do you do when a data set has an even number of members? For
example, in the data set [1, 2, 2, 3, 4, 10] you have 6 members to the data set. To calculate
21
Dr. Murtadha Alshareifi e-Library
the m
median you need to find the two midd
dle members
s (in this case
e, 2 and 3) thhen average
e
them
m. So, the me
edian would be 2.5.
Altho
ough it is human nature to want
w
to “throw
w out” an outlying piece of da
ata from a dataa set, it is not
prop
per to do so un
nder most circumstance or a
at least withou
ut rigorous statistical analysiis. Using median
as a summary stattistic allows yo
ou to use all o
of the results in
n a data set an
nd still get an idea of the ce
entral
tendency of the re
esults.
The mode is the value
v
in the data set that occcurs most ofte
en. It is not as commonly useed in the
pharrmaceutical scciences but it has
h particular value in describing the mos
st common occ
ccurrences of
results that tend to
o center aroun
nd more than o
one value (e.g
g., a bimodal distribution
d
thaat has two
comm
monly occurrin
ng values). Fo
or example, in the data set [1,
[ 2, 4, 4, 5, 5,
5 5, 6, 9, 10] tthe mode valu
ue is
equa
al to 5. Howevver, we sometimes see a da
ata set that has
s two “clusters
s” of results raather than one
e. For
exam
mple, the data set [1, 2, 4, 4,
4 5, 5, 5, 6, 9, 10, 11, 11, 11
1, 11, 13, 14] is bimodal andd thus has two
o
modes (one mode
e is 5 and the other
o
is 11). T
Taking the arith
hmetic mean of
o the data sett would not giv
ve an
indiccation of the biimodal behaviior. Neither wo
ould the media
an.
Varriability: Measure
es of Disp
persion
In orrder to fully un
nderstand the properties of tthe data set th
hat you are analyzing, it is nnecessary to
convvey a sense off the dispersio
on or scatter a round the cen
ntral value. This is done so tthat an estima
ate of
the vvariation in the
e data set can be calculated
d. This variabillity is usually expressed
e
as the range,
the m
mean deviation, or thestand
dard deviation.. Another usefful measure off dispersion coommonly used
d in
the p
pharmaceutica
al sciences is the coefficientt of variation (CV), which is a dimensionleess parameterr.
Since much of thiss will be a review for many o
of the students
s using this text, only the moost pertinent
featu
ures will be disscussed. The results obtain ed in the phys
sical, chemica
al, and biologiccal aspects of
pharrmacy have diffferent charac
cteristics. In th e physical sciences, for exa
ample, instrum
ment
measurements are
e often not perfectly reprodu
ucible. In othe
er words, varia
ability may ressult from rando
om
measurement erro
ors or may be due to errors in observation
ns. In the biolo
ogical sciencees, however, th
he
sourrce of variation
n is viewed slig
ghtly differentlly since memb
bers of a popu
ulation differ grreatly. In othe
er
word
ds, biological variations
v
that we typically o
observe are intrinsic to the individual, orgaanism, or
biolo
ogical process.
The range is the difference
d
betw
ween the large
est and the sm
mallest value in
n a group of d ata and gives a
rough idea of the dispersion.
d
It sometimes
s
lea
ads to ambiguous results, however, whenn the maximum
m
and minimum valu
ues are not in line with the re
est of the data
a. The range will
w not be connsidered furthe
er.
The average dista
ance of all the hits from the b
bull's eye wou
uld serve as a convenient m
measure of the
scattter on the targ
get. The average spread abo
out the arithm
metic mean of a large series of weighings or
analyyses is the me
ean deviation δ of the popullation. The sum of the positive and negattive deviations
s
abou
ut the mean eq
quals zero; he
ence, the algeb
braic signs are
e disregarded to obtain a m
measure of the
dispe
ersion. The mean
m
deviation d for a samplle, that is, the deviation of an
a individual obbservation fro
om
the a
arithmetic mea
an of the samp
ple, is obtaine
ed by taking the difference between
b
each individual
value
e Xi and the arithmetic mean [X with bar a
above], adding
g the differenc
ces without reggard to the
algeb
braic signs, an
nd dividing the
e sum by the n
number of valu
ues to obtain the
t average. T
The mean
devia
ation of a sam
mple is express
sed as
in wh
hich Σ|Xi-[X with bar above]| is the sum off the absolute deviations fro
om the mean. The vertical lines
on either side of th
he term in the numerator ind
dicate that the
e algebraic sign of the deviaation should be
e
disre
egarded.
Youd
den6 discoura
ages the use of
o the mean de
eviation becau
use it gives a biased
b
estimat
ate that sugges
sts a
grea
ater precision than
t
actually exists
e
when a small numberr of values are
e used in the ccomputation.
Furth
hermore, the mean
m
deviation of small sub
bsets may be widely
w
scattered around thee average of th
he
estim
mates, and acccordingly, d is not particularrly efficient as a measure off precision.
22
Dr. Murtadha Alshareifi e-Library
The standard deviiation σ (the Greek
G
lowercasse letter sigma
a) is the squarre root of the m
mean of the
squa
ares of the devviations. This parameter is u
used to measu
ure the dispersion or variabbility of a large
number of measurrements, for example, the w
weights of the contents of se
everal million ccapsules. This
s set
of ite
ems or measurements appro
oximates the p
population and σ is, therefore, called the population
stand
dard deviation
n. Population standard
s
devia
ations are sho
own in Figure 1-1.
As p
previously note
ed, any finite group
g
of experrimental data may
m be consid
dered as a subbset or sample of
the p
population; the
e statistic or ch
haracteristic o
of a sample fro
om the univers
se used to exppress the varia
ability
of a subset and su
upply an estim
mate of the sta ndard deviatio
on of the popu
ulation is know
wn as the
P.10
0
samp
ple standard deviation
d
and is designated by the letter s.
s The formula
a is
For a small sample, the equation is written
The term (N - 1) iss known as the
e number of d
degrees of free
edom. It replac
ces N to reducce the bias of the
stand
dard deviation
n s, which on the
t average iss lower than th
he universe sta
andard deviatiion.
The reason for intrroducing (N - 1) is as follow
ws. When a sta
atistician selec
cts a sample aand makes a single
s
measurement or observation,
o
he
e or she obtai ns at least a rough
r
estimate
e of the mean of the parent
popu
ulation. This siingle observattion, however,, can give no hint
h as to the degree
d
of vari ability in the
popu
ulation. When a second mea
asurement is ttaken, howeve
er, a first basis
s for estimatinng the population
varia
ability is obtain
ned. The statis
stician states tthis fact by saying that two observations
o
ssupply one de
egree
of fre
eedom for estiimating variatiions in the uniiverse. Three values provide
e two degreess of freedom, four
f
value
es provide three degrees off freedom, and
d so on. There
efore, we do not have accesss to all N valu
ues
of a sample for ob
btaining an esttimate of the sstandard devia
ation of the population. Insteead, we must use 1
less than N, or (N - 1), as shown
n in equation ((1-9). When N is large, say N > 100, we ccan use N insttead
of (N
N - 1) to estima
ate the popula
ation standard deviation bec
cause the diffe
erence betweeen the two is
negliigible.
Modern statistical methods hand
dle small sam
mples quite well; however, th
he investigatorr should recog
gnize
that the estimate of
o the standard
d deviation be
ecomes less re
eproducible an
nd, on the aveerage, become
es
lowe
er than the pop
pulation standard deviation as fewer samples are used to compute thhe estimate.
How
wever, for manyy students stu
udying pharma
acy there is no
o compelling re
eason to view
w standard
devia
ation in highlyy technical term
ms. So, we willl simply refer to standard deviation as “S
SD” from this point
p
forwa
ard.
A sa
ample calculatiion involving the arithmetic mean, the me
ean deviation, and the estim
mate of the
stand
dard deviation
n follows.
Exa
ample 1-8
A ph
harmacist recceives a pres
scription for a patient with rheumatoid
d arthritis callling for seve
en
divid
ded powderss, each of wh
hich is to weig
gh 1.00 g. To
o check his skill
s in filling tthe powders
s, he
removes the con
ntents from each
e
paper a
after filling the
e prescription
n by the blocck-and-divide
e
meth
hod and then
n weighs the
e powders ca
arefully. The results of the
e weighings aare given in the
first column of Table 1-2; the
e deviations o
of each value
e from the arrithmetic meaan, disregard
ding
the ssign, are give
en in column
n 2, and the ssquares of th
he deviations
s are shown in the last
colu
umn. Based on
o the use off the mean d
deviation, the
e weight of th
he powders ccan be expre
essed
as 0
0.98 ± 0.046 g. The variability of a sin
ngle powder can
c also be expressed
e
inn terms of
perccentage deviation by divid
ding the mea
an deviation by the arithm
metic mean aand multiplyin
ng
23
Dr. Murtadha Alshareifi e-Library
by 100. The resu
ult is 0.98% ± 4.6%; of co
ourse, it inclu
udes errors due
d to removving the powd
ders
from
m the papers and weighin
ng the powde
ers in the ana
alysis.
Table 1-2 Statistical
S
Analysis
A
off Divided Powder Com
mpoundingg Techniquee
Weight of
Powd
der
Contentts (g)
Deviation (Sign
Ign
nored), |Xi-[X
- with Square
S
of thhe Deviatio
on,
bar abov
ve]|
(X
Xi-[X with bbar above]])2
1.00
0..02
0.0004
0.98
0..00
0.0000
1.00
0..02
0.0004
1.05
0..07
0.0049
0.81
0..17
0.0289
0.98
0..00
0.0000
1.02
0..04
0.0016
Total
Σ = 6.84
4
Σ = 0.32
Σ = 0.03622
Average
0.98
0..046
The standard deviiation is used more frequenttly than the mean deviation in research. F
For large sets of
data, it is approxim
mean deviatio
mately 25% larger than the m
on, that is, σ = 1.25δ.
Statisticians have estimated tha
at owing to cha
ance errors, about 68% of all
a results in a large set will fall
f
within one standarrd deviation on
n either side o
of the arithmettic mean, 95.5
5% within ±2 st
standard
devia
ations, and 99
9.7% within ±3
3 standard devviations, as se
een in Figure 1-1.
1
Gold
dstein7 selecte
ed 1.73δ as an
n equitable tollerance standa
ard for prescription productss, whereas
Saun
nders and Fleming8 advoca
ated the use o
of ±3σ as appro
oximate limits of error for a single result. In
pharrmaceutical wo
ork, it should be
b considered
d permissible to
t accept ±2s as a measuree of the variability
or “sspread” of the data in small samples.
s
The n, roughly 5%
% to 10% of the
e individual ressults will be
expe
ected to fall ou
utside this rang
ge if only chan
nce errors occ
cur.
The estimate of th
he standard de
eviation in Exa
ample 1-8 is ca
alculated as fo
ollows:
and ±2s is equal to
o ±0.156 g. Th
hat is, based u
upon the analy
ysis of this exp
periment, the pharmacist sh
hould
expe
ect that roughly 90% to 95%
% of the sample
e values would fall within ±0
0.156 g of the sample mean
n.
24
Dr. Murtadha Alshareifi e-Library
The smaller the sttandard deviattion estimate ((or the mean deviation),
d
the
e more precisee is the operation.
In the filling of cap
psules, precisio
on is a measu
ure of the abilitty of the pharm
macist to put tthe same amo
ount
of drrug in each capsule and to reproduce
r
the result in subs
sequent opera
ations. Statisticcal techniques
s for
predicting the prob
bability of occu
urrence of a
P.11
speccific deviation in future opera
ations, althoug
gh important in pharmacy, require
r
methodds that are ou
utside
the sscope of this book.
b
The interested readerr is referred to treatises on statistical
s
analyysis.
Whe
ereas the average deviation and the stand
dard deviation
n can be used as measures of the precisio
on of
a me
ethod, the diffe
erence betwee
en the arithme
etic mean and the true or ab
bsolute value eexpresses the
e
errorr that can often be used as a measure of the accuracy of the method
d.
The true or absolu
ute value is ord
dinarily regard
ded as the universe mean µ—that
µ
is, the mean for an
infiniitely large set—
—because it is
s assumed tha
at the true value is approached as the saample size
beco
omes progresssively larger. The
T universe m
mean does no
ot, however, co
oincide with thhe true value of
o the
quan
ntity measured
d in those case
es in which de
eterminate errrors are inhere
ent in the meas
asurements.
The difference bettween the sam
mple arithmeticc mean and th
he true value gives
g
a measuure of the accu
uracy
of an
n operation; it is known as th
he mean errorr.
In Exxample 1-8, th
he true value is
s 1.00 g, the a
amount requested by the ph
hysician. The aapparent error
invollved in compo
ounding this prrescription is
in wh
hich the positivve sign signifies that the tru
ue value is gre
eater than the mean value. A
An analysis off
these
e results show
ws, however, that
t
this differe
ence is not sta
atistically signiificant but rathher is most like
ely
due to accidental errors.
e
Hence, the accuracyy of the operattion in Example 1-8 is sufficciently great th
hat no
syste
emic error can
n be presumed
d. However, o
on further analy
ysis it is found
d that one or sseveral results
s are
quesstionable. Thiss possibility is considered la
ater. If the arith
hmetic mean inExample 1-88 were 0.90 ins
stead
of 0.98, the differe
ence could be stated with asssurance to ha
ave statistical significance bbecause the
prob
bability that succh a result cou
uld occur by cchance alone would
w
be small.
The mean error in this case is
y dividing the m
mean error by
y the true value
e. It can be exxpressed as a
The relative error is obtained by
ultiplying by 10
00 or in parts p
per thousand by
b multiplying by 1000. It is easier to com
mpare
percentage by mu
eral sets of ressults by using the relative errror rather than the absolute
e mean error. T
The relative error
e
seve
in the
e case just citted is
e for a result to
The reader should
d recognize tha
at it is possible
t be precise without
w
being accurate, thatt is, a
consstant error is present.
p
If the capsule
c
conte
ents in Example 1-8 had yielded an averaage weight of 0.60
0
g witth a mean devviation of 0.5%
%, the results w
would have be
een accepted as precise. Thhe degree of
accu
uracy, howeve
er, would have
e been low beccause the average weight would
w
have difffered from the
e true
value
e by 40%. Con
nversely, the fact
f
that the re
esult may be accurate
a
does not necessarrily mean that it is
also precise. The situation can arise
a
in which the mean value is close to the true valuee, but the scatter
due to chance is la
g8 observed, “it is better to be roughly acccurate than
arge. Saunderrs and Fleming
precisely wrong.”
A stu
udy of the individual values of a set often throws additio
onal light on th
he exactitude of the
comp
pounding operations. Returrning to the da
ata of Example
e 1-8 (Table 1-2), we note oone rather
disco
ordant value, namely,
n
0.81 g.
g If the arithm
metic mean is recalculated ig
gnoring this m
measurement, we
obtain a mean of 1.01
1
g. The mean deviation without the doubtful result is 0.02 g. It is now seen tha
at the
diverrgent result is 0.20 g smalle
er than the new
w average or, in other words, its deviationn is 10 times
grea
ater than the mean
m
deviation
n. A deviation greater than four
f
times the mean deviatioon will occur purely
p
by ch
hance only ab
bout once or tw
wice in 1000 m
measurements
s; hence, the discrepancy
d
inn this case is
25
Dr. Murtadha Alshareifi e-Library
prob
bably caused by
b some definite error in tecchnique. Statis
sticians rightly question this rule, but it is a
usefu
ul though not always reliable criterion for finding discre
epant results.
Having uncovered
d the variable weight
w
among
g the units, one
e can proceed
d to investigatee the cause off the
determinate error. The pharmac
cist may find th
hat some of th
he powder was
s left on the siides of the mo
ortar
or on
n the weighing
g paper or pos
ssibly was lostt during trituration. If several of the powdeer weights dev
viated
wide
ely from the me
ean, a serious
s deficiency in the compoun
nder's techniqu
ue would be suuspected. Suc
ch
apprraisals as thesse in the colleg
ge laboratory w
will aid the stu
udent in locatin
ng and correct
cting errors and will
help the pharmacist become a safe
s
and proficcient compoun
nder before en
ntering the praactice of
pharrmacy.
The CV is a dimen
nsionless para
ameter that is quite useful. The
T CV relates
s the standardd deviation to the
mean and is defined as
en the mean is
s nonzero. It iss also commonly reported as
a a percentagge (%CV is CV
V
It is vvalid only whe
multiiplied by 100). For example, if SD = 2 and
d mean = 3, th
hen the %CV is 67%. The C
CV is useful
beca
ause the stand
dard deviation of data must always be understood in the context of thhe mean of the
results. The CV sh
hould be used instead of the
e standard dev
viation to asse
ess the differeence between data
sets with dissimila
ar units or very
y different mea
ans.
Vis
sualizing Results:: Graphic
c Method
ds, Lines
Scientists are not usually so forttunate as to b
begin each pro
oblem with an equation at haand relating th
he
varia
ables under study. Instead, the investigato
or must collec
ct raw data and
d
P.12
2
put them in the forrm of a table or
o graph to bettter observe th
he relationship
ps. Constructinng a graph witth
the d
data plotted in a manner so as to form a ssmooth curve often permits the investigattor to observe the
relationship more clearly and pe
erhaps will allo
ow expression
n of the connection in the forrm of a
math
hematical equation. The pro
ocedure of obttaining an emp
pirical equation from a plot oof the data is
know
wn as curve fittting and is tre
eated in bookss on statistics and
a graphic analysis.
The magnitude of the independe
ent variable iss customarily measured
m
alon
ng the horizonntal coordinate
e
scale
e, called the x axis. The dep
pendent variab
ble is measure
ed along the vertical
v
scale, oor the y axis. The
data are plotted on
n the graph, and
a a smooth lline is drawn through
t
the po
oints. The x vaalue of each po
oint
is kn
nown as the xccoordinate or the
t abscissa; the y value is known as the
e y coordinate or the ordinatte.
The intersection of the x axis an
nd the y axis iss referred to as
a the origin. The
T x and yvallues may be either
e
nega
ative or positivve.
We w
will first go through some off the technical aspects of lin
nes and linear relationships.. The simplestt
relationship betwe
een two variab
bles, in which tthe variables contain
c
no exp
ponents other than 1, yields
sa
straig
ght line when plotted using rectangular co
oordinates. Th
he straight-line
e or linear relaationship is
exprressed as
26
Dr. Murtadha Alshareifi e-Library
inn which y iss the depend
dent variablle, x is the in
ndependent variable,
annd a and b are
a constantts. The consstant b is th
he slopeof th
he line; the ggreater the
vvalue of b, thhe steeper iss the slope. It is expresssed as the change
c
in y w
with the
chhange in x, or
is also the ttangent of the
t angle thaat the line m
makes with
thhe x axis. Thhe slope maay be positivve or negatiive depending on wheth
ther the line
sllants upwarrd or downw
ward to the rright, respecctively. Wh
hen b = 1, thhe line makees
ann angle of 45°
4 with thee x axis and the equatio
on of the line can be wrritten as
foollows:
Whe
en b = 0, the lin
ne is horizonta
al (in other wo
ords, parallel to
o the x axis), and
a the equattion reduces to
o
The constant a is known as the y intercept an
nd denotes the
e point at whic
ch the line crossses the y axis.
If a iss positive, the
e line crosses the
t y axis abo
ove the x axis; if it is negativ
ve, the line inteersects the y axis
a
below
w the x axis. When
W
a is zero
o, equation (1 -11) may be written
w
as
and the line passe
es through the origin.
The results of the determination
n of the refracttive index of a benzene solu
ution containinng increasing
conccentrations of carbon tetrach
hloride are sho
own in Table 1-3 The data are
a plotted in F
Figure 1-2 and are
seen
n to produce a straight line with
w a negative
e slope. The equation
e
of the
e line may be obtained by using
u
the two-point form
m of the linear equation
e
Tab
ble 1-3 Refrractive Ind
dices of Mixxtures of Beenzene and
d Carbon T
Tetrachlorid
de
Concentratioon of CCl4 (x) (Volum
me %)Refra
active Index (y)
10.0
1.4
497
25.0
1.4
491
33.0
1.4
488
50.0
1.4
481
60.0
1.4
477
The method involvves selecting two
t
widely sep
parated points
s (x1, y1) and (x2, y2) on the line and
subsstituting into th
he two-point eq
quation.
27
Dr. Murtadha Alshareifi e-Library
Exa
ample 1-9
Refe
erring to Figu
ure 1-2, let 10.0% be x1 a
and its corres
sponding y value
v
1.497 bbe y1; let 60.0
0%
be x2 and let 1.4
477 be y2. The equation th
hen become
es
The value -4.00 × 10-4 is the slo
ope of the stra ight line and corresponds
c
to
o b in equationn (1-11). A
ative value for b indicates th
hat y decrease
es with increas
sing values of x, as observeed in Figure 1--2.
nega
The value 1.501 iss the y intercept and corresp
ponds to a in equation
e
(1-11
1). It can be obbtained
3
P.13
from the plot in Fig
gure 1-2 by ex
xtrapolating (e
extending) the line upward to
o the left until it intersects
the y axis. It will allso be observe
ed that
Fig.. 1-2. Refractive index of the systeem benzenee–carbon tettrachloride aat 20°C.
Table 1-44 Emulsion Stability aas a Functio
on of Emullsifier Conccentration
28
Dr. Murtadha Alshareifi e-Library
Emulsifier (x) (%
Con
ncentration
n)
Oil Separation
n (y)
((mL/month
h)
Log
garithm off Oil
Sep
paration (loog y)
0.50
5.10
0.708
0
1.00
3.60
0.556
0
1.50
2.60
0.415
0
2.00
2.00
0.301
0
2.50
1.40
0.146
0
3.00
1.00
0.000
0
and this simple forrmula allows one
o to computte the slope off a straight line
e.
2
Not a
all experimenttal data form straight
s
lines. E
Equations con
ntaining x or y2 are known aas seconddegrree or quadrattic equations, and
a graphs off these equatio
ons yield para
abolas, hyperbbolas, ellipses,, and
circle
es. The graphs and their corresponding e
equations can be found in sttandard textboooks on analytic
geom
metry.
Loga
arithmic relatio
onships occur frequently in sscientific work
k. Data relating
g the amount oof oil separating
from an emulsion per month (de
ependent varia
able, y) as a fu
unction of the emulsifier conncentration
ependent varia
able, x) are co
ollected in Tab
ble 1-4.
(inde
The data from thiss experiment may
m be plotted
d in several wa
ays. In Figure 1-3, the oil seeparation y is
plotte
ed as ordinate
e against the emulsifier
e
conccentration x as abscissa on a rectangularr coordinate grid.
In Figure 1-4, the logarithm of th
he oil separati on is plotted against
a
the concentration. Inn Figure 1-5, the
t
data are plotted ussing semilogarithmic scale, consisting of a logarithmic scale on the vvertical axis an
nd a
linea
ar scale on the
e horizontal ax
xis.
29
Dr. Murtadha Alshareifi e-Library
Fig.. 1-3. Emulssion stabilitty data plottted on a rectangular coordinate griid.
30
Dr. Murtadha Alshareifi e-Library
Fig.. 1-4. A plot of the logaarithm of oiil separation
n of an emu
ulsion versuss concentration
on a rectangulaar grid.
Altho
ough Figure 1-3 provides a direct reading
g of oil separattion, difficulties arise when oone attempts to
draw
w a smooth line
e through the points or to exxtrapolate the curve beyond
d the experimeental data.
Furth
hermore, the equation
e
for th
he curve canno
ot be obtained
d readily from Figure 1-3. W
When the logarrithm
of oill separation iss plotted as the
e ordinate, as in
P.14
4
Figure 1-4, a straig
ght line results
s, indicating th
hat the phenom
menon follows
s a logarithmicc or exponentiial
relationship. The slope
s
and the y intercept are
e obtained from the graph, and
a the equattion for the line
e is
subssequently foun
nd by use of th
he two-point fo
ormula:
31
Dr. Murtadha Alshareifi e-Library
Fig.. 1-5. Emulssion stabilitty plotted onn a semilogarithmic griid.
Figure 1-4 requires that we obta
ain the logarith
hms of the oil--separation da
ata before the graph is
consstructed and, conversely,
c
tha
at we obtain th
he antilogarith
hm of the ordin
nates to read ooil separation from
the g
graph. These inconvenience
i
es of convertin
ng to logarithm
ms and antilogarithms can bbe overcome by
b
plottiing on a semillogarithmic sc
cale. The x and
d y values of Table
T
1-4 are plotted directlyy on the graph
h to
yield
d a straight line
e, as seen in Figure
F
1-5. Altthough such a plot ordinarily
y is not used tto obtain the
equa
ation of the line, it is conven
nient for readin
ng the oil sepa
aration directly
y from the grapph. It is well to
o
reme
ember that the
e ln of a number is simply 2..303 times the
e log of the number. Therefoore, logarithmic
grap
ph scales may be used for ln
n as well as fo
or log plots. In fact, today the
e natural logarrithm is more
comm
monly used th
han the base 10
1 log.
Not e
every pharmacy student will have the nee
ed to calculate
e slopes and in
ntercepts of linnes. In fact, once
the b
basics are und
derstood many
y of these ope
erations can be
e performed quite
q
easily witth modern
calcu
ulators. Howevver, every pha
armacy studen
nt should at le
east be able to
o look at a visuual representa
ation
of da
ata and get so
ome sense of what
w
it is tellin
ng you and wh
hy it is important. For exampple, the slope of
o a
plasm
ma drug concentration versus time curve
e is an approximation of the ratio of the inpput rate and th
he
output rate of the drug
d
in the body at that partticular point in
n time (Fig. 1-6
6). At any giveen time point on
o
that curve, the rate
e of change off drug in the b
body is equal to
t the rate of absorption
a
(inpput) minus the
e rate
of eliimination (output or remova
al from the bod
dy). When the two rate proc
cesses are equual, the overall
32
Dr. Murtadha Alshareifi e-Library
slope
e of the curve is zero. This is a very impo
ortant (x,y) point in pharmac
cokinetics becaause it is the time
t
pointt where the pe
eak blood leve
els occur (Cmaxx, tmax). The ra
ate of absorption is greater tthan the rate of
o
elimiination to left of
o the vertical line in Figure 1-6. This is ca
alled the abso
orption phase. When the rate of
elimiination is grea
ater than the ra
ate of absorpt ion, it is called
d the eliminatio
on phase. Thi s region falls to
t
the rright of the verrtical line. The
e steepness off the slope is an
a indicator of the rate. For example, in Figure
F
1-7, three hypothe
etical products
s of the same d
drug are show
wn. As you can
n easily see, thhe rate of
abso
orption of the drug
d
into the bloodstream
b
o
occurs most qu
uickly from Pro
oduct 1 and m
most slowly from
Prod
duct 2 since th
he slope of the
e absorption ph
hase is steepe
est for Produc
ct 1.
Fig.. 1-6. A plot of the abso
orption andd elimination
n phases of a typical pllasma drug
conccentration versus
v
time curve. Duriing the abso
orption phasse, the rate oof input of drug
d
intoo the body iss greater thaan the rate oof output (or eliminatio
on) of drug ffrom the bo
ody.
Thee opposite iss true during
g the eliminnation phasee. At the peaak point (Tmmax, Cmax), th
he
rates of absorpttion and elim
mination arre equal.
33
Dr. Murtadha Alshareifi e-Library
Fig.. 1-7. A plot of plasma drug conceentration versus time fo
or three diffferent produ
ucts
conttaining the same dose of
o a drug. D
Differences in
i the profilles are due tto differencces
in thhe rate of abbsorption reesulting from
m the three types of forrmulations. The slope of
o
the absorption phase
p
is equ
uivalent to tthe rate of drug
d
absorption. The stteeper slopee
(Prooduct A) equuals a fasterr rate of abssorption, wh
hereas a less steep sloppe (Product C)
has a slower abbsorption rate.
Lin
near Regrression Analysis
A
The data given in Table 1-3 and
d plotted in Fig
gure 1-2 clearlly indicate the existence of a linear
relationship betwe
een the refracttive index and the volume percent of carb
bon tetrachloridde in benzene
e.
The straight line th
hat joins virtua
ally all the poin
nts can be dra
awn readily on the figure by sighting the points
p
along
g the edge of a ruler and drrawing a line th
hat can be exttrapolated to the
t y axis withh confidence.
P.15
5
Tab
ble 1-5 Refrractive Ind
dices of Mixxtures of Beenzene and
d Carbon T
Tetrachlorid
de
Concentratioon of CCl4 (x) (Volum
me %)Refra
active Index (y)
10.0
1.4
497
26.0
1.4
493
33.0
1.4
485
50.0
1.4
478
61.0
1.4
477
34
Dr. Murtadha Alshareifi e-Library
Let u
us suppose, however, that the person wh o prepared the solutions an
nd carried out the refractive
indexx measurements was not sk
killed and, as a result of poo
or technique, allowed
a
indeteerminate errorrs to
appe
ear. We might then be prese
ented with the
e data given in
n Table 1-5. When
W
these datta are plotted on
grap
ph paper, an appreciable sca
atter is observved (Fig. 1-8) and we are un
nable, with anyy degree of
confiidence, to draw the line thatt expresses th
he relation bettween refractiv
ve index and cconcentration. It is
here
e that we mustt employ better means of an
nalyzing the av
vailable data.
The first step is to determine wh
hether the data
a in Table 1-5
5 should fit a straight line, annd for this we
calcu
ulate the corre
elation coefficiient, r, using th
he following equation:
Whe
en there is perffect correlation between the
e two variables
s (in other words, a perfect linear
relationship), r = 1. When the tw
wo variables arre completely independent, r= 0. Dependding on the
degrrees of freedom
m and the cho
osen probabilitty level, it is possible to calc
culate values oof r above which
there
e is significantt correlation an
nd below whicch there is no significant corrrelation. Obviiously, in the latter
case
e, it is not profiitable to proce
eed further witth the analysis
s unless the da
ata can be plootted in some other
o
way that will yield a linear relatio
on. An examp
ple of this is sh
hown in Figure
e 1-4, in whichh a linear plot is
obtained by plottin
ng the logarithm of oil separration from an emulsion aga
ainst emulsifierr concentratio
on, as
oppo
osed to Figure
e 1-3, in which the raw data are plotted in the conventio
onal manner.
35
Dr. Murtadha Alshareifi e-Library
Fig.. 1-8. Slopee, intercept, and equatioon of line fo
or data in Taable 1-5 calcculated by
regrression anallysis.
Assu
uming that the
e calculated va
alue of r showss a significantt correlation be
etween x and y, it is then
nece
essary to calcu
ulate the slope
e and intercep
pt of the line us
sing the equattion
inn which is
i the regresssion coefficcient, or slo
ope. By subsstituting thee value
foor b in equaation (1-18), we can obbtain the yin
ntercept:
The following serie
es of calculatio
ons, based on
n the data in Table
T
1-5, will illustrate the uuse of these
equa
ations.
Exa
ample 1-10
Usin
ng the data in
n Table 1-5, calculate the
e correlation coefficient, the
t regressioon coefficientt,
and the intercep
pt on the y ax
xis.
Examination of equations
e
(1--17) through (1-19) show
ws the various
s values we must calcula
ate,
and these are se
et up as follo
ows:
36
Dr. Murtadha Alshareifi e-Library
Subsstituting the re
elevant values into equation
n (1-17) gives
From
m equation (1--18)
and ffinally, from equation (1-19))
P.16
6
Interrcept on the y axis = 1.486
Note
e that for the in
ntercept, we place x equal to
ation (1-17). By
B inserting ann actual value
o zero in equa
of x into equation (1-19), we obttain the value of y that shou
uld be found at that particulaar value of x. Thus,
T
n x = 10,
when
The value agrees with the experimental value
e, and hence this
t
point lies on
o the statisticcally calculate
ed
slope
e drawn in Fig
gure 1-8.
Cha
apter Summ
mary
Mosst of the statisstical calcula
ations review
wed in this ch
hapter will be performed uusing a
calcculator or com
mputer. The objective
o
of tthis chapter was not to in
nundate you with statistic
cal
37
Dr. Murtadha Alshareifi e-Library
formulas or complex equations but rather to give the student a perspective on analyzing data
as well as providing a foundation for the interpretation of results. Numbers alone are not
dynamic and do not give a sense of the behavior of the results. In some situations, equations
or graphic representations were used to give the more advanced student a sense of the
dynamic behavior of the results.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e
References
1. D. L. Sackett, S. E. Straus, W. S. Richardson, W. Rosenberg, and R. B. Haynes, Evidence-Based
Medicine: How to Practice and Teach EBM, 2nd Ed., Churchill Livingstone, Edinburgh, New York, 2000.
2. K. Skau, Am. J. Pharm. Educ. 71, 11, 2007.
3. S. J. Ruberg, Teaching statistics for understanding and practical use. Biopharmaceutical Report,
American Statistical Association, 1, 1992, pp. 14.
4. S. S. Stevens, Science, 103, 677–680, 1946.
5. E. A. Brecht, in Sprowls' American Pharmacy, L. W. Dittert, Ed., 7th Ed., Lippincott Williams &
Wilkins, Philadelphia, 1974, Chapter 2.
6. W. J. Youden, Statistical Methods for Chemists, R. Krieger, Huntington, New York, 1977, p. 9.
7. P. Rowe, in Essential Statistics for the Pharmaceutical Sciences, John Wiley & Sons, Ltd, West
Sussex, England, 2007.
8. S. Bolton, Pharmaceutical Statistics: Preclinical and Clinical Applications, 3rd Ed., Marcel Dekker,
Inc., New York, 1997.
Recommended Readings
S. Bolton, Pharmaceutical Statistics: Preclinical and Clinical Applications, 3rd Ed., Marcel Dekker, Inc.,
New York, 1997.
P. Rowe, Essential Statistics for the Pharmaceutical Sciences, John Wiley & Sons, Inc., West Sussex,
England, 2007.
Chapter Legacy
Fifth Edition: published as Chapter 1 (Introduction). Updated by Patrick Sinko.
Sixth Edition: published as Chapter 1 (Interpretive Tools). Updated by Patrick Sinko.
38
Dr. Murtadha Alshareifi e-Library
2 States of Matter
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Understand the nature of the intra- and intermolecular forces that are involved in
stabilizing molecular and physical structures.
2. Understand the differences in these forces and their relevance to different types
molecules.
3. Discuss supercritical states to illustrate the utility of supercritical fluids for
crystallization and microparticulate formulations.
4. Appreciate the differences in the strengths of the intermolecular forces that are
responsible for the stability of structures in the different states of matter.
5. Perform calculations involving the ideal gas law, molecular weights, vapor pressure,
boiling points, kinetic molecular theory, van der Waals real gases, the Clausius–
Clapeyron equation, heats of fusion and melting points, and the phase rule equations.
6. Understand the properties of the different states of matter.
7. Describe the pharmaceutical relevance of the different states of matter to drug
delivery systems by reference to specific examples given in the text boxes.
8. Describe the solid state, crystallinity, solvates, and polymorphism.
9. Describe and discuss key techniques utilized to characterize solids.
10. Recognize and elucidate the relationship between differential scanning calorimetry,
thermogravimetric, Karl Fisher, and sorption analyses in determining polymorphic
versus solvate detection.
11. Understand phase equilibria and phase transitions between the three main states of
matter.
12. Understand the phase rule and its application to different systems containing multiple
components.
Binding Forces Between Molecules
For molecules to exist as aggregates in gases, liquids, and solids, intermolecular forces must exist. An
understanding of intermolecular forces is important in the study of pharmaceutical systems and follows
logically from a detailed discussion of intramolecular bonding energies. Like intramolecular bonding
energies found in covalent bonds, intermolecular bonding is largely governed by electron orbital
interactions. The key difference is that covalency is not established in theintermolecular state. Cohesion,
or the attraction of like molecules, and adhesion, or the attraction of unlike molecules, are
manifestations of intermolecular forces. Repulsion is a reaction between two molecules that forces them
apart. For molecules to interact, these forces must be balanced in an energetically favored arrangement.
Briefly, the term energetically favored is used to describe the intermolecular distances and
intramolecular conformations where the energy of the interaction is maximized on the basis of the
balancing of attractive and repulsive forces. At this point, if the molecules are moved slightly in any
direction, the stability of the interaction will change by either a decrease in attraction (when moving the
molecules away from one another) or an increase in repulsion (when moving the molecules toward one
another).
Knowledge of these forces and their balance (equilibrium) is important for understanding not only the
properties of gases, liquids, and solids, but also interfacial phenomena, flocculation in suspensions,
stabilization of emulsions, compaction of powders in capsules, dispersion of powders or liquid droplets
in aerosols, and the compression of granules to form tablets. With the rapid increase in biotechnologyderived products, it is important to keep in mind that these same properties are strongly involved in
influencing biomolecular (e.g., proteins, DNA) secondary, tertiary, and quaternary structures, and that
these properties have a profound influence on the stability of these products during production,
formulation, and storage. Further discussion of biomolecular products will be limited in this text, but
39
Dr. Murtadha Alshareifi e-Library
correlations hold between small-molecule and the larger biomolecular therapeutic agents due to the
universality of the physical principles of chemistry.
Repulsive and Attractive Forces
When molecules interact, both repulsive and attractive forces operate. As two atoms or molecules are
brought closer together, the opposite charges and binding forces in the two molecules are closer
together than the similar charges and forces, causing the molecules to attract one another. The
negatively charged electron clouds of the molecules largely govern the balance (equilibrium) of forces
between the two molecules. When the molecules are brought so close that the outer charge clouds
touch, they repel each other like rigid elastic bodies.
Thus, attractive forces are necessary for molecules to cohere, whereas repulsive forces act to prevent
the molecules from interpenetrating and annihilating each other.
P.18
Moelwyn-Hughes1 pointed to the analogy between human behavior and molecular phenomena: Just as
the actions of humans are often influenced by a conflict of loyalties, so too is molecular behavior
governed by attractive and repulsive forces.
Repulsion is due to the interpenetration of the electronic clouds of molecules and increases
exponentially with a decrease in distance between the molecules. At a certain equilibrium distance,
about (3-4) × 10-8 cm (3–4 Å), the repulsive and attractive forces are equal. At this position, the potential
energy of the two molecules is a minimum and the system is most stable (Fig. 2-1). This principle of
minimum potential energy applies not only to molecules but also to atoms and to large objects as well.
The effect of repulsion on the intermolecular three-dimensional structure of a molecule is well illustrated
in considering the conformation of the two terminal methyl groups in butane, where they are
energetically favored in the transconformation because of a minimization of the repulsive forces. It is
important to note that the arrangement of the atoms in a particular stereoisomer gives
the configuration of a molecule. On the other hand, conformation refers to the different arrangements of
atoms resulting from rotations about single bonds.
40
Dr. Murtadha Alshareifi e-Library
Fig.. 2-1. Repullsive and atttractive eneergies and net
n energy ass a functionn of the distaance
betw
ween moleccules. Note that
t a minim
mum occurss in the net energy
e
becaause of the
diffe
ferent characcter of the attraction
a
annd repulsion
n curves.
The various types of attractive intermolecularr forces are dis
scussed in the
e following subbsections.
Key Co
oncept
Van
n Der Waals
s Forces
41
Dr. Murtadha Alshareifi e-Library
van der Waal interactions are weak forces that involve the dispersion of charge across a
molecule called a dipole. In a permanent dipole, as illustrated by the peptide bond, the
electronegative oxygen draws the pair of electrons in the carbon–oxygen double bond closer
to the oxygen nucleus. The bond then becomes polarized due to the fact that the oxygen
atom is strongly pulling the nitrogen lone pair of electrons toward the carbon atom, thus
creating a partial double bond. Finally, to compensate for valency, the nucleus of the nitrogen
atom pulls the electron pair involved in the nitrogen–hydrogen bond closer to itself and
creates a partial positive charge on the hydrogen. This greatly affects protein structure, which
is beyond the scope of this discussion. In Keesom forces, the permanent dipoles interact with
one another in an ionlike fashion. However, because the charges are partial, the strength of
bonding is much weaker. Debye forces show the ability of a permanent dipole to polarize
charge in a neighboring molecule. In London forces, two neighboring neutral molecules, for
example, aliphatic hydrocarbons, induce partial charge distributions. If one conceptualizes the
aliphatic chains in the lipid core of a membrane like a biologic membrane or a liposome, one
can imagine the neighboring chains in the interior as inducing a network of these partial
charges that helps hold the interior intact. Without this polarization, the membrane interior
would be destabilized and lipid bilayers might break down. Therefore, London forces give rise
to the fluidity and cohesiveness of the membrane under normal physiologic conditions.
Van der Waals Forces
Van der Waals forces relate to nonionic interactions between molecules, yet they involve charge–charge
interactions (see Key Concept Box on van der Waals Forces). In organic
P.19
chemistry, numerous reactions like nucleophilic substitutions are introduced where one molecule may
carry a partial positive charge and be attractive for interaction with a partially negatively charged
nucleophilic reactant. These partial charges can be permanent or be induced by neighboring groups,
and they reflect the polarity of the molecule. The converse can be true for electrophilic reactants. The
presence of these polarities in molecules can be similar to those observed with a magnet. For example,
dipolar molecules frequently tend to align themselves with their neighbors so that the negative pole of
one molecule points toward the positive pole of the next. Thus, large groups of molecules may be
associated through weak attractions known as dipole–dipole or Keesom forces. Permanent dipoles are
capable of inducing an electric dipole in nonpolar molecules (which are easily polarizable) to
produce dipole-induced dipole, or Debye, interactions, and nonpolar molecules can induce polarity in
one another by induced dipole-induced dipole, or London, attractions. This latter force deserves
additional comment here.
The weak electrostatic force by which nonpolar molecules such as hydrogen gas, carbon tetrachloride,
and benzene attract one another was first recognized by London in 1930. The dispersion or London
force is sufficient to bring about the condensation of nonpolar gas molecules so as to form liquids and
solids when molecules are brought quite close to one another. In all three types of van der Waals forces,
the potential energy of attraction varies inversely with the distance of separation, r, raised to the sixth
power, r6. The potential energy of repulsion changes more rapidly with distance, as shown in Figure 2-1.
This accounts for the potential energy minimum and the resultant equilibrium distance of separation, re.
A good conceptual analogy to illustrate this point is the interaction of opposite poles of magnets (Fig. 22). If two magnets of the same size are slid on a table so that the opposite poles completely overlap, the
resultant interaction is attractive and the most energetically favored configuration (Fig. 2-2a). If the
magnets are slid further so that the poles of each slide into like-pole regions of the other (Fig. 2-2b), this
leads to repulsion and a force that pushes the magnetic poles back to the energetically favored
configuration (Fig. 2-2a). However, it must be noted that attractive (opposite-pole overlap) and repulsive
(same-pole overlap) forces coexist. If the same-charged poles are slid into the proximity of one another,
the resultant force is complete repulsion (Fig. 2-2c).
42
Dr. Murtadha Alshareifi e-Library
Fig.. 2-2. (a) Thhe attractivee, (b) partiallly repulsive, and (c) fu
ully repulsivve interactio
ons
of tw
wo magnetss being brou
ught togetheer.
Thesse several classses of interac
ctions, known as van der Waals
W
forces* and
a listed in Taable 2-1, are
asso
ociated with the condensatio
on of gases, th
he solubility off some drugs, the formationn of some meta
al
comp
plexes and mo
olecular additiion compound
ds, and certain
n biologic proc
cesses and druug actions. Th
he
enerrgies associate
ed with primarry valence bon
nds are includ
ded for comparrison.
Orb
bital Ove
erlap
An im
mportant dipolle–dipole force
e is the interacction between
n pi-electron orrbitals in systeems. For exam
mple,
arom
matic–aromaticc interactions can occur whe
en the double-bonded pi-orbitals from thee two rings ove
erlap
(Fig. 2-3).2 Aroma
atic rings are dipolar
d
in natu re, having a partial
p
negative
e charge in thee pi-orbital ele
ectron
cloud
d above and below
b
the ring and partial po
ositive charges residing at the equatorial hydrogens, as
s
illusttrated in Figure 2-3a. Therefore, a dipole–
–dipole interac
ction can occu
ur between twoo aromatic
mole
ecules. In fact,, at certain geo
ometries arom
matic–aromatic
c interaction can stabilize innterand/or intramolecu
ular interaction
ns (Fig. 2-3b a
and c), with the highest energy interactionns occurring when
w
the rrings are nearlly perpendicullar to one ano
other.2 This ph
henomena has
s been largelyy studied in
prote
eins, where stacking of 50%
% to 60% of the
e aromatic sid
de chains can often add stabbilizing energy
y to
seco
ondary and terrtiary structure
e (intermolecu lar) and may even
e
participa
ate in stabilizinng quaternary
interractions (intram
molecular). Aromatic stackin
ng can also oc
ccur in the solid state, and w
was first identiified
as a stabilizing forrce in the struc
cture of small organic crysta
als.
It is iimportant to point out that due
d to the natu
ure of these in
nteractions, rep
pulsion is alsoo very plausible
and can be destab
bilizing if the balancing
b
attra
active force is changed. Fina
ally, lone pairss of electrons on
atom
ms like oxygen
n can also interact with arom
matic pi orbitals and lead to attractive or reepulsive
interractions. These interactions are dipole–di pole in nature
e and they are introduced too highlight their
impo
ortance. Stude
ents seeking additional
a
inforrmation on this
s subject shou
uld read the exxcellent review
w by
Meye
er et al.3
P.20
0
43
Dr. Murtadha Alshareifi e-Library
Table 2-1 Intermolecular Forces and Valence Bonds
Bond Energy
(approximately) (kcal/mole)
Bond Type
Van der Waals forces and other intermolecular
attractions
Dipole–dipole interaction, orientation effect,
or Keesom force
Dipole-induced dipole interaction, induction
effect, or Debye force
1–10
Induced dipole–induced dipole interaction,
dispersion effect, or London force
Ion–dipole interaction
Hydrogen bonds: O—H· · ·O
6
C—H· · ·O
2–3
O—H· · ·N
4–7
N—H· · ·O
2–3
F—H· · ·F
7
Primary valence bonds
Electrovalent, ionic, heteropolar
100–200
Covalent, homopolar
50–150
Ion–Dipole and Ion-Induced Dipole Forces
In addition to the dipolar interactions known as van der Waals forces, other attractions occur between
polar or nonpolar molecules and ions. These types of interactions account in part for the solubility of
ionic crystalline substances in water; the cation, for example, attracts the relatively negative oxygen
44
Dr. Murtadha Alshareifi e-Library
atom
m of water and
d the anion attrracts the hydro
rogen atoms of
o the dipolar water
w
moleculees. Ion-induce
ed
dipolle forces are presumably
p
involved in the fformation of th
he iodide complex,
Fig.. 2-3. Schem
matic depictting (a) the dipolar natu
ure of an aromatic ringg, (b) its
prefferred anglee for aromattic–aromaticc interaction
ns between 60° and 90°°, and (c) th
he
less preferred planar
p
interaaction of aroomatic rings. Although
h typically ffound in
protteins, these interactionss can stabiliize states off matter as well.
w See thee excellent
reviiew on this subject with
h respect to biologic reecognition by
b Meyer et al.3
Reacction (2-1) acccounts for the solubility of io
odine in a solution of potassium iodide. Thhis effect can
clearrly influence th
he solubility off a solute and may be important in the dis
ssolution proceess.
Ion
n–Ion Inte
eractions
s
Anotther importantt interaction that involves ch
harge is the ion
n–ion interactiion. An ionic, eelectrovalent bond
betw
ween two coun
nter ions is the
e strongest bon
nding interaction and can persist over thee longest dista
ance.
How
wever, weaker ion–ion intera
actions, in partticular salt form
mations, exist and influencee pharmaceutical
syste
ems. This secction focuses on
o those weakker ion–ion inte
eractions. The
e ion–ion intera
ractions of saltts
and salt forms havve been widely
y discussed in
n prerequisite general chemistry and orgaanic chemistry
y
courrses that utilize
e this text, butt they will brieffly be reviewed here.
It is w
well establishe
ed that ions fo
orm because o
of valency cha
anges in an ato
om. At neutrallity, the numbe
er of
proto
ons and the nu
umber of electtrons in the attom are equal. Imbalance in
n the ratio of pprotons to neuttrons
givess rise to a cha
ange in charge
e state, and th e valency will dictate wheth
her the speciess is cationic or
anion
nic. Ion–ion in
nteractions are
e normally view
wed from the standpoint
s
of attractive force
ces: A cation on
o
one compound willl interact with an anion on a
another compound, giving rise to aninterm
molecular
asso
ociation. Ion–io
on interactions
s can also be repulsive whe
en two ions of like charge arre brought clos
sely
together. The repu
ulsion between
n the like charrges arises fro
om electron clo
oud overlap, w
which causes
the in
ntermolecularr distances to increase, resu
ulting in an energetically fav
vored dispersioon of the
mole
ecules. The illu
ustration of the
e magnetic po
oles in Figure 2-2 offers an excellent
e
coroollary for the
unde
erstanding of the
t attractive cationic
c
(posit ive pole) and anionic (negative pole) inteeractions (pane
el A),
45
Dr. Murtadha Alshareifi e-Library
as well as the need for proper distance to an energetically favored electrovalent interaction (panels A
and B), and the
P.21
repulsive forces that may occur between like charges (panels B and C). Ion–ion interactions may
be intermolecular (e.g., a hydrochloride salt of a drug) orintramolecular (e.g., a salt-bridge interaction
between counter ions in proteins).
Clearly, the strength of ion–ion interactions will vary according to the balancing of attractive and
repulsive forces between the cation- and anion-containing species. It is important to keep in mind that
ion–ion interactions are considerably stronger than many of the forces described in this section and can
even be stronger than covalent bonding when an ionic bond is formed. The strength of ion–ion
interactions has a profound effect on several physical properties of pharmaceutical agents including saltform selection, solid-crystalline habit, solubility, dissolution, pH and p K determination, and solution
stability.
Hydrogen Bonds
The interaction between a molecule containing a hydrogen atom and a strongly electronegative atom
such as fluorine, oxygen, or nitrogen is of particular interest. Because of the small size of the hydrogen
atom and its large electrostatic field, it can move in close to an electronegative atom and form an
electrostatic type of union known as a hydrogen bond or hydrogen bridge. Such a bond, discovered by
Latimer and Rodebush4 in 1920, exists in ice and in liquid water; it accounts for many of the unusual
properties of water including its high dielectric constant, abnormally low vapor pressure, and high boiling
point. The structure of ice is an open but well ordered three-dimensional array of regular tetrahedra with
oxygen in the center of each tetrahedron and hydrogen atoms at the four corners. The hydrogens are
not exactly midway between the oxygens, as may be observed in Figure 2-4. Roughly one sixth of the
hydrogen bonds of ice are broken when water passes into the liquid state, and essentially all the bridges
are destroyed when it vaporizes. Hydrogen bonds can also exist between alcohol molecules, carboxylic
acids, aldehydes, esters, and polypeptides.
The hydrogen bonds of formic acid and acetic acid are sufficiently strong to yield dimers (two molecules
attached together), which can exist even in the vapor state. Hydrogen fluoride in the vapor state exists
as a hydrogen-bonded polymer (F—H …)n, where n can have a value as large as 6. This is largely due
to the high electronegativity of the fluorine atom interacting with the positively charged, electropositive
hydrogen atom (analogous to an ion–ion interaction). Several structures involving hydrogen bonds are
shown in Figure 2-4. The dashed lines represent the hydrogen bridges. It will be noticed that intra- as
well as intermolecular hydrogen bonds may occur (as in salicylic acid).
Bond Energies
Bond energies serve as a measure of the strength of bonds. Hydrogen bonds are relatively weak,
having a bond energy of about 2 to 8 kcal/mole as compared with a value of about 50 to 100 kcal for the
covalent bond and well over 100 kcal for the ionic bond. The metallic bond, representing a third type of
primary valence, will be mentioned in connection with crystalline solids.
46
Dr. Murtadha Alshareifi e-Library
Fig.. 2-4. Repreesentative hy
ydrogen-boonded structtures.
The energies asso
ociated with in
ntermolecular b
bond forces of several compounds are shhown in Table
e 2-2.
It will be observed
d that the total interaction en
nergies betwee
en molecules are contributeed by a
comb
bination of orientation, induction, and disp
persion effects. The nature of the molecuules determine
es
whicch of these facctors is most in
nfluential in the
e attraction. In
n water, a high
hly polar substtance, the
orien
ntation or dipo
ole–dipole interaction predom
minates over the
t other two
P.22
2
force
es, and solubillity of drugs in
n water is influe
enced mainly by the orienta
ation energy oor dipole
interraction. In hydrogen chloride
e, a molecule with about 20% ionic character, the orienntation effect is
s still
signiificant, but the
e dispersion fo
orce contribute
es a large share to the total interaction ennergy between
n
47
Dr. Murtadha Alshareifi e-Library
mole
ecules. Hydrog
gen iodide is predominantly
p
y covalent, with
h its intermole
ecular attractioon supplied
primarily by the Lo
ondon or dispe
ersion force.
Tab
ble 2-2 Energies Assocciated with
h Molecularr and Ionic Interactio ns
Interaction (kcal/molee)
CompoundOrientation
O
nInductionD
DispersionTotal Enerrgy
H2O
8.69
0.46
2.15
11.30
HCl
0.79
0.24
4.02
5.05
Hl
0.006
0.027
6.18
6.21
NaCl
—
—
3.0
183
The ionic crystal sodium
s
chloride is included iin Table 2-2 fo
or comparison
n to show that its stability, as
s
reflected in its larg
ge total energy
y, is much gre
eater than that of molecular aggregates, aand yet the
dispe
ersion force exxists in such io
onic compoun
nds even as it does in molec
cules.
Sta
ates of Matter
Gase
es, liquids, and crystalline solids
s
are the tthree primary states of mattter or phases. The molecule
es,
atom
ms, and ions in
n the solid statte are held in cclose proximitty by intermole
ecular, interatoomic, or ionic
force
es. The atomss in the solid ca
an oscillate on
nly about fixed
d positions. As
s the temperatture of a solid
subsstance is raise
ed, the atoms acquire
a
suffici ent energy to disrupt the orrdered arrangeement of the la
attice
and pass into the liquid form. Finally, when su
ufficient energ
gy is supplied, the atoms or molecules pass
into tthe gaseous state.
s
Solids with
w high vaporr pressures, such as iodine and camphorr, can pass directly
from the solid to th
he gaseous state without me
elting at room
m temperature. This processs is known
ublimation, an
nd the reverse process, thatt is, condensattion to the solid state, may bbe referred to
as su
asde
eposition. Sublimation will not be discusse
ed in detail he
ere but is very important in thhe freeze-dryiing
process, as brieflyy detailed in th
he Key Concep
pt Box on Sub
blimation in Fre
eeze Drying (LLyophilization).
Key Co
oncept
Sub
blimation in
n Freeze Dry
ying (Lyoph
hilization)
Free
eze drying (lyyophilization) is widely ussed in the ph
harmaceutica
al industry forr the
man
nufacturing of
o heat-sensittive drugs. F reeze drying
g is the most common com
mmercial
apprroach to makking a steriliz
zed powder. This is partic
cularly true fo
or injectable formulations
s,
whe
ere a suspend
ded drug mig
ght undergo rapid degrad
dation in solu
ution, and thuus a dried
pow
wder is preferrred. Many protein formullations are also prepared
d as freeze-ddried powders
s to
prevvent chemica
al and physic
cal instability processes that more rap
pidly occur inn a solution state
s
than
n in the solid state. As is implied by itss name, free
eze drying is a process whhere a drug
susp
pended in wa
ater is frozen
n and then drried by a sub
blimation process. The foollowing
proccesses are usually followed in freeze drying: (a) The
T drug is fo
ormulated in a sterile bufffer
form
mulation and placed in a vial
v (it is impo
ortant to note
e that there are
a different types of glas
ss
avaiilable and the
ese types ma
ay have diffe
ering effects on solution stability;
s
(b) a slotted stop
pper
is pa
artially insertted into the vial,
v
with the stopper bein
ng raised abo
ove the vial sso that air can
get iin and out off the vial; (c) the vials are
e loaded onto
o trays and placed
p
in a lyyophilizer, wh
hich
begiins the initial freezing; (d) upon comp
pletion of the primary free
eze, which is conducted at
a a
low temperature
e, vacuum is applied and the water su
ublimes into vapor
v
and is removed fro
om
the ssystem, leavving a powde
er with a high
h water conte
ent (the residual water is more tightly
48
Dr. Murtadha Alshareifi e-Library
boun
nd to the solid powder); (e)
( the tempe
erature is raised (but still maintaining a frozen sta
ate)
to ad
dd more ene
ergy to the sy
ystem, and a secondary freeze-drying
f
g cycle is perrformed under
vacu
uum to pull off
o more of th
he tightly bou
und water; an
nd (f) the stoppers are theen compress
sed
into the vials to seal
s
them an
nd the powde
ers are left re
emaining in a vacuum-seealed container
with no air excha
ange. These vials are su bsequently sealed
s
with a metal cap thhat is crimpe
ed
into place. It is im
mportant to note
n
that therre is often re
esidual waterr left in the poowders upon
n
com
mpletion of lyo
ophilization. In addition, i f the caps were not air tig
ght, humidityy could enterr the
vial and cause th
he powders to
t absorb atm
mospheric water
w
(the measurement oof the ability of
o a
pow
wder/solid ma
aterial to abso
orb water is called its hyg
groscopicity)), which couldd lead to gre
eater
insta
ability. Some
e lyophilized powders are
e so hygrosco
opic that they
y will absorbb enough watter to
form
m a solution; this is called deliquescen
nce and is co
ommon in lyo
ophilized pow
wders. Finally
y,
beca
ause the watter is remove
ed by sublima
ation and the
e compound is not crystaalized out, the
e
resid
dual powder is commonly
y amorphouss.
Certa
ain moleculess frequently ex
xhibit a fourth p
phase, more properly
p
terme
ed a mesophaase (Greek me
esos,
midd
dle), which liess between the liquid and cryystalline states
s. This so-calle
ed liquid crysttalline state is
discu
ussed later. Supercritical flu
uids are also cconsidered a mesophase,
m
in
n this case a sttate of matter that
existts under high pressure
p
and temperature a
and has prope
erties that are intermediate bbetween those
e of
liquid
ds and gases.. Supercritical fluids will also
o be discussed
d later becaus
se of their incrreased utilizatiion in
pharrmaceutical ag
gent processin
ng.
The
e Gaseou
us State
Owin
ng to vigorouss and rapid mo
otion and resu ltant collisions
s, gas molecules travel in raandom paths and
a
collid
de not only witth one another but also with
h the walls of the
t container in
i which they are confined.
Hencce, they exert a pressure—a
a force per un
nit area—expre
essed in dyne
es/cm2. Pressuure is also
recorded in atmosspheres or in millimeters
m
of m
mercury because of the use
e of the barom
meter in pressu
ure
measurement. Another importan
nt characteristtic of a gas, its
s volume, is usually expresssed in liters orr
cubicc centimeters (1 cm3 = 1 mL
L). The tempe
erature involve
ed in the gas equations
e
is givven according
g the
abso
olute or Kelvin scale. Zero degrees
d
on the
e centigrade scale is equal to
t 273.15 Kelvvin (K).
P.23
3
The
e Ideal Gas
G Law
The student may recall
r
from gen
neral chemistrry that the gas
s laws formula
ated by Boyle, Charles, and GayLusssac refer to an
n ideal situation where no inttermolecular interactions ex
xist and collisioons are perfec
ctly
elasttic, and thus no
n energy is ex
xchanged upo
on collision. Ideality allows for certain ass umptions to be
made to derive the
ese laws. Boyle's law relate
es the volume and pressure of a given maass of gas at
consstant temperatture,
or
ussac and Cha
arles states th
hat the volume
e and absolute
e temperature of a given ma
ass of
The law of Gay-Lu
at constant prressure are dirrectly proportio
onal,
gas a
Thesse equations can
c be combin
ned to obtain tthe familiar rellationship
49
Dr. Murtadha Alshareifi e-Library
In eq
quation (2-4), P1, V1, and T1 are the value
es under one set
s of conditions and P2, V2, and T2 the va
alues
unde
er another set.
Exa
ample 2-1
The
e Effect of Pressure
P
Ch
hanges on tthe Volume
e of an Ideal Gas
In th
he assay of ethyl
e
nitrite sp
pirit, the nitricc oxide gas that
t
is liberatted from a deefinite quantity of
spiriit and collectted in a gas burette
b
occu pies a volum
me of 30.0 mL
L at a tempeerature of 20°°C
and a pressure of
o 740 mm Hg.
H Assuming
g the gas is ideal, what is
s the volume at 0°C and 760
7
mm Hg? Write
From
m equation (2--4) it is seen th
hat PV/T unde
er one set of conditions is eq
qual to PV/T uunder another set,
and so on. Thus, one
o reasons th
hat although P
P, V, and Tcha
ange, the ratio
o PV/T is consstant and can be
exprressed mathem
matically as
or
in wh
hich R is the constant
c
value
e for the PV/T ratio of an ide
eal gas. This equation
e
is corrrect only for 1
mole
e (i.e., 1 g molecular weight) of gas; for n moles it beco
omes
Equa
ation (2-6) is known
k
as the general
g
ideal g
gas law, and because
b
it rela
ates the speciffic conditions or
state
e, that is, the pressure,
p
volume, and temp
perature of a given
g
mass of gas, it is calleed the equation
n of
state
e of an ideal gas. Real gase
es do not intera
act without en
nergy exchang
ge, and therefoore do not follo
ow
the la
aws of Boyle and
a of Gay-Lu
ussac and Cha
arles as ideal gases are ass
sumed to do. T
This deviation
n will
be co
onsidered in a later section.
The molar gas con
nstant R is hig
ghly important in physical ch
hemical scienc
ce; it appears in a number of
o
relationships in ele
ectrochemistry
y, solution the
eory, colloid ch
hemistry, and other
o
fields in addition to its
s
earance in the
e gas laws. To obtain a num
merical value fo
or R, let us pro
oceed as follow
ws. If 1 mole of
o an
appe
ideall gas is chosen, its volume under standarrd conditions of
o temperature
e and pressuree (i.e., at 0°C and
760 mm Hg) has been
b
found by
y experiment to
o be 22.414 litters. Substitutting this value in equation (2
2-6),
obtain
we o
The molar gas con
nstant can also be given in e
energy units by
b expressing the pressure in dynes/cm2 (1
6
2
atm = 1.0133 × 10
0 dynes/cm ) and the volum
me in the corre
esponding units of cm3 (22.4414 liters = 22
2,414
3
cm )). Then
or, b
because 1 joule = 107 ergs,
The constant can also be expre
essed in cal/mo
ole deg, employing the equivalent 1 cal = 4.184 joules::
50
Dr. Murtadha Alshareifi e-Library
One must be particularly careful to use the va
alue of R comm
mensurate witth the approprriate units und
der
conssideration in ea
ach problem. In gas law pro
oblems, R is usually
u
express
sed in liter atm
m/mole deg,
wherreas in thermo
odynamic calc
culations it usu
ually appears in the units of cal/mole deg or joule/mole deg.
Exa
ample 2-2
Calc
culation of Volume Us
sing the Ide
eal Gas Law
w
Wha
at is the volume of 2 mole
es of an idea
al gas at 25°C
C and 780 mm
m Hg?
Mo
olecular Weight
W
The approximate molecular
m
weight of a gas ccan be determined by use off the ideal gass law. The num
mber
of moles of gas n is
P.24
4
repla
aced by its equ
uivalent g/M, in
i which g is th
he number of grams of gas and M is the m
molecular weiight:
or
Exa
ample 2-3
Molecular Weig
ght Determ
mination by tthe Ideal Gas Law
If 0.3
30 g of ethyl alcohol in th
he vapor statte occupies 200
2 mL at a pressure of 1 atm and a
temp
perature of 100°C,
1
what is the molec ular weight of
o ethyl alcoh
hol? Assumee that the vap
por
beha
aves as an id
deal gas. Wrrite
The two methods most common
nly used to de
etermine the molecular
m
weight of easily vaaporized liquid
ds
such
h as alcohol an
nd chloroform are the Regn
nault and Victo
or Meyer methods. In the lattter method, th
he
liquid
d is weighed in
n a glass bulb
b; it is then vap
porized and th
he volume is determined at a definite
temp
perature and barometric
b
pre
essure. The va
alues are finally substituted in equation (22-8) to obtain the
t
mole
ecular weight.
Kin
netic Mollecular Th
heory
The equations pre
esented in the previous secttion have been
n formulated from experimeental
conssiderations wh
here the condittions are close
e to ideality. The
T theory thatt was developped to explain the
beha
avior of gases and to lend additional
a
supp
port to the validity of the gas
s laws is calledd the kinetic
mole
ecular theory. Here are som
me of the more
e important sta
atements of the
e theory:
1.
Gases arre composed of
o particles ca
alled atoms or molecules, the total volumee of which is so
s
2.
small as to be negligible in relation to
molecules are
o the volume of the space in which the m
confined.. This condition is approxim ated in actual gases only att low pressurees and high
temperattures, in which
h case the mollecules of the gas are far ap
part.
The particles of the gas do not attracct one another, but instead move with com
mplete
independ
dence; again, this
t
statementt applies only at low pressures.
The particles exhibit co
ontinuous rand
dom motion ow
wing to their kinetic
k
energy.. The average
e
kinetic en
nergy, E, is dirrectly proportio
onal to the absolute temperrature of the g as, or E = (3/2)RT.
3.
51
Dr. Murtadha Alshareifi e-Library
4.
The mole
ecules exhibit perfect elasticcity; that is, the
ere is no net lo
oss of speed oor transfer of
energy affter they collid
de with one an
nother and with
h the molecule
es in the wallss of the confining
vessel, which
w
latter effe
ect accounts ffor the gas pre
essure. Althou
ugh the net veelocity, and
therefore
e the average kinetic energyy, does not cha
ange on collision, the speedd and energy of
o
the individual molecule
es may differ w
widely at any instant. More simply
s
stated, the net velocity
can be an
n average velo
ocity of many molecules; th
hus, a distribution of individuual molecular
velocitiess can be prese
ent in the syste
em.
From
m these and otther postulates, the followin
ng fundamenta
al kinetic equa
ation is derivedd:
wherre P is the pre
essure and V the
t volume occcupied by any
y number n of molecules of mass m havin
ng an
averrage velocity [C
C with bar abo
ove].
U
Using this fuundamental equation, w
we can obtaiin the root mean
m
squaree
vvelocity
(usually
y written µ) of the moleecules by an
n ideal gas.** Solving
foor
in equuation (2-9)) and takingg the square root of both
h sides of thhe equation
leeads to the formula
f
Resttricting this case to 1 mole of
o gas, we find
d that PV beco
omes equal to RT from the eequation of
state
e (2-5), n beco
omes Avogadrro's number N A, and NA mulltiplied by the mass of one m
molecule beco
omes
the m
molecular weig
ght M. The roo
ot mean squarre velocity is therefore
t
given
n by
Exa
ample 2-4
Calc
culation of Root Mean
n Square Ve
elocity
Wha
at is the root mean squarre velocity off oxygen (mo
olecular weight, 32.0) at 225°C (298 K))?
Beca
ause the term nm/V is equa
al to density, w
we can write eq
quation (2-10) as
Rem
membering that density is de
efined as a ma
ass per unit vo
olume, we see
e that the rate of diffusion off a
gas iis inversely prroportional to the
t square roo
ot of its densitty. Such a rela
ation confirms the early findiings
52
Dr. Murtadha Alshareifi e-Library
of Graham, who showed that a lighter gas diff
ffuses more ra
apidly through a porous mem
mbrane than does
d
a heavier one.
P.25
5
The
e van derr Waals Equation
E
n for Reall Gases
The fundamental kinetic
k
equatio
on (2-9) is foun
nd to compare
e with the idea
al gas equationn because the
e
ased on the as
ssumptions off the ideal statte. However, real
r
gases aree not compose
ed of
kinettic theory is ba
infiniitely small and
d perfectly elas
stic nonattractting spheres. Instead, they are composedd of molecules
s of a
finite
e volume that tend
t
to attractt one another. These factors
s affect the vo
olume and presssure terms in
n the
ideall equation so that
t
certain re
efinements mu
ust be incorporrated if equatio
on (2-5) is to pprovide results
s that
checck with experim
ment. A number of such exp
pressions have been suggested, the van der Waals
equa
ation being the
e best known of
o these. For 1 mole of gas,, the van der Waals
W
equatioon is written as
s
For tthe more gene
eral case of n moles of gas in a containerr of volume V, equation (2-1 3) becomes
The term a/V2 acccounts for the internal
i
pressu
ure per mole resulting
r
from the intermoleecular forces of
o
attra
action between
n the molecule
es; b accountss for the incom
mpressibility off the molecules
es, that is,
the e
excluded volum
me, which is about
a
four time
es the molecular volume. Th
his relationshi p holds true fo
or all
gase
es; however, th
he influence of
o nonideality i s greater whe
en the gas is compressed. P
Polar liquids ha
ave
high internal presssures and serv
ve as solventss only for subs
stances of similar internal prressures. Non
npolar
mole
ecules have lo
ow internal pre
essures and arre not able to overcome the
e powerful cohhesive forces of
o the
polarr solvent mole
ecules. Minera
al oil is immisccible with wate
er for this reason.
Whe
en the volume of a gas is larrge, the moleccules are well dispersed. Un
nder these
cond
ditions, a/V2 an
nd b become insignificant w
with respect to P and V, resp
pectively. Undder these
cond
ditions, the van
n der Waals equation for 1 m
mole of gas re
educes to the ideal gas equ ation, PV = RT,
and at low pressurres, real gases behave in a n ideal manne
er. The values
s of a and b haave been
determined for a number
n
of gas
ses. Some of t hese are listed in Table 2-3
3. The weak vaan der Waals
force
es of attraction
n, expressed by
b the constan
nt a, are those
e referred to in
n Table 2-1.
Tab
ble 2-3 The Van Der Waals
W
Consstants for some
s
Gasess
G
Gas
a (liter2 atm
m/mole2)b (lliter/mole)
H2
0.244
00.0266
O2
1.360
00.0318
CH4
2.253
00.0428
H2O
5.464
00.0305
Cl2
6.493
00.0562
53
Dr. Murtadha Alshareifi e-Library
CHCl3
15.17
00.1022
Exa
ample 2-5
App
plication of the van der Waals Eq uation
A 0.193-mole sa
ample of ethe
er was confin
ned in a 7.35
5-liter vessel at 295 K. Caalculate the
presssure producced using (a) the ideal ga
as equation and
a (b) the va
an der Waalss equation. The
T
2
2
van der Waals a value for ether is 17.38 liter atm/mo
ole ; the b va
alue is 0.13444 liter/mole. To
solvve for pressure, the van der
d Waals eq
quation can be
b rearranged
d as follows::
(a)
(b)
Exa
ample 2-6
Calc
culation of the van derr Waals Co nstants
Calcculate the pre
essure of 0.5
5 mole of CO
O2 gas in a firre extinguisher of 1-liter ccapacity at 27°C
usin
ng the ideal gas
g equation and the van
n der Waals equation.
e
The
e van der Waaals constan
nts
can be calculate
ed from the critical tempe rature Tc and
d the critical pressure Pc (see the
secttion Liquefacction of Gase
es for definitio
ons):
The critical temp
perature and critical presssure of CO2 are 31.0°C and
a 72.9 atm
m, respectively.
Usin
ng the ideal gas
g equation
n, we obtain
54
Dr. Murtadha Alshareifi e-Library
Altho
ough it is beyo
ond the scope of this text, it should be me
entioned that to account for nonideality, th
he
conccept
P.26
6
of fugacity was inttroduced by Le
ewis.5 In gene
eral chemistry
y, the student learns about thhe concept of
mical potentiall. At equilibrium
m in an ideal h
homogeneous
s closed system, intermoleccular interactio
ons
chem
are cconsidered to be nonexisten
nt. However, in
n real gaseous states and in
n multiple-com
mponent syste
ems,
interrmolecular inte
eractions occu
ur. Without goiing into great detail, one can
n say that thesse interactions
s can
influe
ence the chem
mical potential and cause de
eviations from
m the ideal state. These deviiations reflect the
activvities of the component(s) within
w
the syste
em. Simply put, fugacity is a measuremennt of the activity
asso
ociated with no
onideal interac
ctions. For furtther details pe
ertaining to fug
gacity and its eeffects on gas
ses,
the sstudent is directed to any inttroductory phyysical chemisttry text.
The
e Liquid State
Liq
quefactio
on of Gas
ses
Whe
en a gas is coo
oled, it loses some
s
of its kin etic energy in the form of he
eat, and the ve
velocity of the
mole
ecules decreasses. If pressurre is applied to
o the gas, the molecules are
e brought withhin the sphere
e of
the vvan der Waalss interaction fo
orces and passs into the liquiid state. Becau
use of these fo
forces, liquids are
conssiderably densser than gases
s and occupy a definite volu
ume. The trans
sitions from a gas to a liquid
d and
from a liquid to a solid
s
depend not
n only on the
e temperature but also on th
he pressure too which the
subsstance is subje
ected.
If the
e temperature is elevated su
ufficiently, a va
alue is reache
ed above which it is impossiible to liquefy a
gas iirrespective off the pressure applied. Thiss temperature, above which a liquid can nno longer existt, is
know
wn as the criticcal temperaturre. The pressu
ure required to
o liquefy a gas
s at its critical temperature is
the ccritical pressurre, which is als
so the highestt vapor pressu
ure that the liquid can have.. The further a gas
is co
ooled below itss critical tempe
erature, the le
ess pressure is
s required to liquefy it. Baseed on this princ
ciple,
55
Dr. Murtadha Alshareifi e-Library
all known gases have been liquefied. Supercritical fluids, where excessive temperature and pressure
are applied, do exist as a separate/intermediate phase and will be discussed briefly later in this chapter.
The critical temperature of water is 374°C, or 647 K, and its critical pressure is 218 atm, whereas the
corresponding values for helium are 5.2 K and 2.26 atm. The critical temperature serves as a rough
measure of the attractive forces between molecules because at temperatures above the critical value,
the molecules possess sufficient kinetic energy so that no amount of pressure can bring them within the
range of attractive forces that cause the atoms or molecules to “stick” together. The high critical values
for water result from the strong dipolar forces between the molecules and particularly the hydrogen
bonding that exists. Conversely, only the weak London force attracts helium molecules, and,
consequently, this element must be cooled to the extremely low temperature of 5.2 K before it can be
liquefied. Above this critical temperature, helium remains a gas no matter what the pressure.
Methods of Achieving Liquefaction
One of the most obvious ways to liquefy a gas is to subject it to intense cold by the use of freezing
mixtures. Other methods depend on the cooling effect produced in a gas as it expands. Thus, suppose
we allow an ideal gas to expand so rapidly that no heat enters the system. Such an expansion, termed
an adiabatic expansion, can be achieved by carrying out the process in a Dewar, or vacuum, flask,
which effectively insulates the contents of the flask from the external environment. The work done to
bring about expansion therefore must come from the gas itself at the expense of its own heat energy
content (collision frequency). As a result, the temperature of the gas falls. If this procedure is repeated a
sufficient number of times, the total drop in temperature may be sufficient to cause liquefaction of the
gas.
A cooling effect is also observed when a highly compressed nonideal gas expands into a region of low
pressure. In this case, the drop in temperature results from the energy expended in overcoming the
cohesive forces of attraction between the molecules. This cooling effect is known as the Joule–Thomson
effect and differs from the cooling produced in adiabatic expansion, in which the gas does external work.
To bring about liquefaction by the Joule–Thomson effect, it may be necessary to precool the gas before
allowing it to expand. Liquid oxygen and liquid air are obtained by methods based on this effect.
Aerosols
Gases can be liquefied under high pressures in a closed chamber as long as the chamber is maintained
below the critical temperature. When the pressure is reduced, the molecules expand and the liquid
reverts to a gas. This reversible change of state is the basic principle involved in the preparation of
pharmaceutical aerosols. In such products, a drug is dissolved or suspended in a propellant, a material
that is liquid under the pressure conditions existing inside the container but that forms a gas under
normal atmospheric conditions. The container is so designed that, by depressing a valve, some of the
drug–propellant mixture is expelled owing to the excess pressure inside the container. If the drug is
nonvolatile, it forms a fine spray as it leaves the valve orifice; at the same time, the liquid propellant
vaporizes off.
Chlorofluorocarbons and hydrofluorocarbons have traditionally been utilized as propellants in these
products because of their physicochemical properties. However, in the face of increasing environmental
concerns (ozone depletion) and legislation like the Clean Air Act, the use of chlorofluorocarbons and
hydrofluorocarbons is tightly regulated. This has led researchers to identify additional propellants, which
has led to the increased use of other gases such as nitrogen and carbon dioxide. However,
considerable effort is being focused on finding better propellant systems. By varying the proportions of
the various propellants, it is possible to produce pressures within the container ranging from 1 to 6 atm
at room
P.27
temperature. Alternate fluorocarbon propellants that do not deplete the ozone layer of the atmosphere
are under investigation.6
The containers are filled either by cooling the propellant and drug to a low temperature within the
container, which is then sealed with the valve, or by sealing the drug in the container at room
56
Dr. Murtadha Alshareifi e-Library
temperature and then forcing the required amount of propellant into the container under pressure. In
both cases, when the product is at room temperature, part of the propellant is in the gaseous state and
exerts the pressure necessary to extrude the drug, whereas the remainder is in the liquid state and
provides a solution or suspension vehicle for the drug.
The formulation of pharmaceuticals as aerosols is continually increasing because the method frequently
offers distinct advantages over some of the more conventional methods of formulation. Thus, antiseptic
materials can be sprayed onto abraded skin with a minimum of discomfort to the patient. One product,
ethyl chloride, cools sufficiently on expansion so that when sprayed on the skin, it freezes the tissue and
produces a local anesthesia. This procedure is sometimes used in minor surgical operations.
More significant is the increased efficiency often observed and the facility with which medication can be
introduced into body cavities and passages. These and other aspects of aerosols have been considered
by various researchers.7,8 Byron and Clark9a studied drug absorption from inhalation aerosols and
provided a rather complete analysis of the problem. The United States Pharmacopeia (USP)9b includes
a discussion of metered-dose inhalation products and provides standards and test procedures (USP).
The identification of biotechnology-derived products has also dramatically increased the utilization of
aerosolized formulations.10 Proteins, DNA, oligopeptides, and nucleotides all demonstrate poor oral
bioavailability due to the harsh environment of the gastrointestinal tract and their relatively large size and
rapid metabolism. The pulmonary and nasal routes of administration enable higher rates of passage into
systemic circulation than does oral administration.11 It is important to point out that aerosol products are
formulated under high pressure and stress limits. The physical stability of complex biomolecules may be
adversely affected under these conditions (recall that pressure and temperature may influence the
attractive and repulsive inter- and intramolecular forces present).
Vapor Pressure of Liquids
Translational energy of motion (kinetic energy) is not distributed evenly among molecules; some of the
molecules have more energy and hence higher velocities than others at any moment. When a liquid is
placed in an evacuated container at a constant temperature, the molecules with the highest energies
break away from the surface of the liquid and pass into the gaseous state, and some of the molecules
subsequently return to the liquid state, or condense. When the rate of condensation equals the rate of
vaporization at a definite temperature, the vapor becomes saturated and a dynamic equilibrium is
established. The pressure of the saturated vapor* above the liquid is then known as the equilibrium
vapor pressure. If a manometer is fitted to an evacuated vessel containing the liquid, it is possible to
obtain a record of the vapor pressure in millimeters of mercury. The presence of a gas, such as air,
above the liquid decreases the rate of evaporation, but it does not affect the equilibrium pressure of the
vapor.
57
Dr. Murtadha Alshareifi e-Library
Fig.. 2-5. The variation
v
of the
t vapor prressure of some
s
liquidss with tempperature.
As th
he temperaturre of the liquid is elevated, m
more molecule
es approach th
he velocity neccessary for es
scape
and pass into the gaseous state
e. As a result, the vapor pressure increases with rising temperature, as
show
wn in Figure 2-5. Any point on
o one of the curves repres
sents a condition in which thhe liquid and th
he
vapo
or exist together in equilibriu
um. As observved in the diag
gram, if the tem
mperature of aany of the liquids is
incre
eased while th
he pressure is held constantt, or if the pres
ssure is decreased while thee temperature
e is
held constant, all the
t liquid will pass
p
into the vvapor state.
Cla
ausius–C
Clapeyron
n Equatio
on: Heat of Vaporrization
The relationship between the va
apor pressure and the absolute temperatu
ure of a liquid is expressed by
the
P.28
8
Clau
usius–Clapeyro
on equation (tthe Clapeyron and the Clausius–Clapeyro
on equations aare derived
in Ch
hapter 3):
58
Dr. Murtadha Alshareifi e-Library
wherre p1 and p2 are the vapor pressures
p
at a bsolute tempe
eratures T1 and T2, and ∆Hvv is the molar heat
h
of va
aporization, that is, the heat absorbed by 1 mole of liquid when it pas
sses into the vvapor state. He
eats
of va
aporization varry somewhat with
w temperatu
ure. For exam
mple, the heat of vaporizationn of water is 539
5
cal/g
g at 100°C; it iss 478 cal/g at 180°C, and a
at the critical te
emperature, where
w
no distinnction can be made
m
betw
ween liquid and
d gas, the hea
at of vaporizattion becomes zero.
z
Hence, the
t ∆Hvof equuation (215) sshould be reco
ognized as an average valu
ue, and the equation should be consideredd strictly valid only
overr a narrow tem
mperature rang
ge. The equati on contains additional approximations, foor it assumes that
t
the vvapor behavess as an ideal gas
g and that th
he molar volum
me of the liquid is negligiblee with respect to
that of the vapor. These
T
are imp
portant approxximations in lig
ght of the nonideality of reall solutions.
Exa
ample 2-7
App
plication of the Clausiu
us–Clapeyrron Equatio
on
Com
mpute the vapor pressure
e of water at 120°C. The vapor pressu
ure p1 of watter at 100°C is 1
atm,, and ∆Hv ma
ay be taken as
a 9720 cal/ mole for this
s temperature
e range. Thuus,
The Clausius–Clapeyron equ
uation can be
e written in a more generral form,
or in
n natural loga
arithms,
from
m which it is observed
o
tha
at a plot of th e logarithm of
o the vapor pressure agaainst the
recip
procal of the absolute tem
mperature re
esults in a strraight line, en
nabling one tto compute the
t
heatt of vaporization of the liq
quid from the
e slope of the
e line.
Bo
oiling Poin
nt
If a liquid is placed
d in an open container
c
and h
heated until th
he vapor press
sure equals thhe atmospheric
c
pressure, the vapo
or will form bubbles that rise
e rapidly throu
ugh the liquid and
a escape innto the gaseou
us
state
e. The tempera
ature at which
h the vapor pre
essure of the liquid
l
equals the external orr atmospheric
pressure is known
n as theboiling
g point. All the absorbed hea
at is used to ch
hange the liquuid to vapor, and
a
the temperature does not rise until the liquid iis completely vaporized.
v
The atmosphericc pressure at sea
s
levell is approxima
ately 760 mm Hg;
H at higher e
elevations, the
e atmospheric
c pressure deccreases and th
he
boilin
ng point is low
wered. At a pre
essure of 700 mm Hg, water boils at 97.7°C; at 17.5 mm
m Hg, it boils at
20°C
C. The change
e in boiling point with pressu
ure can be com
mputed by using the Clausiuus–Clapeyron
equa
ation.
The heat that is ab
bsorbed when
n water vaporizzes at the normal boiling po
oint (i.e., the heeat of vaporization
at 10
00°C) is 539 cal/g
c
or about 9720
9
cal/mole
e. For benzene
e, the heat of vaporization
v
iss 91.4 cal/g att the
norm
mal boiling poin
nt of 80.2°C. These
T
quantitiies of heat, kn
nown as latentt heats of vapoorization, are taken
t
up w
when the liquid
ds vaporize an
nd are liberated
d when the va
apors condens
se to liquids.
The boiling point may
m be consid
dered the temp
perature at wh
hich thermal agitation can ovvercome the
attra
active forces between the molecules of a lliquid. Therefo
ore, the boiling
g point of a coompound, like the
heat of vaporizatio
on and the vap
por pressure a
at a definite temperature, prrovides a rouggh indication of the
magnitude of the attractive
a
force
es.
The boiling points of normal hyd
drocarbons, si mple alcohols
s, and carboxy
ylic acids increease with
mole
ecular weight because
b
the attractive
a
van d
der Waals forc
ces become greater with inccreasing numb
bers
of atoms. Branchin
ng of the chain
n produces a less compact molecule with
h reduced inteermolecular
59
Dr. Murtadha Alshareifi e-Library
attraction, and a decrease in the boiling point results. In general, however, the alcohols boil at a much
higher temperature than saturated hydrocarbons of the same molecular weight because of association
of the alcohol molecules through hydrogen bonding. The boiling points of carboxylic acids are more
abnormal still because the acids form dimers through hydrogen bonding that can persist even in the
vapor state. The boiling points of straight-chain primary alcohols and carboxylic acids increase about
18°C for each additional methylene group. The rough parallel between the intermolecular forces and the
boiling points or latent heats of vaporization is illustrated inTable 2-4. Nonpolar substances, the
molecules of which are held together predominantly by the London force, have low boiling points and
low heats of vaporization. Polar molecules, particularly those such as ethyl alcohol and water, which are
associated through hydrogen bonds, exhibit high boiling points and high heats of vaporization.
Table 2-4 Normal Boiling Points and Heats of Vaporization
Compound
Boiling Point (°C)Latent Heat of Vaporization (cal/g)
Helium
-268.9
6
Nitrogen
-195.8
47.6
Propane
-42.2
102
Methyl chloride
-24.2
102
Isobutane
-10.2
88
Butane
-0.4
92
Ethyl ether
34.6
90
Carbon disulfide
46.3
85
Ethyl alcohol
78.3
204
Water
100.0
539
P.29
Solids and the Crystalline State
Crystalline Solids
The structural units of crystalline solids, such as ice, sodium chloride, and menthol, are arranged in fixed
geometric patterns or lattices. Crystalline solids, unlike liquids and gases, have definite shapes and an
orderly arrangement of units. Gases are easily compressed, whereas solids, like liquids, are practically
incompressible. Crystalline solids show definite melting points, passing rather sharply from the solid to
60
Dr. Murtadha Alshareifi e-Library
the liiquid state. Crrystallization, as
a is sometim es taught in organic
o
chemis
stry laboratoryy courses, occ
curs
by prrecipitation of the compound out of solutio
on and into an
n ordered arra
ay. Note that thhere are several
impo
ortant variables here, including the solven
nt(s) used, the
e temperature, the pressure,, the crystalline
arrayy pattern, saltss (if crystalliza
ation is occurriing through the formation off insoluble saltt complexes th
hat
precipitate), and so on, that influ
uence the rate
e and stability of the crystal (see the secti on Polymorph
hism)
formation. The varrious crystal fo
orms are divid
ded into six dis
stinct crystal systems basedd on symmetry
y.
Theyy are, togetherr with example
es of each, cu
ubic (sodium chloride),
c
tetrag
gonal (urea), hhexagonal
(iodo
oform), rhombic (iodine), mo
onoclinic (sucrrose), and tric
clinic (boric acid). The morphhology of a
crysttalline form is often referred
d to as its habiit, where the crystal
c
habit is defined as haaving the same
struccture but differrent outward appearance
a
(o
or alternately, the
t collection of faces and ttheir area ratio
os
comp
prising the cryystal).
The units that constitute the crystal structure can be atoms
s, molecules, or
o ions. The soodium chloride
e
crysttal, shown in Figure
F
2-6, consists of a cub
bic lattice of so
odium ions intterpenetrated by a lattice off
chlorride ions, the binding
b
force of
o the crystal b
being the elec
ctrostatic attraction of the opppositely charg
ged
ions.. In diamond and
a graphite, the
t lattice unitts consist of attoms held toge
ether by covallent bonds. So
olid
carbon dioxide, hyydrogen chloride, and naphtthalene form crystals
c
compo
osed of moleccules as the
build
ding units. In organic
o
compo
ounds, the mo lecules are he
eld together by
y van der Waaals forces,
Coullombic forces, and hydrogen bonding, wh
hich account fo
or the weak binding and for the low meltin
ng
pointts of these cryystals. Aliphatiic hydrocarbon
ns crystallize with
w their chains lying in a pparallel
arran
ngement, whe
ereas fatty acid
ds crystallize iin layers of dim
mers with the chains lying pparallel or tilted
d at
an angle with resp
pect to the bas
se plane. Whe
ereas ionic and
d atomic crystals in general are hard and
brittle
e and have high melting points, molecula
ar crystals are soft and have
e relatively low
w melting pointts.
61
Dr. Murtadha Alshareifi e-Library
Fig. 2-6. The crystal lattice of sodium chloride.
Metallic crystals are composed of positively charged ions in a field of freely moving electrons,
sometimes called the electron gas. Metals are good conductors of electricity because of the free
movement of the electrons in the lattice. Metals may be soft or hard and have low or high melting points.
The hardness and strength of metals depend in part on the kind of imperfections, or lattice defects, in
the crystals.
Polymorphism
Some elemental substances, such as carbon and sulfur, may exist in more than one crystalline form and
are said to be allotropic, which is a special case ofpolymorphism. Polymorphs have different stabilities
and may spontaneously convert from the metastable form at a temperature to the stable form. They also
exhibit different melting points, x-ray crystal and diffraction patterns (see later discussion), and
solubilities, even though they are chemically identical. The differences may not always be great or even
large enough to “see” by analytical methods but may sometimes be substantial. Solubility and melting
points are very important in pharmaceutical processes, including dissolution and formulation, explaining
the primary reason we are interested in polymorphs. The formation of polymorphs of a compound may
depend upon several variables pertaining to the crystallization process, including solvent differences
(the packing of a crystal might be different from a polar versus a nonpolar solvent); impurities that may
favor a metastable polymorph because of specific inhibition of growth patterns; the level of
supersaturation from which the material is crystallized (generally the higher the concentration above the
solubility, the more chance a metastable form is seen); the temperature at which the crystallization is
carried out; geometry of the covalent bonds (are the molecules rigid and planar or free and flexible?);
attraction and repulsion of cations and anions (later you will see how x-ray crystallography is used to
define an electron density map of a compound); fit of cations into coordinates that are energetically
favorable in the crystal lattice; temperature; and pressure.
Perhaps the most common example of polymorphism is the contrast between a diamond and graphite,
both of which are composed of crystalline carbon. In this case, high pressure and temperature lead to
the formation of a diamond from elemental carbon. When contrasting an engagement ring with a pencil,
it is quite apparent that a diamond has a distinct crystal habit from that of graphite. It should be noted
that a diamond is a less stable (metastable) crystalline form of carbon than is graphite. Actually, the
imperfections in diamonds continue to occur with time and represent the diamond converting, very
slowly at the low ambient temperature and pressure, into the more stable graphite polymorph.
P.30
Nearly all long-chain organic compounds exhibit polymorphism. In fatty acids, this results from different
types of attachment between the carboxyl groups of adjacent molecules, which in turn modify the angle
of tilt of the chains in the crystal. The triglyceride tristearin proceeds from the low-melting metastable
alpha (α) form through the beta prime (β′) form and finally to the stable beta (β) form, having a high
melting point. The transition cannot occur in the opposite direction.
Theobroma oil, or cacao butter, is a polymorphous natural fat. Because it consists mainly of a single
glyceride, it melts to a large degree over a narrow temperature range (34°C–36°C). Theobroma oil is
capable of existing in four polymorphic forms: the unstable gamma form, melting at 18°C; the alpha
form, melting at 22°C; the beta prime form, melting at 28°C; and the stable beta form, melting at 34.5°C.
Riegelman12 pointed out the relationship between polymorphism and the preparation of cacao butter
suppositories. If theobroma oil is heated to the point at which it is completely liquefied (about 35°C), the
nuclei of the stable beta crystals are destroyed and the mass does not crystallize until it is supercooled
to about 15°C. The crystals that form are the metastable gamma, alpha, and beta prime forms, and the
suppositories melt at 23°C to 24°C or at ordinary room temperature. The proper method of preparation
involves melting cacao butter at the lowest possible temperature, about 33°C. The mass is sufficiently
62
Dr. Murtadha Alshareifi e-Library
fluid to pour, yet the crystal nuclei of the stable beta form are not lost. When the mass is chilled in the
mold, a stable suppository, consisting of beta crystals and melting at 34.5°C, is produced.
Polymorphism has achieved significance in last decade because different polymorphs exhibit different
solubilities. In the case of slightly soluble drugs, this may affect the rate of dissolution. As a result, one
polymorph may be more active therapeutically than another polymorph of the same drug. Aguiar et
al.13 showed that the polymorphic state of chloramphenicol palmitate has a significant influence on the
biologic availability of the drug. Khalil et al.14 reported that form II of sulfameter, an antibacterial agent,
was more active orally in humans than form III, although marketed pharmaceutical preparations were
found to contain mainly form III. Another case is that of the AIDS drug ritonavir, which was marketed in a
dissolved formulation until a previously unknown, more stable and less soluble polymorph appeared.
This resulted in a voluntary recall and reformulation of the product before it could be reintroduced to the
market.
Polymorphism can also be a factor in suspension technology. Cortisone acetate exists in at least five
different forms, four of which are unstable in the presence of water and change to a stable
form.15 Because this transformation is usually accompanied by appreciable caking of the crystals, these
should all be in the form of the stable polymorph before the suspension is prepared. Heating, grinding
under water, and suspension in water are all factors that affect the interconversion of the different
cortisone acetate forms.16
63
Dr. Murtadha Alshareifi e-Library
Fig.. 2-7. (a) Sttructure and
d numberingg of spiperon
ne. (b) Molecular confo
formation off
two polymorphhs, I and II, of spiperonne. (Modifieed from J. W.
W Moncrieff and W. H.
Jonees, Elementts of Physica
al Chemistry
ry, Addison-Wesley, Reading, Maass., 1977, p.
p
93; R. Chang, Physical
P
Ch
hemistry witth Applicatiions to Biolo
ogical Systeems, 2nd Ed
d.,
Maccmillan, Neew York, 19
977, p. 162.)) (From M. Azibi, M. Draguet-Bru
D
ughmans, R.
R
Bouuche, B. Tinnant, G. Gerrmain, J. P. Declercq, and
a M. Van Meersschee, J. Pharm.
Sci. 72, 232, 19983. With permission.)
p
)
Altho
ough crystal sttructure determination has b
become quite routine with th
he advent of ffast, highresolution diffracto
ometer system
ms as well as ssoftware allow
wing solution frrom powder x--ray diffraction
n
data, it can be cha
allenging to de
etermine the ccrystal structurre of highly unstable polymoorphs of a drug
g.
Azibi et al.17 studied two polym
morphs of spipe
erone, a poten
nt antipsychotic agent used mainly in the
treattment of schizophrenia. The
e chemical stru
ucture of spipe
erone is shown in Figure 2-77aand the
mole
ecular conform
mations of the two polymorp
phs, I and II, arre shown in Figure 2-7b. Thhe difference
betw
ween the two polymorphs
p
is in the position
ning of the ato
oms in the side
e chains, as seeen in Figure 2-7b,
64
Dr. Murtadha Alshareifi e-Library
together with the manner
m
in which each moleccule binds to neighboring
n
sp
piperone moleecules in the
crysttal. The results of the investtigation showe
ed that the cry
ystal of polymo
orph II is madee up of dimers
s
(molecules in pairss), whereas po
olymorph crysstal I is constru
ucted of nondiimerized moleecules of
spipe
erone. In a latter study, Azib
bi et al.18 exam
mined the poly
ymorphism of a number of ddrugs to ascerrtain
whatt properties ca
ause a compound to exist in
n more than on
ne crystalline form.
f
Differennces in
interrmolecular van
n der Waals fo
orces and hydrrogen bonds were
w
found to produce differrent crystal
strucctures
P.31
in an
ntipsychotic co
ompounds suc
ch as haloperid
dol (Fig. 2-8) and bromperid
dol. Variabilityy in hydrogen
bond
ding also contrributes to poly
ymorphism in tthe sulfonamid
des.19
Fig.. 2-8. Halopperidol.
Gold
dberg and Beccker20 studied
d the crystallin e forms of tam
moxifen citrate
e, an antiestroggenic and
antin
neoplastic drug
g used in the treatment
t
of b
breast cancer and postmeno
opausal sympttoms. The
strucctural formula of tamoxifen is shown in Fig
gure 2-9. Of th
he two forms found,
f
the paccking in the sta
able
polym
morph, referre
ed to as form B,
B is dominate
ed by hydrogen bonding. On
ne carboxyl grroup of the citrric
acid moiety donate
es its proton to
o the nitrogen
n atom on an adjacent
a
tamoxifen moleculee to bring about
the h
hydrogen-bond
ding network responsible
r
fo
or the stabiliza
ation of form B. The other poolymorph, kno
own
as fo
orm A, is a me
etastable polym
morph of tamo
oxifen citrate, its molecular structure
s
beingg less organiz
zed
than that of the sta
able B form. An
A ethanolic su
uspension of polymorph
p
A spontaneously
s
y rearranges in
nto
polym
morph B.
Lowe
es et al.21 perrformed physical, chemical,, and x-ray stu
udies on carba
amazepine. Caarbamazepine
e is
used
d in the treatm
ment of epileps
sy and trigemin
nal neuralgia (severe
(
pain in
n the face, lipss, and tongue).
The β polymorph of
o the drug can be crystallizzed from solve
ents of high die
electric constaant, such as th
he
aliph
hatic alcohols. The α polymo
orph is crystal lized from solv
vents of low dielectric consttant, such as
carbon tetrachlorid
de and cyclohexane. A rathe
er thorough sttudy of the two
o polymorphicc forms of
carbamazepine wa
as made using
g infrared spe
ectroscopy, the
ermogravimetrric analysis (T
TGA), hot-stage
micro
oscopy, dissolution rate, an
nd x-ray powde
er diffraction. The
T hydrogen
n-bonded struccture of
the α polymorph of
o carbamazep
pine is shown in Figure 2-10
0a and its mole
ecular formulaa in Figure 2-10b.
Estro
ogens are esssential hormon
nes for the devvelopment of female
f
sex characteristics. W
When the pote
ent
synth
hetic estrogen
n ethynylestrad
diol is crystalliized from the solvents aceto
onitrile, methaanol, and
chlorroform saturatted with waterr, four differentt crystalline so
olvates are forrmed. Ethynyleestradiol had been
repo
orted to exist in
n several polymorphic formss. However, Is
shida et al.22 showed
s
from tthermal analysis,
65
Dr. Murtadha Alshareifi e-Library
infrared spectrosccopy, and x-ray
y studies that these forms are
a crystals co
ontaining solveent molecules and
thus should be cla
assified as solv
vates rather th
han as polymo
orphs. Solvate
es are sometim
mes
calle
ed pseudopolyymorphs. Of co
ourse, solvate
es may also ex
xhibit polymorp
phism as longg as one comp
pares
“like”” solvation-sta
ate crystal stru
uctures.23 Oth
her related esttradiol compou
unds may exisst in true polym
meric
forms.
Fig.. 2-9. Tamooxifen citrate.
66
Dr. Murtadha Alshareifi e-Library
67
Dr. Murtadha Alshareifi e-Library
Fig. 2-10. (a) Two molecules of the polymorph α-carbamazepine joined together by
hydrogen bonds. (From M. M. J. Lowes, M. R. Caira, A. P. Lotter, and J. G. Van Der
Watt, J. Pharm. Sci. 76, 744, 1987. With permission.) (b) Carbamazepine.
Behme et al.24 reviewed the principles of polymorphism with emphasis on the changes that the
polymorphic forms may undergo. When the change from one form to another is reversible, it is said to
be enantiotropic. When the transition takes place in one direction only—for example, from a metastable
to a stable form—the change is said to be monotropic. Enantiotropism and monotropism are important
properties of polymorphs as described by Behme et al.24
The transition temperature in polymorphism is important because it helps characterize the system and
determine the more stable form at temperatures of interest. At their transition temperatures, polymorphs
have the same free energy (i.e., the forms are in equilibrium with each other), identical solubilities in a
particular solvent, and identical vapor pressures. Accordingly, plots of logarithmic solubility of two
polymorphic forms against 1/T provide the transition temperature at the intersection of the extrapolated
curves. Often, the plots are nonlinear and cannot be extrapolated with accuracy. For dilute solutions, in
which Henry's law applies, the logarithm of the solubility ratios of two polymorphs can be plotted
P.32
against 1/T, and the intersection at a ratio equal to unity gives the transition temperature.25 This
temperature can also be obtained from the phase diagram of pressure versus temperature and by using
differential scanning calorimetry (DSC).26
Solvates
Because many pharmaceutical solids are often synthesized by standard organic chemical methods,
purified, and then crystallized out of different solvents, residual solvents can be trapped in the crystalline
lattice. This creates a cocrystal, as described previously, termed a solvate. The presence of the residual
solvent may dramatically affect the crystalline structure of the solid depending on the types of
intermolecular interactions that the solvent may have with the crystalline solid. In the following sections
we highlight the influence of solvates and how they can be detected using standard solid
characterization analyses.
Biles27 and Haleblian and McCrone28 discussed in some detail the significance of polymorphism and
solvation in pharmaceutical practice.
Amorphous Solids
Amorphous solids as a first approximation may be considered supercooled liquids in which the
molecules are arranged in a somewhat random manner as in the liquid state. Substances such as glass,
pitch, and many synthetic plastics are amorphous solids. They differ from crystalline solids in that they
tend to flow when subjected to sufficient pressure over a period of time, and they do not have definite
melting points. In the Rheology chapter, a solid is characterized as any substance that must be
subjected to a definite shearing force before it fractures or begins to flow. This force, below which the
body shows elastic properties, is known as theyield value.
Amorphous substances, as well as cubic crystals, are usually isotropic, that is, they exhibit similar
properties in all directions. Crystals other than cubic areanisotropic, showing different characteristics
(electric conductance, refractive index, crystal growth, rate of solubility) in various directions along the
crystal.
It is not always possible to determine by casual observation whether a substance is crystalline or
amorphous. Beeswax and paraffin, although they appear to be amorphous, assume crystalline
arrangements when heated and then allowed to cool slowly. Petrolatum contains both crystalline and
amorphous constituents. Some amorphous materials, such as glass, may crystallize after long standing.
Whether a drug is amorphous or crystalline has been shown to affect its therapeutic activity. Thus, the
crystalline form of the antibiotic novobiocin acid is poorly absorbed and has no activity, whereas the
68
Dr. Murtadha Alshareifi e-Library
amorphous form is readily absorbed and therapeutically active.29 This is due to the differences in the
rate of dissolution. Once dissolved, the molecules exhibit no memory of their origin.
X-Ray Diffraction
X-rays are a form of electromagnetic radiation (Chapter 4) having a wavelength on the order of
interatomic distances (about 1.54 Å for most laboratory instruments using Cu Kα radiation; the C—C
bond is about 1.5 Å). X-rays are diffracted by the electrons surrounding the individual atoms in the
molecules of the crystals. The regular array of atoms in the crystal (periodicity) causes certain directions
to constructively interfere in some directions and destructively interfere in others, just as water waves
interfere when you drop two stones at the same time into still water (due to the similarity of the
wavelengths to the distance between the atoms or molecules of crystals mentioned). The x-ray
diffraction pattern on modern instruments is detected on a sensitive plate arranged behind the crystal
and is a “shadow” of the crystal lattice that produced it. Using computational methods, it is possible to
determine the conformation of the molecules as well as their relationship to others in the structure. This
results in a full description of the structure including the smallest building block, called the unit cell.
The electron density and, accordingly, the position of the atoms in complex structures, such as penicillin,
may be determined from a comprehensive mathematical study of the x-ray diffraction pattern. The
electron density map of crystalline potassium benzylpenicillin is shown in Figure 2-11. The elucidation of
this structure by x-ray crystallography paved the way for the later synthesis of penicillin by organic
chemists. Aspects of x-ray crystallography of pharmaceutical interest are reviewed by Biles30 and Lien
and Kennon.31
Where “single” crystals are unavailable or unsuitable for analysis, a powder of the substance may be
investigated. The powder x-ray diffraction pattern may be thought of as a fingerprint of the single-crystal
structure. Comparing the position and intensity of the lines (the same constructive interference
discussed previously) on such a pattern with corresponding lines on the pattern of a known sample
allows one to conduct a qualitative and a quantitative analysis. It is important to note that two
polymorphs will provide two distinct powder x-ray diffraction patterns. The presence of a solvate will also
influence the powder x-ray diffraction pattern because the solvate will have its own unique crystal
structure. This may lead to a single polymorphic form appearing as changeable or two distinct
polymorphs. One way to determine whether the presence of a change in a powder x-ray diffraction
pattern is due to a solvate or is a separate polymorph is to measure the powder x-ray diffraction patterns
at various temperatures. Because solvents tend to be driven out of the structure below the melting point,
measuring the powder x-ray diffraction patterns at several temperatures may eliminate the solvent and
reveal an unsolvated form. Lack of a change in the powder x-ray diffraction patterns at the different
temperatures is a strong indication that the form is not really solvated, or minor changes may indicate a
structure that maintains its packing motif without the solvent preset (see Fig. 2-12 for spirapril). This can
be confirmed by other methods as described later.
P.33
69
Dr. Murtadha Alshareifi e-Library
Fig.. 2-11. (a) Electron
E
den
nsity map off potassium
m benzylpenicillin. (Moodified from
m G.
L. P
Pitt, Acta Crrystallogr. 5,
5 770, 19522.) (b) A mo
odel of the structure
s
thaat can be bu
uilt
from
m analysis of
o the electron density pprojection.
Me
elting Poiint and Heat
H
of Fu
usion
The temperature at
a which a liqu
uid passes into
o the solid statte is known as
s the freezing point. It is also
o
the m
melting point of
o a pure crysttalline compou
und. The freez
zing point or melting
m
point off a pure crysta
alline
solid
d is strictly defiined as the temperature at w
which the pure
e liquid and so
olid exist in eqquilibrium. In
practice, it is taken
n as the tempe
erature of the equilibrium mixture
m
at an ex
xternal pressuure of 1 atm; th
his is
some
etimes known
n as the norma
al freezing or m
melting point. The student is
s reminded thaat different
interrmolecular forcces are involve
ed in holding tthe crystalline
e solid together and that the addition of he
eat to
melt the crystal is actually the addition of ene
ergy. Recall tha
at in a liquid, molecular
m
mottion occurs at a
much
h greater rate than in a solid
d.
70
Dr. Murtadha Alshareifi e-Library
Fig.. 2-12. Pow
wder x-ray diiffraction paatterns for spirapril
s
hyd
drochloridee. The
monnohydrate (ttop) and thee sample dehhydrated at 75°C for 106 and 228 hr (middle and
botttom, respecttively) demonstrating tthat the stru
uctural motiff is essentiaally unchang
ged
in thhe “dehydraated hydratee.” (From W
W. Xu, Invesstigation off Solid State Stability off
Seleected Bioacttive Compounds, unpubblished disssertation, Pu
urdue Univeersity, Purdue,
Ind.., 1997. Witth permissio
on.)
The heat (energy) absorbed when 1 g of a so
olid melts or th
he heat liberate
ed when it freeezes is known
n as
the la
atent heat of fusion,
f
and forr water at 0°C it is about 80 cal/g (1436 cal/mole). The heat added during
the m
melting processs does not brring about a ch
hange in temp
perature until all
a of the solid has disappea
ared
beca
ause this heat is converted into
i
the potenttial energy of the molecules
s that have esccaped from th
he
solid
d into the liquid
d state. The no
ormal melting points of som
me compounds
s are collectedd in Table 25 tog
gether with the
e molar heats of fusion.
Chan
nges of the fre
eezing or meltting point with pressure can be obtained by
b using a form
m of the Clape
eyron
equa
ation, written as
a
Table 2--5 Normal Melting Pooints and Molar
M
Heats of Fusionn of some
C
Compound
ds
Su
ubstance Melting
M
Poiint (K)Mollar Heat of Fusion, ΔH
Hf (cal/molee)
H2O
273.15
14440
H2S
187.61
5668
NH3
195.3
14424
PH3
139.4
2668
CH4
90.5
2226
C2H6
90
6883
n-C3H8
85.5
8442
C6H6
278.5
23348
C10H8
353.2
45550
P.34
4
71
Dr. Murtadha Alshareifi e-Library
wherre Vl and Vs are the molar volumes
v
of the
e liquid and so
olid, respective
ely. Molar voluume (volume in
n
3
unitss of cm /mole)) is computed by dividing the
e gram molecular weight by
y the density oof the
comp
pound. ∆Hf is the molar hea
at of fusion, th
hat is, the amo
ount of heat ab
bsorbed when 1 mole of the
e
solid
d changes into
o 1 mole of liqu
uid, and ∆T is the change of melting point brought abouut by a pressu
ure
chan
nge of ∆P.
Wate
er is unusual in that it has a larger molar vvolume in the solid state tha
an in the liquidd state (Vs > Vl) at
the m
melting point. Therefore, ∆T
T/∆P is negativve, signifying that
t
the meltin
ng point is low
wered by an
incre
ease in pressu
ure. This phenomenon can b
be rationalized
d in terms of Le
L Chatelier's pprinciple, whic
ch
state
es that a syste
em at equilibriu
um readjusts sso as to reduc
ce the effect of an external sstress.
Acco
ordingly, if a pressure is app
plied to ice at 0
0°C, it will be transformed in
nto liquid wateer, that is, into
o the
state
e of lower volu
ume, and the freezing
f
point will be lowere
ed.
Exa
ample 2-8
Dem
monstration
n of Le Chatelier's Prin
nciple
Wha
at is the effecct of an incre
ease of presssure of 1 atm
m on the freez
zing point of water (melting
poin
nt of ice)?
At 0°C, T = 273.16 K, ∆Hf [co
ongruent] 14
440 cal/mole,, the molar volume of watter is 18.018
8,
and the molar vo
olume of ice is 19.651, orr Vl - Vs = -1..633 cm3/mole. To obtainn the result in
n
deg//atm using equation (2-18
8), we first co
onvert ∆Hf in
n cal/mole intto units of errgs/mole by
multtiplying by the factor 4.18
84 × 107 ergss/cal:
Then multiplying
g equation (2-18) by the cconversion fa
actor (1.013 × 106 dynes//cm2)/atm (w
which
is pe
ermissible be
ecause the numerator
n
an
nd denominator of this fac
ctor are equivvalent, so the
facto
or equals 1) gives the res
sult in the de
esired units:
Hencce, an increasse of pressure of 1 atm lowe
ers the freezin
ng point of water by about 0..0075 deg, or an
incre
ease in pressu
ure of about 13
33 atm would be required to
o lower the fre
eezing point off water by 1 de
eg.
(Whe
en the ice–wa
ater equilibrium
m mixture is sa
aturated with air
a under a total pressure off 1 atm, the
temp
perature is low
wered by an ad
dditional 0.002
23 deg.) Press
sure has only a slight effectt on the equilib
brium
temp
perature of con
ndensed syste
ems (i.e., of liq
quids and solids). The large
e molar volumee or low densiity of
ice (0
0.9168 g/cm3 as compared with 0.9988 g
g/cm3 for water at 0°C) acco
ounts for the faact that the ice
e
floatss on liquid water. The lowerring of the me lting point with
h increasing pressure is takken advantage
e of in
ice-sskating. The pressure of the
e skate lowers the melting point and thus causes the icee to melt below
w the
skate
e. This thin layyer of liquid prrovides lubrica
ating action an
nd allows the skate
s
to glide over the hard
surfa
ace. Of course
e, the friction of
o the skate alsso contributes
s greatly to the
e melting and lubricating action.
Exa
ample 2-9
Pres
ssure Effec
cts on Freez
zing Points
s
Acco
ording to Exa
ample 2-8, an increase o
of pressure of
o 1 atm reduces the freezzing (melting
g)
poin
nt of ice by 0..0075 deg. To
T what temp
perature is th
he melting po
oint reduced w
when a 90-lb
b
boy skates acrosss the ice? The
T area of th
he skate blad
des in contac
ct with the icce is 0.085 cm
m2.
In ad
ddition to the
e atmospheriic pressure, w
which may be
b disregarde
ed, the presssure of the
skattes on the ice
e is the mass
s (90 lb = 40 .8 kg) multip
plied by the acceleration cconstant of
gravvity (981 cm/sec2) and div
vided by the area of the skate
s
blades
s (0.085 cm2)):
72
Dr. Murtadha Alshareifi e-Library
Cha
anging to atm
mospheres (1 atm = 1.013
325 × 106 dy
ynes/cm2) yie
elds a pressuure of 464.7 atm.
a
The change in volume ∆V fro
om water to ice is 0.018 - 0.01963 lite
er/mole, or -00.00163
liter//mole for the
e transition fro
om ice to liqu
uid water.
Use equation (2--18) in the fo
orm of a derivvative:
For a pressure change
c
of 1 atm
a to 464.7 atm when th
he skates of the 90-lb boyy touch the ice,
the m
melting temp
perature will drop from 27
73.15 K (0°C
C) toT, the final melting teemperature of
o the
ice u
under the ska
ate blades, which
w
converrts the ice to liquid water and facilitatees the
lubriication. For such
s
a proble
em, we mustt put the equation in the form
f
of an inttegral; that is
s,
integ
grating betwe
een 273.15 K and T caussed by a pressure change under the skate blades
s
from
m 1 atm to (46
64.7 + 1) atm
m:
In th
his integrated
d equation, 1440 cal/mole
e is the heat of fusion ∆H
Hf of water in the region of
o
0°C, and 24.2 ca
al/liter atm is a conversio
on factor (see
e the front lea
af of the boook) for converrting
cal tto liter atm. We
W now have
e
The melting temperature has
s been reducced from 273
3.15 to 269.69 K, or a redduction in me
elting
poin
nt of 3.46 K by
b the pressu
ure of the ska
ates on the ic
ce.
A sim
mpler way to
o do the ice-s
skating probllem is to realize that the small
s
changee in tempera
ature,
-3.46 K, occurs over
o
a large pressure cha
ange of abou
ut 465 atm. Therefore,
T
wee need not
integ
grate but rath
her may obta
ain the tempe
erature chan
nge ∆T per unit atmospheere change, ∆P,
and multiply thiss value by the
e actual presssure, 464.7 atm. Of courrse, the heatt of fusion of
wate
er, 1440 cal/mole, must be
b multiplied by the conversion factorr, 1 liter atm/224.2 cal, to yield
y
59.5
504 liter atm. We have
For a pressure change
c
of 464.7 atm, the decrease in temperature
e is
as ccompared witth the more accurate
a
valu
ue, -3.46 K.
P.35
5
73
Dr. Murtadha Alshareifi e-Library
Fig.. 2-13. The melting poiints of alkannes and carb
boxylic acid
ds as a funct
ction of carb
bon
chaiin length. (M
Modified fro
om C. R. N
Noller,Chemistry of Org
ganic Comppounds, 2nd
Ed.,, Saunders, Philadelphiia, 1957, ppp. 40, 149.)
Me
elting Poiint and In
ntermolec
cular Forrces
The heat of fusion
n may be cons
sidered as the heat required
d to increase th
he interatomicc or intermolec
cular
dista
ances in crysta
als, thus allow
wing melting (in
ncreased mole
ecular motion)) to occur. A ccrystal that is
boun
nd together byy weak forces generally hass a low heat off fusion and a low melting pooint, whereas one
boun
nd together byy strong forces
s has a high he
eat of fusion and
a a high me
elting point.
Beca
ause polymorp
phic forms rep
present differe
ent molecular arrangements
a
leading to difffferent crystalline
forms of the same
e compound, itt is obvious tha
at different inttermolecular fo
orces will accoount for these
differrent forms. Th
hen consider polymorph
p
A, w
which is held together
t
by higher attractivee forces than is
i
polym
morph B. It is obvious that more
m
heat will be required to
o break down the attractive forces in
polym
morph A, and thus its meltin
ng temperaturre will be highe
er than that off polymorph B .
Para
affins crystallizze as thin leaflets composed
d of zigzag chains packed in
n a parallel arrrangement. Th
he
meltiing points of normal
n
saturatted hydrocarbo
ons increase with
w molecular weight becauuse the van der
Waa
als forces betw
ween the molecules of the c rystal become
e greater with an increasing number of ca
arbon
atom
ms. The melting points of the
e alkanes with
h an even num
mber of carbon
n atoms are higgher than thos
se of
the h
hydrocarbons with an odd number of carb
bon atoms, as shown in Figu
ure 2-13. Thiss phenomenon
n
presumably is beccause alkanes with an odd n
number of carbon atoms are
e packed in thhe crystal less
efficiiently.
Tab
ble 2-6 Meltting Points and Solub
bilities of some Xanthiines*
74
Dr. Murtadha Alshareifi e-Library
Compound
M
Melting Poin
nt (°C
Un
ncorrected)
So
olubility in Water at
30
0°C (mole/lliter × 102)
Theophylliine (R = H)
270–274
4.5
Caffeine (R
R = CH3)
238
13.3
7-Ethyitheeophylline (R
R=
CH2CH3)
156–157
17.6
7-Propylthheophylline (R =
CH2CH2CH
H3
99–100
104.0
*From D. Guttman
G
an
nd T. Higuchhi, J. Am. Pharm.
P
Asso
oc. Sci. Ed. 46, 4,
1957.
Fiig. 2-14. Coonfiguration
n of fatty accid moleculees in the cry
ystalline statte. (Modifieed
from A. E. Bailey, Meltiing and Soliidification of
o Fats, Inteerscience, N
New York,
19950, p. 120..)
The melting pointss of normal carboxylic acidss also show this alternation, as seen in Figgure 2-13. This
can b
be explained as
a follows. Fatty acids crysttallize in molecular chains, one
o segment oof which is shown
in Fig
gure 2-14. The even-numbe
ered carbon a
acids are arran
nged in the cry
ystal as seen iin the more
symm
metric structurre I, whereas the odd-numb
bered acids arre arranged ac
ccording to strructure II. The
carboxyl groups are joined at tw
wo points in the
e even-carbon
n compound; hence,
h
the cryystal lattice is more
stablle and the melting point is higher.
h
The melting pointss and solubilities of the xantthines of pharm
maceutical interest, determiined by Guttm
man
and Higuchi,32 furrther exemplify
y the relationsship between melting
m
point and
a molecularr structure.
Solubilities, like melting points, are
a strongly in
nfluenced by in
ntermolecular forces. This iss readily obse
erved
75
Dr. Murtadha Alshareifi e-Library
in Ta
able 2-6, wherre the methyla
ation of theoph
hylline to form caffeine and the
t lengtheninng of the side chain
from methyl (caffe
eine) to propyl in the 7 positiion result in a decrease of the melting poiint and an
incre
ease in solubility. These effe
ects presumab
bly are due to a progressive
e weakening oof intermolecullar
force
es.
The
e Liquid Crystalline State
Thre
ee states of ma
atter have bee
en discussed tthus far in this
s chapter: gas, liquid, and soolid. A fourth state
s
of matter is the
P.36
6
liquid
d crystalline sttate or mesop
phase. The terrm liquid crysta
al is an appare
ent contradictiion, but it is us
seful
in a d
descriptive se
ense because materials in th
his state are in
n many ways intermediate bbetween the liq
quid
and solid states.
Fig.. 2-15. Liquuid crystallin
ne state. (a)) Smectic sttructure; (b)) nematic strructure.
Strructure of
o Liquid Crystals
As seen earlier, molecules
m
in the liquid state a
are mobile in three
t
direction
ns and can alsso rotate abou
ut
three
e axes perpen
ndicular to one
e another. In th
he solid state, on the other hand, the mol ecules are
immo
obile, and rota
ations are not as readily posssible.
It is n
not unreasona
able to suppos
se, therefore, tthat intermediiate states of mobility
m
and rootation should
existt, as in fact the
ey do. It is these intermedia
ate states that constitute the
e liquid crystallline phase, or
meso
ophase, as the liquid crysta
alline phase is called.
The two main type
es of liquid cry
ystals are term
med smectic (s
soaplike or gre
easelike)
and nematic (threa
adlike). In the smectic state
e, molecules are mobile in tw
wo directions aand can rotate
e
abou
ut one axis (Fig. 2-15a). In the nematic sta
ate, the molec
cules again rotate only abouut one axis but are
mobile in three dim
mensions (Fig. 2-15b). A thi rd type of crys
stal (cholesterric) exists but ccan be consid
dered
as a special case of the nematic
c type. In athe
erosclerosis, itt is the incorpo
oration of choleesterol and lip
pids
in hu
uman subendo
othelial macrophages that le
eads to an insoluble liquid crystalline
c
bioloogic
mem
mbrane33 that ultimately res
sults in plaque
e formation.
76
Dr. Murtadha Alshareifi e-Library
The smectic mesophase is probably of most pharmaceutical significance because it is this phase that
usually forms in ternary (or more complex) mixtures containing a surfactant, water, and a weakly
amphiphilic or nonpolar additive.
In general, molecules that form mesophases (a) are organic, (b) are elongated and rectilinear in shape,
(c) are rigid, and (d) possess strong dipoles and easily polarizable groups. The liquid crystalline state
may result either from the heating of solids (thermotropic liquid crystals) or from the action of certain
solvents on solids (lyotropic liquid crystals). The first recorded observation of a thermotropic liquid
crystal was made by Reinitzer in 1888 when he heated cholesteryl benzoate. At 145°C, the solid formed
a turbid liquid (the thermotropic liquid crystal), which only became clear, to give the conventional liquid
state, at 179°C.
Properties and Significance of Liquid Crystals
Because of their intermediate nature, liquid crystals have some of the properties of liquids and some of
the properties of solids. For example, liquid crystals are mobile and thus can be considered to have the
flow properties of liquids. At the same time they possess the property of being birefringent, a property
associated with crystals. In birefringence, the light passing through a material is divided into two
components with different velocities and hence different refractive indices.
Some liquid crystals show consistent color changes with temperature, and this characteristic has
resulted in their being used to detect areas of elevated temperature under the skin that may be due to a
disease process. Nematic liquid crystals may be sensitive to electric fields, a property used to
advantage in developing display systems. The smectic mesophase has application in the solubilization
of water-insoluble materials. It also appears that liquid crystalline phases of this type are frequently
present in emulsions and may be responsible for enhanced physical stability owing to their highly
viscous nature.
The liquid crystalline state is widespread in nature, with lipoidal forms found in nerves, brain tissue, and
blood vessels. Atherosclerosis may be related to the laying down of
P.37
lipid in the liquid crystalline state on the walls of blood vessels. The three components of bile
(cholesterol, a bile acid salt, and water), in the correct proportions, can form a smectic mesophase, and
this may be involved in the formation of gallstones. Bogardus34 applied the principle of liquid crystal
formation to the solubilization and dissolution of cholesterol, the major constituent of gallstones.
Cholesterol is converted to a liquid crystalline phase in the presence of sodium oleate and water, and
the cholesterol rapidly dissolves from the surface of the gallstones.
Nonaqueous liquid crystals may be formed from triethanolamine and oleic acid with a series of
polyethylene glycols or various organic acids such as isopropyl myristate, squalane, squalene, and
naphthenic oil as the solvents to replace the water of aqueous mesomorphs. Triangular plots or tertiary
phase diagrams were used by Friberg et al.35a,b to show the regions of the liquid crystalline phase
when either polar (polyethylene glycols) or nonpolar (squalene, etc.) compounds were present as the
solvent.
Ibrahim36 studied the release of salicylic acid as a model drug from lyotropic liquid crystalline systems
across lipoidal barriers and into an aqueous buffered solution.
Finally, liquid crystals have structures that are believed to be similar to those in cell membranes. As
such, liquid crystals may function as useful biophysical models for the structure and functionality of cell
membranes.
Friberg wrote a monograph on liquid crystals.35b For a more detailed discussion of the liquid crystalline
state, refer to the review by Brown,37 which serves as a convenient entry into the literature.
The Supercritical Fluid State
Supercritical fluids were first described more than 100 years ago and can be formed by many different
normal gases such as carbon dioxide. Supercritical fluids have properties that are intermediate between
those of liquids and gases, having better ability to permeate solid substances (gaslike) and having high
densities that can be regulated by pressure (liquidlike). A supercritical fluid is a mesophase formed from
77
Dr. Murtadha Alshareifi e-Library
the g
gaseous state where the ga
as is held unde
er a combination of tempera
atures and preessures that
exce
eed the criticall point of a sub
bstance (Fig. 2
2-16). Briefly, a gas that is brought
b
abovee its critical
temp
perature Tc willl still behave as
a a gas irres pective of the applied press
sure; the critica
cal pressure (P
Pc) is
the m
minimum pressure required to liquefy a ga
as at a given temperature.
t
As
A the pressu re is raised higher,
the d
density of the gas can increa
ase without a significant inc
crease in the viscosity
v
whilee the ability of the
supe
ercritical fluid to
t dissolve com
mpounds also
o increases. A gas that may have little to nno ability to
disso
olve a compou
und under ambient condition
ns can comple
etely dissolve the compoundd under high
pressure in the supercritical range. Figure 2-1
17 illustrates this phenomen
non with CO2, where CO2 he
eld at
the ssame tempera
ature can disso
olve different cchemical class
ses from a nattural product ssource when the
t
pressure is increased.38 It is als
so important to
o note that mo
ore than one gas
g or even thhe addition of a
solve
ent (termed a cosolvent) such as water a nd/or ethanol can be made to increase thhe ability of the
e
supe
ercritical fluid to
t dissolve com
mpounds.
Fig.. 2-16. Phasse diagram showing
s
thee supercriticcal region of a compresssed gas. Keey:
CP = critical pooint. (Modiffied from M
Milton Roy, Co., Superccritical Fluiid Extractio
on:
Prinnciples and Application
ns [Bulletinn 37.020], Milton
M
Roy Co.,
C Ivydalee, Pa.)
Supe
ercritical fluid applications in
n the pharmacceutical scienc
ces were exce
ellently revieweed by Kaiser et
e
al.39
9 There are se
everal common uses for sup
percritical fluid
ds including
extra
action,40,41 crrystallization,4
42 and the pre
eparation of fo
ormulations (th
hey are increassingly being used
,
,
to prrepare polyme
er mixtures43 44
4 45 and for tthe formation of micro- and nanoparticless46). Supercritical
fluidss offer several advantages over
o
traditiona
al methodolog
gies: the potential for low-tem
mperature
extra
actions and pu
urification of co
ompounds (co
onsider heat added
a
for distillation proceduures), solvent
volattility under am
mbient conditions, selectivityy of the extractted compound
ds (Fig. 2-17), and lower ene
ergy
requirement and lo
ower viscosity
y than solventss.38 Most imp
portant is the re
educed toxicitty of the gases
s and
78
Dr. Murtadha Alshareifi e-Library
the rreduced need for hazardous
s solvents thatt require expe
ensive disposa
al. For examplee, supercritica
al
CO2 can be simplyy disposed of by opening a valve and rele
easing the CO
O2 into the atm
mosphere.
One of the best exxamples of the
e use of superrcritical fluids is
i in the decafffeination of
coffe
ee.39,47 Traditionally, solve
ents like methyylene chloride have been us
sed in the decaaffeination
process. This lead
ds to great exp
pense in the p
purchase and disposal
d
of the
e residual solvvents and
incre
eases the chan
nce for toxicity
y. Supercritica
al CO2has now
w been utilized
d for the decafffeination of co
offee
and tea. Interestin
ngly, initial sup
percritical CO2 resulted in the removal of the
t caffeine annd
P.38
8
impo
ortant flavor-ad
dding compou
unds from coffe
fee. The loss of
o the flavor-ad
dding compouunds resulted in a
poorr taste and an unacceptable
e product. Add
ditional studies
s demonstrated that adding water to the
supe
ercritical CO2 significantly
s
re
educed the losss of the flavorr. However, in this process a sample run
using
g coffee beans is performed
d through the ssystem to satu
urate the wate
er with the flavvor-enhancing
comp
pounds. The supercritical
s
CO
C 2 is passed over an extraction column upon
u
release oof the pressurre
and the residual caffeine it carries is collected
d on the colum
mn. The sample run of beanns is removed and
dispo
osed of, and then the first batch of marke
etable coffee beans
b
is passe
ed through thee system as th
he
wate
er is recirculate
ed. Several ba
atches of deca
affeinated cofffee are preparred in this mannner before the
wate
er is discarded
d. The process
s leads to app roximately 97% of caffeine being removeed from the be
eans.
This is an excellen
nt example of how cosolven
nts may be add
ded to improve the quality oof the supercritical
CO2--processed material.
Fig.. 2-17. The effect of prressure on thhe ability off supercriticcal fluids to selectively
extrract differennt compound
ds. (Modifieed from Miilton Roy Co
o., Supercriitical Fluid
Extrraction: Priinciples and
d Applicatioons [Bulletin
n 37.020], Milton
M
Roy Co., Ivylan
nd,
Pa.))
The
ermal An
nalysis
As noted earlier in
n this chapter, a number of p
physical and chemical
c
effec
cts can be prodduced by
temp
perature chang
ges, and meth
hods for chara
acterizing thes
se alterations upon
u
heating oor cooling a
samp
ple of the matterial are referred to as therm
rmal analysis. The most com
mmon types off thermal analy
ysis
are D
DSC, differenttial thermal an
nalysis (DTA), TGA, and the
ermomechanic
cal analysis (T
TMA). These
79
Dr. Murtadha Alshareifi e-Library
methods have proved to be valuable in pharmaceutical research and quality control for the
characterization and identification of compounds, the determination of purity,
polymorphism20,21 solvent, and moisture content, amorphous content, stability, and compatibility with
excipients.
In general, thermal methods involve heating a sample under controlled conditions and observing the
physical and chemical changes that occur. These methods measure a number of different properties,
such as melting point, heat capacity, heats of reaction, kinetics of decomposition, and changes in the
flow (rheologic) properties of biochemical, pharmaceutical, and agricultural materials and food. The
methods are briefly described with examples of applications. Differential scanning calorimetry is the
most commonly used method and is generally a more useful technique because its measurements can
be related more directly to thermodynamic properties. It appears that any analysis that can be carried
out with DTA can be performed with DSC, the latter being the more versatile technique.
Differential Scanning Calorimetry
In DSC, heat flows and temperatures are measured that relate to thermal transitions in materials.
Typically, a sample and a reference material are placed in separate pans and the temperature of each
pan is increased or decreased at a predetermined rate. When the sample, for example, benzoic acid,
reaches its melting point, in this case 122.4°C, it remains at this temperature until all the material has
passed into the liquid state
P.39
because of the endothermic process of melting. A temperature difference therefore exists between
benzoic acid and a reference, indium (melting point [mp] = 156.6°C), as the temperature of the two
materials is raised gradually through the range 122°C to 123°C. A second temperature circuit is used in
DSC to provide a heat input to overcome this temperature difference. In this way the temperature of the
sample, benzoic acid, is maintained at the same value as that of the reference, indium. The difference is
heat input to the sample, and the reference per unit time is fed to a computer and plotted
as dH/dt versus the average temperature to which the sample and reference are being raised. The data
collected in a DSC run for a compound such as benzoic acid are shown in the thermogram in Figure 218. There are a wide variety of features in DSCs such as autosamplers, mass flow controllers, and builtin computers. An example of a modern DSC, the Q200, is shown in Figure 2-19. The differential heat
input is recorded with a sensitivity of ± 0.2 µW, and the temperature range over which the instrument
operates is -180°C to 725°C.
80
Dr. Murtadha Alshareifi e-Library
Fig.. 2-18. Therrmogram off a drug com
mpound. Endothermic transitions
t
((heat
absoorption) aree shown in the upward ddirection an
nd exotherm
mic transitioons (heat losss)
are pplotted dow
wnward. Meelting is an eendothermicc process, whereas
w
crysstallization or
freeezing is an exothermic
e
process.
p
Thhe area of th
he melting peak
p
is propoortional to the
t
heatt of fusion, ΔH
Δ f.
Diffe
erential scanning calorimetry
y is a measure
ement of heat flow into and out of the sysstem. In generral,
an endothermic (th
he material is absorbing hea
at) reaction on
n a DSC arises from desolvvations, melting
g,
glasss transitions, and,
a
more rare
ely, decompossitions. An exo
othermic reacttion measuredd by DSC is us
sually
indiccative of a deccomposition (e
energy is relea
ased from the bond breaking
g) process andd molecular
reorg
ganizations su
uch as crystallization. Differe
ential scannin
ng calorimetry has found inccreasing use in
n
stand
dardization off the lyophilization process.4
48 Crystal cha
anges and eutectic formationn in the frozen
n
state
e as well as am
morphous cha
aracter can be detected by DSC
D
(and by DTA)
D
when thee instruments are
operrated below ro
oom temperatu
ure.
Altho
ough DSC is used
u
most wid
dely in pharma
acy to establish identity and purity, it is alm
most as comm
monly
used
d to obtain hea
at capacities and
a heats of fu
usion and capacities. It
P.40
0
is alsso useful for constructing
c
ph
hase diagramss to study the polymorphs discussed
d
in thhis chapter and for
carryying out studie
es on the kinettics of decomp
position of solids.
81
Dr. Murtadha Alshareifi e-Library
Fig.. 2-19. Perkkin Elmer diifferential sccanning callorimeter 7 (DSC
(
7).
Diffe
erential scanning calorimetry
y and other the
ermal analytic
c methods hav
ve a number oof applications in
biom
medical research and food te
echnology. Gu
uillory and ass
sociates49,50 explored the aapplications off
therm
mal analysis, DSC,
D
and DTA
A in particularr, in conjunctio
on with infrared
d spectroscoppy and x-ray
diffra
action. Using these
t
techniqu
ues, they charracterized variious solid form
ms of drugs, suuch as
sulfo
onamides, and
d correlated a number of phyysical properties of crystalline materials w
with interaction
ns
betw
ween solids, dissolution rates, and stabilitiies in the crystalline and am
morphous statees.
For a
additional refe
erences to the use of DSC in
n research and technology, contact the m
manufacturers of
differrential therma
al equipment fo
or complete biibliographies.
Diffferential Thermall Analysis
is
In DT
TA, both the sample
s
and the
e reference m
material are heated by a com
mmon heat souurce (Fig. 2-20
0)
rathe
er than the ind
dividual heaterrs used in DSC
C (Fig. 2-21). Thermocouple
es are placed in contact witth the
samp
ple and the re
eference material in DTA to monitor the diifference in tem
mperature bettween the sam
mple
and the reference material as th
hey are heated
d at a constan
nt rate. The tem
mperature diffference betwe
een
the ssample and the reference material
m
is plottted against tim
me, and the en
ndotherm as m
melting occurs
s (or
exoth
ome decompossition reaction
ns) is represen
nted by a peakk in the
herm as obtained during so
therm
mogram.
Altho
ough DTA is a useful tool, a number of fa
actors may affe
ect the results. The temperaature
differrence, ∆T, depends, among
g other factorss, on the resisttance to heat flow, R. In turn
rn, R depends on
temp
perature, nature of the samp
ple, and packi ng of the mate
erial in the pan
ns. Therefore,, it is not poss
sible
to dirrectly calculatte energies of melting, subli mation, and decomposition, and DTA is uused as a
82
Dr. Murtadha Alshareifi e-Library
qualiitative or semiiquantitative method
m
for calo
orimetric measurements. Th
he DSC, althoough more
expe
ensive, is need
ded for accura
ate and precise
e results.
Fig.. 2-20. Com
mmon heat source of diffferential th
hermal analy
yzer with thhermocouplees
in contact with the samplee and the refference material.
A rellated techniqu
ue is dielectric analysis, also
o discussed in
n Chapter 4. The concept is that molecule
es
move
e in response to an applied electric field a
according to th
heir size, dipo
ole moments, aand environme
ent
(whicch changes with
w temperaturre). In this tec hnique, the sa
ample serves as the dielectrric medium in a
capa
acitor, and as the
t ac electric
c field oscillate
es, the motion of permanentt dipoles is sennsed as a pha
ase
shift relative to the
e timescale of the frequencie
es used. Each
h sample will have
h
a characcteristic respon
nse
or pe
ermittivity at a given temperrature and freq
quency. As the
e sample is he
eated or cooleed, the response
will vvary because the
t mobility off the molecula
ar dipoles will change.
c
On go
oing through a first-order
transsition such as melting, the mobility
m
will exxhibit a drastic frequency-ind
dependent chaange giving ris
se to
a fam
mily of curves with different intensities butt a common maximum
m
occu
urring at the tra
ransition
temp
perature. For glass
g
transitions (pseudoseccond order) where
w
the trans
sition occurs oover a broad
temp
perature range
e, there will be
e a distinct varriation in both intensity and temperature, allowing extre
emely
senssitive detection
n of such
P.41
even
nts. Dielectric analysis detec
cts the microsscopic “viscosity” of the system and can yyield informatio
on on
activvation energiess of changes as
a well as the homogeneity
y of samples. The
T techniquee does, however,
require significantlly more data analysis
a
than tthe other therm
mal methods discussed.
d
83
Dr. Murtadha Alshareifi e-Library
Fig.. 2-21. Sepaarate heat so
ources and pplatinum heeat sensors used
u
in diffe
ferential
scannning caloriimetry.
Key Co
oncept
Prac
ctical Appliication of Thermal
T
Ana
alysis
Usin
ng TGA combined with DSC or DTA, we can class
sify endothermic and exoothermic
reacctions. For exxample, if a DSC
D
thermog
gram contain
ns an endoth
hermic reacti on at 120°C,
anotther endothe
erm at 190°C
C, and an exo
otherm at 260
0°C, the scie
entist must deetermine the
e
causse of each off these heat exchange prrocesses. With the utiliza
ation of a TGA
A instrument,
the cchange in the weight of a sample is m
measured as
s a function of
o temperaturre. If the sam
me
mate
erial is subje
ected to TGA
A analysis, the
e transitions can be class
sified accordding to weigh
ht
chan
nges. Thereffore, if a 4% weight loss iis observed at
a 120°C, no
o weight channge is observ
ved
at 19
90°C, and th
he weight is lo
ost at 260°C
C, certain ass
sumptions ca
an be made. The 4% weig
ght
loss was associa
ated with an endothermicc change, wh
hich can corrrespond to a desolvation,,
makking it a likelyy choice for the endotherrmic response. The lack of
o a weight cchange at 190
0°C
wou
uld most likelyy suggest a melt due to tthe endotherrm observed on the DSC . The final lo
oss of
all o
of the remaining mass at 260°C is mo
ost likely due to decomposition; the exxotherm on the
DSC
C would supp
port that conc
clusion. In ad
ddition, Karl Fisher analy
ysis, describeed in the nex
xt
subssection, could be used to
o determine w
whether the solid
s
is a water solvate too determine
whe
ether the 4% loss in mass
s at 120°C iss due to wate
er.
The
ermogravimetric and Therrmomech
hanical Analyses
A
Chan
nges in weight with tempera
ature (thermog
gravimetric an
nalysis, TGA) and
a changes iin mechanical
prop
perties with tem
mperature (the
ermomechaniccal analysis, TMA)
T
are used
d in pharmace utical enginee
ering
research and in industrial quality
y control. In T
TGA, a vacuum
m recording ba
alance with a ssensitivity of 0.1
0
µg iss used to record the sample
e weight underr pressures of 10-4 mm to 1 atm.
Therrmogravimetricc analysis insttruments have
e now begun to
t be coupled with infrared oor mass
specctrometers to measure
m
the chemical
c
naturre of the evolv
ved gases bein
ng lost from thhe sample. The
next subsection de
escribes Karl Fisher analysiis, which can also help in de
etermining whhether the
deso
olvation may be
b attributed to
o water or resiidual solvents from chemica
al processing. The changes with
temp
perature in hyd
drated salts su
uch as calcium
m oxalate, CaC
C2O4·H2O, are
e evaluated ussing TGA, as
discu
ussed by Simo
ons and Newk
kirk.51
84
Dr. Murtadha Alshareifi e-Library
The characterization by TGA of bone tissue associated with dental structures was reported by Civjan et
al.52 Thermogravimetric analysis also can be used to study drug stability and the kinetics of
decomposition.
Thermomechanical analysis measures the expansion and extension of materials or changes in
viscoelastic properties and heat distortions, such as shrinking, as a function of temperature. By use of a
probe assembly in contact with the test material, any motion due to expansion, melting, or other physical
change delivers an electric signal to a recorder. The furnace, in which are placed a sample and a probe,
controls the temperature, which can be programmed over a range from -150°C to 700°C. The apparatus
serves essentially as a penetrometer, dilatometer, or tensile tester over a wide range of programmed
temperatures. Humphries et al.53used TMA in studies on the mechanical and viscoelastic properties of
hair and the stratum corneum of the skin. Thermomechanical analysis is also widely used to look at
polymer films and coatings used in pharmaceutical processes.
Karl Fisher Method
The Karl Fisher method is typically performed as a potentiometric titration method commonly used to
determine the amount of water associated with a solid material. The method follows the reaction of
iodine (generated electrolytically at the anode in the reagent bath) and sulfur with water. One mole of
iodine reacts with 1 mole of water, so the amount of water is directly proportional to the electricity
produced. As mentioned, a DSC measurement may indicate an endothermic reaction at 120°C. This
endothermic reaction may constitute an actual melt of the crystalline material or may be due to either
desolvation or a polymorphic conversion. If one measured the same material using TGA and found a
weight loss of about 4% at the same temperature as the endotherm, one could determine that the
endotherm arose from a desolvation process.
Utilizing Karl Fisher analysis, one can add the solid material to the titration unit and determine the
amount of water by mixing of reagents and the potentiometric electrodes. The Karl Fisher method is an
aid in that it can determine whether the desolvation is all water (showing a 4% water content) or arises
from the loss of a separate solvent trapped in the crystalline lattice. This method is routinely used for
pharmaceutical applications, including the study of humidity effects in solids undergoing water sorption
from the air and in quality control efforts to demonstrate the amount of water associated in different lots
of manufactured solid products.
Vapor Sorption/Desorption Analysis
This technique is similar to that of TGA in that it measures weight changes in solids as they are exposed
to different solvent vapors and humidity and/or temperature conditions, although it is typically operated
isothermally. Greenspan54
P.42
published a definitive list of saturated salt solutions that can be used to control the relative humidity and
are widely used to study many physicochemical properties of drugs. For example, if a selected saturated
salt solution providing a high relative humidity is placed in a sealed container, the hygroscopicity of a
drug can be assessed by determining the weight change in the solid under that humidity. A positive
change in the weight would indicate that the solid material is absorbing (collectively called sorption) the
solvent, in this case water, from the atmosphere inside the container. The ability of a solid to
continuously absorb water until it goes into solution is called deliquescence. A weight loss could also be
measured under low relative humidities controlled with different salts, which is termed desorption. Water
vapor sorption/desorption can be used to study changes in the solvate state of a crystalline
material.55,56 Variations of commercial instruments also allow the determination of the
sorption/desorption of other solvents. The degree of solvation of a crystalline form could have an
adverse effect on its chemical stability57and/or its manufacturability.
Generally, the less sensitive a solid material or formulation is to changes in the relative humidity, the
more stable will be the pharmaceutical shelf life and product performance. The pharmaceutical industry
supplies products throughout the world, with considerable variation in climate. Therefore, the
85
Dr. Murtadha Alshareifi e-Library
measurement of moisture
m
sorption/desorption
n rates and ex
xtents is very im
mportant for thhe prediction of
o
the sstability of drug
gs.
Phase Equiilibria and the Pha
ase Rule
e
The three primary phases (solid
d, liquid, and g
gaseous) of matter are often
n defined indivvidually under
differrent conditions, but in most systems we u
usually encoun
nter phases in
n coexistence.. For example, a
glasss of ice water on a hot summ
mer day comp
prises three co
oexisting phas
ses: ice (solid)), water (liquid),
and vvapor (gaseou
us). The amou
unt of ice in th e drink depen
nds heavily on several variabbles including the
amount of ice placced in the glas
ss, the temperrature of the water
w
in which it was placed,, and the
temp
perature of the
e surrounding air. The longe
er the drink is exposed to the warm air, thhe more the
amount of ice in th
he drink will de
ecrease, and tthe more wate
er the melting ice will producce. However,
evap
poration of the
e water into vapor that is rele
eased into the
e large volume
e of air will alsoo decrease the
liquid
d volume. For this system, there
t
is no esttablishment off equilibrium because
b
the voolume for vapo
or is
infiniite in contrast to the ice and
d liquid volume
es.
If the
e ice water is sealed
s
in a bo
ottle, evaporatiion effects are
e limited to the
e available heaadspace, the ice
i
meltss to liquid, and
d evaporation becomes time
e and tempera
ature depende
ent. For exampple, if the
conta
ainer is placed
d in a freezer, only one pha
ase, ice, may be
b present afte
er long-term st
storage. Heatin
ng of
the ccontainer, provvided the volume stays fixed
d, could poten
ntially cause th
he formation oof only a vaporr
phasse. Opening and closing of the
t container w
would change
e the vapor phase compositiion and thus affect
a
equillibrium. This one-componen
o
nt example ca
an be extended
d to the two-co
omponent sysstem of a drug
susp
pension where
e solid drug is suspended an
nd dissolved in
n solution and
d evaporation m
may take plac
ce in
the h
headspace of the container.. The suspend
ded system will sit at equilibrium until the container is
open
ned for adminiistration of the
e drug, and the
en equilibrium
m would have to be reestablisshed for the new
n
syste
em. A new equilibrium or no
onequilibrium sstate is establlished because dispensing oof the suspension
will d
decrease the volume
v
of the liquid and sol id in the conta
ainer. Thereforre, a new systtem is created
d
afterr each opening
g, dispensing of the dose, a
and then resea
aling. This will be discussedd in more detail
laterr.
Befo
ore we get into
o detail about the
t individual phases, it is im
mportant to un
nderstand how
w the phases
coexxist, the rules that
t
govern their coexistencce, and the number of variab
bles required tto define the
state
e(s) of matter present
p
underr defined cond
ditions.
The
e Phase Rule
In ea
ach of the examples just giv
ven, each phasse can be defined by a serie
es of independdent variables
s
(e.g., temperature) and their coe
existence can only occur ov
ver a limited ra
ange. For exam
mple, ice does
s not
last a
as long in boiling water as itt does in cold water. Thereffore, to unders
stand and defiine the state of
o
each
h phase, know
wledge of seve
eral variables i s required. J. Willard Gibbs
s formulated thhe phase rule,
whicch is a relationship for determ
mining the lea
ast number of intensive varia
ables (indepenndent variable
es
that do not depend
d on the volum
me or size of th
he phase, e.g., temperature
e, pressure, deensity, and
conccentration) tha
at can be chan
nged without cchanging the equilibrium
e
state of the systeem, or, alterna
ately,
the le
east number required
r
to define the state of the system. This critical number
n
is callled F, the num
mber
of de
egrees of freed
dom of the sys
stem, and the
e rule is expres
ssed as follow
ws:
wherre C is the num
mber of components and P is the number of phases prresent.
Lookking at these terms in more detail, we can
n define a pha
ase as a homo
ogeneous, phyysically distinct
portion of a system
m that is separated from oth
her portions off the system by bounding suurfaces. Thus,, a
syste
em containing
g water and its vapor is a two
wo-phase syste
em. An equilibrium mixture oof ice, liquid water,
w
and water vapor iss a three-phas
se system.
The number of com
mponents is th
he smallest nu
umber of cons
stituents by wh
hich the compo
position of each
phasse in the syste
em at equilibriu
um can be exp
pressed in the
e form of a che
emical formulaa or equation. The
number of compon
nents in the eq
quilibrium mixxture of ice, liquid water, and
d water vapor is one becaus
se
the ccomposition off all three phases is describ
bed by the che
emical formula
a H2O. In the thhree-phase
syste
em CaCO3 = CaO
C
+ CO2, th
he composition
n of each phase can be exp
pressed by a ccombination of any
two o
of the chemica
al species present. For exam
mple, if we ch
hoose to use CaCO
C
O2, we can
3 and CO
86
Dr. Murtadha Alshareifi e-Library
P.43
write CaO as (CaCO3 - CO2). Accordingly, the number of components in this system is two.
Table 2-7 Application of the Phase Rule to Single-Component Systems*
System
Number Degrees of
of Phases Freedom
Comments
Gas,
liquid, or
solid
1
F=CP+ 2
=1-1+
2=2
System is bivariant (F = 2) and lies
anywhere within the area marked
vapor, liquid, or solid in Figure 2-22.
We must fix two variables,
e.g., P2 and t2, to define system D.
Gas–
liquid,
liquid–
solid, or
gas–solid
2
F=CP+ 2
=1-2+
2=1
System is univariant (F = 1) and lies
anywhere along a line between twophase regions, i.e., AO, BO, or CO
in Figure 2-22. We must fix one
variable, e.g., either P1 or t2, to
define system E.
Gas–
liquid–
solid
3
F=CP+ 2
=1-3+
2=0
System is invariant (F = 0) and can
lie only at the point of intersection of
the lines bounding the three-phase
regions, i.e., point O in Figure 2-22.
*Key: C = number of components; P = number of phases.
The number of degrees of freedom is the least number of intensive variables that must be fixed/known
to describe the system completely. Herein lies the utility of the phase rule. Although a large number of
intensive properties are associated with any system, it is not necessary to report all of these to define
the system. For example, let us consider a given mass of a gas, say, water vapor, confined to a
particular volume. Using the phase rule only two independent variables are required to define the
system, F = 1 - 1 + 2 = 2. Because we need to know two of the variables to define the gaseous system
completely, we say that the system has two degrees of freedom. Therefore, even though this volume is
known, it would be impossible for one to duplicate this system exactly (except by pure chance) unless
the temperature, pressure, or another variable is known that may be varied independent of the volume
of the gas. Similarly, if the temperature of the gas is defined, it is necessary to know the volume,
pressure, or some other variable to define the system completely.
Next, consider a system comprising a liquid, say water, in equilibrium with its vapor. By stating the
temperature, we define the system completely because the pressure under which liquid and vapor can
coexist is also defined. If we decide to work instead at a particular pressure, then the temperature of the
system is automatically defined. Again, this agrees with the phase rule because equation (2-18) now
gives F = 1 - 2 + 2 = 1.
As a third example, suppose we cool liquid water and its vapor until a third phase (ice) separates out.
Under these conditions, the state of the three-phase ice–water–vapor system is completely defined, and
the rule gives F = 1 - 3 + 2 = 0; in other words, there are no degrees of freedom. If we attempt to vary
87
Dr. Murtadha Alshareifi e-Library
the p
particular cond
ditions of temp
perature or pre
essure necess
sary to mainta
ain this system
m, we will lose a
phasse. Thus, if we
e wish to prepa
are the three-p
phase system
m of ice–water–
–vapor, we haave no choice as to
the temperature or pressure at which
w
we will w
work; the com
mbination is fixed and uniquee. This is know
wn as
the ccritical point, and
a later we diiscuss it in mo
ore detail.
The relation betwe
een the numbe
er of phases a
and the degree
es of freedom in one-compoonent systems
s is
summ
marized in Ta
able 2-7. The student
s
should
d confirm these data by refe
erence to Figurre 2-22, which
h
show
ws the phase equilibria
e
of water at modera
rate pressures
s.
It is iimportant to appreciate thatt as the numbe
er of compone
ents increases
s, so do the reequired degree
es of
freed
dom needed to
o define the sy
ystem. Thereffore, as the sy
ystem become
es more compllex, it become
es
nece
essary to fix more
m
variables to define the ssystem. The greater
g
the number of phasees in equilibriu
um,
howe
ever, the fewe
er are the degrrees of freedo
om. Thus:



Liquid wa
ater + vapor
Liquid eth
hyl alcohol + vapor
v
Liquid wa
ater + liquid etthyl alcohol + vvapor mixture
e
(Note
e: Ethyl alcohol and water are
a completelyy miscible both
h as vapors an
nd liquids.)

Liquid wa
ater + liquid be
enzyl alcohol + vapor mixture
(Note
e: Benzyl alco
ohol and waterr form two sep
parate liquid phases and one vapor phasee. Gases are
misccible in all prop
portions; water and benzyl a
alcohol are on
nly partially mis
scible. It is theerefore necess
sary
to de
efine the two variables
v
in the
e completely m
miscible [one-phase] ethyl alcohol–water
a
system but on
nly
one vvariable in the
e partially misc
cible [two-pha
ase] benzyl–wa
ater system.)
Fig.. 2-22. Phasse diagram for
f water att moderate pressures.
p
88
Dr. Murtadha Alshareifi e-Library
P.44
4
Sys
stems Co
ontaining
g One Co
omponen
nt
We h
have already considered
c
a system
s
contai ning one com
mponent, name
ely, that for waater, which is
illusttrated in Figure 2-22 (not drawn to scale).. The curve OA in theP–T (p
pressure–tem perature) diag
gram
in Fig
gure 2-22 is known
k
as the vapor
v
pressure
e curve. Its up
pper limit is at the critical tem
mperature, 374°C
for w
water, and its lower end term
minates at 0.00
098°C, called the triple poin
nt. Along the vvapor pressure
e
curve
e, vapor and liquid coexist in equilibrium. This curve is analogous to the curve for water seen
in Fig
gure 2-5. Curvve OC is the sublimation
s
cu
urve, and here vapor and so
olid exist togethher in equilibrium.
Curvve OB is the melting
m
point cu
urve, at which
h liquid and solid are in equilibrium. The nnegative slope of
OB sshows that the
e freezing poin
nt of water deccreases with in
ncreasing exte
ernal pressuree, as we already
found in Example 2-8.
The result of chan
nges in pressure (at fixed tem
mperature) orr changes in te
emperature (at
at fixed pressure)
beco
omes evident by
b referring to
o the phase dia
agram. If the temperature
t
is
s held constannt at t1, where
wate
er is in the gasseous state ab
bove the critica
al temperature
e, no matter ho
ow much the ppressure is raised
(verttically along th
he dashed line
e), the system remains as a gas. At a temperature t2 beelow the critica
al
temp
perature, wate
er vapor is con
nverted into liq
quid water by an
a increase off pressure beccause the
comp
pression bring
gs the molecules within the range of the attractive
a
intermolecular forcces. It is intere
esting
to ob
bserve that at a temperature
e below the tri ple point, say t3, an increase of pressure on water in th
he
vapo
or state converts the vapor first
f
to ice and
d then at highe
er pressure into liquid water.. This sequence,
vapo
or → ice → liquid, is due to the
t fact that icce occupies a larger volume
e than liquid w
water below the
e
triple
e point. At the triple point, alll three phasess are in equilib
brium, that is, the only equillibrium is at this
pressure at this temperature of 0.0098°C (or w
with respect to
o the phase ru
ule, F = 0).
was seen in Ta
able 2-7, in any one of the th
hree regions in which pure solid,
s
liquid, oor vapor exists
As w
and P = 1, the pha
ase rule gives
Therrefore, we musst fix two cond
ditions, namelyy temperature
e and pressure
e, to specify orr describe the
syste
em completelyy. This statement means tha
at if we were to
t record the results of a sciientific experim
ment
invollving a given quantity
q
of watter, it would no
ot be sufficien
nt to state that the water wass kept at, say,,
76°C
C. The pressurre would also have to be sp
pecified to define the system
m completely. IIf the system were
w
open
n to the atmossphere, the atm
mospheric pre
essure obtainin
ng at the time of the experim
ment would be
e
recorded. Converssely, it would not
n be sufficie
ent to state tha
at liquid water was present aat a certain
pressure without also
a
stating the
e temperature
e. The phase rule
r
tells us tha
at the experim
menter may altter
two cconditions with
hout causing the
t appearancce or disappea
arance of the liquid phase. H
Hence, we say
that liquid water exxhibits two degrees of freed
dom.
Along any three off the curves where
w
two phasses exist in eq
quilibrium, F = 1 (see Table 2-7). Hence, only
one condition need be given to define the sysstem. If we sta
ate that the sys
stem containss both liquid water
and water vapor in
n equilibrium at
a 100°C, we n
need not spec
cify the pressure, for the vappor pressure can
c
have
e no other valu
ue than 760 mm
m Hg at 100°°C under these
e conditions. Similarly,
S
only one variable is
required to define the system along line OB o
or OC. Finally, at the triple point
p
where thee three phases—
ice, lliquid water, and
a water vapo
or—are in equ
uilibrium, we saw that F = 0.
As a
already noted, the triple poin
nt for air-free w
water is 0.0098
8°C, whereas the freezing ppoint (i.e., the point
at wh
hich liquid watter saturated with
w air is in eq
quilibrium with
h ice at a total pressure of 1 atm) is 0°C. In
incre
easing the pressure from 4.5
58 mm to 1 attm, we lower the freezing po
oint by about 00.0075 deg
(Exa
ample 2-8). Th
he freezing poiint is then lowe
wered an additiional 0.0023 deg
d by the preesence of disso
olved
air in
n water at 1 attm. Hence, the
e normal freezzing point of water
w
is 0.0075
5 deg + 0.00233 deg = 0.0098
deg below the triple point. In summary, the te
emperature at which a solid melts dependds (weakly) on the
pressure. If the pre
essure is that of the liquid a
and solid in equilibrium with the vapor, thee temperature is
know
wn as the triple
e point; howev
ver, if the presssure is 1 atm, the temperatture is the norrmal freezing point.
p
Co
ondensed
d Systems
89
Dr. Murtadha Alshareifi e-Library
We have seen from the phase rule that in a single-component system the maximum number of degrees
of freedom is two. This situation arises when only one phase is present, that is, F = 1 - 1 + 2 = 2. As will
become apparent in the next section, a maximum of three degrees of freedom is possible in a twocomponent system, for example, temperature, pressure, and concentration. To represent the effect of all
these variables upon the phase equilibria of such a system, it would be necessary to use a threedimensional model rather than the planar figure used in the case of water. Because in practice we are
primarily concerned with liquid and/or solid phases in the particular system under examination, we
frequently choose to disregard the vapor phase and work under normal conditions of 1 atm pressure. In
this manner, we reduce the number of degrees of freedom by one. In a two-component system,
therefore, only two variables (temperature and concentration) remain, and we are able to portray the
interaction of these variables by the use of planar figures on rectangular-coordinate graph paper.
Systems in which the vapor phase is ignored and only solid and/or liquid phases are considered are
termed condensed systems. We shall see in the later discussion of three-component systems that it is
again more convenient to work with condensed systems.
It is important to realize that in aerosol and gaseous systems, vapor cannot be ignored. Condensed
systems are most appropriate for solid and liquid dosage forms. As will be discussed in this and later
chapters, solids can also have liquid phase(s) associated with them, and the converse is true.
Therefore, even in an apparently dry tablet form, small amounts of “solution” can be present. For
example, it will be
P.45
shown in the chapter on stability that solvolysis is a primary mechanism of solid drug degradation.
Two-Component Systems Containing Liquid Phases
We know from experience that ethyl alcohol and water are miscible in all proportions, whereas water
and mercury are, for all practical purposes, completely immiscible regardless of the relative amounts of
each present. Between these two extremes lies a whole range of systems that exhibit partial miscibility
(or immiscibility). One such system is phenol and water, and a portion of the condensed phase diagram
is plotted in Figure 2-23. The curve gbhci shows the limits of temperature and concentration within which
two liquid phases exist in equilibrium. The region outside this curve contains systems having but one
liquid phase. Starting at the point a, equivalent to a system containing 100% water (i.e., pure water) at
50°C, adding known increments of phenol to a fixed weight of water, the whole being maintained at
50°C, will result in the formation of a single liquid phase until the point b is reached, at which point a
minute amount of a second phase appears. The concentration of phenol and water at which this occurs
is 11% by weight of phenol in water. Analysis of the second phase, which separates out on the bottom,
shows it to contain 63% by weight of phenol in water. This phenol-rich phase is denoted by the
point c on the phase diagram. As we prepare mixtures containing increasing quantities of phenol, that is,
as we proceed across the diagram from point b to point c, we form systems in which the amount of the
phenol-rich phase (B) continually increases, as denoted by the test tubes drawn in Figure 2-23. At the
same time, the amount of the water-rich phase (A) decreases. Once the total concentration of phenol
exceeds 63% at 50°C, a single phenol-rich liquid phase is formed.
90
Dr. Murtadha Alshareifi e-Library
Fig.. 2-23. Tem
mperature–co
omposition diagram forr the system
m consistingg of water an
nd
phennol. (From A. N. Camp
pbell and A
A. J. R. Cam
mpbell, J. Am
m. Chem. Seec. 59, 2481
1,
1937.)
The maximum tem
mperature at which
w
the two-p
phase region exists is terme
ed the critical solution, or up
pper
conssolute, temperrature. In the case
c
of the phe
enol–water sy
ystem, this is 66.8°C
6
(point h in Fig. 2-23)). All
comb
binations of ph
henol and watter above this temperature are
a completely
y miscible andd yield one-phase
liquid
d systems.
The line bc drawn across the region containin
ng two phases
s is termed a tie line; it is alw
ways parallel to
t the
base
e line in two-co
omponent systems. An impo
ortant feature of phase diag
grams is that aall systems
prep
pared on a tie line,
l
at equilibrium, will sepa
arate into phases of constan
nt compositionn. These phas
ses
are ttermed conjug
gate phases. For
F example, a
any system represented by a point on thee line bc at 50°°C
sepa
arates to give a pair of conju
ugate phases whose compo
ositions are b and
a c. Therelaative amounts
s of
the two layers or phases
p
vary, however,
h
as se
een in Figure 2-23. Thus, if we prepare a system conta
aining
24% by weight of phenol and 76
6% by weight of water (poin
nt d), at equilib
brium we havee two liquid phases
present in the tube
e. The upper one,
o
A, has a ccomposition of
o 11% phenol in water (poinnt b on the
diagrram), whereass the lower lay
yer, B, contain
ns 63% pheno
ol (point c on th
he diagram). P
Phase B will lie
e
below
w phase A because it is rich
h in phenol, an
nd phenol has
s a higher density than wateer. In terms of the
91
Dr. Murtadha Alshareifi e-Library
relative weights off the two phas
ses, there will b
be more of the
e water-rich phase A than thhe phenol-rich
h
phasse B at point d.
d Thus:
The right-hand terrm might appe
ear at first glan
nce to be the reciprocal
r
of th
he proportion oone should wrrite.
The weight of phase A is greate
er than that of phase B, how
wever, because
e point d is clooser to point b than
it is tto point c. The
e lengths dc an
nd bd can be measured with a ruler in centimeters or innches from the
phasse diagram, bu
ut it is frequen
ntly more convvenient to use the units of pe
ercent weight of phenol as found
f
on th
he abscissa off Figure 2-23. For example, because poin
nt b = 11%, po
oint c = 63%, aand point d = 24%,
2
the rratio dc/bd = (6
63 - 24)/(24 - 11) = 39/13 = 3/1. In other words,
w
for eve
ery 10 g of a liqquid system in
n
equillibrium represented by point d, one finds 7.5 g of phase
e A and 2.5 g of phase B. Iff, on the otherr
hand
d, we prepare a system containing 50% b
by weight of ph
henol (point f, Fig. 2-23), thee ratio of phas
se A
to ph
hase B is fc/bff = (63 - 50)/(5
50 - 11) = 13/3
39 = 1/3. Acco
ordingly, for ev
very 10 g of syystem f preparred,
we o
obtain an equillibrium mixture
e of 2.5 g of p
phase A and 7.5 g of phase B. It should bee apparent tha
at a
syste
em containing
g 37% by weight of phenol w
will, under equilibrium condittions at 50°C, give equal
weig
ghts of phase A and phase B.
B
Workking on a tie line in a phase diagram enab
bles us to calc
culate the com
mposition of eaach phase in
addittion to the weiight of the pha
ases. Thus, it becomes a sim
mple matter to
o calculate thee distribution of
o
phen
nol (or water) throughout
t
the
e system as a whole. As an
n example, let us suppose thhat we mixed 24
2 g
of ph
henol with 76 g of water, wa
armed the mixtture to 50°C,
P.46
6
and allowed it to re
each equilibriu
um at this tem
mperature. On separation of the two phasees, we would find
f
75 g of phase A (ccontaining 11%
% by weight off phenol) and 25 g of phase
e B (containingg 63% by weig
ght of
phen
nol). Phase A therefore conttains a total off (11 × 75)/100
0 = 8.25 g of phenol,
p
whereeas phase B
conta
ains a total of (63 × 25)/100
0 = 15.75 g of phenol. This gives
g
a sum to
otal of 24 g of phenol in the
wholle system. Thiis equals the amount
a
of phe
enol originally added and the
erefore confirm
ms our
assu
umptions and calculations.
c
It is left to the reader to conffirm that phase A contains 666.75 g of watter
and phase B 9.25 g of water. Th
he phases are
e shown at ban
nd c in Figure 2-23.
Applying the phase rule to Figurre 2-23 showss that with a tw
wo-componentt condensed ssystem having
g one
liquid
d phase, F = 3.
3 Because the
e pressure is ffixed, F is reduced to 2, and
d it is necessaary to fix both
temp
perature and concentration
c
to define the ssystem. When
n two liquid phases are pressent, F = 2; ag
gain,
pressure is fixed. We
W need only
y define tempe
erature to com
mpletely define the system beecause F is
om Figure 2-23
3, it is seen tha
at if the tempe
erature is give
en, the compossitions of the two
t
reduced to 1.* Fro
b the points at
a the ends of the tie lines, for example, points b and c at 50°C. The
phasses are fixed by
comp
positions (rela
ative amounts of phenol and
d water) of the
e two liquid lay
yers are then ccalculated by the
t
meth
hod already discussed.
The phase diagram
m is used in practice
p
to form
mulate system
ms containing more
m
than onee component
wherre it may be advantageous to achieve a ssingle liquid-ph
hase product. For example,, the handling of
solid
d phenol, a neccrotic agent, is
s facilitated in the pharmacy
y if a solution of phenol andd water is used
d. A
number of solution
ns containing different
d
conce
entrations of phenol
p
are official in severaal pharmacope
eias.
Unle
ess the freezin
ng point of the phenol–waterr mixture is su
ufficiently low, however, som
me solidificatio
on
may occur at a low
w ambient tem
mperature. Thiss will lead to in
naccuracies in
n dispensing aas well as a loss of
convvenience. Mullley58 determin
ned the releva
ant portion of the
t phenol–wa
ater phase diaagram and
sugg
gested that the
e most conven
nient formulati on of a single liquid phase solution
s
was 880% w/v,
equivvalent to abou
ut 76% w/w. This mixture ha
as a freezing point
p
of about 3.5°C comparred with liqueffied
phen
nol, USP, whicch contains ap
pproximately 9
90% w/w of ph
henol and freezes at about 117°C. It is not
posssible, therefore
e, to use the official
o
prepara
ation much bellow 20°C, or room temperatture; the
formulation propossed by Mulley from a consid
deration of the
e phenol–wate
er phase diagrram therefore is to
be preferred. A nu
umber of otherr binary liquid systems of the same type as
a phenol and water have been
b
92
Dr. Murtadha Alshareifi e-Library
studiied, although few
f
have prac
ctical applicatio
on in pharmac
cy. Some of th
hese are wateer–aniline, carb
bon
disullfide–methyl alcohol,
a
isopen
ntane–phenol, methyl alcohol–cyclohexan
ne, and isobuttyl alcohol–wa
ater.
Fig.. 2-24. Phasse diagram for
f the systeem triethylaamine–wateer showing llower conso
olute
temp
mperature.
Figure 2-24 illustra
ates a liquid mixture
m
that sh ows no upperr consolute tem
mperature, butt instead has
a low
wer consolute temperature below
b
which th
he componentts are miscible
e in all proporttions. The exa
ample
show
wn is the trieth
hylamine–wate
er system. Fig ure 2-25 show
ws the phase diagram
d
for thhe nicotine–wa
ater
syste
em, which hass both a lower and an upperr consolute tem
mperature. Lo
ower consolutee temperature
es
arise
e presumably because of an
n interaction b
between the co
omponents tha
at brings abouut complete
misccibility only at lower
l
tempera
atures.
Tw
wo-Compo
onent Sy
ystems C
Containing
g Solid and
a Liquiid Phases
s:
Eutectic Mix
ixtures
We rrestrict our disscussion, in the main, to tho
ose solid–liquid
d mixtures in which
w
the two components are
comp
pletely miscible in the liquid
d state and com
mpletely immiscible as solid
ds, that is, the solid phases that
form consist of pure crystalline
P.47
7
comp
mples of such systems are ssalol–thymol, salol–campho
or, and acetam
minophen–
ponents. Exam
prop
pyphenazone.
93
Dr. Murtadha Alshareifi e-Library
Fig.. 2-25. Nicootine–water system shoowing upperr and lower consolute t emperaturees.
94
Dr. Murtadha Alshareifi e-Library
Fig.. 2-26. Phasse diagram for
f the thym
mol–salol sy
ystem showiing the euteectic point.
(Daata from A. Siedell,
S
Solubilities of Organic Co
ompounds, 3rd
3 Ed., Vool. 2, Van
Nosstrand, New
w York, 1941, p. 723.)
The phase diagram
m for the salol–thymol syste
em is shown in
n Figure 2-26. Notice that thhere are four
ons: (i) a single liquid phase
e, (ii) a region containing solid salol and a conjugate liq uid phase, (iii) a
regio
regio
on in which so
olid thymol is in
n equilibrium w
with a conjuga
ate liquid phas
se, and (iv) a rregion in which
h
both components are present as
s pure solid ph
hases. Those regions conta
aining two phaases (ii, iii, and
d iv)
are ccomparable to
o the two-phas
se region of th e phenol–watter system sho
own inFigure 22-23. Thus it is
s
posssible to calcula
ate both the co
omposition an d relative amo
ount of each phase
p
from knoowledge of the
e tie
liness and the phasse boundaries.
Supp
pose we prepa
are a system containing
c
60%
% by weight of
o thymol in salol and raise tthe temperature of
the m
mixture to 50°C
C. Such a sys
stem is represe
ented by pointt xin Figure 2--26. On coolingg the system, we
obse
erve the follow
wing sequence
e of phase cha
anges. The sys
stem remains as a single liqquid until the
temp
perature falls to
t 29°C, at wh
hich point a mi nute amount of
o solid thymo
ol separates ouut to form a tw
wophasse solid–liquid system. At 25
5°C (room tem
mperature), sy
ystem x (denotted in Fig. 2-266 as x1) is
comp
posed of a liqu
uid phase, a1 (composition 53% thymol in
n salol), and pure solid thym
mol, b1. The we
eight
ratio of a1 to b1 is (100 - 60)/(60 - 53) = 40/7, that is, a1:b1 = 5.71:1. When the temperaature is reduce
ed to
20°C
C (point x2), the composition
n of the liquid p
phase is a2 (45% by weight of thymol in ssalol), whereas
s the
solid
d phase is still pure thymol, b2. The phase
e ratio is a2:b2 = (100 - 60)/(6
60 - 45) = 40/ 15 = 2.67:1. At
A
15°C
C (point x3), the composition
n of the liquid p
phase is now 37% thymol in
n salol (a3) andd the weight ratio
of liq
quid phase to pure
p
solid thym
mol (a3:b3) is ((100 - 60)/(60 - 37) - 40/23 = 1.74:1. Beloow 13°C, the liquid
phasse disappears altogether an
nd the system contains two solid phases of
o pure salol aand pure thymol.
Thuss, at 10°C (poiint x4), the sys
stem contains an equilibrium
m mixture of pure solid saloll (a4) and pure
e
solid
d thymol (b4) in
n a weight ratio of (100 - 60))/(60 - 0) = 40
0/60 = 0.67:1. As
A system x iss progressivelly
95
Dr. Murtadha Alshareifi e-Library
cooled, the results indicate that more and more of the thymol separates as solid. A similar sequence of
phase changes is observed if system y is cooled in a like manner. In this case, however, the solid phase
that separates at 22°C is pure salol.
The lowest temperature at which a liquid phase can exist in the salol–thymol system is 13°C, and this
occurs in a mixture containing 34% thymol in salol. This point on the phase diagram is known as
the eutectic point. At the eutectic point, three phases (liquid, solid salol, and solid thymol) coexist. The
eutectic point therefore denotes an invariant system because, in a condensed system, F = 2 - 3 + 1 = 0.
The eutectic point is the point at which the liquid and solid phases have the same composition (the
eutectic composition). The solid phase is an intimate mixture of fine crystals of the two compounds. The
intimacy of the mixture gives rise to the phenomenon of “contact melting,” which results in the lowest
melting temperature over a composition range. Alternately explained, a eutectic composition is the
composition of two or more compounds that exhibits a melting temperature
P.48
lower than that of any other mixture of the compounds. Mixtures of salol and camphor show similar
behavior. In this combination, the eutectic point occurs in a system containing 56% by weight of salol in
camphor at a temperature of 6°C. Many other substances form eutectic mixtures (e.g., camphor, chloral
hydrate, menthol, and betanaphthol). The primary criterion for eutectic formation is the mutual solubility
of the components in the liquid or melt phase.
In the thermal analysis section in this chapter, we showed that calorimetry can be used to study phase
transitions. Eutectic points are often determined by studying freezing point (melting point if one is adding
heat) depression. Note that the freezing point in a one-component system is influenced simply by the
temperature. In systems of two or more components, interactions between the components can occur,
and depending on the concentrations of the components, the absolute freezing point may change. A
eutectic point is the component ratio that exhibits the lowest observed melting point. This relationship is
often used to provide information about how solutes interact in solution, with the eutectic point providing
the favored composition for the solutes in solution, as illustrated in salol–thymol example. Lidocaine and
prilocaine, two local anesthetic agents, form a 1:1 mixture having a eutectic temperature of 18°C. The
mixture is therefore liquid at room temperature and forms a mixed local anesthetic that may be used for
topical application. The liquid eutectic can be emulsified in water, opening the possibility for topical
bioabsorption of the two local anesthetics.59,60
Solid Dispersions
Eutectic systems are examples of solid dispersions. The solid phases constituting the eutectic each
contain only one component and the system may be regarded as an intimate crystalline mixture of one
component in the other. A second major group of solid dispersions are the solid solutions, in which each
solid phase contains both components, that is, a solid solute is dissolved in a solid solvent to give a
mixed crystal. Solid solutions are typically not stoichiometric, and the minor component or “guest” inserts
itself into the structure of the “host” crystal taking advantage of molecular similarities and/or open
spaces in the host lattice. Solid solutions may exhibit higher, lower, or unchanged melting behavior
depending upon the degree of interaction of the guest in the crystal structure. A third common
dispersion is the molecular dispersion of one component in another where the overall solid is
amorphous. Such mixed amorphous or glass solutions exhibit an intermediate glass transition
temperature between those of the pure amorphous solids. The dispersion of solid particles in semisolids
is also a common dispersion strategy in which crystalline or amorphous solids are dispersed to aid
delivery, as in some topical products (e.g., [tioconazole vaginal] Monistat-1).
There is widespread interest in solid dispersions because they may offer a means of facilitating the
dissolution and frequently, therefore, the bioavailability of poorly soluble drugs when combined with
freely soluble “carriers” such as urea or polyethylene glycol. This increase in dissolution rate is achieved
by a combination of effects, the most significant of which is reduction of particle size to an extent that
cannot be readily achieved by conventional comminution approaches. Other contributing factors include
increased wettability of the material, reduced aggregation and agglomeration, and a likely increase in
96
Dr. Murtadha Alshareifi e-Library
solubility of the drug owing to the presence of the water-soluble carrier. Consult the reviews by Chiou
and Riegelman61 and Goldberg62 for further details.
Phase Equilibria in Three-Component Systems
In systems containing three components but only one phase, F = 3 - 1 + 2 = 4 for a noncondensed
system. The four degrees of freedom are temperature, pressure, and the concentrations of two of the
three components. Only two concentration terms are required because the sum of these subtracted from
the total will give the concentration of the third component. If we regard the system as condensed and
hold the temperature constant, then F = 2, and we can again use a planar diagram to illustrate the
phase equilibria. Because we are dealing with a three-component system, it is more convenient to use
triangular coordinate graphs, although it is possible to use rectangular coordinates.
The various phase equilibria that exist in three-component systems containing liquid and/or solid phases
are frequently complex and beyond the scope of the present text. Certain typical three-component
systems are discussed here, however, because they are of pharmaceutical interest. For example,
several areas of pharmaceutical processing such as crystallization, salt form selection, and
chromatographic analyses rely on the use of ternary systems for optimization.
Rules Relating to Triangular Diagrams
Before discussing phase equilibria in ternary systems, it is essential that the reader becomes familiar
with certain “rules” that relate to the use of triangular coordinates. It should have been apparent in
discussing two-component systems that all concentrations were expressed on a weight–weight basis.
This is because, although it is an easy and direct method of preparing dispersions, such an approach
also allows the concentration to be expressed in terms of the mole fraction or the molality. The
concentrations in ternary systems are accordingly expressed on a weight basis. The following
statements should be studied in conjunction with Figure 2-27:
1.
Each of the three corners or apexes of the triangle represent 100% by weight of one
component (A, B, or C). As a result, that same apex will represent 0% of the other two
components. For example, the top corner point in Figure 2-27 represents 100% of B.
2.
The three lines joining the corner points represent two-component mixtures of the three
possible combinations of A, B, and C. Thus the lines AB, BC, and CAare used for
P.49
two-component mixtures of A and B, B and C, and C and A, respectively. By dividing each line
into 100 equal units, we can directly relate the location of a point along the line to the percent
concentration of one component in a two-component system. For example, point y, midway
between A and B on the line AB, represents a system containing 50% of B (and hence 50%
of A also). Point z, three fourths of the way along BC, signifies a system containing 75%
of C in B.
97
Dr. Murtadha Alshareifi e-Library
Fig. 2-227. The trian
ngular diagrram for threee-componeent systems..
In going along
a
a line bo
ounding the trriangle so as to represent th
he concentratioon in a twocompone
ent system, it does
d
not matte
er whether we
e proceed in a clockwise or a
countercllockwise direc
ction around th
he triangle, pro
ovided we are
e consistent. T
The more usua
al
conventio
on is clockwise
e and has bee
en adopted he
ere. Hence, as
s we move aloong AB in the
direction of B, we are signifying
s
systtems of A and B containing increasing conncentrations of
o B,
and corre
espondingly sm
maller amountts of A. Movin
ng along BC to
oward C will reepresent syste
ems
of B and C containing more and morre of C; the clo
oser we appro
oach A on the line CA, the
greater will
w be the conc
centration of A in systems of
o A and C.
3.
The area
a within the tria
angle represen
nts all the pos
ssible combina
ations of A, B, and C to give
e
three-com
mponent syste
ems. The loca
ation of a partic
cular three-component systeem within the
triangle, for
f example, point
p
x, can be
e undertaken as
a follows.
The line AC
A opposite apex
a
B represe
ents systems containing A and
a C. Compoonent B is abs
sent,
that is, B = 0. The horiz
zontal lines ru
unning across the triangle pa
arallel to AC ddenote increas
sing
percentages of B from B = 0 (on line
e AC) to B = 100 (at point B)). The line parrallel to AC tha
at
cuts poin
nt x is equivale
ent to 15%B; cconsequently, the system co
ontains 15% oof B and 85%
98
Dr. Murtadha Alshareifi e-Library
of A and C together. Applying similar arguments to the other two components in the system, we
can say that along the line AB, C = 0. As we proceed from the line AB toward C across the
diagram, the concentration of C increases until at the apex, C = 100%. The point x lies on the
line parallel to AB that is equivalent to 30% of C. It follows, therefore, that the concentration
of A is 100 - (B + C) = 100 - (15 + 30) = 55%. This is readily confirmed by proceeding across
the diagram from the line BC toward apex A; point x lies on the line equivalent to 55% of A.
4.
5.
If a line is drawn through any apex to a point on the opposite side (e.g., line DC in Fig. 2-27),
then all systems represented by points on such a line have a constant ratio of two components,
in this case A and B. Furthermore, the continual addition of C to a mixture of A and B will
produce systems that lie progressively closer to apex C (100% of component C). This effect is
illustrated in Table 2-8, in which increasing weights of C are added to a constant-weight
mixture of A and B. Note that in all three systems, the ratio of A to B is constant and identical to
that existing in the original mixture.
Any line drawn parallel to one side of the triangle, for example, line HI in Figure 2-27,
represents ternary system in which the proportion (or percent by weight) of one component is
constant. In this instance, all systems prepared along HI will contain 20% of C and varying
concentrations of A and B.
Ternary Systems with One Pair of Partially Miscible Liquids
Water and benzene are miscible only to a slight extent, and so a mixture of the two usually produces a
two-phase system. The heavier of the two phases consists of water saturated with
P.50
benzene, while the lighter phase is benzene saturated with water. On the other hand, alcohol is
completely miscible with both benzene and water. It is to be expected, therefore, that the addition of
sufficient alcohol to a two-phase system of benzene and water would produce a single liquid phase in
which all three components are miscible. This situation is illustrated in Figure 2-28, which depicts such a
ternary system. It might be helpful to consider the alcohol as acting in a manner comparable to that of
temperature in the binary phenol–water system considered earlier. Raising the temperature of the
phenol–water system led to complete miscibility of the two conjugate phases and the formation of one
liquid phase. The addition of alcohol to the benzene–water system achieves the same end but by
different means, namely, a solvent effect in place of a temperature effect. There is a strong similarity
between the use of heat to break cohesive forces between molecules and the use of solvents to achieve
the same result. The effect of alcohol will be better understood when we introduce dielectric constants of
solutions and solvent polarity in later chapters. In this case, alcohol serves as an intermediate polar
solvent that shifts the electronic equilibrium of the dramatically opposed highly polar water and nonpolar
benzene solutions to provide solvation.
Table 2-8 Effect of Adding a third Component (C) to a Binary System
of A (5.0 G) and B (15.0 G)
99
Dr. Murtadha Alshareifi e-Library
Final System
Weight of Third
Component C Added
Weight
(g)
ComponentWeight (g) (%)
10.0
100.0
1000.0
A
5.0
16.67
B
15.0
50.00
C
10.0
33.33
A
5.0
4.17
B
15.0
12.50
C
100.0
83.33
A
5.0
0.49
B
15.0
1.47
C
1000.0
98.04
100
Location of
System
Ratio in Figure 2of A toB 27
3:1
Point E
3:1
Point F
3:1
Point G
Dr. Murtadha Alshareifi e-Library
Fiig. 2-28. A system of th
hree liquidss, one pair of
o which is
paartially misccible.
In Figure 2-28, let us suppose th
hat A, B, and C represent water,
w
alcohol, and benzenee, respectively. The
line A
AC therefore depicts
d
binary
y mixtures of A and C, and the points a an
nd c are the lim
mits of solubility
of C in A and of A in C, respectively, at the pa
articular tempe
erature being used. The currve afdeic,
frequ
uently termed a binodal curv
ve or binodal, marks the exttent of the two
o-phase regionn. The remainder
of the triangle contains one liquid phase. The
e tie lines within the binodal are not necesssarily parallel to
one another or to the
t base line, AC, as was th
he case in the
e two-phase re
egion of binaryy systems. In fact,
f
the d
directions of th
he tie lines are
e related to the
e shape of the
e binodal, whic
ch in turn depeends on the
relative solubility of
o the third com
mponent (in th
his case, alcoh
hol) in the othe
er two componnents. Only when
the a
added compon
nent acts equa
ally on the oth
her two compo
onents to bring
g them into sollution will the
binod
dal be perfecttly symmetric and
a the tie line
es run parallel to the baselin
ne.
The properties of tie
t lines discus
ssed earlier sttill apply, and systems g and h prepared aalong the tie
line ffi both give risse to two phas
ses having the
e compositions
s denoted by the points f an d i. The relativ
ve
amounts, by weigh
ht, of the two conjugate
c
pha
ases will depend on the position of the oriiginal system along
the tie line. For exxample, system
m g, after reacching equilibriu
um, will separa
ate into two phhases, f and i: The
ratio of phase f to phase i, on a weight basis, is given by th
he ratio gi:fg. Mixture
M
h, halfw
way along the
e tie
line, will contain eq
qual weights of
o the two pha
ases at equilibrium.
The phase equilibria depicted in
n Figure 2-28 sshow that the addition of co
omponent B too a 50 : 50 mix
xture
of co
omponents A and
a C will prod
duce a phase change from a two-liquid system to a on e-liquid system
m at
pointt d. With a 25::75 mixture of A
P.51
101
Dr. Murtadha Alshareifi e-Library
and C, shown as point
p
j, the add
dition of B lead
ds to a phase change at point e. Naturallyy, all mixtures
s
lying
g along dB and
d eB will be on
ne-phase syste
ems.
Fig.. 2-29. Alterrations of th
he binodal ccurves with changes in temperaturre. (a) Curves
on tthe triangulaar diagramss at temperaatures t1, t2, and t3. (b) The
T three-diimensional
arraangement off the diagram
ms in the orrder of increeasing temp
perature. (c)) The view one
o
wouuld obtain by
b looking down
d
from tthe top of (b
b).
As w
we saw earlier, F = 2 in a sin
ngle-phase reg
gion, and so we
w must define
e two concenttrations to fix the
particular system. Along the binodal curve afd
deic, F = 1, an
nd we need on
nly know one cconcentration term
ause this will allow
a
the comp
position of one
e phase to be fixed on the binodal
b
curve. From the tie line,
beca
we ccan then obtain
n the composition of the con
njugate phase
e.
Efffect of Te
emperatu
ure
Figure 2-29 showss the phase eq
quilibria in a th
hree-compone
ent system und
der isothermaal conditions.
Chan
nges in tempe
erature will cau
use the area o
of immiscibility
y, bounded by the binodal cuurve, to chang
ge. In
gene
eral, the area of
o the binodal decreases ass the temperatture is raised and
a miscibilityy is promoted.
Even
ntually, a pointt is reached at which compllete miscibility
y is obtained and the binodaal vanishes. To
o
studyy the effect off temperature on
o the phase equilibria of th
hree-compone
ent systems, a three-dimens
sional
figurre, the triangullar prism, is fre
equently used
d (Fig. 2-29b). Alternatively, a family of cuurves represen
nting
the vvarious temperatures may be
b used, as sh
hown in Figure
e 2-29c. The th
hree planar siddes of the pris
sm
are ssimply three-p
phase diagram
ms of binary-co
omponent systtems. Figure 2-29
2
illustratess the case of a
terna
ary-componen
nt system conttaining one pa
air of partially immiscible
i
liqu
uids (A and C)). As the
temp
perature is raissed, the region of immiscibi lity decreases
s. The volume outside the shhaded region of
the p
prism consistss of a single ho
omogeneous lliquid phase.
102
Dr. Murtadha Alshareifi e-Library
Fig.. 2-30. Effect of temperature changges on the binodal
b
curv
ves represennting a systeem
of tw
wo pairs of partially miscible liquiids.
Terrnary Sys
stems wiith Two o
or Three Pairs of Partially
P
Miscible
e
Liq
quids
All th
he previous co
onsiderations for
f ternary sysstems containing one pair of
o partially imm
miscible liquids
s still
applyy. With two pa
airs of partially
y miscible liquiids, there are two binodal cu
urves. The situuation is show
wn
in Fig
gure 2-30b, in
n which A and C as well as B and C show
w partial miscib
bility; A and B
P.52
2
are ccompletely misscible at the te
emperature ussed. Increasin
ng the tempera
ature generallyy leads to a
reduction in the arreas of the two
o binodal curvves and their eventual
e
disappearance (Figg. 2-30c).
Redu
uction of the te
emperature ex
xpands the bin
nodal curves, and, at a sufficiently low tem
mperature, the
ey
meet and fuse to form
f
a single band
b
of immisscibility as sho
own in Figure 2-30a.
2
Tie linees still exist in this
regio
on, and the ussual rules apply. Nor do the number of degrees of freed
dom change—
—whenP = 1, F = 2;
when
n P = 2, F = 1.
103
Dr. Murtadha Alshareifi e-Library
Fig. 2-31. Temperature effects on a system of three pairs of partially miscible liquids.
Systems containing three pairs of partially miscible liquids are of interest. Should the three binodal
curves meet (Fig. 2-31a), a central region appears in which threeconjugate liquid phases exist in
equilibrium. In this region, D, which is triangular, F = 0 for a condensed system under isothermal
conditions. As a result, all systems lying within this region consist of three phases whose compositions
are always given by the points x, y, and z. The only quantities that vary are the relative amounts of these
three conjugate phases. Increasing the temperature alters the shapes and sizes of the regions, as seen
in Figures 2-31b and c.
The application and discussion of phase phenomena and their application in certain pharmaceutical
systems will be discussed in later chapters.
Chapter Summary
As one of the foundational chapters of this text, many important subject areas have been
covered from the examination of the binding forces between molecules to the various states
of matter. Many of these subjects in this chapter are aimed at the more experienced
pharmacy student or graduate student who is interested in understanding the fundamental
physical aspects of the pharmaceutical sciences.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. E. A. Moelwyn-Hughes, Physical Chemistry, 2nd Ed., Pergamon, New York, 1961, p. 297.
2. L. Serrano, M. Bycroft, and A. Fersht, J. Mol. Biol. 218, 465, 1991.
3. E. A. Meyer, R. K. Castellano, and F. Diederich, Angew. Chem. 42, 1210, 2003.
4. W. M. Latimer and W. H. Rodebush, J. Am. Chem. Soc. 42, 1419, 1920. According to P. Schuster, in
P. Schuster, G. Zundel, and C. Sandorfy, (Eds.), The Hydrogen Bond, Vol. III, North-Holland, New York,
1976, the concept of the hydrogen bond was introduced by T. S. Moore and T. F. Winmill, J. Chem.
Soc. 101, 1635, 1912.
5. G. N. Lewis and M. Randall, Thermodynamics, Rev. K. S. Pitzer and L. Brewer, McGraw-Hill, New
York, 1961.
6. P. R. Byron, J. Pharm. Sci. 75, 433, 1986; R. N. Dalby, P. R. Byron, H. R. Shepherd, and E.
Popadopoulous, Pharm. Technol. 14, 26, 1990.
7. J. Pickthall, Pharm. J. 193, 391, 1964.
8. J. J. Sciarra, J. Pharm. Sci. 63, 1815, 1974.
9. (a) P. R. Byron and A. R. Clark, J. Pharm. Sci. 74, 934, 939, 1985; (b) United States
Pharmacopeia, The United States Pharmacopeia, 27th Rev., U.S. Pharmacopeial Convention,
Rockville, Md., 2004.
10. C. Henry, Chem. Eng. News 78, 49, 2000.
11. DeFelippis, M. R., Overcoming the challenges of noninvasive protein and peptide delivery. American
Pharmaceutical Review 6, 21, 2003.
12. S. Riegelman, in R. A. Lyman and J. B. Sprowls (Eds.), American Pharmacy, 4th Ed., Lippincott
Williams & Wilkins, Philadelphia, 1955, Chapter 18.
13. A. J. Aguiar, J. Kro, A. W. Kinkle, and J. Samyn, J. Pharm. Sci. 56, 847, 1967.
14. S. A. Khalil, M. A. Moustafa, A. R. Ebian, and M. M. Motawi, J. Pharm. Sci. 61, 1615, 1972.
15. R. K. Callow and O. Kennard, J. Pharm. Pharmacol. 13, 723, 1961.
16. J. E. Carless, M. A. Moustafa, and H. D. C. Rapson, J. Pharm. Pharmacol. 18, 1905, 1966.
17. M. Azibi, M. Draguet-Brughmans, R. Bouche, B. Tinant, G. Germain, J. P. Declercq, and M. Van
Meerssche, J. Pharm. Sci. 72, 232, 1983.
18. M. Azibi, M. Draguet-Brughmans, and R. Bouche, J. Pharm. Sci. 73, 512, 1984.
19. S. S. Yank and J. K. Guillory, J. Pharm. Sci. 61, 26, 1972.
20. I. Goldberg and Y. Becker, J. Pharm. Sci. 76, 259, 1987.
104
Dr. Murtadha Alshareifi e-Library
21. M. M. J. Lowes, M. R. Caira, A. P. Lotter, and J. G. Van Der Watt, J. Pharm. Sci. 76, 744, 1987.
22. T. Ishida, M. Doi, M. Shimamoto, N. Minamino, K. Nonaka, and M. Inque, J. Pharm. Sci. 78, 274,
1989.
23. M. Stoltz, A. P. Lötter, and J. G. Van Der Watt, J. Pharm. Sci. 77, 1047, 1988.
24. R. J. Behme, D. Brooks, R. F. Farney, and T. T. Kensler, J. Pharm. Sci. 74, 1041, 1985.
25. W. I. Higuchi, P. K. Lau, T. Higuchi, and J. W. Shell, J. Pharm. Sci. 52, 150, 1963.
26. J. K. Guillory, J. Pharm. Sci. 56, 72, 1967.
27. J. A. Biles, J. Pharm. Sci. 51, 601, 1962.
28. J. Haleblian and W. McCrone, J. Pharm. Sci. 58, 911, 1969; J. Haleblian, J. Pharm. Sci. 64, 1269,
1975.
29. J. D. Mullins and T. J. Macek, J. Am. Pharm. Assoc. Sci. Ed. 49, 245, 1960.
30. J. A. Biles, J. Pharm. Sci. 51, 499, 1962.
31. E. J. Lien and L. Kennon, in Remington's Pharmaceutical Sciences, 17th Ed., Mack, Easton, Pa.,
1975, pp. 176–178.
P.53
32. D. Guttman and T. Higuchi, J. Am. Pharm. Assoc. Sci. Ed. 46, 4, 1957.
33. B. Lundberg, Atherosclerosis 56, 93, 1985.
34. J. B. Bogardus, J. Pharm. Sci. 72, 338, 1983.
35. (a) S. E. Friberg, C. S. Wohn, and F. E. Lockwood, J. Pharm. Sci. 74, 771, 1985; (b) S. E. Friberg,
Lyotropic Liquid Crystals, American Chemical Society, Washington, D.C., 1976.
36. H. G. Ibrahim, J. Pharm. Sci. 78, 683, 1989.
37. G. H. Brown, Am. Sci. 60, 64, 1972.
38. Milton Roy Co., Supercritical Fluid Extraction: Principles and Applications (Bulletin 37.020), Milton
Roy Co., Ivyland, Pa.
39. C. S. Kaiser, H. Rompp, and P. C. Schmidt, Pharmazie 56, 907, 2001.
40. D. E. LaCroix and W. R. Wolf, J. AOAC Int. 86, 86, 2000.
41. C.-C. Chen and C.-T. Ho, J. Agric. Food Chem. 36, 322, 1988.
42. H. H. Tong, B. Y. Shekunov, P. York, and A. H. Chow, Pharm. Res. 18, 852, 2001.
43. V. Krukonis, Polymer News 11, 7, 1985.
44. A. Breitenbach, D. Mohr, and T. Kissel, J. Control. Rel. 63, 53, 2000.
45. M. Moneghini, I. Kikic, D. Voinovich, B. Perissutti, and J. Filipovic-Grcic, Int. J. Pharm. 222, 129,
2001.
46. E. Reverchon, J. Supercritical Fluids 15, 1, 1999.
47. H. Peker, M. P. Srinvasan, J. M. Smith, and B. J. McCoy, AIChE J. 38, 761, 1992.
48. J. B. Borgadus, J. Pharm. Sci. 71, 105, 1982; L. Gatlin and P. P. DeLuca, J. Parenter Drug
Assoc. 34, 398, 1980.
49. J. K. Guillory, S. C. Hwang, and J. L. Lach, J. Pharm. Sci. 58, 301, 1969; H. H. Lin and J. K. Guillory,
J. Pharm. Sci. 59, 972, 1970.
50. S. S. Yang and J. K. Guillory, J. Pharm. Sci. 61, 26, 1972; W. C. Stagner and J. K. Guillory, J.
Pharm. Sci. 68, 1005, 1979.
51. E. L. Simons and A. E. Newkirk, Talanta 11, 549, 1964.
52. S. Civjan, W. J. Selting, L. B. De Simon, G. C. Battistone, and M. F. Grower, J. Dent. Res. 51, 539,
1972.
53. W. T. Humphries, D. L. Millier, and R. H. Wildnauer, J. Soc. Cosmet. Chem. 23, 359, 1972; W. T.
Humphries and R. H. Wildnauer, J. Invest. Dermatol. 59, 9, 1972.
54. L. Greenspan, J. Res. NBS 81A, 89, 1977.
55. K. R. Morris, in H. G. Brittain (Ed.), Polymorphism in Pharmaceutical Solids, Marcel Dekker, New
York, 1999, pp. 125–181.
105
Dr. Murtadha Alshareifi e-Library
56. R
R. V. Manek and
a W. M. Kolling, AAPS Ph
harmSciTech. 5, article 14, 2004.
2
Availablle at
www
w.aapspharmscitech.org/deffault/IssueView
w.asp? vol = 058issue
0
= 01.
57. C
C. Ahlneck and G. Zografi, Int.
I J. Pharm. 62, 87, 1990.
58. B
B. A. Mulley, Drug
D
Stand. 27
7, 108, 1959.
59. A
A. Brodin, A. Nyqvist-Mayer
N
r, T. Wadsten, B. Forslund, and F. Broberrg, J. Pharm. S
Sci. 73, 481, 1984.
1
60. A
A. Nyqvist-Mayyer, A. F. Brod
din, and S. G. Frank, J. Pha
arm. Sci. 74, 1192, 1985.
61. W
W. L. Chiou an
nd S. Riegelm
man, J. Pharm.. Sci. 60, 1281
1, 1971.
62. A
A. H. Goldberg
g, in L. J. Lees
son and J. T. C
Carstensen (E
Eds.), Dissoluttion Technologgy, Academy of
Pharrmaceutical Sccience, Washington, D.C., 1
1974, Chapterr 5.
Recommen
nded Readings
J. Liu
u, Pharm. Devv. Technol. 11, 3–28, 2006.
D. Singhal and W.. Curatolo, Adv. Drug Deliv. Rev. 56, 335–347, 2004.
e term van derr Waals forces
s is often used
d loosely. Som
metimes all com
mbinations of intermolecular
*The
force
es among ionss, permanent dipoles,
d
and in
nduced dipole
es are referred to as van derr Waals forces
s. On
the o
other hand, the
e London forc
ce alone is freq
quently referre
ed to as the va
an der Waals fforce because
e it
acco
ounts for the attraction between nonpolar gas molecule
es, as expressed by the a/V22 term in the van
v
der W
Waals gas equ
uation. In this book, the thre
ee dipolar forc
ces of Keesom
m, Debye, and London are called
c
van d
der Waals forcces. The other forces such as the ion-dip
pole interaction
n and the hydrrogen bond (w
which
have
e characteristiccs similar both
h to ionic and dipolar forces
s) are designatted appropriattely where
nece
essary.
*N
Note that thhe root mean square veelocity
is not thee same as thhe average
vvelocity, [C with bar above]. This ccan be show
wn by a simp
ple examplee: Let c hav
ve
thhe three valuues 2, 3, an
nd 4. Then [C
C with bar above]
a
= (2 + 3 + 4)/3 = 3, whereaas
is thee square roo
ot of the meaan of the su
um of the sq
quares,
or
and µ = 3.11.
*A ga
as is known as a vapor belo
ow its critical ttemperature. A less rigorous
s definition of a vapor is a
subsstance that is a liquid or a so
olid at room te
emperature an
nd passes into
o the gaseous state when he
eated
to a sufficiently hig
gh temperature. A gas is a ssubstance tha
at exists in the gaseous statee even at room
m
temp
perature. Mentthol and ethan
nol are vaporss at sufficiently
y high tempera
atures; oxygenn and carbon
dioxiide are gases.
*The
e number of de
egrees of freedom calculate
ed from the ph
hase rule if the
e system is not
ot condensed is still
the ssame. Thus, when
w
one liquid
d phase and itts vapor are present,
p
F = 2 - 2 + 2 = 2; it is therefore
nece
essary to defin
ne two conditio
ons: temperatu
ure and conce
entration. Whe
en two liquids and the vaporr
phasse exist, F = 2 - 3 + 2 = 1, and only tempe
erature need be
b defined.
Cha
apter Legac
cy
Fifth
h Edition: pu
ublished as Chapter
C
2. U
Updated by Gregory
G
Knipp, Susan BoogdanowichKnip
pp and Kenneth Morris.
Sixtth Edition: published
p
as Chapter 2. U
Updated by Patrick
P
Sinko
o.
106
Dr. Murtadha Alshareifi e-Library
3T
Thermo
odynam
mics
Cha
apter Objec
ctives
At th
he conclusion of this chapter the s
student sho
ould be able to:
1. Understand the theo
ory of thermo
odynamics an
nd its use forr describing eenergy-relate
ed
changess in reactions
s.
2. Understand the first law of therm
modynamics and
a its use.
3. Understand the seco
ond law of th
hermodynamics and its us
se.
4. Understand the third
d law of therm
modynamics and its use.
5. Define and
a calculate
e free energyy functions an
nd apply them to pharmaaceutically
relevantt issues.
6. Understand the basic principles o
of the impac
ct of thermodynamics on ppharmaceutiically
relevantt applications
s.
7. Define th
he chemical potential and
d equilibrium
m processes.
Therrmodynamics deals with the
e quantitative rrelationships of
o interconvers
sion of the varrious forms
of en
nergy, includin
ng mechanicall, chemical, ele
ectric, and rad
diant energy. Although
A
therm
modynamics was
w
origin
nally develope
ed by physicis
sts and engine
eers interested
d in the efficien
ncies of steam
m engines, the
e
conccepts formulated from it hav
ve proven to be
e extremely useful in the ch
hemical sciencces and related
disciplines like pha
armacy. As illu
ustrated later in this chapterr, the property
y calledenergyy is broadly
appliicable, from determining the
e fate of simpl e chemical prrocesses to de
escribing the vvery complex
beha
avior of biologic cells.
Therrmodynamics is based on th
hree “laws” or facts of experrience that hav
ve never beenn proven in a direct
d
way,, in part due to
o the ideal con
nditions for wh
hich they were
e derived. Various conclusioons, usually
exprressed in the form
f
of mathem
matical equatiions, howeverr, may be dedu
uced from thes
ese three
princciples, and the
e results consistently agree with observattions. Consequ
uently, the law
ws of
therm
modynamics, from
f
which the
ese equationss are obtained, are accepted
d as valid for ssystems involv
ving
large
e numbers of molecules.
m
It is u
useful at this point
p
to disting
guish the attrib
butes of the th
hree types of systems
s
that aare frequently used
to de
escribe thermo
odynamic prop
perties. Figure
e 3-1a shows an open syste
em in which ennergy and mattter
can b
be exchanged
d with the surroundings. In ccontrast, Figurre 3-1b and c are exampless
of clo
osed systems, in which therre is no excha
ange of matterr with the surro
oundings, thatt is, the system
m's
masss is constant. However, ene
ergy can be tra
ansferred by work
w
(Fig. 3-1b
b) or heat (Figg. 3-1c) throug
gh
the cclosed system
m's boundaries. The last exa
ample (Fig. 3-1
1d) is a system
m in which neitther matter no
or
enerrgy can be excchanged with the
t surroundin
ngs; this is callled an isolated system.
For instance, if two
o immiscible solvents,
s
wate
er and carbon tetrachloride, are confined iin a closed
conta
ainer and iodine is distribute
ed between th
he two phases
s, each phase is an open syystem, yet the total
syste
em made up of
o the two phases is closed because it does not exchan
nge matter witth its surround
dings.
Key Co
oncept
Bas
sic Definitio
ons of Therm
modynamic
cs
Befo
ore beginning
g with details
s about the o
origin and concepts involv
ving these thhree laws, we
e
defin
ne the language common
nly used in th
hermodynam
mics, which has precise sscientific
mea
anings. A sysstem in therm
modynamics is a well-deffined part of the
t universe that one is
interrested in stud
dying. The system is sep
parated from surrounding
gs, the rest off the univers
se
and from which the
t observattions are mad
de, by physic
cal (or virtual) barriers deefined
asbo
oundaries. Work
W
(W) and
d heat (Q) alsso have prec
cise thermodynamic meaanings. Work
k is a
transfer of energ
gy that can be used to ch
hange the height of a weig
ght somewheere in the
surroundings, an
nd heat is a transfer
t
of en
nergy resultin
ng from a tem
mperature diifference
betw
ween the sysstem and the surrounding
gs. It is imporrtant to consider that
107
Dr. Murtadha Alshareifi e-Library
both
h work and heat appear only
o
at the syystem's boun
ndaries where the energyy is being
transferred.
The
e First La
aw of The
ermodyn
namics
The first law is a statement
s
of th
he conservatio
on of energy. It states that, although
a
energgy can be
transsformed from one kind into another,
a
it can
nnot be create
ed or destroyed. Put in anothher way, the total
enerrgy of a system
m and its imme
ediate surroun
ndings remain
ns constant during any operaation. This
state
ement follows from the fact that
t
the variou
us forms of en
nergy are equiv
valent, and whhen one kind is
formed, an equal amount
a
of ano
other kind musst disappear. The
T relativistic
c picture of thee universe
exprressed by Einsstein's equatio
on
sugg
gests that mattter can be con
nsidered as an
nother form off energy, 1 g being
b
equivaleent to 9 ×
1013 joules. These
e enormous qu
uantities of en ergy are involved in nuclear transformatioons but are no
ot
impo
ortant in ordina
ary chemical reactions.
r
P.55
5
108
Dr. Murtadha Alshareifi e-Library
Fig.. 3-1. Exam
mples of therrmodynamicc systems. (a)
( An open
n system excchanging mass
m
withh its surrounndings; (b) a closed sysstem exchan
nging work with its surrroundings; (c)
a cloosed system
m exchangin
ng heat withh its surroun
ndings; (d) an
a isolated ssystem, in
which neither work
w
nor heeat can be exxchanged th
hrough boun
ndaries.
Acco
ording to the fiirst law, the efffects of Q and
d W in a given
n system during a transformaation from an
initia
al thermodynam
mic state to a final thermodyynamic state are
a related to an intrinsic prroperty of the
syste
em called the internal energ
gy, defined as
wherre E2 is the intternal energy of the system in its final sta
ate and E1 is th
he internal eneergy of the sys
stem
in itss initial state, Q is the heat, and
a W is the w
work. The cha
ange in interna
al energy ∆E iss related
to Q and W transfe
erred between
n the system a
and its surroun
ndings. Equation (3-1) also expresses the
e fact
nt ways of cha
anging the inte
ernal energy of
o the system.
that work and heatare equivalen
gy is related to
o the microsco
opic motion of the atoms, ion
ns, or moleculles of which th
he
The internal energ
em is composed. Knowledg
ge of its absolu
ute value would tell us some
ething about thhe microscopiic
syste
motio
on of the vibra
ational, rotational, and transslational comp
ponents. In add
dition, the abssolute value would
w
also provide inform
mation about the
t kinetic and
d potential ene
ergies of their electrons andd nuclear elem
ments,
whicch in practice is extremely difficult to attain
n. Therefore, change
c
of inte
ernal energy raather than abs
solute
enerrgy value is the
e concern of thermodynamiics.
Exa
ample 3-1
The
ermodynam
mic State
Con
nsider the exa
ample of tran
nsporting a b
box of equipm
ment from a camp
c
in a vaalley to one at
a
the ttop of a mou
untain. The main
m
concern
n is with the potential
p
energy rather thhan the intern
nal
enerrgy of a syste
em, but the principle
p
is th
he same. On
ne can haul th
he box to thee top of the
mou
untain by a block and tackle suspende
ed from an overhanging
o
cliff and prodduce little he
eat
by th
his means. One
O can also
o drag the bo
ox up a path, but more wo
ork will be reequired and
conssiderably mo
ore heat will be
b produced owing to the
e frictional re
esistance. Thhe box can be
e
carried to the ne
earest airportt, flown over the appropriate spot, and
d dropped byy parachute. It is
read
dily seen thatt each of these methods involves a different amount of heat aand work. The
chan
nge in potential energy depends onlyy on the difference in the height of thee camp in the
e
valle
ey and the on
ne at the top
p of the moun
ntain, and it is independent of the pathh used to
transport the boxx.
By using equation (3-1) (the first law), one ca
an evaluate the
e change of internal energy by
ate. However, it is useful to relate
r
the channge of interna
al
measuring Q and W during the change of sta
asurable prope
erties of the syystem: P, V, and T. Any two
o of these variaables must be
e
enerrgy to the mea
speccified to define
e the internal energy.
e
For an
n infinitesimal change in the energy dE, eequation (3-1) is
writte
en as
109
Dr. Murtadha Alshareifi e-Library
w
where q is the heat absorbed and w is the work
w
done during
d
the innfinitesimall
chhange of the system. Capital
C
letterrs Qand W are
a used forr heat and w
work in
eqquation (3-11) to signify
y finite channges in thesse quantitiess. The symbbol in
eqquation(3-22) signifies infinitesima
i
al changes of
o propertiess that depennd on the
“ppath,” also called inexaact different
ntials. Hencee, qand w are not inn these
ciircumstancees thermody
ynamic propperties.
The infinitesimal change
c
of any state propertyy like dE, also called an exa
act differential,, can be generally
writte
en, for instancce, as a functio
on of T and V as in the follo
owing equation
n for a closed system (i.e.,
consstant mass):
The partial derivattives of the energy in equatiion (3-3) are im
mportant prop
perties of the ssystem and sh
how
e in energy witth the change in T at consta
ant volume or with
w the changge of V at con
nstant
the rrate of change
temp
perature. Therrefore, it is use
eful to find the
eir expression in terms of me
easurable prooperties. This can
c
be done by combining equations (3-2) and (3
3-3) into
ome properties of E as a fun
nction of T andd V.
This equation will be used later to describe so
Iso
othermal and Adia
abatic Prrocesses
W
When the tem
mperature is kept consttant during a process, the
t reactionn is said to be
b
coonducted issothermally.. An isotherrmal reactio
on may be carried out bby placing
thhe system inn a large con
nstant-tempperature bath so that heeat is drawnn from or
reeturned to itt without afffecting the temperaturee significan
ntly. When hheat is
nneither lost nor
n gained during
d
a proocess, the reeaction is saaid to
occur adiabaatically. A reaction
r
carr
rried on inside a sealed Dewar flassk or
“vvacuum botttle” is adiab
batic becauuse the systeem is thermaally insulateed from its
suurroundingss. In thermo
odynamic teerms, it can be said thatt an adiabattic process is
i
one in whichh q = 0, an
nd the first llaw under adiabatic
a
con
nditions redduces to
P.56
6
Key Co
oncept
110
Dr. Murtadha Alshareifi e-Library
The
ermodynam
mic State
The term thermo
odynamic sta
ate means th
he condition in which the measurable properties of
o the
systtem have a definite
d
value
e. The state o
of 1 g of wate
er atE1 may be
b specified by the
cond
ditions of, sa
ay, 1 atm pressure and 10
0°C, and the
e state E2 by the conditionns of 5 atm and
a
150°°C. Hence, the states of most interesst to the chem
mist ordinarily
y are definedd by specifying
any two of the th
hree variables, temperatu
ure (T), press
sure (P), and
d volume (V);; however,
addiitional indepe
endent variables sometim
mes are need
ded to speciffy the state oof the system
m.
Any equation rellating the nec
cessary varia
ables—for ex
xample, V = f (T,P)—is aan equation of
o
state
e. The ideal gas law and the van der Waals equa
ation describe
ed in Chapteer 2 are
equa
ations of statte. Thus,V, P,
P and T are variables tha
at define a sttate, so they are called state
varia
ables. The va
ariables of a thermodyna
amic state arre independe
ent of how thee state has been
b
reacched.
A feature of the change of internal energ y, ∆E, discov
vered by the first law is thhat it depend
ds
onlyy on the initia
al and final th
hermodynam ic states, tha
at is, it is a va
ariable of staate; it is a
thermodynamic property of the system. O
On the other hand, both Q and W deppend on the
man
nner in which
h the change is conducted
d. Hence, Q and W are not
n variabless of state or
thermodynamic properties; th
hey are said to depend on
o the “path” of the transfformation.
Acco
ording to equa
ation (3-5), when work is don
ne by the systtem, the intern
nal energy deccreases, and
beca
ause heat cannot be absorb
bed in an adiab
batic process, the temperature must fall. Here, the worrk
done
e becomes a thermodynami
t
ic property de pendent only on the initial and
a final statess of the system
m.
Wo
ork of Exp
xpansion Against a Consta
ant Press
sure
We ffirst discuss th
he work term. Because of itss importance in thermodyna
amics, initial foocus is on the work
produced by varying the volume
e of a system (i.e., expansio
on work or com
mpression worrk) against
a con
nstant opposin
ng external prressure, Pex. Im
magine a vapo
or confined in a hypotheticaal cylinder fitte
ed
with a weightless, frictionless piston of area A
A, as shown in
n Figure 3-2. Iff a constant exxternal
pressure Pex is exe
erted on the piston,
p
the tota
al force is Pex × A because P = Force/Areaa. The vapor in
i the
cylinder is now ma
ade to expand
d by increasing
g the temperatture, and the piston
p
moves a distance h. The
workk done againstt the opposing
g pressure in o
one single stage is
Now
w A × h is the in
ncrease in volume, ∆V = V2 - V1, so that, at constant pressure,
Reversible Processes
Now
w let us imagine the hypothetical case of w
water at its boiiling point contained in a cyllinder fitted wiith a
weig
ghtless and fricctionless pisto
on (Fig. 3-3a). The apparatu
us is immersed
d in a constantt-temperature
e bath
main
ntained at the same tempera
ature as the w
water in the cylinder. By definition, the vappor pressure of
o
wate
er at its boiling
g point is equal to the atmosspheric pressu
ure, represente
ed in Figure 3--3 by a set of
weig
ghts equivalent to the atmos
spheric pressu
ure of 1 atm; th
herefore, the temperature iss 100°C. The
process is an isoth
hermal one, th
hat is, it is carrried out at con
nstant tempera
ature. Now, if tthe external
pressure is decrea
ased slightly by
b removing on
ne of the infiniitesimally sma
all weights (Figg. 3-3b), the
volum
me of the systtem increases
s and the vapo
or pressure falls infinitesima
ally. Water thenn evaporates to
main
ntain the vapor pressure con
nstant at its orriginal value, and
a heat is extracted from thhe bath to kee
ep
the temperature co
onstant and bring about the
e vaporization.. During this process, a heaat exchange
betw
ween the syste
em and the tem
mperature bat h will occur.
On the other hand
d, if the externa
al pressure is increased slig
ghtly by adding an infinitesim
mally small we
eight
(Fig. 3-3c), the sysstem is comprressed and the
e vapor pressure also rises infinitesimallyy. Some of the
e
wate
er condenses to
t reestablish the equilibrium
m vapor press
sure, and the liberated heat
P.57
7
is ab
bsorbed by the
e constant-tem
mperature bath
h. If the proces
ss could be co
onducted infin itely slowly so
o that
111
Dr. Murtadha Alshareifi e-Library
no w
work is expend
ded in supplyin
ng kinetic enerrgy to the piston, and if the piston is conssidered to be
frictio
onless so thatt no work is do
one against th e force of fricttion, all the wo
ork is used to eexpand or
comp
press the vapor. Then, beca
ause this proccess is always
s in a state of virtual
v
thermoddynamic
equillibrium, being reversed by an
a infinitesima
al change of prressure, it is said
s
to bereverrsible. If the
pressure on the syystem is increa
ased or decre
eased rapidly or
o if the tempe
erature of the bbath cannot adjust
insta
antaneously to
o the change in
n the system, the system is not in the sam
me thermodynnamic state at each
mom
ment, and the process
p
is irre
eversible.
Fig.. 3-2. A cyliinder with a weightlesss and frictio
onless piston
n.
112
Dr. Murtadha Alshareifi e-Library
Fig.. 3-3. A reversible proccess: evaporration and condensation
c
n of water aat 1 atm in a
clossed system. (a) System at equilibriium with Pex
=
1
atm;
(b)
(
expansioon is
e
infinnitesimal; (cc) compresssion is infinnitesimal.
Altho
ough no real system
s
can be
e made strictlyy reversible, so
ome are nearly
y so. One of thhe best examples
of re
eversibility is th
hat involved in
n the measure
ement of the po
otential of an electrochemic
e
cal cell using the
potentiometric method.
Ma
aximum Work
W
The work done byy a system in an
a isothermal e
expansion pro
ocess is at a maximum
m
wheen it is done
reversibly. This sta
atement can be
b shown to be
e true by the following
f
argument. No work
rk is accomplis
shed
if an ideal gas exp
pands freely in
nto a vacuum, where P = 0, because any work accompllished depend
ds on
the e
external pressure. As the ex
xternal pressu re becomes greater,
g
more work
w
is done bby the system, and
it rise
es to a maxim
mum when the external presssure is infinite
esimally less th
han the pressuure of the gas
s, that
is, w
when the proce
ess is reversible. Of course, if the externa
al pressure is continually
c
inccreased, the gas is
comp
pressed rathe
er than expand
ded, and workk is done on the system rather than by thee system in an
n
isoth
hermal reversible process.
Then
n the maximum
m work done for
f a system th
hat is expande
ed in reversible fashion is
wherre Pex was rep
placed by P be
ecause the exxternal pressurre is only infinitesimally smaaller than the
pressure of the syystem. In simila
ar fashion, it ccan be deduce
ed that the min
nimum work inn a reversible
comp
pression of the system will also
a
lead to eq
quation (3-8), because at ea
ach stage Pex is only
infiniitesimally larger than P. The
e right term in equation (3-8
8) is depicted in the shaded area in Figure
e 3-4,
whicch represents the
t maximum expansion wo
ork or the minimum compression work in a reversible
process.
Exa
ample 3-2
A ga
as expands by
b 0.5 liter ag
gainst a consstant pressure of 0.5 atm
m at 25°C. Whhat is the wo
ork in
ergss and in joule
es done by th
he system?
113
Dr. Murtadha Alshareifi e-Library
The following exxample demo
onstrates the
e kind of prob
blem that can
n be solved bby an applica
ation
of th
he first law off thermodyna
amics.
The external presssure in equatio
on (3-8) can b
be replaced by
y the pressure of an ideal gaas, P = nRT/V
V, and
nsuring that th
he temperature of the gas re
emains consta
ant during the change of staate (isotherma
al
by en
process); then one
e can take nRT outside the integral, giving the equation
n
Note
e that in expan
nsion, V2 > V1, and ln(V2/V1) is a positive quantity; there
efore, the workk is done by th
he
syste
em, so that itss energy decre
eases (negativve sign). When
n the opposite
e is true, V2 < V 1, and ln(V2/V1) is
nega
ative due to ga
as compressio
on, work is don
ne by the systtem, so that its
s energy increeases
P.58
8
(positive sign). The process itse
elf determines the sign of W and ∆E.
114
Dr. Murtadha Alshareifi e-Library
Fig.. 3-4. Reverrsible expan
nsion of a gaas.
Equa
ation (3-10) gives the maxim
mum work don
ne in the expansion as well as the heat abbsorbed,
beca
ause Q = ∆E - W, and, as will
w be shown la
ater, ∆E is equ
ual to zero for an ideal gas in an isotherm
mal
process. The maxximum work in an isotherma
al reversible ex
xpansion may also be expreessed in terms
s of
pressure because, from Boyle's
s law, V2/V1 = P 1/P2 at consttant temperatu
ure. Thereforee, equation (310) ccan be written as
Exa
ample 3-3
One
e mole of watter in equilibrrium with its vapor is con
nverted into steam
s
at 1000°C and 1 atm
m.
The heat absorb
bed in the pro
ocess (i.e., th
he heat of va
aporization of water at 1000°C) is abou
ut
9720
0 cal/mole. What
W
are the values of th e three first-law terms Q, W, and ∆E??
The amount of heat
h
absorbe
ed is the heatt of vaporizattion, given as
s 9720 cal/m
mole. Therefo
ore,
Q = 9720cal/mole
The work W perfformed again
nst the consttant atmosph
heric pressurre is obtainedd by using
equa
ation (3-10), W = -nRT ln
n(V2/V1). Now
w, V1 is the volume
v
of 1 mole
m
of liquidd water at 100°C,
or about 0.018 liter. The volu
ume V2 of 1 m
mole of steam
m at 100°C and
a 1 atm is given by the
e gas
law, assuming th
hat the vaporr behaves ide
eally:
115
Dr. Murtadha Alshareifi e-Library
It is now possible
e to obtain th
he work,
W = -(1 mole)(1..9872 cal/K mole)(398.15
m
5 K) ln (30.6//0.018)
W = -5883 cal
The internal ene
ergy change ∆E
∆ is obtaine
ed from the first-law
f
exprression,
∆E = 9720 - 588
83 = 3837 cal
Therrefore, of the 9720
9
cal of heat absorbed b
by 1 mole of water,
w
5883 cal is employed in doing the work
w
of exxpansion, or “P
PV work,” aga
ainst an extern
nal pressure off 1 atm. The re
emaining 38377 cal increase
es the
interrnal energy of the system. This
T
quantity o
of heat supplies potential energy to the vaapor molecules
s,
that is, it represents the work do
one against th e noncovalent forces of attrraction.
Exa
ample 3-4
Wha
at is the maxximum work done
d
in the issothermal re
eversible expansion of 1 m
mole of an id
deal
gas from 1 to 1.5
5 liters at 25°°C?
The conditions of
o this problem are simila
ar to those of Example 3–
–2, except thaat equation (3(
10) can now be used to obta
ain the (maxim
mum) work involved in ex
xpanding revversibly this gas
g
by 0
0.5 liters; thuss,
W = -(1 mole)(8..3143 joules//K mole)(298
8.15 K) ln (1.5/1.0)
W = -1005.3 joules
P.59
9
Whe
en the solution
n to Example 3-4
3 (work done
e in a reversib
ble fashion) is contrasted witth the results
from Example 3-3 (work done in
n one stage ag
gainst a fixed pressure), it becomes
b
appaarent that the
amount of work re
equired for exp
pansion betwe
een separate pathways
p
can dramatically ddiffer.
Ch
hanges off State att Constan
nt Volum
me
If the
e volume of the system is ke
ept constant d
during a chang
ge of state, dV
V = 0, the first law can be
exprressed as
pt V indicates that volume iss constant. Similarly, under these conditioons the combined
wherre the subscrip
equa
ation (3-4) is re
educed to
T
This equation relates thee heat trans ferred durin
ng the proceess at constaant
vvolume, Qv, with the change in ttemperature,dT. The rattio betweenn these
qquantities deefines the molar
m
heat caapacity at co
onstant volu
ume:
116
Dr. Murtadha Alshareifi e-Library
Ide
eal Gases
s and the
e First La
aw
A
An ideal gass has no inteernal pressuure, and hence no work needs be peerformed to
o
seeparate the molecules from
f
their ccohesive forrces when th
he gas expan
ands.
T
Therefore, w = 0, and the first law
w becomes
Thuss, the work done by the systtem in the isotthermal expan
nsion of an ide
eal gas is equaal to the heat
abso
orbed by the gas.
g
Because the
t process iss done isotherm
mally, there is
s no temperatuure change in the
surro
oundings, dT = 0, and q = 0. Equation (3--5) is reduced to
In this equation, dV
d ≠ 0 because
e there has be
een an expans
sion, so that we
w can write
Equa
ation (3-17) su
uggests that th
he internal ene
ergy of an ideal gas is a fun
nction of the teemperature on
nly,
whicch is one of the
e conditions needed to defin
ne an ideal ga
as in thermody
ynamic terms.
Ch
hanges off State att Constan
nt Pressu
ure
Whe
en the work of expansion is done at consttant pressure, W = -P ∆V = -P(V2 - V1) byy equation (3–7
7),
and under these conditions,
c
the
e first law can b
be written as
eat absorbed at
a constant prressure. Rearrranging the eq
quation resultss in
wherre QP is the he
The term E + PV is called the en
nthalpy, H. Th
he increase in enthalpy, ∆H,, is equal to thhe heat absorb
bed
at co
onstant pressu
ure by the systtem. It is the h
heat required to
t increase the
e internal eneergy and to perrform
the w
work of expansion, as seen by substitutin g H in equatio
on (3-19),
and writing equatio
on (3-18) as
For a
an infinitesima
al change, one
e can write as
117
Dr. Murtadha Alshareifi e-Library
The heat absorbed
d in a reaction
n carried out a
at atmospheric
c pressure is in
ndependent off the number of
o
stepss and the mecchanism of the
e reaction. It d
depends only on
o the initial and final condittions. This fac
ct will
be used in the secction on thermochemistry.
It sho
ould also be stressed
s
that ∆H
∆ = QP only w
when nonatmo
ospheric work
k (i.e., work othher than that
again
nst the atmosphere) is ruled
d out. When e
electric work, work
w
against surfaces,
s
or ceentrifugal force
es
are cconsidered, on
ne must write as
The function H is a composite of
o state properrties, and there
efore it is also a state propeerty that can be
defin
ned as an exact differential. If T and P are
e chosen as variables, dH can
c be written as
Whe
en the pressure is held cons
stant, as, for exxample, when
n a reaction proceeds in an oopen containe
er in
the la
aboratory at essentially
e
con
nstant atmosph
heric pressure
e, equation (3--24) becomess
B
Because qP
P = dH at co
onstant presssure accord
ding to equaation (3–22)), the molarr
hheat capacityy CP at consstant pressuure is defined as
P.60
0
and ffor a change in
i enthalpy be
etween produccts and reactants,
∆ H = Hproducts - Hreactants
r
Tab
ble 3-1 Mod
dified Firstt-Law Equaations for Processes
P
Occurring
O
U
Under Various
Conditionss
118
Dr. Murtadha Alshareifi e-Library
equa
ation (3-26) may be written as
a
wherre ∆CP = (CP)products
- (CP)reaactants. Equation
n (3-27) is kno
own as the Kirrchhoff equatioon.
p
Summary
Som
me of the speciial restrictions that have bee
en placed on the
t first law up
p to this point in the chapterr,
together with the resultant
r
modifications of the
e law, are bro
ought together in Table 3-1. A comparison
n of
the e
entries in Table 3-1 with the material that has gone before will serve as a comprehhensive review
w of
the ffirst law.
The
ermoche
emistry
Many chemical an
nd physical pro
ocesses of inte
erest are carriied out at atmospheric (esseentially consta
ant)
pressure. Under th
his condition, the
t heat excha
anged during the process equals
e
the chaange in enthalp
py
acco
ording to equation (3-20), QP = ∆H. Thus, the change in
n enthalpy acc
companying a chemical reaction
rema
ains a function
n only of temperature as sta
ated in equatio
on (3-27). A ne
egative ∆H annd QP means that
heat is released (e
exothermic); a positive value
e of ∆H and QP means that heat is absorrbed
(endothermic). It iss also possible
e that a reactio
on takes place
e in a closed container;
c
in ssuch a case the
heat exchanged equals the change in interna
al energy (i.e., Qv = ∆E).
Therrmochemistry deals with the
e heat change
es accompanying isothermal chemical reaactions at cons
stant
pressure or volum
me, from which values of ∆H
H or ∆E can be
e obtained. These thermodyynamic propertties
are, of course, rela
ated by the de
efinition of H, e
equation (3-21
1). In solution reactions, thee P ∆ V terms are
not ssignificant, so that ∆H [cong
gruent] ∆E. Th
his close appro
oximation does not hold, how
owever, for
reactions involving
g gases.
Heat of Forrmation
For a
any reaction re
epresented by
y the chemica l equation
aA + bB = cC + dD
D
the e
enthalpy change can be writtten as
119
Dr. Murtadha Alshareifi e-Library
w
where
= enthalpy
e
peer mole (callled the molaar enthalpy)) and a, b, cc, and d are
sttoichiometrric coefficients. It is knnown that on
nly the molaar enthalpiees of
coompounds, either as reeactants or pproducts, co
ontribute to the change of enthalpy
y
of a chemicaal reaction. Enthalpies
E
aare all relatiive magnitu
udes, so it iss useful to
ddefine the heeat involved
d in the form
mation of ch
hemical com
mpounds.
One can, for instance, choose the reaction off formation of carbon
c
dioxide
e from its elem
ments,
e
Here
The subscripts rep
present the ph
hysical states, (s) standing for
f solid and (g
g) for gas. Addditional symbo
ols,
(l) fo
or liquid and (a
aq) for dilute aqueous solutio
on, will be found in subsequ
uent thermochhemical equatiions.
P.61
Key Co
oncept
Stan
ndard Enthalpy
In th
hermodynam
mics, the conv
vention of asssigning zero
o enthalpy to all elementss in their mos
st
stab
ble physical state
s
at 1 atm
m of pressure
e and 25°C is
s known as choosing
c
a sstandard or
reference state:
Iff the
vallues for the elements C (s) and O2(g) are chosen arbitrarily to be zero,
thhen accordinng to equatiion (3-31), tthe molar enthalpy of the
t compouund, (CO2,
gg, 1 atm), is equal to thee enthalpy oof the formaation reactio
on, ΔH°f(25°CC), for the
process in eqquation (3-3
30) at 1 atm
m of pressuree and 25°C.
120
Dr. Murtadha Alshareifi e-Library
The standard heat of formation of gaseous carbon dioxide is ∆H°f(25°C) = -94,052 cal. The negative sign
accompanying the value for ∆H signifies that heat is evolved, that is, the reaction is exothermic. The
state of matter or allotropic form of the elements also must be specified in defining the standard state.
Equation (3-30) states that when 1 mole of solid carbon (graphite) reacts with 1 mole of gaseous oxygen
to produce 1 mole of gaseous carbon dioxide at 25°C, 94,052 cal is liberated. This means that the
reactants contain 94,052 cal in excess of the product, so that this quantity of heat is evolved during the
reaction. If the reaction were reversed and CO2were converted to carbon and oxygen, the reaction
would be endothermic. It would involve the absorption of 94,052 cal, and ∆H would have a positive
value.
The standard heats of formation of thousands of compounds have been determined, and some of these
are given in Table 3-2.
Table 3-2 Standard Heats of Formation at 25°C*
Substance
ΔH°
(kcal/mole)
H2(g)
0
H(g)
52.09
O2(g)
0
O(g)
l2(g)
Substance
ΔH°
(kcal/mole)
Methane(g)
-17.889
Ethane(g)
-20.236
59.16
Ethylene(g)
12.496
14.88
Benzene(g)
19.820
H2O(g)
-57.798
Benzene(l)
11.718
H2O(l)
-68.317
Acetaldehyde(g)
-39.76
HCl(g)
-22.063
Ethyl alcohol(l)
-66.356
Glycine(g)
-126.33
Acetic acid(l)
-116.4
Hl(g)
CO2(g)
6.20
-94.052
*From F. D. Rossini, K. S. Pitzer, W. J. Taylor, et al., Selected Values of
Properties of Hydrocarbons (Circular of the National Bureau of Standards
461), U.S. Government Printing Office, Washington, D.C., 1947; F. D.
Rossini, D. D. Wagman, W. H. Evans, et al., Selected Values of Chemical
Thermodynamic Properties (Circular of the National Bureau of Standards
121
Dr. Murtadha Alshareifi e-Library
500), U.S. Governmen
nt Printing O
Office, Wasshington, D.C.,
D
1952.
Hes
ss's Law
w and Hea
at of Com
mbustion
It is n
not possible to
o directly measure the heatss of formation of every know
wn compound as in equation (330). Incomplete orr side reaction
ns often compllicate such determinations. However, as eearly as 1840,
es of a system
m, thermochem
mical
Hesss showed thatt because ∆H depends onlyy on the initial and final state
equa
ations for seve
eral steps in a reaction could
d be added an
nd subtracted to obtain the hheat of the ov
verall
reaction. The princciple is known
n as Hess's law
w of constant heat summatiion and is usedd to obtain he
eats
of re
eaction that are
e not easily measured direcctly.
Exa
ample 3-5
It is extremely im
mportant to use the heat o
of combustio
on, that is, the heat involvved in the
com
mplete oxidatiion of 1 mole
e of a compo
ound at 1 atm
m pressure, to
o convert thee compound to
its p
products. Forr instance, the formation o
of methane is
i written as
and the combustion of metha
ane is written
n as
The enthalpies of
o formation of
o both CO2(gg) and H2O(l) have been measured
m
witth extreme
accu
uracy; thereffore, ∆H°f(25°CC) for methan
ne gas can be obtained by subtractingg ∆H°f(25°C) off
CO22(g) and twice that of H2O(l)
eat of combu
ustion:
( from the he
122
Dr. Murtadha Alshareifi e-Library
The heat of form
mation of CH4(g)
e, is reported in Table 3-2
2 as -17.889..
4 , methane
Heats of Re
eaction frrom Bond
d Energiies
In Ch
hapter 2, enerrgies of bondin
ng were discu ssed in terms of binding forrces that hold molecules
(intra
amolecular bo
onding) or state
es of matter (n
noncovalent fo
orces) togethe
er. For examplle, the energie
es
asso
ociated with co
ovalent bondin
ng between ato
oms range fro
om 50 to 200 kcal/mole.
k
Theese figures
enco
ompass many of the types of
o bonding tha
at you learned about in gene
eral and organnic chemistry
courrses, including
g double and trriple bonds tha
at arise from π-electron
π
orb
bital overlap, bbut they are we
eaker
than covalent σ-electron orbital bonds. Electro
rovalent or ionic bonds occu
urring betweenn atoms with
oppo
osing permane
ent charges ca
an be much sttronger than a covalent bond. In a molecuule, the forces
s
enab
bling a bond
P.62
2
to form between atoms
a
arise fro
om a combinattion of attractiv
ve and repulsive energies aassociated with
each
h atom in the molecule.
m
These forces varyy in strength because
b
of inte
erplay betweeen attraction an
nd
repulsion. Therefo
ore, the net bonding energy that holds a series
s
of atoms
s together in a molecule is the
t
addittive result of all
a the individual bonding en ergies.
In a chemical reacction, bonds may
m be broken and new bon
nds may be forrmed to give riise to the prod
duct.
The net energy asssociated with the reaction, the heat of reaction, can be
e estimated froom the bond
enerrgies that are broken
b
and the bond energiies that are formed during th
he reaction prrocess. Many of
the ccommon covalent and otherr bonding-type
e energies can
n be commonly found in boooks on
therm
modynamics, like the ones listed
l
in the fo
ootnote in the opening
o
of this
s chapter.
Exa
ample 3-6
The covalent bonding energy
y in the reacttion
can be calculate
ed from know
wing a C==C bond is brok
ken (requiring
g 130 kcal), a Cl—Cl bon
nd is
brokken (requiring
g 57 kcal), a C—C bond is formed (lib
berating 80 kcal),
k
and tw
wo C—Cl bon
nds
are fformed (liberrating 2 × 78
8 or 156 kcal)). Thus, the energy
e
∆H of the reactionn is
∆ H = 130 + 57 - 80 -156 = -49 kcal
Because 1 cal = 4.184 joules
s, -49 kcal is expressed in SI units as
s -2.05 × 105 joules.
The foregoing prrocess is nott an idealized
d situation; strictly
s
speaking, it appliess only to
reacctions in a ga
as phase. If there is a cha
ange from a gas into a co
ondensed phhase, then
cond
densation, so
olidification, or crystalliza
ation heats must
m
be involv
ved in the ovverall
123
Dr. Murtadha Alshareifi e-Library
calcculations. This approach may
m find som
me use in esttimating the energy assoociated with
chem
mical instability.
Ad
dditional Applicati
A
ions of T
Thermoch
hemistry
Therrmochemical data
d
are imporrtant in many cchemical calculations. Heatt-of-mixing datta can be used to
determine whether a reaction su
uch as precipi tation is occurrring during the mixing of tw
wo salt solution
ns. If
no re
eaction takes place when diilute solutions of the salts are mixed, the heat of reactioon is zero.
The constancy of the
t heats of neutralization o
obtained expe
erimentally whe
en dilute aqueeous solutions
s of
vario
ous strong acid
ds and strong bases are mi xed shows tha
at the reaction
n involves onlyy
No ccombination occcurs between
n any of the otther species in
n a reaction su
uch as
beca
ause HCl, NaO
OH, and NaCl are complete ly ionized in water.
w
In the ne
eutralization oof a weak
electtrolyte by a strrong acid or base, howeverr, the reaction involves ionization in additioon to
neutralization, and
d the heat of re
eaction is no l onger constan
nt at about -13
3.6 kcal/mole. Because som
me
heat is absorbed in the ionizatio
on of the weakk electrolyte, th
he heat evolve
ed falls below the value for the
neutralization of co
ompletely ionized species. T
Thus, knowled
dge of ∆H of neutralization
n
aallows one to
differrentiate betwe
een strong and
d weak electro
olytes.
Anotther importantt application off thermochem
mistry is the determination off the number oof calories
obtained from variious foods. Th
he subject is d iscussed in biiochemistry texts as well ass books on foo
od
and nutrition.
The
e Second
d Law of Thermod
dynamics
s
In orrdinary experie
ence, most na
atural phenome
ena are obserrved as occurrring only in onne direction. For
insta
ance, heat flow
ws spontaneou
usly only from hotter to cold
der bodies, whereas two boddies having the
same
e temperature
e do not evolve
e into two bod
dies having diffferent temperatures, even tthough the firs
st law
of thermodynamiccs does not pro
ohibit such a p
possibility. Sim
milarly, gases expand naturaally from highe
er to
lowe
er pressures, and
a solute molecules diffuse
e from a region of higher to one of lower cconcentration.
Thesse spontaneou
us processes will not procee
ed in reverse without
w
the inttervention of ssome external
force
e to facilitate th
heir occurrenc
ce. Although sspontaneous processes
p
are not thermodyynamically
reversible, they ca
an be carried out
o in a nearlyy reversible ma
anner by an outside force. M
Maximum work is
obtained by condu
ucting a sponta
aneous proce
ess reversibly; however, the frictional lossses and the
nece
essity of carryiing out the pro
ocess at an inffinitely slow ra
ate preclude th
he possibility oof complete
reversibility in reall processes.
The first law of the
ermodynamics
s simply obserrves that energy must be co
onserved whenn it is converte
ed
from one form to another.
a
It has
s nothing to sa
ay about the probability that a process wil l occur. The
seco
ond law refers to the probab
bility of the occcurrence of a process
p
based
d on the obserrved tendency
y of a
syste
em to approacch a state of energy
e
equilibrrium.
The historical deve
elopment of th
he thermodyna
amic property that explains the natural teendency of the
e
processes to occu
ur, now called entropy, has iits origins in sttudies of the efficiency
e
of stteam engines,, from
whicch the following
g observation was made: “A
A steam engin
ne can do work
k only with a fa
fall in temperature
and a flow of heat to the lower temperature. N
No useful work
k can be obtained from heat
at at constant
temp
perature.” Thiss is one way to
o state the seccond law of th
hermodynamic
cs.
The
e Efficien
ncy of a Heat
H
Eng
gine
An im
mportant conssideration is th
hat of the posssibility of conve
erting heat into work. Not onnly is heat
isoth
hermally unava
ailable for worrk, it can neve
er be converted
d completely into
i
work.
P.63
3
The spontaneous character of natural
n
processses and the limitations on the conversionn of heat into work
w
consstitute the seco
ond law of the
ermodynamicss. Falling water can be made
e to do work oowing to the
124
Dr. Murtadha Alshareifi e-Library
differrence in the potential energ
gy at two differrent levels, and electric work
k can be donee because of the
differrence in electrric potential (e
emf). A heat e ngine (such as a steam eng
gine) likewise can do useful
workk by using two heat reservoiirs, a “source”” and a “sink,” at two differen
nt temperaturees. Only part of
o the
heat at the source
e is converted into work, with
h the remainder being returned to the sin k (which, in
practical operation
ns, is often the
e surroundingss) at the lowerr temperature. The fraction oof the heat, Q,
Q at
the ssource converrted into work, W, is known a
as the efficien
ncy of the engine:
The efficiency of even
e
a hypothe
etical heat eng
gine operating
g without frictio
on cannot be uunity because
e W is
alwa
ays less than Q in a continuo
ous conversio
on of heat into work accordin
ng to the secoond law of
therm
modynamics.
Imag
gine a hypothe
etical steam engine operatin
ng reversibly between
b
an up
pper temperatture Thot and a
lowe
er temperature
e Tcold. It absorrbs heat Qhot ffrom the hot bo
oiler or source
e, and by meaans of the work
king
subsstance, steam, it converts th
he quantity W into work and returns heat Qcold to the coold reservoir orr
sink.. Carnot, in 18
824, proved that the efficienccy of such an engine, opera
ating reversiblyy at every stage
and returning to itss initial state (cyclic processs), could be given by the expression
eat flow in the operation of th
ows the tempe
erature gradieent, so that the
e
It is kknown that he
he engine follo
heat absorbed and
d rejected can
n be related di rectly to temperatures. Lord
d Kelvin used tthe ratios of th
he
two h
heat quantitiess Qhot and Qcoold of the Carno
ot cycle to esttablish the Kelvin temperatuure scale:
By combining equations (3-33) through
t
(3-35)), we can describe the efficiiency by
observed from
m equation (3-3
36) that the hiigher Thot beco
omes and the lower Tcold beecomes, the
It is o
grea
ater is the efficciency of the engine. When Tcold reaches absolute
a
zero on the Kelvin scale, the
reversible heat engine converts heat complettely into work, and its theore
etical efficiency
cy becomes un
nity.
This can be seen by setting Tcold = 0 in equatiion (3-36). Be
ecause absolutte zero is conssidered
unattainable, howe
ever, an efficie
ency of 1 is im
mpossible, and
d heat can nev
ver be compleetely converted
d to
workk. This stateme
ent can be written using the
e notation of lim
mits as follows
s:
If Thoot = Tcold in equ
uation (3-36), the cycle is issothermal and the efficiency
y is zero, confi rming the earlier
state
ement that hea
at is isotherma
ally unavailablle for conversiion into work.
Exa
ample 3-7
A steam engine operates between the te
emperatures of 373 and 298
2 K. (a) Whhat is the
theo
oretical efficie
ency of the engine?
e
(b) Iff the engine is supplied with
w 1000 call of heat Qhot,
wha
at is the theorretical work in ergs?
(a)
(b)
125
Dr. Murtadha Alshareifi e-Library
Entropy
In an
nalyzing the prroperties of re
eversible cycle
es, one feature
e was unveiled
d; the sum of tthe quantities Qrev,
s zero for the cycle,
c
that is,
i/Ti is
This behavior is siimilar to that of
o other state ffunctions such
h as ∆E and ∆H; Carnot reco
cognized that
when
n Qrev, a path--dependent prroperty, is divid
ded by T, a ne
ew path-independent prope rty is generate
ed,
calle
ed entropy. It iss defined as
and ffor an infinitessimal change,
Thuss, the term Qhoot/Thot is known as the entro
opy change of the reversible
e process at Thhot,
and Qcold/Tcold is th
he entropy cha
ange of the revversible proce
ess at Tcold. Th
he entropy chaange ∆Shot during
the a
absorption of heat
h
from Thot is positive, ho
owever, becau
use Qhot is pos
sitive. At the loower
temp
perature, Qcold is negative and the entropyy change ∆Scoold is negative.. The total enttropy change
∆Scyycle in the reverrsible cyclic prrocess is zero
o. Let us use th
he data from Example
E
3-7 too demonstrate
e
this. It can be seen
n that only part of the heat Q hot (1000 cal) is converted to work (200 cal) in the eng
gine.
The difference in energy
e
(800 ca
al) is the heat Qcold that is re
eturned to the sink at the low
wer temperatu
ure
and is unavailable
e for work. When the entropyy changes at the
t two tempe
eratures are caalculated, they
y are
both found to equa
al 2.7 cal/deg::
Therrefore,
P.64
4
It ma
ay also be noted that if Qhot is the heat ab
bsorbed by an engine at Thot, then -Qhot m
must be the heat
lost b
by the surroun
ndings (the ho
ot reservoir) att Thot, and the entropy of the
e surroundingss is
Hencce, for any sysstem and its surroundings o
or universe,
Therrefore, there are
a two cases in which ∆S = 0: (a) a syste
em in a reversible cyclic proocess and (b) a
syste
em and its surrroundings und
dergoing any reversible pro
ocess.
In an
n irreversible process,
p
the entropy change
e of the total system
s
or univ
verse (a system
m and its
surro
oundings) is always positive
e because ∆Sssurr is always le
ess than ∆Ssyst in an irreverrsible process. In
math
hematical sym
mbols, we write
e,
and this can serve
e as a criterion
n of spontaneitty of a real pro
ocess.
126
Dr. Murtadha Alshareifi e-Library
Equa
ations (3-42) and
a (3-43) sum
mmarize all off the possibilities for the entrropy; for irreveersible proces
sses
(real transformatio
ons) entropy always increasses until it reac
ches its maxim
mum at equilibbrium, at which
h
pointt it remains invvariable (i.e., change
c
equalss zero).
Two examples of entropy
e
calcullations will now
w be given, firrst, a reversible process, annd second, an
irreversible processs.
Exa
ample 3-8
Wha
at is the entro
opy change accompanyin
a
ng the vaporrization of 1 mole
m
of wateer in equilibriu
um
with its vapor at 25°C? In this
s reversible isothermal process,
p
the heat
h
of vaporrization
∆Hv required to convert
c
the liquid to the vvapor state is
s 10,500 cal//mole.
The process is carried
c
out att a constant p
pressure, so
o that QP = ∆H
Hv, and becaause it is a
reve
ersible process, the entro
opy change ccan be written as
T
The entropy change inv
volved as thee temperatu
ure changes is often dessired in
thhermodynam
mics; this reelationship iis needed fo
or the next example.
e
Thhe heat
abbsorbed at constant
c
preessure is givven by equaation (3-26), qP = CP, dT, and fo
or
a reversible process
p
Integ
grating betwee
en T1 and T2 yields
y
Exa
ample 3-9
Com
mpute the entropy change
e in the (irrevversible) tran
nsition of 1 mole
m
of liquid water to
crysstalline waterr at -10°C, at constant pre
essure. The entropy is ob
btained by caalculating the
e
entropy changess for several reversible stteps.
Firstt consider the reversible conversion o
of supercoole
ed liquid watter at -10°C tto liquid wate
er at
0°C, then change this to ice at 0°C, and ffinally cool th
he ice revers
sibly to -10°C
C. The sum of
o the
entropy changess of these ste
eps gives ∆S
Swater. To this, add the enttropy changee undergone by
the ssurroundingss so as to ob
btain the tota l entropy cha
ange. If the process
p
is sppontaneous, the
resu
ult will be a positive
p
value
e. If the syste
em is at equilibrium, that is, if liquid w
water is in
equiilibrium with ice at -10°C,, there is no tendency forr the transitio
on to occur, aand the total
entropy change will be zero. Finally, if the
e process is not spontaneous, the tottal entropy
chan
nge will be negative.
The heat capacity of water is
s 18 cal/deg and that of ic
ce is 9 cal/de
eg within thiss temperature
e
rang
ge.
The reversible change of water at -10°C to ice at -10°C is carried out as follow
ws:
127
Dr. Murtadha Alshareifi e-Library
The entropy of water
w
decreases during th
he process because
b
∆S is
i negative, bbut the
spon
ntaneity of th
he process ca
annot be jud
dged until one
e also calculates ∆S of thhe surroundings.
For the entropy change
c
of the surroundin
ngs, the wate
er needs to be
b consideredd to be in
equiilibrium with a large bath at -10°C, an
nd the heat liberated whe
en the water ffreezes is
abso
orbed by the
e bath without a significan
nt temperature increase. Thus, the rev
eversible
abso
orption of he
eat by the batth is given byy
whe
ere 1343 cal/mole is the heat
h
of fusion
n of water at -10°C. Thus
s,
The process in Exxample 3-8 is spontaneous
s
b
because ∆S > 0. This criterion of spontanneity is not a
convvenient one, however, because it requiress a calculation
n of the entropy change bothh in the system
m
and the surroundin
ngs. The free energy functio
ons, to be trea
ated in a later section, do noot require
inforrmation concerning the surro
oundings and are more suittable criteria of
o spontaneity..
Entropy and Disord
der
The second law prrovides a crite
erion for decid ing whether a process follow
ws
the n
natural or spon
ntaneous direc
ction. Even th ough the caus
ses for the pre
eference for a particular dire
ection
of ch
hange of state or the imposs
sibility for the reverse directtion are unkno
own, the underrlying reason is not
that the reverse prrocess is impo
ossible, but ratther that it is extraordinarily
e
improbable.
Therrmodynamic systems described by macro
oscopic properties such as T,
T P, or compoosition can als
so be
desccribed in termss of microscop
pic quantities ssuch as molec
cular random motions.
m
The m
microscopic or
o
mole
ecular interpre
etation of
P.65
5
entro
opy is commonly stated as a measure of disorder owing to molecular motion. As ddisorder increa
ases,
entro
opy will also in
ncrease. The impossibility o
of converting all
a thermal ene
ergy into work results from the
“diso
orderliness” off the molecules existing in th
he system. Dis
sorder can be seen as the nnumber of way
ys
the in
nside of a sysstem can be arrranged so tha
at from the outside the syste
em looks the ssame. A
quan
ntitative interpretation of enttropy was give
en long before
e by Boltzmann; the equatioon for the entro
opy
of a system in a giiven thermody
ynamic state iss
128
Dr. Murtadha Alshareifi e-Library
Fig.. 3-5. Free expansion
e
of
o a gas in a closed systtem.
wherre k is the Bolttzmann consta
ant and O is th
he number of microscopic states
s
or configguration that the
t
syste
em may adoptt. Unlike the th
hermodynamicc definition of entropy in equ
uation (3-39) oor (3-40), to
unde
erstand what a configuration
n means (i.e., O), a structurral model of the system is neeeded.
Let u
us imagine tha
at the system consists
c
of a g
gas in a closed container that is allowed tto expand free
ely
into a vacuum (Fig
g. 3-5). The initial state corre
responds to V1and the final state correspoonds to V2, wh
hich
is larrger than V1: V1 = ½V2.
If only one gas particle is in the initial state, sttatistically there are two equ
ual possibilitiees for a particle
e to
occu
upy V2, wherea
as only one ex
xists for V1. In other words, the probability
y P for that paarticle to stay
in V1 is P = ½ (Fig
g. 3-6a). Now consider
c
two i ndependent particles;
p
the probability
p
of fi nding both
in V1 after V2 is avvailable is P = (½)2 = ¼ (Fig
g. 3-6b). For th
hree particles there are eighht possible
confiigurations, so the probability
y for all particlles to stay in V1 is only 1/8. For a system composed
ofN p
particles the probability
p
bec
comes P = (½))N. Now, if N is
s on the order of regular moolecular quantiities,
that is, ~1023 mole
ecules, the pro
obability of find
ding all of them
m in the initial state is extraoordinarily
1023
smalll, P = ½ .
Thuss, in this mode
el system, by changing
c
from
m the initial sta
ate V1 to a final state V2, we significantly
incre
ease the numb
ber of distributtion possibilitie
es of the N pa
articles. Once the
t system haas expanded, it is
extre
emely improba
able that they will be found b
by chance onlly in V1. For th
his simple systtem, the proba
ability
exprressed by the number of con
nfigurations affter an irrevers
sible process from
f
state 1 too state 2 is
gene
erally related to
t the volumes
s as
and, therefore, the
e change of en
ntropy for 1 mo
ole of gas is given
g
by
m the standpoiint of this simp
ple model, the
e isothermal ex
xpansion of an
n ideal gas inccreases the
From
entro
opy because of
o
P.66
6
the e
enhanced num
mber of configu
urations in a la
arger volume compared to a smaller one.. The larger th
he
number of configu
urations, the more
m
disordere
ed a system is considered. Thus,
T
the increease in entrop
py
with increasing nu
umber of config
gurations is niicely described by Boltzman
nn concept givven in equation (346).
129
Dr. Murtadha Alshareifi e-Library
Fig.. 3-6. (a) Thhe two possible states oof a system of one moleecule. (b) T
The four
posssible states of a system
m of two inddependent molecules.
m
The
e Third Law
L
of Th
hermodyn
namics
The third law of thermodynamic
cs states that tthe entropy of a pure crystalline substancce is zero at
abso
olute zero because the crysttal arrangeme
ent must show
w the greatest orderliness
o
at this temperatture.
In other words, a pure
p
perfect crystal has onlyy one possible
e configuration
n, and accordiing to equation
n (346), its entropy is zero
z
[i.e., S = k ln(1) = 0]. A
As a conseque
ence of the thirrd law, the tem
mperature of
olute zero (0 K)
K is not possib
ble to reach evven though so
ophisticated prrocesses that use the orienttation
abso
of ele
ectron spins and
a nuclear sp
pins can reach
h very low tem
mperatures of 2 × 10-3 and 100-5 K, respectively.
The third law makes it possible to calculate th
he absolute en
ntropies of purre substancess from equation (344) rrearranged as
130
Dr. Murtadha Alshareifi e-Library
w
where S0 is the
t molar en
ntropy at abbsolute zero and ST is th
he absolute m
molar
enntropy at anny temperatture. The maagnitude qrev can be reeplaced by tthe
coorrespondinng CPdT vallues at consstant pressurre according
g to equatioon (3-26), so
o
thhat STmay be
b determineed from knoowledge of the heat cap
pacities andd the entropy
y
chhanges duriing a phase change as tthe temperaature rises frrom 0 K to T. The
foollowing eqquation show
ws ST for a ssubstance th
hat undergo
oes two phasse changes,
m
melting (m) and vaporizzation (v):
wherre S0 = 0 has been omitted. Each of these
e terms can be evaluated in
ndependently; in particular, the
first integral is calcculated numerrically by plottiing CP/T versu
usT. Below 20
0 K few data aare available, so
s
that it is customaryy to apply the Debye approxximation CP [c
congruent] aT3, joules/mole K in this regio
on.
Fre
ee Energy
y Functio
ons and A
Applicatiions
The criterion for sp
pontaneity giv
ven by the seccond law requires the knowle
edge of the enntropy change
es
both in the system
m and the surro
oundings as sstated in equattion(3-43). It may
m be very usseful, however, to
have
e a property th
hat depends only on the sysstem and neve
ertheless indicates whether a process occ
curs
in the
e natural direcction of change.
Let u
us consider an
n isolated systtem composed
d of a closed container
c
(i.e.,, the system) iin equilibrium with
a tem
mperature bath (i.e., the surrroundings) ass illustrated inF
Figure 3-1d, th
hen accordingg to equations (342) a
and (3-43), the
e resulting equ
uation is
Equilibrium is a co
ondition where
e the transfer o
of heat occurs
s reversibly; therefore, at coonstant
temp
perature, ∆Ssuurr = -∆Ssyst and
d
Now
w, renaming ∆S
Ssyst = ∆S and Qrev = Q, we write equation
n as
whicch can also be written in the following wayy:
For a process at constant
c
volum
me, Qv = ∆E, sso equation (3-53) becomes
s
In the case of a prrocess at cons
stant pressure
e, QP = ∆H, an
nd
131
Dr. Murtadha Alshareifi e-Library
Equa
ations (3-54) and
a (3-55) are
e combinationss of the first an
nd second law
ws of the therm
modynamics and
a
exprress both equilibrium (=) and
d spontaneity (<) conditions
s depending only on the prooperties of the
syste
em; therefore, they are of fu
undamental prractical importtance. The terms on the left of equations (354) a
and (3-55) are
e commonly de
efined as the H
Helmholtz free
e energy or wo
ork function A ,
and the Gibbs free
e energy G,
h are two new
w state propertties because they
t
are comp
posed of state variables E, H,
H T,
respectively, which
pontaneity con
nditions are reduced to only
and S. Thus, equillibrium and sp
Ma
aximum Net
N Work
k
The second law off the thermody
ynamics expre
essed in equation (3-53) for an isothermaal process can be
bined with the
e conservation
n of energy by substituting Q = ∆E - W, yie
elding
comb
P.67
7
T
Table
3-3 Criteria for Spontaneitty and Equ
uilibrium
Fu
unction
Restriction
ns
Sign of Fu
unction
Sp
pontaneousNonspontaneousEquuilibrium
Total system,
ΔSuniverse
ΔE = 0, ΔV = 0
+ or >0
- or <0
0
ΔG
ΔT = 0, ΔP = 0
- or <0
+ or >0
0
ΔA
ΔT = 0, ΔV = 0
- or <0
+ or >0
0
In this case, W rep
presents all po
ossible forms o
of work availa
able and not on
nly PV work (ii.e., expansion
n or
comp
pression of the system). The system doe
es maximum work
w
(Wmax) wh
hen it is workinng under
reversible conditio
ons; therefore, only the equa
ality in equatio
on (3-60) appliies. By includiing the definition
for A in equation (3-60), we obta
ain
The former equation explains th
he meaning off the Helmholtz
z energy (whic
ch also explainns why it is ca
alled
a wo
ork function), that is, the maximum work p
produced in an
n isothermal trransformation is equal to the
e
chan
nge in the Helm
mholtz energy
y.
Now
w if W is separa
ated into the PV
P work and a
all other forms of work exclu
uding expansioon or
comp
pression, Wa (also
(
called us
seful work) forr an isotherma
al process at constant
c
presssure equation (360) ccan be written as
The definition of th
he Gibbs enerrgy given in eq
quation (3-60) can also be written
w
as
132
Dr. Murtadha Alshareifi e-Library
For a reversible prrocess equatio
on, (3-62) tran
nsforms into
Equa
ation (3-64) exxpresses the fact
f
that the ch
hange in Gibb
bs free energy at constant teemperature an
nd
pressure equals th
he useful or maximum
m
net w
work (Wa) that can be obtain
ned from the pprocess. Unde
er
those
e circumstancces in which th
he PV term is insignificant (s
such as in electrochemical ccells and surfa
ace
tensiion measurem
ments), the free
e energy chan
nge is approximately equal to the maximuum work.
Criiteria of Equilibriu
E
um and S
Spontane
eity
Whe
en net work ca
an no longer be
e obtained fro
om a process, G is at a minimum, and ∆G
G = 0. This
state
ement signifiess that the systtem is at equil ibrium. On the
e other hand, from
f
equationn (3-62) a nega
ative
free energy chang
ge written as ∆G
∆ < 0 signifie
es that the process is a spon
ntaneous one.. If ∆G is posittive
(∆G > 0), it indicattes that net wo
ork must be ab
bsorbed for th
he reaction to proceed,
p
and accordingly it is
not sspontaneous.
Whe
en the processs occurs isothe
ermally at con stant volume rather than co
onstant pressuure, ∆A serves
s as
the ccriterion for sp
pontaneity and
d equilibrium. F
From equation
n(3-58) it is ne
egative for a sppontaneous
process and becomes zero at equilibrium.
e
Thesse criteria, tog
gether with the
e entropy criterrion of equilibrrium and spon
ntaneity, are lissted in Table 3-3.
The difference in sign
s
between ∆G and ∆Sunivverse implies th
hat the conditio
on for a processs being
spon
ntaneous has changed from
m an increase o
of the total enttropy, ∆Suniversse > 0, to a deccrease in Gibb
bs
free energy, ∆G < 0. However, Gibbs
G
free ene
ergy is only a composite tha
at expresses tthe total chang
ge in
entro
opy in terms of
o the propertie
es of the syste
em alone. To prove
p
that, let us combine eequations (342) a
and (3-51) for a reversible is
sothermal proccess into
Then
n, if the pressu
ure is constant,
For tthe same proccess,
and therefore
m the former equation
e
it is clear that the o
only criterion fo
or spontaneou
us change is aan increase off the
From
entro
opy of the univverse (i.e., the
e system and iits surrounding
gs). Chemical reactions aree usually carrie
ed
out a
at constant tem
mperature and
d constant pre
essure. Thus, equations
e
invo
olving ∆G are of particular
interrest to the chemist and the pharmacist.
p
It wa
as once though
ht that at cons
stant pressure
e a negative ∆H (evolution of
o heat) was itsself proof of a
spon
ntaneous reacction. Many natural reactionss do occur with an evolution
n of heat; the sspontaneous
meltiing of ice at 10
0°C, however, is accompan
nied by absorp
ption of heat, and
a a number of other exam
mples
can b
be cited to pro
ove the error of
o this assump
ption. The reas
son ∆H is ofte
en thought of aas a criterion of
o
spon
ntaneity can be
e seen from th
he familiar exp
pression (3-67
7).
If T ∆
∆S is small co
ompared with ∆H,
∆ a negative
e ∆H will occu
ur when ∆G is negative (i.e.,, when the pro
ocess
is sp
pontaneous). When
W
T ∆S is large, howeve
er, ∆G may be
e negative, and the processs may be
spon
ntaneous even
n though ∆H is
s positive.
P.68
8
The entropy of all systems, as previously
p
statted, spontaneo
ously tends to
oward randomnness accordin
ng to
the ssecond law, so
o that the morre disordered a system beco
omes, the high
her is its probaability and the
e
grea
ater its entropyy. Hence, equa
ation (3-64) ca
an be written as
a
133
Dr. Murtadha Alshareifi e-Library
One can state thatt ∆G will beco
ome negative a
and the reaction will be spontaneous eithher when the
enthalpy decrease
es or the proba
ability of the ssystem increas
ses at the temperature of th e reaction.
Thuss, although the
e conversion of
o ice into wate
er at 25°C req
quires an abso
orption of heatt of 1650 cal/m
mole,
the rreaction leads to a more pro
obable arrange
ement of the molecules;
m
tha
at is, an increaased freedom of
mole
ecular movement. Hence, th
he entropy inc reases, and ∆S
∆ = 6 cal/mole deg is sufficciently positive
e to
make
e ∆Gnegative, despite the positive
p
value of ∆H.
Many of the complexes of Chap
pter 10 form in
n solution with a concurrent absorption of heat, and the
processes are spo
ontaneous only because the
e entropy chan
nge is positive
e. The increasee in randomne
ess
occu
urs for the follo
owing reason. The dissolutio
on of solutes in
i water may be
b accompaniied by
a decrease in entrropy because both the wate
er molecules and
a the solute molecules losse freedom of
move
ement as hydration occurs. In complexat ion, this highly
y ordered arra
angement is diisrupted as the
e
sepa
arate ions or molecules
m
reac
ct through coo
ordination, and
d the constitue
ents thus exhibbit more freedom
in the
e final comple
ex than they ha
ad in the hydra
rated condition
n. The increase in entropy aassociated with
h this
incre
eased random
mness results in a spontaneo
ous reaction as
a reflected in the negative vvalue of ∆G.
Convversely, some
e association reactions are a
accompanied by a decrease
e in entropy, aand they occurr in
spite
e of the negativve ∆S only be
ecause the hea
at of reaction is sufficiently negative. For example, the
Lewiis acid–base reaction
r
by wh
hich iodine is rrendered solub
ble in aqueous
s solution,
is acccompanied byy a ∆S of -4 ca
al/mole deg. Itt is spontaneo
ous because
In the previous examples the reader should n
not be surprise
ed to find a negative entropyy associated with
w a
spon
ntaneous reacction. The ∆S values
v
conside
ered here are the changes in entropy of tthe substance
e
alone
e, not of the to
otal system, th
hat is, the subsstance and its
s immediate su
urroundings. A
As stated befo
ore,
when
n ∆S is used as
a a test of the
e spontaneity of a reaction, the entropy change of the eentire system must
be co
onsidered. Fo
or reactions at constant temp
perature and pressure,
p
whic
ch are the mos
ost common types,
the cchange in free
e energy is ord
dinarily used a
as the criterion
n in place of ∆S. It is more cconvenient
beca
ause it elimina
ates the need to
t compute an
ny changes in the surroundings.
By re
eferring back to
t Example 3--8, it will be se
een that ∆S wa
as negative fo
or the change ffrom liquid to solid
wate
er at -10°C. Th
his is to be exp
pected becausse the molecu
ules lose some
e of their freeddom when they
y
passs into the crysttalline state. The
T entropy off water plus its
s surroundings
s increases duuring the trans
sition,
howe
ever, and it is a spontaneou
us process. Th
he convenienc
ce of using ∆G
G instead of ∆S
S to obtain the
e
same
e information is apparent fro
om the followi ng example, which
w
may be compared witth the more
elabo
orate analysiss required in Example
E
3-8.
Exa
ample 3-10
∆H a
and ∆S for th
he transition from liquid w
water to ice at
a -10°C and at 1 atm preessure are -1343
cal/m
mole and -4.91 cal/mole deg, respecttively. Compute ∆G for th
he phase chaange at this
temp
perature (-10
0°C = 263.2 K) and indica
ate whether the process is spontaneoous. Write
∆ G = -1343 -[26
63.2 × (-4.91)] = -51 cal/m
mole
= -213 joules
134
Dr. Murtadha Alshareifi e-Library
The process is spontaneous
s
, as reflected
d in the nega
ative value off ∆G.
Pre
essure an
nd Temp
perature C
Coefficients of Frree Energ
gy
By differentiating equation
e
(3-63
3), one obtainss several usefful relationship
ps between freee energy and
d the
pressure and temp
perature. Applying the differrential of a pro
oduct, d(uv) = u dv + V du, tto equation (3-63),
obtain the follo
owing relations
ship:
we o
N
Now, in a revversible pro
ocess in whiich qrev = T dS, the first law, resttricted to
exxpansion work
w
(i.e., dE
E = qrev - P dV), can be
b written as
a
and substituting dE
E of equation (3-74) into eq
quation (3-70) gives
or
onstant tempe
erature, the las
st term becom
mes zero, and equation (3-72
2) reduces to
At co
or
P.69
9
At co
onstant pressu
ure, the first te
erm on the righ
ht side of equa
ation (3-72) be
ecomes zero, and
or
To o
obtain the isoth
hermal change
e of free energ
gy, we integra
ate equation (3
3-73) betweenn states 1 and 2 at
consstant temperatture:
For a
an ideal gas, the
t volume V is equal to nR
RT/P, thus allow
wing the equa
ation to be inteegrated:
135
Dr. Murtadha Alshareifi e-Library
wherre ∆G is the frree energy cha
ange of an ide
eal gas underg
going an isothermal reversibble or irreversible
alterration.
Exa
ample 3-11
Wha
at is the free energy chan
nge when 1 m
mole of an id
deal gas is co
ompressed frrom 1 atm to
o 10
atm at 25°C? We
e write
The change in frree energy of a solute wh
hen the conc
centration is altered
a
is givven by the
equa
ation
in w
which n is the number of moles
m
of solu
ute and a1 an
nd a2 are the initial and finnal activities of
the ssolute, respe
ectively.
Exa
ample 3-12
Borssook and Winegarden1 ro
oughly comp
puted the free
e energy cha
ange when thhe kidneys
transfer various chemical constituents at body tempe
erature (37°C
C or 310.2 K)) from the blo
ood
plassma to the more concentrrated urine. T
The ratio of concentration
c
ns was assu med to be eq
qual
to th
he ratio of acctivities in equ
uation (3-79)). They found
d
The concentratio
on of urea in the plasma is 0.00500 mole/liter;
m
the
e concentratioon in the urin
ne is
0.33
33 mole/liter. Calculate th
he free energ
gy change in transporting
g 0.100 mole of urea from
m the
plassma to the urrine.
Thiss result mean
ns that 259 cal
c of work m
must be done on the syste
em, or this am
mount of net
workk must be pe
erformed by the
t kidneys tto bring abou
ut the transfe
er.
Fug
gacity
For a reversible isothermal proc
cess restricted
d to PV work, the
t Gibbs ene
ergy is describe
bed by equatio
on (373) o
or (3-74). Beca
ause V is alwa
ays a positive magnitude, th
hese relations indicate that aat constant
temp
perature the Gibbs
G
energy varies
v
proporti onal to the ch
hanges in P. Thus, from equuation (3-73) itt is
posssible to evalua
ate the Gibbs free
f
energy ch
hange for a pu
ure substance by integratingg between P°
and P:
136
Dr. Murtadha Alshareifi e-Library
For p
pure solids or liquids the volume has little
e dependence on pressure, so it can be aapproached as
s
consstant, and equation (3-80) is
s reduced to
On the other hand
d, gases have a very strong dependence on pressure; by
b applying thhe ideal gas
equa
ation V = nRT//P, we find tha
at equation (3--80) becomes
Rela
ations (3-81) and (3-82) can be simplified by first assum
ming P° = 1 atm
m as the referrence state. Th
hen,
divid
ding by the am
mount of substa
ance n gives a new property (G/n) called the molar Gibbbs
enerrgy or chemica
al potential, de
efined by the le
etter µ,
Thuss, for an ideal gas we can write
w
wherre the integrattion constant µ°
µ depends on
nly on the temperature and the nature of tthe gas and
repre
esents the che
emical potential of 1 mole o
of the substanc
ce in the refere
ence state whhere P° is equa
al to
1 atm
m. Note that P in equation (3
3-84) is a finite
e number and
d not a function
n. When a reaal gas does no
ot
beha
ave ideally, a function
f
known as the fugaccity (f) can be introduced to replace presssure, just as
activvities are introd
duced to repla
ace concentrattion in nonideal solutions (s
see later discuussion).
Equa
ation (3-84) be
ecomes
P.70
0
Op
pen Syste
ems
The systems conssidered so far have been clo
osed. They ex
xchange heat and
a work with their
surro
oundings, but the processes
s involve no trransfer of mattter, so that the
e amounts of tthe componen
nts of
the ssystem remain
n constant.
The term compone
ent should be clarified beforre proceeding
g. A phase con
nsisting of w2 ggrams of NaC
Cl
disso
olved in w1 gra
ams of water is
i said to conttain two indepe
endently varia
able masses oor two compon
nents.
Altho
ough the phasse contains the
e species Na+ , Cl-, (H2O)n, H3O+, OH-, and so on, they are not all
indep
pendently variable. Because H2O and its various speciies, H3O+, OH-, (H2O)n, and so on, are in
equillibrium, the mass m of wate
er alone is suffficient to spec
cify these spec
cies. All forms can be derive
ed
from the simple sp
pecies H2O. Similarly, all forrms of sodium
m chloride can be representeed by the single
speccies NaCl, and
d the system th
herefore conssists of just two
o components
s, H2O and NaaCl. The numb
ber of
comp
ponents of a system
s
is the smallest
s
numb
ber of indepen
ndently variable chemical suubstances that
mustt be specified to describe th
he phases qua
antitatively.
In an
n open system
m in which the exchange of m
matter among
g phases also must be consiidered, any on
ne of
the e
extensive prop
perties such as
s volume or frree energy bec
comes a functtion of temperrature, pressurre,
and the number off moles of the various comp
ponents.
Ch
hemical Potential
P
137
Dr. Murtadha Alshareifi e-Library
Let u
us consider the change in Gibbs
G
energy fo
for an open sy
ystem composed of a two-coomponent pha
ase
(bina
ary system). An
A infinitesimal reversible ch
hange of state is given by
The partial derivattives (∂G/∂n1)T,P,N2
and (∂G//∂n2)T,P,N2 can be identified as
a the chemicaal potentials (µ) of
T
the ccomponents n1 and n2, resp
pectively, so th
hat equation(3-86) is written more convenniently as
Now
w, relationshipss (3-74) and (3
3-76), (∂G/∂P))T = V and (∂G
G/∂T)P = -S, res
spectively, forr a closed systtem
also apply to an op
pen system, so
s we can write
e equation (3--87)as
ential, also kno
own as the pa
artial molar free energy, can be defined inn terms of othe
er
The chemical pote
nsive properties such as E, H, or A. Howe
ever, what is most
m
useful is the general ddefinition given
n for
exten
Gibb
bs energy: At constant
c
temp
perature and p
pressure, with the amounts of
o the other coomponents (nj)
held constant, the chemical pote
ential of a com
mponent i is eq
qual to the cha
ange in the freee energy brought
abou
ut by an infinite
esimal change
e in the numbe
er of moles ni of the component:
It ma
ay be considerred the change in free energ
rgy, for examp
ple, of an aque
eous sodium cchloride solutio
on
when
n 1 mole of Na
aCl is added to a large quan
ntity of the solution so that the
t compositioon does not
unde
ergo a measurrable change.
At co
onstant tempe
erature and pre
essure, the firs
rst two right-ha
and terms of equation
e
(3–888) become zerro,
and
or, in
n abbreviated notation,
whicch, upon integrration, gives
for a system of constant composition N = n1 + n2 + …. In eq
quation (3-92)), the sum of thhe right-hand
terms equals the total
t
free energ
gy of the syste
em at constan
nt pressure, temperature, annd composition.
Therrefore, µ1, µ2, …, µn can be considered ass the contributtions per mole
e of each compponent to the total
free energy. The chemical
c
poten
ntial, like any o
other partial molar
m
quantity,, is an intensivve property; in
n
other words, it is in
ndependent of the number o
of moles of the components
s of the system
m.
For a closed syste
em at equilibriu
um and consta
ant temperatu
ure and pressu
ure, the free ennergy change
e is
zero, dGT,P = 0, an
nd equation (3
3-91) becomess
all the phases of the overall system, which
h are closed.
for a
Equilibrium
m in a Hetterogene
eous Systtem
We b
begin with an example sugg
gested by Klottz.2 For a two--phase system
m consisting of
of, say, iodine
distriibuted betwee
en water and an
a organic pha
ase, the overa
all system is a closed one, w
whereas the
sepa
arate aqueouss and organic solutions
s
of io
odine are open
n. The chemica
al potential of iodine in the
aque
eous phase is written as µIw and that in th
he organic pha
ase as µIo. When the two phhases are in
equillibrium at consstant tempera
ature and presssure, the resp
pective free en
nergy changess dGwand dGo of
the two phases mu
ust be equal because
b
the frree energy of the
t overall sys
stem is zero. T
Therefore, the
e
chem
mical potentialls of iodine in both phases a
are identical. This
T
can be sh
hown by allow
wing an infinites
simal
amount of iodine to
t pass from th
he water to the
e organic phase, in which, at
a equilibrium,, according to
equa
ation (3-93),
P.71
138
Dr. Murtadha Alshareifi e-Library
Now
w, a decrease of
o iodine in the
e water is exa
actly equal to an
a increase of iodine in the oorganic phase
e:
Subsstituting equattion (3-95) into
o (3-94) gives
and ffinally
This conclusion may
m be generalized by statin g that the che
emical potentia
al of a componnent is identica
al in
all ph
hases of a hetterogeneous system
s
when tthe phases arre in equilibrium
m at a fixed teemperature an
nd
pressure. Hence,
wherre α, β, γ, … are
a various phases among w
which the subs
stance i is disttributed. For eexample, in a
saturrated aqueouss solution of sulfadiazine, th
he chemical po
otential of the drug in the soolid phase is th
he
same
e as its chemiical potential in the solution phase.
Whe
en two phases are not in equ
uilibrium at co
onstant temperrature and pre
essure, the tottal free energy
y of
the ssystem tends to
t decrease, and
a the substa
ance passes spontaneously
s
y from a phasee of higher
chem
mical potentiall to one of low
wer chemical p
potential until the potentials are
a equal. Hennce, the chem
mical
potential of a subsstance can be used as a me
easure of the escaping
e
tend
dency of the coomponent from
m its
phasse. The concept of escaping
g tendency willl be used in various
v
chapters throughout the book. The
e
analo
ogy between chemical
c
pote
ential and elecctric or gravitattional potentia
al is evident, thhe flow in thes
se
case
es always bein
ng from the hig
gher to the low
wer potential and
a continuing
g until all partss of the system
m are
at a uniform poten
ntial. For a pha
ase consisting
g of a single pu
ure substance
e, the chemicaal potential is the
free energy of the substance pe
er mole defined
d in equation (3-83).
For a two-phase system
s
of a sin
ngle compone
ent, for examplle, liquid water and water vaapor in equilib
brium
at co
onstant temperature and pre
essure, the mo
olar free energ
gy G/n is identical in all phaases. This
state
ement can be verified by combining equattions (3-98) an
nd (3-83).
Cla
ausius–C
Clapeyron
n Equatio
on
If the
e temperature and pressure
e of a two-phasse system of one
o component, for exampl e, of liquid wa
ater
(l) an
nd water vapo
or (v) in equilib
brium, are cha nged by a small amount, the molar free eenergy change
es
are e
equal and
In a phase change
e, the free ene
ergy changes ffor 1 mole of the
t liquid vapo
or are given byy equation (3--72),
Therrefore, from eq
quations (3-99
9) and (3-72),
or
w, at constant pressure,
p
the heat absorbed
d in the revers
sible process (equilibrium
(
coondition) is equal
Now
to the molar heat of
o vaporization
n, and from th e second law we have
es
Subsstituting equattion (3-101) into (3-100) give
wherre ∆V = Vv - Vl, the differenc
ce in the mola
ar volumes in the
t two phases. This is the Clapeyron
equa
ation.
139
Dr. Murtadha Alshareifi e-Library
The vapor will obe
ey the ideal ga
as law to a goo
od approximattion when the temperature iis far enough away
from the critical po
oint, so that Vv may be repla
aced by RT/P. Furthermore,, Vl is insignificcant compared
with Vv. In the case of water at 100°C,
1
for exa
ample, Vv = 30
0.2 liters and Vl = 0.0188 liteer.
Unde
er these restrictive circumsttances, equatiion (3-102) be
ecomes
whicch is known ass the Clausius–
–Clapeyron eq
quation. It can
n be integrated
d between thee limits of the vapor
v
pressures P1 and P2 and corres
sponding temp
peratures T1an
nd T2, assumin
ng ∆Hv is consstant over the
temp
perature range
e considered:
and ffinally
or
ate the mean h
heat of vaporiz
zation of a liqu
uid if its vapor pressure at tw
wo
This equation is ussed to calcula
peratures is avvailable. Conv
versely, if the m
mean heat of vaporization
v
and
a the vapor pressure at one
temp
temp
perature are known, the vap
por pressure a
at another tem
mperature can be obtained.
The Clapeyron and Clausius–Clapeyron equa
ations are imp
portant in the study
s
of variouus phase
transsitions and in the
t developme
ent of the equ
uations of som
me colligative properties.
p
P.72
2
Exa
ample 3-13
The average hea
at of vaporiza
ation of wate
er can be tak
ken as about 9800 cal/moole within the
e
rang
ge of 20°C to
o 100°C. Wha
at is the vapo
or pressure at
a 95°C? The
e vapor presssure P2 at
temp
perature T2 = 373 K (100
0°C) is 78 cm
m Hg, and R is expressed
d as 1.987 caal/deg mole.
Writte
Activities: Activity
A
Coefficien
C
nts
If the
e vapor above
e a solution can be considerred to behave ideally, the ch
hemical potenttial of the solv
vent
in the
e vapor state in equilibrium with the soluttion can be wrritten in the forrm of equationn (3-84),
If Ra
aoult's law is now
n
introduced
d for the solve
ent, P1 = P1° X1, equation (3-84) becomess
140
Dr. Murtadha Alshareifi e-Library
Com
mbining the firsst and second right-hand terrms into a sing
gle constant gives
for a
an ideal solutio
on. The reference state µ° iss equal to the chemical pote
ential µ1 of thee pure solvent
(i.e., X1 = 1). For nonideal
n
solutiions, equation
n (3-109) is mo
odified by intro
oducing the “eeffective
a
of the solvent to rep
place the mole fraction:
conccentration” or activity
or, fo
or
and γ is referred to
o as the activiity coefficient, we have
t mole fraction scale,
For tthe solute on the
Base
ed on the pracctical (molal an
nd molar) scalles
Equa
ations (3-110) and (3-113) are
a frequently used as defin
nitions of activity.
Gib
bbs–Helm
mholtz Eq
quation
For a
an isothermal process at co
onstant pressu
ure proceeding
g between the initial and finaal states 1 and
d 2,
equa
ation (3-57) yie
elds
Now
w, equation (3-76) may be written as
or
es
Subsstituting equattion (3-118) into (3-119) give
whicch is one form of the Gibbs–
–Helmholtz eq uation
Sta
andard Free
Fr Enerrgy and th
he Equiliibrium Co
onstant
Many of the proce
esses of pharm
maceutical inte
erest such as complexation,, protein bindinng, the
disso
ociation of a weak
w
electrolytte, or the distrribution of a drrug between tw
wo immisciblee phases are
syste
ems at equilibrium and can be described in terms of ch
hanges of the Gibbs free en ergy (∆G).
Conssider a closed
d system at constant pressu re and temperature, such as
a the chemicaal reaction
Beca
ause G is a sta
ate function, the free energyy change of th
he reaction going from reacttants to products is
Equa
ation (3-120) represents
r
a closed
c
system made up of several
s
compo
onents. Thereffore, at
consstant T and P the
t total free energy
e
change
e of the produ
ucts and reacta
ants in equatioon (3-121) is given
g
as th
he sum of the chemical pote
ential µ of each
h component times the num
mber of moles [see equation
n (383)]:
141
Dr. Murtadha Alshareifi e-Library
Whe
en the reactantts and produc
cts are ideal ga
ases, the chem
mical potential of each compponent is
exprressed in terms of partial pre
essure [equatiion (3-84)]. Fo
or nonideal gases, µ is writteen in terms of
fugacities [equatio
on (3-85)]. The
e correspondin
ng expressions for solutions
s are given by equations (3-110)) to (3-116). Le
et us use the more
m
general expression that relates the chemical poteential to the
activvity, equation
P.73
3
(3-11
10). Substituting this equatio
on for each co
omponent in equation
e
(3-121) yields
Rearrranging equa
ation (3-123) gives
µ° is the partial mo
olar free energ
gy change or cchemical pote
ential under sta
andard conditiions. Because
e it is
multiiplied by the number
n
of moles in equation
n (3-124), the algebraic sum
m of the terms involving µ°
repre
esents the tota
al standard fre
ee energy cha
ange of the rea
action and is called
c
∆G°:
or, in
n general,
Using the rules of logarithms, we
w can expresss equation (3-124) as
ckets are calle
ed the reaction
n quotients, de
efined as
The products of acctivities in brac
or, in
n general,
Thuss, equation (3--127) can be written
w
as
Beca
ause ∆G° is a constant at co
onstant P and
d constant T, RT
R is also constant. The conndition for
equillibrium is ∆G = 0, and there
efore equation (3-130) becomes
or
n replaced by K, the equilibrrium constant..
wherre Q has been
Equa
ation (3-132) is a very important expressi on, relating th
he standard fre
ee energy chaange of a reaction
∆G° to the equilibrrium constant K. This expre
ession allows one
o to computte K knowing ∆
∆G° and vice
a.
versa
The equilibrium co
onstant has be
een expressed
d in terms of activities.
a
It als
so can be giveen as the ratio of
partial pressures or
o fugacities (ffor gases) and
d as the ratio of
o the differentt concentrationn expressions
s
used
d in solutions (mole
(
fraction,, molarity, mollality). The equilibrium cons
stant is dimenssionless, the ratio
r
of acctivities or con
ncentration can
nceling the un
nits. However, the numerical value of K difffers dependin
ng on
the u
units used (acttivity, mole fra
action, fugacityy, etc.).
Exa
ample 3-14
142
Dr. Murtadha Alshareifi e-Library
Deriive an expresssion for the free energie
es ∆G and ∆G° of the rea
action
Because the che
emical potential of a solid
d is constant (it does not depend on cconcentration
n),
the e
equilibrium constant
c
depends only on
n the pressurres (or fugac
cities) of the ggases. Using
g
presssures, we obtain
and
The magnitude an
nd sign of ∆G° indicate whetther the reactiion is spontaneous [see equuation (3-59)], but
action is not att equilibrium, ∆G
∆ ≠ 0 and thhe free energy
y
only under standard conditions. When the rea
chan
nge is describe
ed by equation
n (3-130), whe
ere Q, like K, is
i the ratio of activities
a
[equ ation (3-129)],
fugacities, or conccentration units
s of the produ
ucts and reacta
ants but underr different connditions than th
hose
of eq
quilibrium. Q should
s
not be confused
c
with
h K, the ratio of
o activities, fug
gacities, and sso on
unde
er standard co
onditions at eq
quilibrium.
Exa
ample 3-15
Sodium cholate is a bile salt that plays an
n important role
r
in the dis
ssolution or ddispersion off
chollesterol and other
o
lipids in
n the body. S
Sodium chola
ate may exis
st either as m
monomer or as
a
dime
er (or higher n-mers) in aqueous
a
solu
ution. Let us consider the equilibrium monomer–dimer
reacction3:
whicch states tha
at two moles (or molecule
es) of monom
mer form 1 mole (or moleccule) of dime
er.
(a) If the molar concentration
c
n at 25°C of m
monomeric species
s
is 4 × 10-3 mole/l iter and the
conccentration off dimers is 3.52 × 10-5 mo
ole/liter, whatt is the equiliibrium constaant and the
stan
ndard free en
nergy for the dimerization
n process? Write
W
The process is spontaneous
s
under stand
dard conditions.
(b) W
While keepin
ng the concen
ntration of m
monomer constant, suppose that one iis able to rem
move
part of the dimerric species by physical orr chemical means
m
so thatt its concentrration is now
w four
time
es less than the
t original dimer
d
concen
ntration. Com
mpute the free energy chaange. What is
i
the e
effect on the
e equilibrium?
?
The concentratio
on of dimer is
s now
P.74
4
Because the con
nditions are not
n at equilib
brium, equation (3-130) should be useed. First
calcculate Q:
143
Dr. Murtadha Alshareifi e-Library
and from equatio
on (3-130),
∆G is negative, Q is less than K, and the
e reaction shiifts to the right side of thee equation with
w
the fformation of more dimer.
If the
e monomer is
i removed from the solu
ution, the reaction is shifte
ed to the left side, forming
mon
nomer, and ∆G
∆ becomes positive. Su
uppose that monomer
m
con
ncentration iss now 1 × 10
03
-5
mo
ole/liter and dimer
d
concen
ntration is 3.5
52 × 10 mo
ole/liter:
∆ G = -466.6 + (1.9872
(
× 298 × 3.561) = +1642 cal/m
mole
The positive sign
n of ∆G indic
cates that the
e reaction do
oes not proce
eed forward sspontaneous
sly.
The
e van't Hoff
H
Equa
ation
The effect of temp
perature on eq
quilibrium consstants is obtain
ned by writing the equation
and differentiating with respect to
t temperature
e to give
The Gibbs–Helmh
holtz equation may be writte
en in the form (see one of the thermodynaamics texts cited in
the R
Recommended Readings se
ection at the e
end of this cha
apter)
Exprressing equation (3-135) in a form for the reactants and
d products in their
t
standard states, in which
∆G b
becomes equa
al to ∆G°, and substituting in
nto equation (3-134)
(
yields
wherre ∆H° is the standard
s
enthalpy of reactio
on. Equation (3
3-136) is know
wn as the van''t Hoff equatio
on. It
may be integrated
d, assuming ∆H
H° to be consttant over the temperature
t
ra
ange considerred; it become
es
ation (3-137) allows
a
one to compute
c
the e
enthalpy of a reaction
r
if the equilibrium coonstants
Equa
at T1 and T2 are avvailable. Conv
versely, it can be used to su
upply the equilibrium constaant at a definite
e
temp
perature if it is known at ano
other temperatture. Because
e ∆H° varies with
w temperatuure and
equa
ation (3-137) gives
g
only an approximate
a
a
answer, more elaborate equ
uations are reqquired to obtaiin
accu
urate results. The
T solubility of
o a solid in an
n ideal solution
n is a special type of equilibbrium, and it is
s not
surprising that the solubility can be written ass
whicch closely rese
embles equatio
on (3-137). Th
hese expressio
ons will be encountered in l ater chapters..
Com
mbining equatio
ons (3-117) an
nd (3-133) yie
elds yet anothe
er form of the van't Hoff equuation, namely
y
or
e intercept on the ln K axis of a plot of ln K versus 1/T.
wherre ∆S°/R is the
144
Dr. Murtadha Alshareifi e-Library
Whereas equation (3-137) provides a value of ∆H° based on the use of two K values at their
corresponding absolute temperatures T1 and T2, equations (3-139) and(3-140) give the values of ∆H°
and ∆S°, and therefore the value of ∆G° = ∆H° - T∆S°. In the least squares linear regression
equations (3-139) and (3-140), one uses as many ln K and corresponding 1/T values as are available
from experiment.
Example 3-16
In a study of the transport of pilocarpine across the corneal membrane of the eye, Mitra and
Mikkelson4 presented a van't Hoff plot of the log of the ionization constant, Ka, of pilocarpine
versus the reciprocal of the absolute temperature, T-1 = 1/T.
Using the data in Table 3-4, regress Ka versus T-1. With reference to the van't Hoff equation,
equation (3-139), obtain the standard heat (enthalpy), ∆H°, of ionization for pilocarpine and
the standard entropy for the ionization process. From ∆H° and ∆S° calculate ∆G° at 25°C.
What is the significance of the signs and the magnitudes of ∆H°, ∆S°, and ∆G°?
Table 3-4 Ionization Constants of Pilocarpine at Various Temperatures*†
T(°C)
T(K)
1/T × 103
Ka × 107
log Ka
15
288
3.47
0.74
-7.13
20
293
3.41
1.07
-6.97
25
298
3.35
1.26
-6.90
30
303
3.30
1.58
-6.80
35
308
3.24
2.14
-6.67
40
313
3.19
2.95
-6.53
45
318
3.14
3.98
-6.40
*For the column headed 1/T × 103 the numbers are 1000 (i.e., 103)
times larger than the actual numbers. Thus, the first entry in column 3 has
the value 3.47 × 10-3 or 0.00347. Likewise, in the next column Ka ×
107 signifies that the number 0.74 and the other entries in this column are to
be accompanied by the exponential value 10-7, not 10+7. Thus, the first value
in the fourth column should be read as 0.74 × 10-7 and the last value 3.98 ×
10-7.
†
From A. K. Mitra and T. J. Mikkelson, J. Pharm. Sci. 77, 772, 1988. With
permission.
Answers:
145
Dr. Murtadha Alshareifi e-Library
P.75
5
Thesse thermodyna
amic values have the follow
wing significance. ∆H° is a la
arge positive vvalue that indic
cates
that the ionization of pilocarpine
e (as its conjug
gate acid) sho
ould increase as
a the temperaature is elevatted.
The increasing values of Ka in th
he table show
w this to be a fa
act. The stand
dard entropy inncrease, ∆S°=
=
1.30 entropy units, although small, provides a force for the reaction of the pilocarpinium
m ions to form
m
piloccarpine (Fig. 3-7). The positively charged pilocarpine molecules,
m
because of their ioonic nature, are
a
prob
bably held in a more orderly arrangement than the pred
dominantly non
nionic pilocarppine in the aqu
ueous
envirronment. This increase in disorder in the dissociation process
p
accounts for the inccreased entrop
py,
whicch, however, iss a small value
e: ∆S° = 1.30 entropy units.. Note that a positive
p
∆H° dooes not mean that
the io
onization will not
n occur; rath
her, it signifiess that the equilibrium consta
ant for the forw
ward reaction
-7
(ionizzation) will have a small vallue, say Ka [co
ongruent] 1 × 10 , as obserrved in Table 33-4. A further
expla
anation regard
ding the sign of
o ∆H° is helpfful here. Maha
an5 pointed ou
ut that in the fiirst stage of
ionizzation of phosp
phoric acid, fo
or example,
the h
hydration reacction of the ion
ns being bound
d to the water molecules is sufficiently exxothermic to
produce the necesssary energy for
f ionization, that is, enoug
gh energy to re
emove the prooton from the acid,
a
H3PO
O4. For this re
eason, ∆H° in the
t first stage of ionization is
i negative and K1 = 7.5 × 110-3 at 25°C. In
n the
seco
ond stage,
Fig.. 3-7. Reacttion of piloccarpinium ioon to yield pilocarpine
p
base.
∆H° is now positivve, the reaction
n is endotherm
mic, and K2 = 6.2 × 10-8. Fin
nally, in the thiird stage,
∆H° is a relatively large positive
e value and K3 = 2.1 × 10-13. These ∆H° and
a Ka values show that
incre
easing energy is needed to remove the po
ositively charg
ged proton as the negative ccharge increases
in the
e acid from the first to the th
hird stage of io
onization. Pos
sitive ∆H° (end
dothermic reacction) values do
d
146
Dr. Murtadha Alshareifi e-Library
not signal nonionization of the acid—that is, that the process is nonspontaneous—but rather simply
show that the forward reaction, represented by its ionization constant, becomes smaller and smaller.
Chapter Summary
In this chapter, the quantitative relationships among different forms of energy were reviewed
and expressed in the three laws of thermodynamics. Energy can be considered as the
product of an intensity factor and a capacity factor; thus, the various types of energy may be
represented as a product of an intensive property (i.e., independent of the quantity of
material) and the differential of an extensive property that is proportional to the mass of the
system. For example, mechanical work done by a gas on its surroundings is P dV. Some of
the forms of energy, together with these factors and their accompanying units, are given
in Table 3-5.
Table 3-5 Intensity and Capacity Factors of Energy
Intensity or Potential
Factor
Energy Form (Intensive Property)
Capacity or
Energy Unit
Quantity Factor
Commonly
(Extensive Property) Used
Heat
(thermal)
Temperature (deg)
Entropy change
(cal/deg)
Calories
Expansion
Pressure (dyne/cm2)
Volume change
(cm3)
Ergs
Surface
Surface tension
(dyne/cm)
Area change
(cm2)
Ergs
Electric
Electromotive force or
potential difference
(volts)
Quantity of
electricity
(coulombs)
Joules
Chemical
Chemical potential
(cal/mole)
Number of moles
Calories
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
P.76
References
1. H. Borsook and H. M. Winegarden, Proc. Natl. Acad. Sci. USA 17, 3, 1931.
1. J. M. Klotz, Chemical Thermodynamics, Prentice Hall, Englewood Cliffs, N.J., 1950, p. 226.
3. M. Vadnere and S. Lindenbaum, J. Pharm. Sci. 71, 875, 1982.
4. A. K. Mitra and T. J. Mikkelson, J. Pharm. Sci. 77, 771, 1988.
5. B. H. Mahan, Elementary Chemical Thermodynamics, Benjamin/ Cummings, Menlo Park, Calif.,
1963, pp. 139–142.
Recommended Readings
147
Dr. Murtadha Alshareifi e-Library
W. Atkins and J. W. Locke, Atkins's Physical Chemistry, 7th Ed., Oxford University Press, Oxford, 2002.
G. W. Castellan, Physical Chemistry, 3rd Ed., Addison-Wesley, New York, 1983.
R. Chang, Physical Chemistry for the Chemical and Biological Sciences, 3rd Ed., University Science
Books, Sausalito, Calif., 2000.
I. M. Klotz and R. M. Rosenberg, Chemical Thermodynamics: Basic Theory and Methods, 6th Ed.,
Wiley-Interscience, New York, 2000.
I. Tinoco, K. Sauer, J. C. Wang, and J. D. Puglisi, Physical Chemistry: Principles and Applications in
Biological Sciences, 4th Ed., Prentice Hall, Englewood Cliffs, N.J., 2001.
Chapter Legacy
Fifth Edition: published as Chapter 3. Updated by Gregory Knipp, Hugo Morales-Rojas, and
Dea Herrera-Ruiz.
Sixth Edition: published as Chapter 3. Updated by Patrick Sinko.
148
Dr. Murtadha Alshareifi e-Library
4 Determination of the Physical Properties of
Molecules
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Understand the nature of intra- and intermolecular forces that are involved in
stabilizing molecular and physical structures.
2. Understand the differences in the energetics of these forces and their relevance to
different molecules.
3. Understand the differences in energies between the vibrational, translational, and
rotational levels and define their meaning.
4. Understand the differences between atomic and molecular spectroscopic techniques
and the information they provide.
5. Appreciate the differences in the strengths of selected spectroscopic techniques used
in the identification and detection of pharmaceutical agents.
6. Define the electromagnetic radiation spectrum in terms of wavelength, wave number,
frequency, and the energy associated with each range.
7. Define and understand the relationships between atomic and molecular forces and
their response to electromagnetic energy sources.
8. Define and understand ultraviolet and visible light spectroscopy in terms of electronic
structure.
9. Define and understand fluorescence and phosphorescence in terms of electronic
structure.
10. Understand electron and nuclear precession in atoms subjected to electromagnetic
radiation and its role in the determination of atomic structure in a molecule.
11. Understand polarization of light beams and the ability to use polarized light to study
chiral molecules.
12. Understand fundamental principles of refraction of electron and neutron beams and
how these beams are used to determine molecular properties.
Molecular Structure, Energy, and Resulting Physical Properties
An atom consists of a nucleus, made up of neutrons (neutral in charge) and protons (positively
charged), with each particle carrying a weight of approximately 1 g/mole. In addition, electrons
(negatively charged) exist in atomic orbits surrounding the nucleus and have a significantly lower weight.
Charged atoms arise from an imbalance in the number of electrons and protons and can lead to ionic
interactions (discussed in Chapter 2). The atomic mass is derived from counting the number of protons
and neutrons in a nucleus. Isotopes may also exist for a given type of atom. For example, carbon has an
atomic number of 6, which describes the number of protons, and there are several carbon isotopes with
different numbers of neutrons in the nucleus: 11C (with five neutrons), 12C (with six neutrons), 13C (with
seven neutrons), 14C (with eight neutrons), and 15C (with nine neutrons).1 Carbon-13, 13C, is a common
isotope used in nuclear magnetic resonance (NMR) and kinetic isotope effect studies on rates of
reaction, and 14C is radioactive and used as a tracer for studies that require high sensitivity and for
carbon dating. Both11C and 15C are very short-lived, having half-lives of 20.3 min and 2.5 sec,
respectively, and are not used in practical applications.
Molecules arise when interatomic bonding occurs. The molecular structure is reflected by the array of
atoms within a molecule and is held together by bonding energy, which relies heavily on electron orbital
orientation and overlap. This is illustrated in the Atomic Structure and Bonding Key Concept Box. Each
bond in a complex molecule has an intrinsic energy and will have different properties, such as reactivity.
The properties within a molecule depend on intramolecular interactions, and each molecule will possess
a net energy of bonding that is defined by its unique composition of atoms. It is also important to note
149
Dr. Murtadha Alshareifi e-Library
that for macromole
ecules like pro
oteins or synth
hetic polymers
s, the presence of neighboriing charges orr
dipolle interactionss, steric hindra
ance and repu lsion, Debye forces,
f
and so
o on, within thee backbone orr
functtional groups of the molecule can also givve rise to disto
ortions in the primary
p
intram
molecular bond
ding
enerrgies and can even give rise
e to additional covalent bond
ding, for exam
mple, disulfide bridges in a
prote
ein like insulin.
The more bonding
g that an atom participates i n, the lower is
s the density of
o the electron cloud around the
nucle
eus and the greater is the shared
s
distribu
ution of atomic
c electrons in the
t interatomicc bonded space.
For e
example, beca
ause of differe
ences in atomiic properties such
s
as electro
onegativity, th e nature of ea
ach
interratomic bond can
c vary greattly. Here, the ““nature” refers
s to the energy
y of the bond, the rotation
around the bond, the
t vibration and
a motion of atoms in the bond,
b
the rotation of a nucleeus of an atom
m in a
bond
d, and so on.
As a
an example of the nature of a bond, consid
der the carbon
nyl–amide bon
nd between tw
wo amino acids in
a pro
otein
P.78
8
(Fig. 4-1), as desccribed by Math
hews and van Holde.3 The oxygen
o
in the carbonyl is veery electroneg
gative
and pulls electronss toward itselff. The lone paiir of electrons on the nitroge
en remains unnbound, but
nitrogen has a con
nsiderably low
wer electronega
ativity than ox
xygen. Recall that
t
the covaleent bond occu
urs
betw
ween the carbo
on (which also
o has a low ele
ectronegativity
y) in the carbo
onyl group andd the amine off the
next amino acid. However,
H
the electrons
e
in th
he carbon–oxy
ygen double bo
ond in the carrbonyl are pulled
much
h closer to the
e oxygen due to its greater e
electronegativ
vity, causing a partially negaatively charged
d
oxyg
gen. The carbo
on then carries
s a partial possitive charge and
a pulls the lo
one pair of eleectrons from th
he
nitrogen. This form
ms a partial do
ouble bond, wh
hich causes th
he nitrogen to carry a partiaal positive charrge,
yield
ding a net dipo
ole (see Perma
anent Dipole M
Moment of Polar Molecules section). Interrestingly, in
exten
nded seconda
ary structures, in particular α
α-helices, the effective dipole of the peptiide bond can have
h
a strrongly stabilizing effect on secondary stru
ucture.4 This reduces rotatio
on around the carbon–nitrog
gen
bond
d, making onlyycis/trans isom
mers allowable
e. Although thiis is not discus
ssed here, thee student shou
uld
recall discussions about energe
etically favored
d bond rotation
ns in organic chemistry.
c
Thee transpeptide
e
bond
d orientation iss energetically
y preferred, wi th the cis confformer appearring in only sppecialized case
es, in
particular, around the conformationally constrrained proline. As discussed
d in general biiochemistry
courrses, the secon
ndary structurre of a peptide
e is then determined by rotation around thheφ and Ψ ang
gles
only.. Finally, amin
no acids natura
ally occur in th
he L-conforme
er, with the exc
ception of glyccine. This chirrality
is folllowed in almo
ost all peptides
s and proteinss, resulting in a profound efffect on functioon. An example of
chira
ality with an L-- and a D-amin
no acid is pressented in the Key
K Concept Box
B Chirality oof Amino Acids as
a reffresher.
Key Co
oncept
Atomic Structu
ure and Bon
nding
150
Dr. Murtadha Alshareifi e-Library
Con
nsider two ca
arbon atoms that
t
are not b
bound. Each
h nucleus, ca
arrying six prootons (6P) and
six n
neutrons (6n) and a net positive
p
charg
ge, is surrounded by a fa
airly uniform, six-electron (6e)
clou
ud. Consider what happen
ns when the two carbons
s form a cova
alent bond. T
Two electrons,
one from each carbon,
c
fill the
e hybrid orbittal space between the attoms. The nuucleus no lon
nger
is su
urrounded byy a uniform cloud
c
of equa
al charge. Th
he nucleus is now surrounnded closely
y by
five electrons an
nd has an ele
ectron that iss shared in th
he bond. The
e charge distrribution arou
und
each
h nucleus is similar to tha
at of the atom
m alone, yet its shape is different.
d
As we will see in
this chapter, the difference in
n this distribu
ution relates to the physic
cal property oof the molec
cule.
Whe
en the atomss alone or in the molecule
e are expose
ed to external energy fieldds, they will
beha
ave differenttly. Therefore
e, changes in
n atomic orde
er when mole
ecular bondi ng occurs ca
an
be m
measured byy different tec
chniques usin
ng different levels of exte
ernal energy..
Key Co
oncept
Chirality of Am
mino Acids
All α
α-amino acids, with the ex
xception of g
glycine, have
e a chiral α-carbon. The LL-conformer is
prefferred, yet the
e D-conformer is also fou
und. To dete
ermine the co
onformer, Ricchardson2 an
nd
othe
ers proposed
d the “corn crrib” structure . In connection with the figure shown here, if one
hold
ds the α-helixx in front of one's
o
eyes an
nd looks dow
wn the bond with
w the α-caarbon, readin
ng
clocckwise around the α-carbon, one seess the word “C
CORN” pointing into the ppage for the Lamin
no acid confo
ormer.
151
Dr. Murtadha Alshareifi e-Library
Additive an
nd Constiitutive Prroperties
s
A stu
udy of the phyysical propertie
es of drug mollecules is a prrerequisite for product formuulation and lea
ads
to a better understtanding of the relationship b
between a dru
ug's molecular
P.79
9
and physicochemiical properties
s and its structture and action
n. These prop
perties come frrom the molec
cular
bond
ding order of th
he atoms in th
he molecule an
nd may be tho
ought of as either additive (dderived from the
sum of the propertties of the indiividual atoms or functional groups
g
within the molecule))
or co
onstitutive (dependent on th
he structural arrrangement off the atoms within the moleccule). For
exam
mple, mass is an additive prroperty (the m olecular weigh
ht is the sum of
o the massess of each atom
m in
the m
molecule), whe
ereas optical rotation
r
may b
be thought of as
a a constitutive property beecause it depe
ends
on th
he chirality of the
t molecule, which is dete rmined by ato
omic bonding order.
o
Fig.. 4-1. The nature
n
of an amide bondd within a peptide/prote
p
ein and its eeffect on
152
Dr. Murtadha Alshareifi e-Library
struucture. As mentioned
m
in
n the text, thhetransisom
mer of a pepttide is energgetically
favoored due to steric hindeerance of thhe amino aciids. Proline (tertiary am
mine group)) and
glyccine (two hyydrogens, liittle steric hhinderence from
f
the R group)
g
can m
more readily
y
adappt a transisoomer conforrmation in a peptide orr protein thaan can the oother amino
acidds.
Many physical pro
operties are co
onstitutive and
d yet have som
me measure off additivity. Thhe molar refrac
ction
of a compound, fo
or example, is the sum of the
e refraction off the atoms and the array off functional gro
oups
makiing up the com
mpound. Beca
ause the arran
ngements of attoms in each group
g
are diffeerent, the
refra
active index off two molecule
es will also be different. That is, the individ
dual groups inn two different
mole
ecules contribu
ute different amounts to the
e overall refrac
ction of the mo
olecules.
A sa
ample calculatiion will clarify the principle o
of additivity an
nd constitutivity. The molar rrefractions of the
two ccompounds
and
whicch have exactlyy the same nu
umber of carbo
on, hydrogen, and oxygen atoms,
a
are callculated
using
g Table 4-1. We
W obtain the following resu
ults:
Thuss, although the
ese two compounds have th
he same numb
ber and type of
o atoms, theirr molar refractions
are d
different. The molar refractio
ons of the atom
ms are additiv
ve, but the carrbon and oxyggen atoms are
consstitutive in refra
action. Refrac
ction of a singl e-bonded carbon does not add equally too that of a dou
ublebond
ded carbon, an
nd a carbonyl oxygen (C–O
O) is not the sa
ame as a hydro
oxyl oxygen; ttherefore, the two
comp
pounds exhibiit additive–con
nstitutive prop
perties and hav
ve different mo
olar refractionns. These will be
b
more
e comprehenssively discusse
ed in this chap
pter and in oth
her sections off the book.
Mole
ecular perturba
ations result frrom external ssources that exert energy on
n molecules. A
Actually, every
y
mole
ecule that exissts does so under atmosphe
eric conditions
s provided thatt ideality is exccluded. As suc
ch, it
existts in a perturbed natural sta
ate. Hence, ph ysical propertties encompas
ss the specificc relations betw
ween
the a
atoms in molecules and well-defined form
ms of energy or other external “yardsticks”” of measurem
ment.
For
P.80
0
exam
mple, the conccept of weight uses the force
e of gravity as
s an external measure
m
to coompare the ma
ass of
objeccts, whereas that
t
of optical rotation uses plane-polarize
ed light to des
scribe the opticcal rotation of
mole
ecules organizzed in a particu
ular bonding p
pattern. Therefore, a physical property shhould be easily
y
153
Dr. Murtadha Alshareifi e-Library
measured or calculated, and such measurements should be reproducible for an individual molecule
under optimal circumstances.
Table 4-1 Atomic and Group Contributions to Molar Refraction*
C— (single)
2.418
—C== (double)
1.733
—C[triple bond] (triple)
2.398
Phenyl (C6H5)
25.463
H
1.100
O (C==O)
2.211
O (O—H)
1.525
O (ether, ester, C—O)
1.643
Cl
5.967
Br
8.865
L
13.900
*These values are reported for the D line of sodium as the light source. From
J. Dean (Ed.), Lange's Handbook, 12th Ed. McGraw-Hill, New York, 1979,
pp. 10–94. See also Bower et al., in A. Weissberger (Ed.), Physical Methods
of Organic Chemistry, Vol. 1, Part II, 3rd Ed., Wiley-Interscience, New
York, 1960, Chapter 28.
When one carefully associates specific physical properties with the chemical nature of closely related
molecules, one can (a) describe the spatial arrangement of atoms in drug molecules, (b) provide
evidence for the relative chemical or physical behavior of a molecule, and (c) suggest methods for the
qualitative and quantitative analysis of a particular pharmaceutical agent. The first and second of these
often lead to implications about the chemical nature and potential action that are necessary for the
creation of new molecules with selective pharmacologic activity. The third provides the researcher with
tools for drug design and manufacturing and offers the analyst a wide range of methods for assessing
the quality of drug products. The level of understanding of the physical properties of molecules has
154
Dr. Murtadha Alshareifi e-Library
expa
anded greatly and, with incre
easing advancces in technology, the deve
elopment of coomputer-based
d
comp
putational tools to develop molecules
m
with
h ideal physic
cal properties has
h become ccommonplace..
Now
where has the impact
i
of com
mputational mo
odeling been of
o broader imp
portance than in drug discov
very
and design, where
e computational tools offer g
great promise for enhancing
g the speed off drug design and
a
selecction. For example, computer modeling o
of therapeutic targets
t
(e.g., HIV
H protease) is widely utiliz
zed.
This approach is based
b
on the screening
s
of kknown or prediicted physical properties (e..g., the protein
n
confo
ormation of th
he HIV proteas
se) of a therap
peutic target in
n a computer to
t elucidate arrea(s) where a
mole
ecule can be synthesized
s
to
o optimally bind
d and either block
b
(antagonist) or enhancce (agonist) the
actio
ons of the target. This type of
o modeling ussually reveals a limited num
mber of atomic spatial
arran
ngements and
d types, such as
a a functiona
al group, that can
c be utilized in the designn of a molecule
e to
elicitt the desired pharmacologic
p
c response. Otther computattional program
ms can then bee utilized to
comp
pute the numb
ber of moleculles that might make reasonable drugs based on these molecular
restrrictions.
It is iimportant to note that these
e models are re
relatively new but offer great potential for the future of drug
d
disco
overy and dessign. It is also important to n
note that all of these computtational approoaches are bas
sed
solelly on the physsical properties
s of molecule((s) that have been
b
determined through yeears of researc
ch.
The ability to desig
gn and test the
ese moleculess using a computer and to rapidly synthessize chemical
librarries has also dramatically
d
in
ncreased the n
need for techn
nologically sup
perior equipmeent to measure the
physsical propertiess of these molecules to con
nfirm that the actual
a
properties match thosse that are
predicted. Toward
d that end, che
emical librariess can reach well
w over 10,000 individual m
molecules and can
be te
ested for nume
erous molecullar properties in modern hig
gh-throughput systems. In faact, “virtual”
librarries routinely contain
c
million
ns of compoun
nds. This aids in the ability of
o scientists too select one le
ead
comp
pound whose potential to make
m
it into clin
nical use is gre
eater than all of the other m
molecules
scree
ened. Finally, the ability to measure
m
the p
physical prope
erties of a lead
d compound iss critical to
assu
uring that durin
ng the transitio
on from compu
uter to large-s
scale manufac
cturing, the inittially identified
d
mole
ecule remains physically the
e same. This iss a guideline that
t
the Food and Drug Adm
ministration
requires to ensure
e human safety
y and drug effficacy.
The remainder of this
t
chapter describes some
e of the well-d
defined interac
ctions that aree important for
determining the ph
hysical properrties of molecu
ules. All of the quantities pre
esented are exxpressed in
Système Internatio
onal (SI) units
s for all practiccal cases.
Die
electric Constant
C
and Indu
uced Pola
arization
Electricity is relate
ed to the naturre of charges iin a dynamic state
s
(e.g., ele
ectron flow). W
When a charge
ed
mole
ecule is at restt, its properties
s are defined by electrostattics. For two ch
harges separaated by a
dista
ance r, their po
otential energy
y is defined byy Coulomb's la
aw,
wherre the chargess q1 and q2 are
e in coulombss (C), r is in me
eters, ε0 (the permittivity
p
connstant) = 8.854 ×
-12
2
-1
-2
10 C2 N m , and
a the potential energy is in
n joules. Coulo
omb's law can
n be used to ddescribe both
attra
active and repu
ulsive interactions. From eq
quation (4-1), it is clear that electrostatic innteractions als
so
rely o
on the permitttivity of the me
edium in which
h the charges exist. Differen
nt solvents havve differing
perm
mittivities due to
t their chemic
cal nature, po lar or nonpola
ar. For a more comprehensivve discussion, see
Berg
gethon and Sim
mons.5
Whe
en no permane
ent charges ex
xist in moleculles, a measure of their polarity (i.e., the eelectronic
distriibution within the molecule) is given by th
he property we
e called the dip
pole moment (µ). In the sim
mplest
desccription, a dipo
ole is a separa
ation of two op
pposing charge
es over a dista
ance r (Fig. 4--2) and is gene
erally
desccribed by a vecctor whose ma
agnitude is givven by µ = qr. Dipoles do no
ot have a net ccharge, but this
charrge separation
n can often cre
eate chargelike
e interactions and influence
e several physsical and chem
mical
prop
perties. In com
mplex molecule
es, the electro n distribution can be approx
ximated so thaat overall dipo
ole
mom
ments can be estimated
e
from
m partial atomiic charges. A key feature off this calculatioon is the symm
metry
of the molecule. A symmetric molecule will dissplay no dipolle moment eith
her because tthere is no cha
arge
155
Dr. Murtadha Alshareifi e-Library
sepa
aration (e.g., H2, O2, N2) or as
a a conseque
ence of cance
ellation of the dipole
d
vectors (e.g., benzen
ne,
meth
hane).
A mo
olecule can maintain a sepa
aration of elecctric charge eitther through in
nduction by ann external elec
ctric
field or by a perma
anent charge separation
s
witthin a polar molecule. The
P.81
dipollar nature of a peptide bond
d is an examplle of a fixed orr permanent dipole,
d
and its effects on the
e
bond
ding structure were discusse
ed earlier in th
his chapter. In proteins, the permanent dippolar nature of
o the
peptide bond and side chains ca
an stabilize se
econdary struc
ctures like α-helices and cann also influenc
ce
the h
higher-order conformation of
o a protein. Ind
duced dipoles
s in a protein can
c also influeence hydropho
obic
core protein structtures, so both types of dipolle play a prom
minent role in th
he stabilizatioon of proteinderivved therapeutiics. Permanen
nt dipoles in a molecule are discussed in the next sectioon.
Fig.. 4-2. Vectoorial nature of permaneent dipole moments
m
for (a) water, ((b) carbon
dioxxide, and (c) a peptide bond.
b
The ddistance r iss given by th
he dashed liine for each
h
mollecule. The arrow repreesents the coonventionall direction th
hat a dipolee moment
vecttor is drawnn, from negaative to posiitive.
To p
properly discusss dipoles and
d the effects off solvation, on
ne must understand the conccepts of polarrity
and dielectric consstant. Placing a molecule in
n an electric fie
eld is one way
y to induce a ddipole. Consider
two p
parallel condu
ucting plates, such
s
as the pla
ates of an electric condense
er, which are sseparated by
some
e medium acrross a distance
e r, and apply a potential ac
cross the plate
es (Fig. 4-3). E
Electricity will flow
f
from the left plate to the right pla
ate through th
he battery untill the potential difference of tthe plates equ
uals
that of the battery supplying the initial potentia
al difference. The capacitan
nce (C, in faraads [F]) is equa
al to
the q
quantity of electric charge (q
q, in coulombss) stored on th
he plates divided by the poteential differenc
ce
(V, in
n volts [V]) bettween the plattes:
156
Dr. Murtadha Alshareifi e-Library
Fig.. 4-3. Paralllel-plate con
ndenser.
The capacitance of
o the condens
ser in Figure 4
4-3 depends on
o the type of medium
m
separrating the plates as
well as on the thicckness r. When
n a vacuum fillls the space between
b
the plates, the cappacitance is C0.
This value is used
d as a referenc
ce to compare
e capacitances
s when other substances
s
filll the space. If
wate
er fills the spacce, the capacitance is increa
ased because
e the water mo
olecule can oriientate itself so
s
that its negative en
nd lies neares
st the positive condenser pla
ate and its pos
sitive end lies nearest the
nega
ative plate (see Fig. 4-3). Th
his alignment p
provides addittional moveme
ent of charge because of the
incre
eased ease wiith which electtrons can flow
w between the plates. Thus, additional chaarge can be placed
on th
he plates per unit
u of applied voltage.
The capacitance of
o the condens
ser filled with ssome material, Cx, divided by
b the referencce standard, C0, is
referrred to as the dielectric cons
stant, ε:
The dielectric consstant ordinarily
y has no dime
ensions becau
use it is the rattio of two capaacitances. By
defin
nition, the diele
ectric constant of a vacuum
m is unity. Diele
ectric constants of some liquuids are listed
d
in Ta
able 4-2. The dielectric
d
cons
stants of solve
ent mixtures ca
an be related to drug solubiility as describ
bed
by G
Gorman and Hall,6and ε for drug
d
vehicles can be related to drug plasma concentraation as reported
by P
Pagay et al.7
The dielectric consstant is a mea
asure of the ab
bility of molecu
ule to resist ch
harge separatiion. If the ratio
os of
the ccapacitances are
a close, then
n there is grea
ater resistance
e to a charge separation.
s
Ass the ratios
incre
ease, so too does the molec
cule's ability to
o separate cha
arges. Note that field strenggth and
temp
perature play a role in these
e measuremen
nts. If a molec
cule has a perm
manent dipolee, it will have
minim
mal resistance
e to charge se
eparation and will migrate in
n the field. How
wever, the connverse is also true.
Conssider a water molecule
m
at th
he atomic leve
el with its posittively charged nuclei drawn toward the
nega
ative plate and
d its negatively
y charged elecctron cloud pu
ulled and orien
nted toward th e positive platte.
Oxyg
gen is stronglyy
157
Dr. Murtadha Alshareifi e-Library
P.82
electronegative and will have a stronger attraction to the positive pole than will a less electronegative
atom. Likewise, hydrogen atoms are more electropositive and will move further toward the positive plate
than an atom that is more electronegative. Conceptually, this will reflect less resistance to the field.
Table 4-2 Dielectric Constants of Some Liquids at 25°C
Substance
Dielectric Constant, ε
N-Methylformamide
182
Hydrogen cyanide
114
Formamide
110
Water
78.5
Glycerol
42.5
Methanol
32.6
Tetramethylurea
23.1
Acetone
20.7
n-Propanol
20.1
Isopropanol
18.3
Isopentanol
14.7
l-Pentanol
13.9
Benzyl alcohol
13.1
Phenol
9.8
Ethyl acetate
6.02
(60°C)
158
Dr. Murtadha Alshareifi e-Library
Chloroform
4.80
Hydrochloric acid
4.60
Diethyl ether
4.34
Acetonitrile
3.92
Carbon disulfide
2.64
Triethylamine
2.42
Toluene
2.38
Beeswax (solid)
2.8
Benzene
2.27
Carbon tetrachloride
2.23
l,4-Dioxane
2.21
Pentane
1.84
(20°C)
Furfural
41
(20°C)
Pyridine
12.3
Methyl salicylate
9.41
(20°C)
(30°C)
A molecule placed in that field will align itself in the same orientation as the water molecules even
though the extent of the alignment and induced charge separation will dramatically differ due to atomic
structure. For this discussion, consider a molecule like pentane. Pentane is an aliphatic hydrocarbon
that is not charged and relies on van der Waals interactions for its primary attractive energies. Pentane
is not a polar molecule, and carbon is not strongly electronegative. Carbon–hydrogen bonds are much
stronger (less acidic) than oxygen–hydrogen bonds; therefore, the electrons in the σ bonds are more
shared. The electronic structure would not favor a large charge separation in the molecule and should
159
Dr. Murtadha Alshareifi e-Library
have
e a lower diele
ectric constantt. If a polar fun
nctional group such as an alcohol moiety is added to
gene
erate 1-pentan
nol, an enhanc
ced ability to u
undergo charg
ge separation and
a an increaased dielectric
consstant is the ressult. This is rea
adily observed
d in Table 4-2, where the dielectric constaant for pentane (at
20°C
C) is 1.84, thatt for 1-pentano
ol (at 25°C) is 13.9, and tha
at for isopentan
nol is 14.7 (at 25°C).
Key Co
oncept
Pola
arizability
Pola
arizability is defined
d
as th
he ease with which an ion
n or molecule
e can be polaarized by any
y
exte
ernal force, whether
w
it be an electric fiield or light energy
e
or thro
ough interact
ction with ano
other
mole
ecule. Large anions have
e large polariizabilities because of their loosely heeld outer
elecctrons. Polarizabilities for molecules a
are given in the table. The
e units for αpp are Å3 or 1024
cm
m3.
Polarizaabilities
M
Molecule αp × 1024 (cm3)
H2O
1.68
N2
1.79
HCl
3
3.01
HBr
3
3.5
HI
5
5.6
HCN
5
5.9
Whe
en nonpolar molecules like pentane
p
are pllaced in a suittable solvent between
b
the pplates of a charged
capa
acitor, an induced polarizatio
on of the mole
ecules can occ
cur. Again, this induced dipoole occurs
beca
ause of the separation of ele
ectric charge w
within the molecule due to the electric fielld generated
betw
ween the plates. The displac
cement of elecctrons and nuc
clei from their original positioons is the main
result of the inducttion process. This
T
temporarry induced dip
pole moment is
s proportional to the field
ngth of the cap
pacitor and the
e induced pola
arizability, αp, which is a cha
aracteristic prooperty of the
stren
particular molecule
e.
From
m electromagn
netic theory, it is possible to obtain the relationship
wherre n is the num
mber of molec
cules per unit vvolume. Equattion (4-4) is kn
nown as the C
Clausius–Moss
sotti
equa
ation. Multiplyiing both sides
s by the moleccular weight off the substance, M, and dividding both side
es by
the ssolvent densityy, ρ, we obtain
n
wherre N is Avogad
dro's number, 6.023 × 1023 mole-1, and Pi is known as the induced m
molar
polarrization. Pi rep
presents the in
nduced dipole moment per mole of nonpo
olar substancee when the ele
ectric
field strength of the condenser, V/m in volts p
per meter, is unity.
160
Dr. Murtadha Alshareifi e-Library
P.83
3
Exa
ample 4-1
Pola
arizability of
o Chloroforrm
Chlo
oroform has a molecular weight of 11 9 g/mole and
d a density of
o 1.43 g/cm33 at 25°C. What
is itss induced mo
olar polarizab
bility? We ha
ave
The concept of ind
duced dipole moments
m
can be extended from
f
the condenser model j ust discussed
d to
the m
model of a non
npolar molecu
ule in solution ssurrounded by
y ions. In this case, an anioon would repel
mole
ecular electron
ns, whereas a cation would attract them. This would ca
ause an interacction of the
mole
ecule in relatio
on to the ions in
i solution and
d produce an induced dipole
e. The distribuution and ease
e of
attra
action or repulssion of electro
ons in the nonp
polar molecule
e will affect the
e magnitude oof this induced
d
dipolle, as would th
he applied extternal electric field strength.
Perrmanent Dipole Moment
M
o
of Polar Molecules
M
s
In a polar molecule, the separattion of positive
ely and negativ
vely charged regions
r
can bee permanent, and
the m
molecule will possess
p
a perm
rmanent dipole
e moment, µ. This
T
is a nonio
onic phenomeenon, and altho
ough
regio
ons of the molecule may possess partial ccharges, these
e charges bala
ance each othher so that the
e
mole
ecule does nott have a net charge. Again, we can relate
e this to atomic
c structure by considering th
he
electtronegativity of
o the atoms in
n a bond. Wate
er molecules possess
p
a perrmanent dipolee due to the
differrences in the oxygen and th
he hydrogen. T
The magnitude of the perma
anent dipole m
moment, µ, is
indep
pendent of an
ny induced dipole from an ellectric field. It is defined as the
t vector sum
m of the individ
dual
charrge moments within
w
the mole
ecule itself, in cluding those from bonds and
a lone-pair eelectrons. The
e
vecto
ors depend on
n the distance of separation
n between the charges. Figu
ure 4-2 providees an illustration of
dipolle moment vectors for water, carbon dioxxide, and a pep
ptide bond. Th
he unit of µ is the debye, witth 1
debyye equal to 10-18 electrostatiic unit (esu) cm
m. The esu is the measure of electrostatiic charge, defiined
as a charge in a vacuum that re
epels a like cha
arge 1 cm away with a force
e of 1 dyne. Inn SI units, 1 de
ebye
= 3.3
34 × 10-30 C-m
m. This is deriv
ved from the ccharge on the electron (abou
ut 10-10 esu) m
multiplied by th
he
averrage distance between charged centers o
on a molecule (about 10-8 cm
m or 1 Å).
In an
n electric field,, molecules with permanentt dipole mome
ents can also have
h
induced dipoles. The polar
p
mole
ecule, howeve
er, tends to orie
ent itself with its negatively charged cente
ers closest to positively cha
arged
cente
ers on other molecules
m
befo
ore the electricc field is applie
ed. When the applied field iss present, the
e
orien
ntation is in the
e direction of the
t field. Maxiimum dipole moments
m
occu
ur when the moolecules are
almo
ost perfectly orriented with re
espect to the ffields. Absolutely perfect orientation can nnever occur ow
wing
to thermal energy of the molecu
ules, which con
ntributes to molecular motio
on that opposees the molecular
align
nment. The tottal molar polarrization,P, is th
he sum of indu
uction and perrmanent dipol e effects:
wherre P0 is the orientation polarrization of the permanent diipoles. P0 is eq
qual to 4πNµ22/9 kT, in whic
ch k is
-23
the B
Boltzmann con
nstant, 1.38 × 10 joule/K. Because P0depends on the
e temperaturee, T, equation (46) ca
an be rewritten
n in a linear fo
orm as
wherre the slope A is 4π Nµ2/9k and Pi is the y intercept. If P is obtained at several tem
mperatures and
plotte
ed against 1/T
T, the slope off the graph can
n be used to calculate
c
µ and
d the intercept
pt can be applied to
comp
pute αp. The values
v
of P can be obtained
d from equation (4-6) by mea
asuring the dieelectric consta
ant
and the density off the polar com
mpound at variious temperattures. The dipo
ole moments oof several
comp
pounds are lissted in Table 4-3.
4
In so
olution, the permanent dipole of a solventt such as wate
er can strongly
y interact with solute molecu
ules.
This interaction co
ontributes to th
he solvent effe
ect and is asso
ociated, in the
e case of wateer, with the
161
Dr. Murtadha Alshareifi e-Library
hydration of ions and molecules. The symmetry of the molecule can also be associated with its dipole
moment, which is observed with carbon dioxide (no net dipole) in Figure 4-2. Likewise, benzene and pdichlorobenzene are symmetric planar molecules and have dipole moments of zero. Meta and ortho
derivatives of benzene, however, are not symmetric and have significant dipole moments, as listed
in Table 4-3.
The importance of dipole interactions should not be underestimated. For ionic solutes and nonpolar
solvents, ion–induced dipole interactions have an essential role in solubility phenomena (Chapter 9). For
drug–receptor binding, dipole–dipole interactions are essential noncovalent forces that contribute to
enhance the pharmacologic effect, as described by Kollman.8 For solids composed of molecules with
permanent dipole moments, the dipole interactions contribute to the crystalline arrangement and overall
structural nature of the solid. For instance, water molecules in ice crystals are organized through their
dipole forces. Additional interpretations of the significance of dipole moments are given by Minkin et al.9
Electromagnetic Radiation
Electromagnetic radiation is a form of energy that propagates through space as oscillating electric and
magnetic fields at right angles to each other and to the direction of the propagation, shown in Figure 44a. Both electric and magnetic fields can be described by sinusoidal waves with characteristic
amplitude, A, and frequency, v. The common representation of the electric field in two dimensions is
shown in
P.84
Figure 4-4b. This frequency, v, is the number of waves passing a fixed point in 1 sec. The
wavelength, λ, is the extent of a single wave of radiation, that is, the distance between two successive
maxima of the wave, and is related to frequency by the velocity of propagation, v:
Table 4-3 Dipole Moments of Some Compounds
Compound
Dipole Moment (Debye Units)
p-Dichlorobenzene
0
H2
0
Carbon dioxide
0
Benzene
0
l,4-Dioxane
0
Carbon monoxide
0.12
Hydrogen iodide
0.38
Hydrogen bromide
0.78
162
Dr. Murtadha Alshareifi e-Library
Hydrogen chloride
1.03
Dimethylamine
1.03
Barbital
1.10
Phenobarbital
1.16
Ethylamine
1.22
Formic acid
1.4
Acetic acid
1.4
Phenol
1.45
Ammonia
1.46
m-Dichlorobenzene
1.5
Tetrahydrofuran
1.63
n-Propanol
1.68
Chlorobenzene
1.69
Ethanol
1.69
Methanol
1.70
Dehydrocholesterol
1.81
Water
1.84
163
Dr. Murtadha Alshareifi e-Library
Chloroform
m
1.86
Cholesterool
1.99
Ethylenediiamine
1.99
Acetylsaliccylic acid
2.07
o-Dichloroobenzene
2.3
Acetone
2.88
Hydrogen cyanide
2.93
Nitromethaane
3.46
Acetanilide
3.55
Androsteroone
3.70
Acetonitrille
3.92
Methyltesttosterone
4.17
Testosteronne
4.32
Urea
4.56
Sulfanilam
mide
5.37
The frequency of the
t radiation depends
d
on th e source and remains cons
stant; howeverr, the velocity
depe
ends on the co
omposition of the medium th
hrough which it passes. In a vacuum, thee wave of radia
ation
trave
els at its maxim
mum, the spee
ed of light, c = 2.99792 × 10
08 m/sec. Thus, the frequenncy can be deffined
as
164
Dr. Murtadha Alshareifi e-Library
The wave numberr, [v with bar above],
a
defined
d as the reciprrocal of the wa
avelength, is aanother way of
o
desccribing the elecctromagnetic radiation:
The wave numberr (in cm-1) reprresents the nu
umber of wave
elengths found
d in 1 cm of raddiation in a
vacu
uum. It is widely used becau
use it is directl y proportionall to the frequency, and thuss to the energy
y of
radia
ation, E, given
n by the famou
us and fundam
mental relationship of Planck
k and Einsteinn for the light
quan
ntum or energyy of a photon:
wherre h is Planck''s constant, which is equal tto 6.6261 × 10
0-34 joules.
The electromagne
etic spectrum is classified acccording to its wavelength or
o its correspoonding wave
number, as illustra
ated in Table 4-4.
4 The wave
elength becom
mes shorter as the corresponnding radiant
enerrgy increases. Most of our knowledge
k
abo
out atomic and
d molecular structure and prroperties comes
from the interactio
on of electroma
agnetic radiatiion with matte
er. The electric
c field componnent is mostly
responsible for phenomena suc
ch as transmisssion, reflection, refraction, absorption,
a
annd emission off
electtromagnetic ra
adiation, which
h give rise to m
many of the sp
pectroscopic techniques disscussed in this
s
chap
pter. The magn
netic compone
ent is responssible for the ab
bsorption of en
nergy in electrron paramagnetic
resonance (EPR) and nuclear magnetic
m
reson
nance (NMR),, techniques described at thhe end of the
chap
pter.
Ato
omic and
d Molecullar Specttra
All atoms and molecules absorb
b electromagn
netic radiation at certain cha
aracteristic freqquencies.
Acco
ording to elem
mentary quantu
um theory, ion s and molecules can exist only
o
in certain discrete state
es
charracterized by finite
f
amounts of energy (i.e
e., energy leve
els or electronic states) becaause electrons
s are
in mo
otion around the
t positively charged
c
nucle
eus in atoms. Thus,
T
the radiant energy abbsorbed by a
chem
mical species has only certa
ain discrete va
alues correspo
onding to the individual transsitions from on
ne
enerrgy state (E1) to
t a second, upper
u
energy sstate (E2) that can occur in an
a atom or moolecule (Fig. 4-5).
4
The pattern of abssorption freque
encies is calle
ed an absorptio
on spectrum and
a is produceed if radiation of a
particular wavelen
ngth is passed through a sam
mple and mea
asured. If the energy
e
of the radiation is
decrreased upon exit,
e
the change in the intenssity of the radiiation is due to
o electronic exxcitation.
In so
ome instancess, electromagn
netic radiation can also be produced
p
(emiission) when thhe excited spe
ecies
(atom
ms, ions, or molecules
m
excitted by means of the heat off a flame, an electric
e
P.85
5
curre
ent spark, abssorption of ligh
ht, or some oth
her energy sou
urce) relax to lower energy levels by relea
asing
enerrgy as photonss (hv). The em
mitted radiation
n as a function
n of the wavele
ength or frequuency is called
d the
emisssion spectrum
m.
165
Dr. Murtadha Alshareifi e-Library
Fig.. 4-4. The oscillating
o
ellectric and m
magnetic fieelds associaated with eleectromagneetic
radiiation.
Atom
mic spectra are
e the simplestt to describe a
and usually pre
esent a series of lines correesponding to th
he
frequ
uencies of spe
ecific electronic transition sta
ates. An exac
ct description of
o these transiitions is possib
ble
only for the hydrog
gen atom, for which a comp
plete quantum mechanical solution
s
exists..10 Even so, the
t
abso
orption and em
mission spectra
a for the hydro
ogen atom als
so can be expllained using a less sophistic
cated
theory such as the
e Bohr's mode
el. In this appro
oach, the energy of an electron moving inn a definite orb
bit
around a positivelyy charged nuc
cleus in an ato
om is given by
y
on (9.1 × 10-31 kg), n is an innteger numberr
wherre Z is the atomic number, m is the masss of the electro
corre
esponding to the
t principal quantum numb
ber of the orbitt, e is the charrge on the elecctron (1.602 × 1019
-14
3/2
2
1/2
-1
co
oulomb or 1.51
19 × 10 m kg sec ), a
and h is Planc
ck's constant.
166
Dr. Murtadha Alshareifi e-Library
Bohrr's model also assumes thatt electrons mo
oving in these orbits are stable (i.e., anguular momentum
m is
quan
ntized), and th
herefore absorrption or emisssion occurs on
nly when electtrons change oorbit. Thus, the
charracteristic frequency of the photon
p
absorb
bed or emitted corresponds to the absolutte value of the
e
differrence of the energy
e
states involved in an electronic transition in an atom.
a
Thus, thhe difference
betw
ween electron energy levels,, E2 - E1, havin
ng respective quantum num
mbers n2and n11, is given by the
t
exprression
hoton of electromagnetic ra
adiation E is re
elated to the frequency of thhe radiation by
y
The energy of a ph
equa
ation (4-12). Substituting
S
the
e energy differrence by the corresponding
c
P.86
6
P.87
7
charracteristic wavve number acc
cording to equa
ation (4-11) in
n equation (4-1
13), we obtainn
T
Table 4-4 Ellectromagn
netic Radiaation Rangees Accordin
ng to Waveelength, Wa
ave
nu
umber, Freequency, an
nd Energy,, and Techn
niques Tha
at Utilize Thhese Formss of
radiation*
*
Wavve
Num
mber
[v w
with
bar
Represenntative
E
Electomagn
netic Wavelengt abovve] Frequ
uenc Energ
gy Techniquues (Not
R
Radiation
h λ (m
m)
(cm--1) y (Hz))
(joule)) Inclusive))
Gamma raays
<1 ×
10-11
>11
×
1009
>3 ×
10199
>2
×
10
-14
X-rays
1 × 1011
–1
– ×
10-8
1×
1009
–11
×
1006
3×
10199–
3×
10166
2
×
10
14
Microsccopy,
photogrraphy,
radioacctivity
tracer
Microsccopy,
photogrraphy,
radioacctivity
tracer, ddiffraction,
ionizatiion
–2
×
10
-17
Vacuum
ultraviolet
1 × 108
–2
2×
10-7
1×
1006
–55
3×
10166–
1.5 ×
167
2
×
10
Spectroophotometr
y
Dr. Murtadha Alshareifi e-Library
×
104
1015
17
–
9.
9
×
10
-19
Nearultraviolet
2 × 107
–4 ×
10-7
5×
104
–
2.5
×
104
1.5 ×
1015–
7.5 ×
1014
9.
9
×
10
Spectrophotometr
y, fluorescence,
circular dichroism
19
–5
×
10
-19
Visible light
4 × 107
–7 ×
10-7
2.5
×
104
–
1.4
×
104
7.5 ×
1014–
4×
1014
5
×
10
-
Spectrophotometr
y, fluorescence,
light scattering,
optical rotation
19
–3
×
10
-19
Nearinfrared
7 × 107
–5 ×
10-6
1.4
×
104
–2
×
103
4×
1014–
6×
1013
3
×
10
19
Vibrational and
rotational
spectroscopy,
solid–state
characterization
–4
×
10
-20
Mid
(intermediat
e) infrared
5 × 106
–3 ×
10-5
2×
103
–
6×
1013–
1×
168
4
×
10
Vibrational and
rotational
spectroscopy
Dr. Murtadha Alshareifi e-Library
3.3
3×
102
1013
20
–
6.
6
×
10
-21
Far infrared
3 × 105
–1 ×
10-3
3.3
3×
102
–10
1×
1013–
3×
1011
6.
6
×
10
Vibrational and
rotational
spectroscopy
21
–2
×
10
-22
Microwave
1 × 103
–0.1
10–
0.1
3×
1011–
3×
109
2
×
10
Electron
paramagnetic
resonance
22
–2
×
10
-24
Radio
>0.1
<0.
1
<3 ×
109
<2
×
10
Nuclear magnetic
resonance
-24
*Information on the values was derived in part from the NASA Web site
http://imagine.gsfc.nasa.gov/docs/science/know_11/spectrum_chart.html
and Campbell and Dweb.12
169
Dr. Murtadha Alshareifi e-Library
Fiig. 4-5. Com
mparison off (a) atomic and (b) mo
olecular energy levels.
wherre n1 and n2 are the principa
al quantum nu
umbers for the
e orbital states involved in ann electronic
transsition of the attom from level E1 to E2.
The first term on th
he right in equ
uation (4-14) iss composed of
o fundamental constants annd the atomic
e hydrogen ato
om Z = 1, and substitution of
o the appropriate values affo
fords a single
number Z. For the
-1
ue 109,737.3 cm . This val ue is calculate
ed from the Bo
ohr's theory annd assumes th
hat
consstant R8 of valu
the n
nucleus of infin
nite mass was
s stationary wiith respect to the
t electron orbit. A more aaccurate analysis
invollves the replaccement of the mass of the e
electron by the
e reduced mas
ss of the electtron in the
hydrrogen atom, µH = 0.9994558
8m. The correccted theoretical value of R8, now called R H, is (109737.3)
(0.99
994558) = 109
9,677.6 cm-1. The
T experime ntal value for this constant, also known aas the Rydberg
g
consstant, is RH = 109,678.76
1
cm
m-1. The remarrkable agreem
ment between the theoreticaal and experim
mental
value
es of RH gave initial supportt to Bohr's the
eory, although it was later ab
bandoned for tthe more com
mplete
quan
ntum mechanical treatment..10Neverthele
ess, the absorp
ption and emis
ssion spectra for the hydrog
gen
atom
m are fully explained by Bohr's theory. Thu
us, introducing
g RH in equation (4-14), we find the
charracteristic wavve number only
y from the valu
ues of n, the principal
p
quantum number oof the electron's
orbit for the transittion, given by the simple rel ation
Exa
ample 4-2
Ene
ergy of Exciitation
(a) W
What is the energy
e
(in jou
ules and cm--1) of a quanttum of radiation absorbedd to promote
e the
elecctron in the hydrogen atom
m from its grround state (n1 = 1) to the
e second orbbital, n2 = 2?
We can solve this problem by using equa
ation (4-13) and
a replacing
g the values of all physic
cal
consstants involved and using
g Z = 1; we fifind
Note
e that joule = kg × m2 × sec
s -2.
170
Dr. Murtadha Alshareifi e-Library
To o
obtain the an
nswer in cm-1, we can use
e equation (4
4-11) to conv
vert joules intto wave num
mber,
or directly use eq
quation (4-15
5):
Usin
ng this equattion we obtain,
and therefore, from equation
n (4-10), the w
wavelength of
o the spectral line when an electron
passses from the n = 1 orbitall to the n = 2 orbital is
This is the first line
e of the Lyman
n ultraviolet se
eries of the ato
omic spectra for
f hydrogen. Note that, from
m
equa
ation (4-12), th
he electron of the hydrogen atom in the ground state, n = 1, has a low
wer energy (-2
2.18
× 10-18 joule) than that in the next highest elecctron state, n = 2 (-0.55 × 10-18 joule).
P.88
8
Whe
en the electron
n acquires suffficient energy to leave the atom,
a
it is rega
arded as infinittely distant fro
om
the n
nucleus, and the nucleus is considered no
o longer to afffect the electro
on. The energyy required for this
process, which ressults in the ion
nization of the atom according to
is alsso known as the ionization potential.
p
If we
e consider this
s process as occurring
o
whenn n = ∞, then the
ionizzation potentia
al from the ground state (n = 1) to n = ∞ is
s
Beca
ause 1/∞ [cong
gruent] 0,
This is equivalent to -E for n = 1, according to
o equation (4-1
13). Thus, the ionization pottential exactly
y
als the negativve energy of th
he electron in the ground state with a valu
ue of 10,9677 .6 cm-1 (equalls
equa
13.6 eV).
ually all atoms but hydrogen have more th
han one electrron. Therefore, approximatee methods are
Virtu
used
d to evaluate electronic
e
transition states fo
or most atoms
s. In these atoms, spin pairinng energy and
d
electtron–electron repulsion arise, so that desscription of the
e electronic sta
ates is not as sstraightforward as
in the
e hydrogen attom. For practtical purposes , most elemen
nts of the perio
odic table dispplay a
charracteristic atom
mic absorption
n and emission
n spectrum as
ssociated with the electronicc transition sta
ates
that can be used to
t identify and quantify speccific elements. Some of the more sensitivee spectral
wave
elengths associated with pa
articular atomss are given in Table 4-5. For instance, thee simplified en
nergylevell diagram for the
t sodium ato
om in Figure 4
4-5a illustrates
s the source off the lines in a typical emiss
sion
specctrum of gas-p
phase sodium. The single-ou
uter-electron ground
g
state fo
or sodium is loocated in the
3s orrbital. After exxcitation (i.e., absorption
a
of e
energy as sho
own by wavy lines), two emiission lines (about
330 and 590 nm) result
r
from the
e transitions 4
4p → 3s and 3p → 3s, respe
ectively. Manyy types of atom
mic
specctrometers are
e available dep
pending on the
e methods use
ed for atomiza
ation and introoduction of the
e
samp
ple. Further de
etailed informa
ation can be fo
found in the re
eferences cited
d in this chaptter.
Taable 4-5 Speectral Wav
velengths A
Associated with
w Electronic Transsitions Used
d in
the Detection
n of Particu
ular Elemen
nts
171
Dr. Murtadha Alshareifi e-Library
ElementWavelength (nm)
As
193.7
CA
422.7
Na
589.0
Cu
324.8
Hg
253.7
Li
670.8
Pb
405.8
Zn
213.9
K
766.5
Atomic spectroscopy has pharmaceutical applications in analysis of metal ions from drug products and
in the quality control of parenteral electrolyte solutions. For example, blood levels of lithium, used to
treat bipolar disorder (manic depressive disorder), can be analyzed by atomic spectroscopy to check for
overdosing of lithium salts.
In addition to having electronic states, molecules have quantized vibrational states, which are
associated with energies due to interatomic vibrations (e.g., stretching and bending), and rotational
states, which are related to the rotation of molecules around their center of gravity. These additional
energy states available for electron transitions make the spectra of molecules more complex than those
of atoms. In the case of vibration, the interatomic bonds may be thought of as springs between atoms
(see Fig. 4-14) that can vibrate in various stretching or bending configurations depending on their
energy levels. In rotation, the motion is similar to that of a top spinning according to its energy level. In
addition, the molecule may have some kinetic energy associated with its translational (straight-line)
motion in a particular direction.
The energy levels associated with these various transitions differ greatly from one another. The energy
-18
associated with movement of an electron from one orbital to another is typically about 10 joule
(electronic transitions absorb in the ultraviolet and visible light region between 180 and 780 nm;
see Example 4-2), where the energy involved in vibrational changes is about 10-19 to 10-20 joule (infrared
region) depending on the atoms involved, and the energy for rotational change is about 10-21 joule. The
energy associated with translational change is even smaller, about 10-35 joule. The precise energies
associated with these individual transitions depend on the atoms and bonds that compose the molecule.
Each electronic energy state of a molecule normally has several possible vibrational states, and each of
these has several rotational states. The rotational energy levels are lower than the vibrational levels and
are drawn in a similar manner to the quantized vibrational levels in the electronic states, as shown
172
Dr. Murtadha Alshareifi e-Library
in Figure 4-5b. The translational states are so numerous and the energy levels between translational
states are so small that they are normally considered a continuous form of energy and are not treated as
quantized. The total energy of a molecule is the sum of its electronic, vibrational, rotational, and
translational energies.
When a molecule absorbs electromagnetic radiation, it can undergo certain transitions that depend on
the quantized amount of energy absorbed. In Figure 4-5, the absorption of radiation (wavy lines) leads
to two different energy transitions, ∆E, which result in the electronic transition from the lowest level of
the ground state (S0) to an excited electronic state (S1 or S2). Electronic transitions of molecules involve
energies corresponding to ultraviolet or visible radiation.
P.89
Purely vibrational transitions may occur within the same electronic state (e.g., a change from level 1 to 2
in S0) and involve near-infrared (IR) radiation. Rotational transitions (not shown in Fig. 4-5) are
associated with low-energy radiation over the entire infrared wavelength region. The relatively large
energy associated with an electronic transition usually leads to a variety of concurrent, different
vibrational and rotational changes. Slight differences in the vibrational and rotational nature of the
excited electronic state complicate the spectrum. These differences lead to broad bands, characteristic
of the ultraviolet and visible regions, rather than the sharp, narrow lines characteristic of individual
vibrational or rotational changes in the infrared region.
The energy absorbed by a molecule may be found only at a few discrete wavelengths in the ultraviolet,
visible, and infrared regions, or the absorptions may be numerous and at longer wavelengths than
originally expected. The latter case, involving longer-wavelength radiation, is normally found for
molecules that have resonance structures, such as benzene, in which the bonds are elongated by the
resonance and have lower energy transitions than would be expected otherwise. Electromagnetic
energy may also be absorbed by a molecule from the microwave and radio wave regions (see Table 44). Low-energy transitions involve the spin of electrons in the microwave region and the spin of nuclei in
the radio wave region. The study of these transitions constitutes the fields of EPR and NMR
spectroscopy. These various forms of molecular spectroscopy are discussed in the following sections.
Ultraviolet and Visible Spectrophotometry
Electromagnetic radiation in the ultraviolet (UV) and visible (Vis) regions of the spectrum fits the energy
of electronic transitions of a wide variety of organic and inorganic molecules and ions. Absorbing
species are usually classified according to the type of molecular energy levels involved in the electronic
transition, which depends on the electronic bonding within the molecule.11,12 Commonly, covalent
bonding occurs as a result of a pair of electrons moving around the nuclei in a way that minimizes both
internuclear and interelectronic Coulombic repulsions. Combinations of atomic orbitals (i.e., overlap)
give rise to molecular orbitals, which are locations in space with an associated energy in which bonding
electrons within a molecule can be found.
173
Dr. Murtadha Alshareifi e-Library
Fig.. 4-6. Moleccular orbitall electronic transitions commonly found in ulltraviolet–
visibble spectrosscopy.
For instance, the molecular
m
orbiitals commonl y associated with
w single cov
valent bonds iin organic spe
ecies
are ccalled sigma orbitals
o
(σ), wh
hereas a doub
ble bond is usu
ually described
d as containinng two types of
o
mole
ecular orbitals: one sigma and one pi (π). Molecular orb
bital energy levels usually foollow the orde
er
show
wn in Figure 4-6.
Thuss, when organ
nic molecules are
a exposed to
o light in the UV–Vis
U
regions of the specttrum (see Tab
ble 44), th
hey absorb light of particula
ar wavelengthss depending on
o the type of electronic
e
trannsition that is
asso
ociated with the absorption. For example, hydrocarbons
s that contain σ-type bonds can undergo
only σ → σ* electrronic transition
ns from their g
ground state. The
T asterisk (**) indicates thee antibonding
mole
ecular orbital occupied
o
by th
he electron in tthe excited sta
ate after absorption of a quaantized amoun
nt of
enerrgy. These ele
ectronic transittions occur at short-wavelen
ngth radiation in the vacuum
m ultraviolet re
egion
(wavvelengths typiccally between 100 and 150 nm). If a carbonyl group is present
p
in a m
molecule, howe
ever,
the o
oxygen atom of
o this function
nal group posssesses a pair of nonbonding
g (n) electronss that can
unde
ergo n → π* or n → σ* electtronic orbital trransitions. These transitions
s require a low
wer energy tha
an
do σ → σ* transitio
ons and thereffore occur from
m the absorpttion of longer wavelengths
w
oof radiation
(see Fig. 4-6). Forr acetone, n → π* and n → σ* transitions occur at 280 and 190 nm, rrespectively. For
F
aldeh
hydes and kettones, the region of the ultra
aviolet spectru
um between 270 and 290 nm
m is associate
ed
with their carbonyll
P.90
0
n → π* electronic transitions, an
nd this fact ca n be used for their identifica
ation. Thus, thhe types of
electtronic orbitals present in the
e ground state
e of the molecu
ule dictate the
e region of the spectrum in which
w
abso
orption can takke place. Thos
se parts of a m
molecule that can
c be directly
y associated w
with an absorp
ption
of ulttraviolet or vissible light, such as the carbo
onyl group, are
e called chrom
mophores.
Mostt applications of absorption spectroscopyy to organic mo
olecules rely on
o transitions ffrom n or π
electtrons to π* because the ene
ergies associa
ated with these
e electronic tra
ansitions fall inn an
expe
erimentally con
nvenient regio
on from 200 to
o 700 nm.
The amount of ligh
ht absorbed by
y a sample is based on the measurement of the transm
mittance, T, orr the
abso
orbance, A, in transparent cuvettes or cel ls having a pa
ath length b (in
n cm) accordinng to equation
n (416):
174
Dr. Murtadha Alshareifi e-Library
wherre I0 is the inte
ensity of the in
ncident light be
eam and I is in
n the intensity of light after iit emerges from
the ssample.
Equa
ation (4-16) is also known as
a Beer's law, and relates th
he amount of light absorbed (A) to the
conccentration of absorbing subs
stance (C in m
mole/liter), the length of the path
p
of radiatioon passing
throu
ugh the samplle (b in cm), and the consta nt ε, known as
s the molar ab
bsorptivity for a particular
abso
orbing speciess (in units of litter/mole cm). T
The molar abs
sorptivity depe
ends not only oon the molecu
ule
whosse absorbance
e is being dete
ermined but a
also on the type of solvent being used, thee temperature, and
the w
wavelength of light used for the analysis.
Exa
ample 4-3
Abs
sorbance Maxima
M
Changes Due to
o Chemical Instability
a. A solutio
on of c = 2 × 10-5 mole/lite
er of chlordia
azepoxide dissolved in 0 .1 N sodium
hydroxid
de was place
ed in a fused silica cell ha
aving an optical path of 1 cm. The
absorba
ance A was fo
ound to be 0
0.648 at a wa
avelength of 260
2 nm. Whaat is the molar
absorptivity?
b. If a soluttion of chlord
diazepoxide had an abso
orbance of 0.298 in a 1-cm
m cell at 260
0 nm,
what is its
i concentra
ation?
c.
d.
The large value off ε indicates th
hat chlordiazep
poxide absorb
bs strongly at this
t
wavelengtth. This molarr
abso
orptivity is characteristic of the drug disso
olved in 0.1 N NaOH
N
at this wavelength
w
annd is not the same
s
as it would be in 0.1
0 N HCl. A la
actam is forme
ed from the dru
ug under acid conditions thaat has an
abso
orbance maxim
mum at 245 ra
ather than 260
0 nm and a corrrespondingly different ε vallue. Large values
for m
molar absorptivvities are usua
ally found in m
molecules haviing a high deg
gree of conjugaation of
chromophores, su
uch as the conjugated doublle bonds in 1,3
3-butadiene, or
o the combinaation of carbon
nyl or
carboxylic acids with
w double bon
nds as in α, β unsaturated ketones.
k
In particular, highlyy conjugated
ecules such ass aromatic hyd
drocarbons an
nd heterocycle
es typically dis
splay absorptioon bands
mole
charracteristic ofπ → π* transitio
ons and have ssignificantly higher ε values
s.
Exa
ample 4-4
Abs
sorption Ma
axima Calcu
ulation
Aminacrine is an
n anti-infectiv
ve agent with
h the followin
ng molecular structure:
Its h
highly conjugated acridine
e ring producces a comple
ex ultraviolet spectrum inn dilute sulfurric
acid
d that include
es absorption
n maxima at 2
260, 313, 32
26, 381, 400, and 422 nm
m. The molar
abso
orptivities of the absorbances at 260 and 313 nm are 63,900 and
a 1130 liteer/mole cm,
resp
pectively. Wh
hat is the min
nimum amou
unt of aminac
crine that can
n be detectedd at each of
thesse two wavelengths?
If we
e assume an
n absorbance
e level of A = 0.002 corre
esponding to a minimum ddetectable
conccentration off the drug, the
en, at 260 nm
m,
and at 381 nm
Nearrly 100 times greater
g
sensitivity in detecti ng the drug is
s possible with the 260-nm aabsorption ban
nd,
whicch is reflective of a higher nu
umber of electtronic transitio
ons that can oc
ccur at the higgher energy. The
T
abso
orbance level of
o 0.002 was chosen
c
by jud
dging this value to be a significant signal aabove instrum
mental
noise
e (i.e., interferrence generate
ed by the specctrophotomete
er in the absen
nce of the druug). For a partiicular
175
Dr. Murtadha Alshareifi e-Library
analyysis, the minim
mum absorban
nce level for d
detection of a compound
c
depends on bothh the instrume
ental
cond
ditions and the
e sample state
e, for example , the solvent chosen
c
for diss
solution of thee sample.
Instrrumental conditions vary acrross spectroph
hotometers, and
a will be disc
cussed later. W
With regard to
o the
samp
ple conditionss, solvents may have drama
atically differen
nt polarities, which
w
are addre
ressed in the
Diele
ectric Constan
nt and Induced
d Polarization section. The polarity
p
of a so
olvent may inffluence the
bond
ding state of a molecule by changing
c
the a
attractive and repulsive forc
ces in the moleecule through
solva
ation. For example, a polar compound disssolved in watter will have different capaccities to underg
go
electtronic transitio
ons than if it were dissolved in a nonpolarr, aprotic solution.
As illlustrated in Exxample 4-4, a molecule mayy have more than one chara
acteristic absoorption wavele
ength
band
d, and the com
mplete UV–Vis
s wavelength a
absorption spe
ectrum can provide informat
ation for the
posittive identification of a compound. Each fu
unctional group has different associated m
molecular bonding
and electronic stru
ucture, which will
w have differrent waveleng
gths that maxim
mally absorb eenergy. There
efore,
the rresulting specttrum for a mollecule can be used as a “mo
olecular fingerrprint” to deterrmine the
mole
ecular structurre. Changes in
n the order of b
bonding or in electronic stru
ucture will
P.91
alterr the UV–Vis spectrum,
s
as illustrated with chlordiazepox
xide and its la
actam. As a reesult, many
differrent molecular properties ca
an be monitore
red by using a UV–Vis spectrophotometerr including
chem
mical reactionss, complexatio
on, and degrad
dation, making
g it a powerfull tool for pharm
maceutical
scien
ntists.
Fig.. 4-7. A douuble-beam ultraviolet–v
u
visible specctrophotomeeter. (Courteesy of Variaan,
Inc., Palo Alto,, CA.)
Abso
orption spectro
oscopy is one of the most w
widely used me
ethods for qua
antitative analyysis due to its
s wide
appliicability to botth organic and
d inorganic sysstems, modera
ate to high sen
nsitivity and seelectivity, and
good
d accuracy and convenience
e. Modern spe
ectroscopic ins
strumentation for measuringg the complete
e
mole
ecular UV–Vis absorption sp
pectra is comm
monplace in both industrial and academicc settings. Two
o
main
n types of specctrophotomete
ers, usually co
oupled to personal computers for data anaalysis, are
comm
mercially available: double--beam and dio
ode-array instrruments.
176
Dr. Murtadha Alshareifi e-Library
A schematic diagram of a traditional double-beam UV–Vis spectrophotometer is shown in Figure 4-7.
The beam of light from the source, usually a deuterium lamp, passes through a prism or grating
monochromator to sort the light according to wavelength and spread the wavelengths over a wide
range. This permits a particular wavelength region to be easily selected by passing it through the
appropriate slits. The selected light is then split into two separate beams by a rotating mirror, or
“chopper,” with one beam passed through the reference, which is typically the blank solvent used to
dissolve the sample, and the other through the sample cell containing the test molecule. After each
beam passes through its respective cell, it is reflected onto a second mirror in another chopper
assembly, which alternatively selects either the reference or the combined beams to focus onto the
photomultiplier detector. The rapidly changing current signal from the detector is proportional to the
intensity of the particular beam, and this is fed into an amplifier, which electronically separates the
signals of the reference beam from those of the sample beam. The final difference in beam signals is
automatically recorded. In addition, it is common practice to first put the solvent in the sample cell and
set the spectrophotometer's absorbance to zero to serve as the baseline reference. The samples can
then be placed in the sample cell and measured, with the difference from the baseline being reported.
The sample data are reported as a plot of the intensity, usually as absorbance, against the wavelength,
as shown for the chlordiazepam lactam in Figure 4-8.
Figure 4-9 is an illustration of a typical diode-array spectrophotometer. Note that these instruments have
simpler optical components, and, as a result, the radiation throughput is much higher than in traditional
double-beam instruments. After the light beam passes the sample, the radiation is focused on an
entrance slit and directed to a grating. The transducer is a diode array from which resolutions of 0.5 nm
can be reached. A single scan from 200 to 1100 nm takes only 0.1 sec, which leads to a significant
improvement in the signal-to-noise ratio by accumulating multiple scans in a short time.
Typically, a calibration curve from a series of standard solutions of known but varying concentration is
used to generate standard curves for quantitative analysis. An absorbance spectrum can be used to
determine one wavelength, typically an absorption maximum, where the absorbance of each sample
can be efficiently measured. The absorbance for standards is measured at this point and plotted against
the concentration, as shown in Figure 4-10, to obtain what is known as a Beer's-law plot. The
concentration of an “unknown” sample can then be determined by interpolation from such a graph.
Spectrophotometry is a useful tool for studying chemical equilibria or determining the rate of chemical
reactions. The chemical species participating in the equilibria must have
P.92
different absorption spectra due to changes in the electronic structure in the associated or dissociated
states (Chapter 8), which influence the bonding structure. This change results in the variation in
absorption at a representative wavelength for each species while the pH or other equilibrium variable is
changed. If one determines the concentrations of the species from Beer's law and knows the pH of the
solution, one can calculate an approximate pKa for a drug. For example, if the drug is a free acid (HA) in
equilibrium with its base (A ), then the pKa is defined by
177
Dr. Murtadha Alshareifi e-Library
Fig.. 4-8. The absorbance
a
maxima
m
of cchlordiazep
poxide lactam
m as a funcction of the
wavvelength. (M
Modified fro
om
httpp://chrom.tuutms.tut.ac.jp
p/JINNO/D
DRUGDATA
A/17chlordiia-zepoxidee.html)
Whe
en [HA] = [A-], as determined
d by their resp
pective absorb
bances in the spectrophotom
s
metric
determination, pKa [congruent] pH.
p
Exa
ample 4-5
Ioniization Statte and Abso
orption
Phenobarbital sh
hows a maximum absorp
ption of 240 nm
n as the mo
onosodium ssalt (A-), whe
ereas
the ffree acid (HA
A) shows no absorption m
maxima in the wavelength region from
m 230 to 290
0 nm.
If the
e free acid in
n water is slo
owly titrated w
with known volumes
v
of dilute NaOH aand the pH of
o the
solu
ution and the absorbance at 240 nm a
are measured
d after each titration, onee reaches a
maxximum absorrbance value at pH 10 aftter the additio
on of 10 mL of titrant. Hoow can pKa be
b
dete
ermined from
m this titration
n?
178
Dr. Murtadha Alshareifi e-Library
Fig.. 4-9. Schem
matic of an HP-8453
H
diiode-array ultraviolet–v
u
visible specctrophotomeeter.
(Courtesy of Hewlett-Pack
H
kard, Palo A
Alto, CA.)
By p
plotting the absorbance
a
against
a
the p H over the titration range
e to pH = 10, one can obttain
the m
midpoint in absorbance,
a
where half th
he free acid has been titrrated, and [H
HA] = [A-] (Fig
g. 48). T
The pH corre
esponding to this absorba
ance midpoin
nt is approxim
mately equall to the pKa,
nam
mely, pKa1 forr the first ionization stage
e of phenobarbital. This midpoint
m
occuurs at a pH of
o
7.3; therefore, pK
Ka [congruen
nt] 7.3. For m
more accurate pKa determ
minations, reffer to the
disccussion in Ch
hapter 8.
Reacction rates can
n also be mea
asured when a particular rea
action species
s has an absorrption spectrum
that is noticeably
P.93
3
differrent from the spectra
s
of other reactants o
or products. One can follow the rate of apppearance or
disap
ppearance of the selected species
s
by reccording its abs
sorbance at sp
pecific times d uring the reac
ction
process. If no othe
er reaction spe
ecies absorbss at the particu
ular wavelength chosen for tthis determina
ation,
the rreaction rate will
w simply be proportional
p
to
o the rate of ch
hange of abso
orbance with reeaction time.
Stan
ndard curves can
c be generated if the sele
ected reactant is available in
n a pure form. It must be notted
that the assumptio
on that the oth
her species or reaction intermediates do not
n interfere w
with the selecte
ed
reactant can also be flawed, as discussed late
er. An example of the use of
o spectrophottometry for the
e
determination of re
eaction rates in
i pharmaceuttics is given by Jivani and Stella,13
S
who uused the
disap
ppearance of para-aminosa
alicylic acid fro
om solution to determine its rate of decarbboxylation.
179
Dr. Murtadha Alshareifi e-Library
Fig.. 4-10. A Beeer's-law plot of absorbbance again
nst the conceentration off
chloordiazepoxide.
Specctrophotometrry can be used
d to study enzyyme reactions
s and to evaluate the effectss of drugs on
enzyymes. For example, the ana
alysis of clavul anic acid can be accomplished by measuuring the ultrav
violet
abso
orption of penicillin G at 240
0 nm, as descrribed by Gutm
man et al.14 Clavulanic acid inhibits the ac
ctivity
of β--lactamase enzymes, which convert peniccillin G to peniicilloic acid, where R is C6H 5CH2—.
The method first re
equires that th
he rate of abso
orbance change at 240 nm be measured with a solutio
on
conta
aining penicillin G and a β-lactamase enzzyme. Duplica
ate experimentts are then peerformed with
incre
easing standarrd concentrations of clavula
anic acid. Thes
se show a dec
crease in absoorbance chang
ge,
equivvalent to the enzyme
e
inhibittion from the d
drug, as the co
oncentration of
o the drug incrreases. The
conccentration of an unknown am
mount of clavu
ulanic acid is measured
m
by comparing
c
its rate of enzym
me
inhib
bition with thatt of the standa
ards.
The primary weakkness with a sp
pectrophotome
eter used in such a mannerr is that it meaasures all of th
he
speccies in a samp
ple even thoug
gh a single wavvelength migh
ht be selected for its molecuular detection
speccificity. UV–Viss spectra gene
erated at a givven wavelengtth(s) cannot properly detectt changes in th
he
speccies that arise during a reaction. For exam
mple, a reactio
on intermediate(s) may be ggenerated thatt has
stron
nger absorptio
on at the wave
elength selecte
ed than the reactant or the product
p
being measured. Finally,
there
e are many diffferent reaction pathways th
hat include mu
ultiple species with overlappping absorption
n, so
the sselection of a single
s
wavelength for detecction does not provide good selectivity. Onne way to corrrect
this iis to generate a UV–Vis abs
sorption scan across a wide
e wavelength range
r
and meeasure change
es in
seve
eral specific ab
bsorption max
xima. Howeverr, the same prroblems can obfuscate the ddata in this
apprroach as well. Therefore, in many cases, the complex reaction
r
milieu
u is not readilyy adaptable to a
singlle spectroscop
pic method. Th
hus, an abilityy to separate th
he individual species
s
and quuantitate theirr
levells would provide enhanced sensitivity. UV
V–Vis spectrop
photometers are
a also used in conjunction
n with
many other metho
ods for detectin
ng molecules,, for example, high-performa
ance liquid ch romatography
y
180
Dr. Murtadha Alshareifi e-Library
(HPLC), which eliminates species interference by separating the compounds before detection occurs.
The major use of spectrophotometry is in the field of quantitative analysis, in which the absorbance of
chromophores is determined. Various applications of spectrophotometry are discussed by Schulman
and Vogt.15
Fluorescence and Phosphorescence
Luminescence is an emission of radiation in the ultraviolet, visible, or near-IR regions from electronically
excited species. An electron in an atom or a molecule can be excited by means of absorbing energy, for
instance a photon of light, to reach an electronic excited state (see prior discussion of orbital transitions).
The excited state is relatively short-lived, and the electron can return to its ground state via radiative and
nonradiative energy emission. If the preferred path to return to the ground state involves releasing
energy through internal conversions by changes in vibrational states or through collisions with the
environment (e.g., solvent molecules), then the molecule will not display luminescence. Many chemical
species, however, emit radiation when returning to the ground state either as fluorescence or as
phosphorescence, depending on the mechanism by which the electron finally returns to the ground
state.
Figure 4-11 shows a simplified energy diagram (also known as a Perrin–Jablonsky diagram) of the
typical mechanisms that a chemical species undergoes after being electronically excited. The absorption
of a photon usually excites an electron from the ground state toward its excited states without changing
its spin (i.e., a singlet ground state will absorb into a singlet excited state, called a spin-allowed
transition).
The triplet state usually cannot be achieved by excitation from the ground state, this being termed a
“forbidden” transition according to quantum theory. It is usually reached through the process
of intersystem crossing (ISC), which is a nonradiative transition between two isoenergetic vibrational
levels belonging to electronic states of different multiplicities. For example, the excited singlet (S1) in the
0 vibrational mode can move to the isoenergetic vibrational level of Tn triplet state, then vibrational
relaxation can bring it to the lowest vibrational level of T1, with the concomitant energy loss (see Fig. 411). From T1, radiative emission to S0 can occur, which is called phosphorescence.
The excited triplet state (T1) is usually considered more stable (i.e., having a longer lifetime) than the
excited singlet state (S1). The length of time during which light will be emitted after the molecule has
become excited depends on the lifetime of the electronic transition. Therefore, we can expect
phosphorescence to occur for a longer period after excitation than after fluorescence. Ordinarily,
fluorescence occurs between 10-10 and 10-7 sec after excitation, whereas the lifetime for
phosphorescence is between 10-7 and 1 sec. Because of its short lifetime, fluorescence is usually
measured while the molecule is being excited. Phosphorescence uses a pulsed excitation source to
allow enough time to detect the emission. It should be noted that measurements of
P.94
fluorescence lifetimes on the order of femtoseconds have been demonstrated to be valuable in studying
transition states, as described by Zewail.16
181
Dr. Murtadha Alshareifi e-Library
Fig.. 4-11. A Peerrin–Jablon
nski diagram
m illustratin
ng the multip
ple pathwayys by which
h
absoorbed moleccular energy
y can be relleased. Relaaxation can occur throuugh a
nonnradiative paathway or raadiative em
mission of flu
uorescence or phosphor
orescence. The
T
singglet (S) and triplet (T) states
s
pertainn to the elecctronic struccture in the molecule after
a
exciitation.
Fluorescence norm
mally has a longer waveleng
gth than the ra
adiation used for the excitattion phase,
princcipally because of internal energy
e
losses within the exc
cited molecule before the fluuorescent
emisssion S1 → S0 occurs (Fig. 4-11).
4
The gap
p between the
e first absorptio
on band and th
the maximum
fluorrescence is ca
alled the Stoke
es shift. Phosp
phorescence ty
ypically has ev
ven longer waavelengths tha
an
fluorrescence owin
ng to the energ
gy difference tthat occurs in ISC as well as
s the loss of eenergy due to
interrnal conversion
n over a longe
er lifetime.
A rep
presentative schematic
s
of a typical fluoro
ometer is show
wn in Figure 4--12. Generallyy, fluorescence
e
inten
nsity is measured in these in
nstruments byy placing the photomultiplier
p
detector at rigght angles to the
t
incid
dent light beam
m that produce
es the excitatio
on. The signal intensity is re
ecorded as rellative fluoresc
cence
again
nst a standard
d solution. Bec
cause photolu
uminescence can
c occur in any direction fro
rom the sample,
the d
detector will se
ense a part of the total emisssion at a charracteristic wav
velength and w
will not be cap
pable
of de
etecting radiattion from the light beam use
ed for excitatio
on.
Fluorometry is a very
v
sensitive technique,
t
up to 1000 times
s more sensitiv
ve than spectrrophotometry.. This
is be
ecause the fluo
orescence inte
ensity is meassured above a low backgrou
und level, wheereas in measu
uring
low a
absorbances, two large sign
nals that are sslightly differen
nt are compare
ed. Recently, advances in
instru
umentation ha
ave made it po
ossible to dete
ect fluorescence at a single--molecule leveel.17
Phottoluminescencce occurs only
y in those mole
ecules that ca
an undergo the
e specified phooton emission
ns
afterr excitation witth consequentt return to the ground state. Many molecu
ules do not posssess any
photoluminescencce, although th
hey can largelyy absorb light. Most often, molecules
m
thatt display
fluorrescence or ph
hosphorescence contain a rrigid conjugate
ed structure su
uch as aromattic hydrocarbo
ons,
rhodamines, coum
marins, oxines, polyenes, an
nd so on, or inorganic lantha
anides ions likke Eu3+ or Tb3++,
whicch also show strong
s
fluoresc
cence. Many d
drugs (e.g., mo
orphine), some natural aminno acids and
cofacctors (e.g., tyrrosine, tryptop
phan, nicotinam
mide adenine dinucleotide, reduced form flavin adenine
e
182
Dr. Murtadha Alshareifi e-Library
dinuccleotide, etc.) fluoresce. Some exampless are given in Table
T
4-6 alon
ng with their chharacteristic
excittation and emiission wavelen
ngths, which ccan be used fo
or qualitative and
a quantitativve analyses.
Fluorescence dete
ection and ana
alysis has beccome a major tool
t
in biopharrmaceutical annd chemical
analyysis, particularly in areas re
elated to manip
pulation of bio
opolymers (nuc
cleic acids andd proteins) an
nd
imag
ging of biologicc membranes and living org
ganisms (cells, bacteria, etc.). For exampple, in peptides
s and
prote
eins the UV ab
bsorbance for the n → π* (εε = 100) and π → π* (ε = 70
000) transitionss in the peptid
de
bond
d occurs from about 210 to 220
2 and 190 n
nm, respective
ely.12 The abs
sorbance of seeveral side-ch
hain
transsitions can ofte
en be obfusca
ated absorban
nce from peptid
de bonds as well
w as other sspecies presen
nt in
the ssolution, and therefore does
s not offer goo
od specificity. Tryptophan
T
(280 nm), tyrosiine (274),
phen
nylalanine (257 nm), and cy
ystine (250 nm
m) have electro
onic
P.95
5
transsitions that occcur above the 220-nm range
e for the peptiide bond, but these transitioons can be
obfuscated as welll.12 However, tryptophan, p
phenylalanine, and tyrosine can also und ergo fluoresce
ence
that can be used to
t discriminate
e peptides or p
proteins from biologic
b
matric
ces. The use oof other dyes and
a
fluorrophores for biiologic applica
ations is also a
an exciting and rapidly deve
eloping area. T
The student is
s
advissed to seek ou
ut Web sites of
o companies ssuch as Molec
cular Probes (www.probes.c
(
com) to look at
a the
wide
e scope and usse of fluoresce
ent labels and
d probes as we
ell as modern instrumentatioon in the field..
Schu
ulman and Stu
urgeon18give a thorough re
eview of the ap
pplications of photoluminesc
p
cence to the
analyysis of traditio
onal pharmace
euticals.
Fig.. 4-12. Scheematic diagrram of a filtter fluorometer. (From G. H. Scheenk, Absorp
ption
of L
Light and Ulltraviolet Ra
adiation, A
Allyn and Baacon, Boston
n, 1973, p. 2260. With
perm
mission.)
183
Dr. Murtadha Alshareifi e-Library
Table 4-6 Fluorescence of Some Drugs
Drug
Excitation
Wavelength
(nm)
Emission
Wavelength
(nm)
Solvent
Phenobarbital
255
410–420
0.1 N NaOH
Hydroflumethiazide
333
393
1 N HCl
Quinine
350
~450
0.1 N H2SO4
Thiamine
365
~440
Isobutanol, after
oxidation with
ferricyanide
Aspirin
280
335
1% acetic acid in
chloroform
Tetracycline
hydrochloride
330
450
0.05 N NaOH(aq)
Fluorescein
493.5
514
Water (pH 2)
Riboflavin
455
520
Ethanol
Hydralazine
320
353
Concentrated H2SO4
Infrared Spectroscopy
The study of the interaction of electromagnetic radiation with vibrational or rotational resonances (i.e.,
the harmonic oscillations associated with the stretching or bending of the bond) within a molecular
structure is termed infrared spectroscopy. Normally, infrared radiation in the region from about 2.5 to
50 µm, equivalent to 4000 to 200 cm-1 in wave number, is used in commercial spectrometers to
determine most of the important vibration or vibration–rotation transitions. The individual masses of the
vibrating or rotating atoms or functional groups, as well as the bond strength and molecular symmetry,
determine the frequency (and, therefore, also the wavelength)
P.96
of the infrared absorption. The absorption of infrared radiation occurs only if the permanent dipole
moment of the molecule changes with a vibrational or rotational resonance. The molecular symmetry
relates directly to the permanent dipole moment, as already discussed. Bond stretching or bending may
affect this symmetry, thereby shifting the dipole moment as found for the normal vibrational modes (2),
(3), and (3′) for CO2 in Figure 4-13. Other resonances, such as (1) for CO2 inFigure 4-13, do not affect
the dipole moment and therefore do not produce infrared absorption. Resonances that shift the dipole
184
Dr. Murtadha Alshareifi e-Library
mom
ments can give
e rise to infrare
ed absorption by molecules, even those considered
c
to have no
perm
manent dipole moment, such
h as benzene or CO2. The frequencies
f
off infrared absoorption bands
corre
espond closelyy to vibrations
s from particula
ar parts of the
e molecule. Th
he bending andd stretching
vibra
ations for acetaldehyde, together with the
e associated in
nfrared frequencies of absorrption, are sho
own
in Fig
gure 4-14. In addition
a
to the
e fundamental absorption ba
ands shown in
n this figure, eeach of which
corre
esponds to a vibration
v
or vib
bration–rotatio
on resonance and a change
e in the dipole moment, wea
aker
overrtone bands may
m be observe
ed for multiple
es of each of th
hese frequenc
cies (in wave nnumbers). Forr
exam
mple, an overttone band may
y appear for a
acetaldehyde at
a 3460 cm-1, which
w
correspponds to twice the
frequ
uency (2 × 173
30 cm-1) for th
he carbonyl strretching band. Because the frequencies aare simply
asso
ociated with ha
armonic motion of the radian
nt energy, the
e overtones ma
ay be thought of as simple
multiiples that are exactly in pha
ase with the fu ndamental fre
equency and can
c therefore ““fit” into the sa
ame
resonant vibration within the mo
olecule.
Fig.. 4-13. The normal vibrrational moodes of CO2 and their reespective w
wave numberrs,
show
wing the dirrections of motion
m
in reeaching the extreme off the harmonnic cycle.
(Moodified from
m W. S. Brey, Physical Chemistry and Its Biological Appplications,
Acaademic Presss, New Yorrk, 1978, p. 316.)
Beca
ause the vibra
ational resonan
nces of a com
mplex molecule
e often can be attributed to pparticular bonds or
groups, they beha
ave as though they result fro
om vibrations in
i a diatomic molecule.
m
Thiss means that
vibra
ations produce
ed by similar bonds
b
and ato ms are associated with infra
ared bands ovver a small
frequ
uency range even
e
though th
hese vibrationss may occur in
n completely different
d
moleccules. A typica
al
infrared spectrum of theophylline is shown in Figure 4-15. The
T spectrum “fingerprints” the drug and
provides one meth
hod of verifying compoundss. The individual bands can be associatedd with particula
ar
groups. For examp
ple, the band at 1660 cm-1, a inFigure 4-1
15, is due to a carbonyl streetching vibratio
on for
theophylline.
185
Dr. Murtadha Alshareifi e-Library
Fig.. 4-14. Bendding and strretching freqquencies fo
or acetaldehy
yde. (From H. H. Willaard,
L. L
L. Merritt, Jrr., and J. A.. Dean,Instrrumental Methods
M
of Analysis, 4thh Ed., Van
Nosstrand, New
w York, 1968
8. With perm
mission.)
Infra
ared spectra ca
an be complex
x, and charactteristic frequencies vary dep
pending on thee physical sta
ate of
the m
molecule
P.97
7
ding between sample molec
g examined. For
F example, hydrogen
h
bond
cules may chaange the spec
ctra.
being
For a
alcohols in dilu
ute carbon tetrachloride sol ution, there is little intermole
ecular hydrogeen bonding, and
a
the h
hydroxyl stretcching vibration
n occurs at abo
out 3600 cm-1. The precise position and sshape of the
infrared band asso
ociated with th
he hydroxyl gro
roup depend on
o the concenttration of the aalcohol and the
degrree of hydroge
en bonding. Stteric effects, th
he size and re
elative charge of neighboringg groups, and
phasse changes ca
an affect simila
ar frequency sshifts.
186
Dr. Murtadha Alshareifi e-Library
Fig.. 4-15. Infraared spectru
um of theophhylline. (Frrom E. G. C. Clarke, Edd., Isolation
n
andd Identificatiion of Drug
gs, Pharmac eutical Presss, London, 1969. Withh permission
n.)
The use of infrared
d spectroscop
py in pharmacyy has centered on its applic
cations for drugg identification
n, as
desccribed by Chap
pman and Moss.19 The devvelopment of Fourier
F
transfo
orm infrared sspectrometry20
0 has
enha
anced infrared
d applications for both qualittative and qua
antitative analy
ysis of drugs oowing to the
grea
ater sensitivity and the enhanced ability to
o analyze aque
eous samples with Fourier ttransform infra
ared
instru
umentation. Thorough
T
surve
eys of the tech
hniques and applications
a
off infrared specctroscopy are
provided by Smith
h21 and Willard
d et al.22
Near-Infrare
ed Spectroscopy
Nearr-IR spectrosccopy is rapidly becoming a vvaluable techn
nique for analy
yzing pharmacceutical
comp
pounds. The near-IR
n
region
n ranges from approximately
y 10,000 to 40
000 cm-1 and ccomprises ma
ainly
of mid-IR overtone
es and combin
nations of thesse overtones arising
a
from he
eavy-atom eleectronic transittions,
inclu
uding O—H, C—H,
C
and N—
—H stretching. T
Thus, near-IR
R spectroscopy
y is typically ussed for analyz
zing
wate
er, alcohols, an
nd amines. In addition, nearr-IR bands arise from forbid
dden transitionns; therefore, the
t
band
ds are far less intense than the typical abssorption band
ds for these ato
oms.23
Som
me advantagess of using nearr-IR spectrosccopy over othe
er techniques include extrem
mely fast analy
ysis
timess, nondestructtive analysis, lack of a need
d for sample preparation,
p
qu
ualitative and qquantitative
results, and the po
ossibility of mu
ulticomponentt analysis. Pha
armaceutical applications
a
innclude polymorph
identtification, wate
er content ana
alysis, and ide ntification and
d/or monitoring
g of an active ccomponent within
a tab
blet or other so
olid dosage fo
orm. Because of the nondes
structive nature
e of the techn ique, online
analyysis for contro
olling large-sca
ale processess is possible.
Ele
ectron Pa
aramagne
etic and N
Nuclear Magnetic
M
c Resona
ance
Spectrosco
opy
Now
w we enter into
o the descriptio
on of two wide
ely used specttroscopic techniques in the ppharmaceutical
scien
nces and relatted fields of ch
hemistry, biolo
ogy, and medicine. Electron paramagneticc resonance and
a
187
Dr. Murtadha Alshareifi e-Library
NMR
R are based on
n the magnetic behavior of atomic particles such as ele
ectrons and nuuclei in the
presence of an exxternal magnettic field. In the
e following, fun
ndamentals of the techniquees will be given,
inclu
uding differencces between electrons
e
and n
nuclei where applicable.
a
Fig.. 4-16. Elecctron and nu
uclear rotatioon.
Charrged particles such as electtrons (negative
ely charged) and
a protons (p
positively chargged) behave as
a if
they were charged
d, spinning top
ps. Accordingl y, they have magnetic
m
mom
ments associatted with the
move
ement around
d their axes, as
s shown in Fig
gure 4-16 (i.e., they behave as small maggnets). In paire
ed
electtrons the spinss are opposite
e, so the magn
netic moments
s cancel each other out. How
wever, single
unpa
aired electronss have a magn
netic moment (Bohr magnetton, µBohr) equ
ual to
wherre e is the elem
mentary charg
ge, h is Planckk's constant, and
a me is the electron's
e
masss.*
Similarly, nuclear motion in prottons and certa
ain nuclei also has an assoc
ciated magnetiic moment, wh
hose
magnitude can be calculated fro
om an equatio
on similar to(4--18) by replacing the mass oof the particle;
proto
ons have a larrger mass than
n electrons, so
o that the nuclear magneton
n is µN = 5.0511 × 10-27 joule/T.
Whe
en the electron
n (or the proton) is placed in
n an external magnetic
m
field,, its magnetic moment is
orien
nted in two diffferent directions relative to tthe magnetic field, B, parallel or antiparalllel, giving rise
e to
two n
new energy le
evels splitting from
f
each oth er by a magniitude ∆E, whic
ch depends onn the applied
field B according to
t
g is ttermed the La
andé splitting factor,
f
or simp
ply the g factorr. For organic free radicals, g is nearly equal
to itss value for a frree electron, 2.0023,
2
which is the ratio of the electron's spin magneticc moment to
its orrbital magneticc moment. The g factor is a
also characteriistic for certain
n metal compleexes
P.98
8
with unpaired elecctrons, from which informatio
on about theirr environment can be obtainned.
188
Dr. Murtadha Alshareifi e-Library
Fig.. 4-17. Elecctron and nu
uclear spin ssplitting wh
hen subjected to an appllied externaal
maggnetic field.
For a
an electron pla
aced in a mag
gnetic field of B = 1 T, ∆E = 18.54 × 10-24 joule; accordiing to equation
n (419). Therefore, a simple
s
calcula
ation of the freq
quency that matches
m
this en
nergy differennce, v = ∆E/h,
e
can be
b excited with
h a radiation of
o 28 GHz, corrresponding too the microwav
ve
indiccates that the electron
regio
on of the specctrum (see Tab
ble 4-4). A sim
milar calculation for protons using
u
with gN = 5.586, afffords an enerrgy splitting off 2.82 × 10-26 joule, correspo
onding to an eelectromagnetic
frequ
uency of 42.58
8 MHz, also in
n the microwavve region.
Tran
nsitions betwee
en electron sp
pin energy leve
els give rise to
o the phenome
enon of EPR, as indicated
in Fig
gure 4-17; the
e correspondin
ng property of the nuclear spin levels lead
ds to NMR. Thhe term
“reso
onance” relate
es to the class
sical interpreta
ation of the phe
enomenon, be
ecause transittions occur only
when
n the frequenccy v of the electromagnetic radiation matc
ches the Larm
mor frequency of precession
around the axis off the applied magnetic
m
field, vL, as display
yed in Figure 4-18.
4
So fa
ar we have co
onsidered only
y electrons and
d protons, parrticles with spin
n ½, which haave only two
posssible energy le
evels. Several nuclei also ha
ave magnetic moments and, therefore, NM
MR signals. Nuclei
N
with odd mass num
mbers have to
otal spin of hallf-integral valu
ue, whereas nu
uclei with odd atomic numb
bers
and even mass nu
umbers have total
t
spin of inttegral value. Examples
E
inclu
ude 13C, 19F, aand 31P, which
h
2
14
have
e spin ½, and H and N, which
w
have spin
n 1.
By considering equations (4-19)) and (4-20), w
we can obtain the resonance signal by gra
radually varyin
ng the
ength, B, while
e keeping the radio frequenc
cy, v, constant. At some parrticular B value, a
magnetic field stre
at flips the ele
ectron or nucle
ei from one spin state to anoother (e.g., fro
om I =
spin transition will take place tha
o +½, as in Fig
gure 4-18). In some spectro
ometers, the experiment
e
can
n be done in thhe reverse
-½ to
fashiion: keeping B constant and
d varying v. Th
his method is generally calle
edcontinuous--wave or field
scan
n, and the partticular value of v depends lin
nearly on B, as
a illustrated in
n Figure 4-19.
189
Dr. Murtadha Alshareifi e-Library
Fig.. 4-18. Elecctronic or nu
uclear spin pprocession around
a
an applied
a
maggnetic field.
In most modern sp
pectrometers, particularly fo
or NMR, all spins of one spe
ecies in the saample are exciited
simu
ultaneously byy using a shortt, high-frequen
ncy pulse of ellectromagnetic
c radiation. Thhen their relax
xation
is me
easured as a function
f
of tim
me and transfo
ormed from the
e time domain to the frequeency domain via
fast Fourier transfo
ormation techniques. The ssignal-to-noise
e ratio can be significantly
s
im
mproved by the
e
accu
umulation of a large numberr of measurem
ments.
As w
we have seen, the theoretica
al basis of EPR
R spectroscop
py is similar to
o that of NMR except that it is
restrricted to param
magnetic spec
cies (i.e., with u
unpaired electtrons). Among
g thousands of types of
subsstances constituting biologic
c systems with
h pharmacolog
gic relevance, only very few
w, namely,
P.99
9
free radicals, inclu
uding molecula
ar oxygen (O2 ) and nitric oxide (NO), and a limited num
mber of transitiion
meta
al ions reveal paramagnetic
p
properties an
nd may be obs
served by EPR
R spectroscoppy. Nevertheless,
EPR
R has many po
otential uses in
n biochemistryy and medicine
e by the intenttional introducction into the
syste
em of exogeno
ous EPR-dete
ectable parama
agnetic probe
es (spin labels)). This methodd often elimina
ates
the n
necessity to pu
urify biologic samples.
s
Thuss, EPR spectro
oscopy applied to biologic tiissues and flu
uids
can iidentify the ch
hanges in redo
ox processes tthat contribute
e to disease. Electron
E
param
magnetic
resonance may alsso be used to characterize certain plant-d
derived products as potentiaally important to
biote
echnology by increasing
i
the
e level of free rradicals and other
o
reactive species
s
produuced during lig
ghtinducced oxidative stress of the cell.
c
Several o
other applicatio
ons can be fou
und in referennces at the end
d of
the cchapter.24,25,26
2
190
Dr. Murtadha Alshareifi e-Library
Fig.. 4-19. Enerrgy differen
nces observeed in electro
onic or nuclear spin proocessions ass a
funcction of maggnetic field strengths.
Nuclear magnetic resonance is a powerful exxperimental me
ethod in chem
mistry and bioloogy because the
t
resonance frequen
ncy of a nucle
eus within a mo
olecule depen
nds on the che
emical environnment. The loc
cal
field experienced by
b an atom in a molecule iss less than the
e external mag
gnetic field by aan amount giv
ven
by
wherre σ is the shie
elding constan
nt for a particu
ular atom (i.e.,, a measure off its susceptibbility to inductio
on
from a magnetic field) and B is the
t external m
magnetic field strength.
s
Inste
ead of measurring the absolu
ute
value
e of σ, it is cusstomary to me
easure the diffference of σ re
elative to a reference.
Exa
ample 4-6
Nuc
clear Spin Energy
E
a. What is the energy change
c
asso ciated with the 1H nuclea
ar spin of chlooroform, whiich
has a sh
hielding cons
stant, σ, of -7
7.25 × 10-6 an
nd a nuclear magnetic m
moment, µN, of
o
5.05082
24 × 10-27 joule/T in a mag
gnetic field, B,
B of 1 T?
We can obtain ∆E = gN µN Blocal ffor this partic
cular nucleus
s, in which thhe corresponding
local fielld experience
ed by the ato
om is given by
b equation (4-21).
(
The loocal magnetic
field Bloccal is replaced
d, and we ob
btain
b. What is the radio frequency at w
which resonan
nce will occur for this nucclear spin
transition under the stated
s
condittions?
191
Dr. Murtadha Alshareifi e-Library
Tetrame
ethylsilane (T
TMS) is often
n used as a reference
r
com
mpound in p roton NMR
because
e the resonan
nce frequenccy of its one proton signa
al from its fouur identical
methyl groups
g
is below that for m
most other co
ompounds. In
n addition, TM
MS is relativ
vely
stable and inert.
Exa
ample 4-7
Res
sonance of Tetramethy
ylsilane
Wha
at is the radio
o frequency at
a which reso
onance occu
urs for TMS in a magneticc field of 1 T?
?
The shielding co
onstant σ is 0.000,
0
and ∆E
E = 2.821243 × 10-26 joule for TMS, sso
From
m the example
es just given, itt is important tto note that th
he difference in
n the resonan ce frequency
betw
ween chloroforrm and TMS at a constant m
magnetic field strength is only 308 Hz or w
waves/sec. Th
his
smalll radio frequency difference
e is equivalentt to more than
n half the total range within w
which 1H NMR
R
signa
als are detecte
ed.
The value of the shielding consttant, σ, for a p
particular nucle
eus will depen
nd on local maagnetic fields,
inclu
uding those pro
oduced by nea
arby electronss within the mo
olecule. This effect
e
is promooted by placing the
mole
ecule within a large externall magnetic fiel d, B. Greater shielding will occur with higgher electron
denssity near a parrticular nucleus, and this red
duces the freq
quency at whic
ch resonance ttakes place. Thus,
T
for T
TMS, the high electron dens
sity from the S i atom produc
ces enhanced shielding and , therefore, a lower
resonance frequen
ncy. The relative difference between a pa
articular NMR signal and a rreference sign
nal
(usually from TMS
S for proton NM
MR) is termed
d the chemicall shift, δ, given
n in parts per m
million (ppm). It is
defin
ned as
wherre σr and σs arre the shieldin
ng constants fo
or the referenc
ce and the sam
mple nucleus,, respectively.
If the
e separation between
b
the sa
ample and refe
erence resona
ance is ∆H or ∆v, then
wherre HR and vR are
a the magne
etic field streng
gth and radio frequency for the nuclei, deepending on
whetther field-swep
pt or frequenc
cy-swept NMR
R is used.
Exa
ample 4-8
Proton Shift Re
eference
Wha
at is the chem
mical shift of the chlorofo
orm proton us
sing TMS as a reference??
Substituting the frequencies obtained fro
om Examples
s 4–6b and 4–7
4 into equaation (4-23), we
w
obta
ain the chemical shift:
This is an approxim
mate value ow
wing to the rela
ative accuracy
y used to dete
ermine each frrequency in the
exam
mple. The accepted experim
mental value fo
or this chemical shift is 7.24
4 ppm. The cheemical shift off a
nucle
eus provides information
i
ab
bout its local m
magnetic envirronment and th
herefore can ““type” a nuclear
speccies. Table 4-7
7 lists some re
epresentative p
proton chemic
cal shifts. Figu
ure 4-20 showss a proton NM
MR
specctrum for benzzyl acetate, CH
H3COOCH2C6 H5, using TMS
S as a reference. Notice thaat each signal band
repre
esents a particcular
P.10
00
type of proton, tha
at is, the proton at δ = 2.0 is from the CH3 group, where
eas that at 5.00 is due to CH2,
and the protons att about 7.3 are
e from the aro
omatic protons
s. Theintegral curve
c
above thhe spectrum is the
sum of the respecctive band area
as, and its ste
epwise height is
i proportional to the numbeer of protons
repre
esented by ea
ach band.
Table 4-7 Proton
P
Cheemical Shiffts for Representativee Chemical Groups orr
C
Compound
ds
192
Dr. Murtadha Alshareifi e-Library
Compound or Group
ppm
Tetramethylsilane (TMS), (CH3)4Si
0.00
Methane
0.23
Cyclohexane
1.44
Acetone
2.08
Methyl chloride
3.06
Chloroform
7.25
Benzene
7.27
Ethylene
5.28
Acetylene
1.80
R—OH (hydrogen bonded)
0.5–5.0
R2—NH
1.2–2.1
Carboxylic acids (R—COOH)
10–13
H2O
~4.7
In Figure 4-20, the signal bands are of simple shape, with little apparent complexity. Such sharp single
bands are known as singlets in NMR terminology. In most NMR spectra, the bands are not as simple
because each particular nucleus can be coupled by spin interactions to neighboring nuclei. If these
neighboring nuclei are in different local magnetic environments, splitting of the bands can occur because
of differences in electron densities. This leads to multiplet patterns with several lines for a single
resonant nucleus. The pattern of splitting in the multiplet can provide valuable information concerning
the nature of the neighboring nuclei.
193
Dr. Murtadha Alshareifi e-Library
Fig.. 4-20. Protoon NMR sp
pectrum of bbenzyl acetaate with tetrramethylsilaane (TMS) as
a a
refeerence. The TMS band appears at tthe far rightt. The upperr curve is ann integration
n of
the spectral bannds, the heig
ght of each step being proportiona
p
al to the areaa under thatt
bandd. (From W.
W S. Brey, Physical
P
Chhemistry and
d Its Biological Applicaations,
Acaademic Presss, New Yorrk, 1978, p. 498. With permission.
p
.)
Figure 4-21 showss the proton NMR
N
spectrum
m of acetaldehy
yde, CH3CHO, in which the doublet of the
e
CH3 group (right side
s
of the figu
ure) is produce
ed from coupling to the neighboring singlee proton, and the
quarrtet of the lone
e proton in the CHO group ( left side of the
e figure) is pro
oduced from cooupling to the
three
e methyl proto
ons. In any mo
olecule, if the rrepresentative
e coupled nuclei are in the pproportion AX:A
AY on
neigh
hboring group
ps, then the resonance band
d for AX will be
e split into (2Y × I + 1) lines, in which I is the
spin quantum num
mber for the nu
uclei, whereass that for AY will be split into (2X × I + 1) linnes, assuming
g the
nucle
ei are in different local envirronments. Forr example, with acetaldehyd
de, AX:AY is CH
H3:CH, and the
resulting proton sp
plitting pattern is AX (CH3) a
as two lines an
nd AY (CH) as four lines. Thiis splitting
produces first-orde
er spectra whe
en the differen
nce in ppm be
etween all the lines of a multtiplet (known as
a
the ccoupling consttant, J) is sma
all compared w
with the differe
ence in the che
emical shift, δ , between the
coup
pled nuclei. Firrst-order spec
ctra produce siimple multiple
ets with intensities determineed by the
coeffficients of the binomial expa
ansion, which can be obtain
ned from Pasc
cal's triangle:
194
Dr. Murtadha Alshareifi e-Library
wherre n is the num
mber of equiva
alent neighborr nuclei. Thus,, a doublet pre
esents intensitties 1:1; a tripllet,
1:2:1
1; and a quarte
et, 1:3:3:1. Fu
urther details o
of multiplicity and
a the interprretation of NM
MR spectra can
n be
found in general te
exts on NMR.2
27,28
The typical range for NMR chem
mical shifts de
epends on the nucleus being
g observed: Foor protons, it is
abou
ut 15 ppm with
h organic compounds, wherreas it is aboutt 400 ppm for 13C and 19F s pectra. Table 48 givves the basic NMR
N
resonance for certain pure isotopes
s, together with
h their natural abundances. As
the n
natural abunda
ance of the iso
otope decreasses, the relativ
ve sensitivity of
o NMR gets pproportionally
smalller.
1
A wid
dely used nuccleus in charac
cterization of n
new chemical compounds is
s 13C. The devvelopment of 13
C
NMR
R spectroscopy in recent years has been influenced by
y the applicatio
on of spin decooupling techniiques
to inttensify and sim
mplify the othe
erwise comple
ex 13C NMR sp
pectra. Decoupling of the prroton spins is
produced by continuously irradia
ating the entirre proton spec
ctral range with
h broadband rradio frequenc
cy
radia
ation. This deccoupling produ
uces the collap
pse of multiple
et signals into simpler and m
more intense
signa
als. It also pro
oduces an effe
ect known as tthe nuclear Ov
verhauser effe
ect (NOE), in
P.10
01
whicch decoupling of the protons
s produces a d
dipole–dipole interaction
i
and
d energy transsfer to the carrbon
nucle
ei, resulting in
n greater relaxation (i.e., rate
e of loss of en
nergy from nuc
clei in the highher spin state to
t
the lo
ower spin statte) and, conse
equently, a gre
eater populatio
on of carbon nuclei
n
in the loower spin state
e
(see Fig. 4-13). Be
ecause this grreater populatiion in the lowe
er spin state permits
p
a greatter absorption
n
signa
al in NMR, the
e NOE can inc
crease the carrbon nuclei sig
gnal by as muc
ch as a factor of 3. These
facto
ors, proton spin decoupling and the NOE,, have enhanc
ced the sensitivity of 13C NM
MR. They
13
comp
pensate for th
he low natural abundance off C as well as
a the smaller magnetic mom
ment, µ, of 13C
comp
pared with tha
at for hydrogen
n.
Fig.. 4-21. Protoon NMR sp
pectrum of aacetaldehyd
de. (From A. S. V. Burggen and J. C.
C
Mettcalfe, J. Phharm. Pharm
macol. 22, 1556, 1970. With
W permisssion.)
A po
owerful extension of this enh
hancement of 13C spectra in
nvolves system
matically decooupling only
speccific protons byy irradiating att particular rad
dio frequencie
es rather than by broadbandd irradiation. Such
S
syste
ematic decoup
pling permits individual carb
bon atoms to show
s
multiplett collapse and signal
195
Dr. Murtadha Alshareifi e-Library
intensification, as mentioned previously, when protons coupled to the particular carbon atom are
irradiated. These signal changes allow particular carbon nuclei in a molecular structure to be associated
with particular protons and to produce what is termed two-dimensional NMR spectrometry.
Farrar29 describes the techniques involved in these experiments.
Table 4-8 Basic Nuclear Magnetic ResonanceS (NMR) and Natural Abundance
of Selected Isotopes*
Isotope
1
1H
NMR Frequency (MHz) at Given Field
Strength
At 1.0000 T
At 2.3487 T
Natural Abundance
(%)
42.57
100.00
99.985
13
6C
10.71
25.14
1.108
15
7N
4.31
10.13
0.365
19
9F
40.05
94.08
100
*From A. J. Gordon and R. A. Ford, The Chemist's Companion, Wiley, New
York, 1972, p. 314. With permission.
Nuclear magnetic resonance is a versatile tool in pharmaceutical research. Nuclear magnetic resonance
spectra can provide powerful evidence for a particular molecular conformation of a drug, including the
distinction between closely related isomeric structures. This identification is normally based on the
relative position of chemical shifts as well peak multiplicity and other parameters associated with spin
coupling. Drug–receptor interactions can be distinguished and characterized through specific changes in
the NMR spectrum of the unbound drug after the addition of a suitable protein binder. These changes
are due to restrictions in drug orientation. Burgen and Metcalfe30 describe applications of NMR to
problems involving drug–membrane and drug–protein interactions. Illustrations of these interactions are
analyzed by Lawrence and Gill31 using electron spin resonance (ESR) and by Tamir and
Lichtenberg32 using proton-NMR techniques. The ESR and proton-NMR results show that the
psychotropic tetrahydrocannabinols reduce the molecular ordering in the bilayer of liposomes used as
simple models of biologic membranes. These results suggest that the cannabinoids exert their
psychotropic effects by way of a nonspecific interaction of the cannabinoid with lipid constituents,
principally cholesterol, of nerve cell membranes. Rackham33 reviewed the use of NMR in
pharmaceutical research, with particular reference to analytical problems.
P.102
Nuclear magnetic resonance characterization has become routine, and specialized techniques have
rapidly expanded its application breadth. Two particularly exciting applications rely on the determination
of the three-dimensional structure of complex biomolecules34 (proteins and nucleic acids) and the
imaging of whole organisms (tomography). The paramount importance of these developments
employing NMR has been recognized by the awarding of the Nobel Prize in Chemistry and Medicine in
2002 and 2003 for this work.
Refractive Index and Molar Refraction
196
Dr. Murtadha Alshareifi e-Library
Lightt passes more
e slowly throug
gh a substancce than through a vacuum. As
A light enterss a denser
subsstance, the advvancing waves interact with
h the atoms in the substance
e at the interfaace and
throu
ughout the thicckness of the substance. Th
hese interactio
ons modify the
e light waves bby absorbing
enerrgy, resulting in the waves being
b
closer to
ogether by reducing the speed and shorteening the
wave
elength, as sh
hown in Figure
e 4-22. If the lig
ght enters the
e denser substtance at an anngle, as shown
n,
one part of the wa
ave slows dow
wn more quickl y as it passes
s the interface, and this prodduces a bending of
the w
wave toward th
he interface. This
T
phenome
enon is calledrrefraction. If lig
ght enters a leess dense
subsstance, it is reffracted away from
f
the interfface rather tha
an toward it. However,
H
in booth cases there
e is
an im
mportant obse
ervation: The liight travels in the same dire
ection after inte
eraction as beefore it. In
reflection, the lightt travels in the
e opposite dire
ection after inc
cidence with th
he medium. Thhe relative vallue of
the e
effect of refracction between two substanc es is given by
y the refractive
e index, n:
wherre sin i is the sine
s
of the ang
gle of the incid
dent ray of ligh
ht, sin r is the sine of the anngle of the
refra
acted ray, and c1 and c2 are the speeds off light of the re
espective med
dia. Normally, the numerator is
taken
n as the veloccity of light in air
a and the den
nominator is th
hat in the material being invvestigated. The
refra
active index, by this conventtion, is greaterr than 1 for su
ubstances denser than air. T
Theoretically, the
t
referrence state wh
here n = 1 sho
ould be for ligh
ht passing thro
ough a vacuum
m; however, thhe use of air as
a a
referrence produce
es a difference
e in n of only 0
0.03% from tha
at in a vacuum
m and is more commonly us
sed.
Fig.. 4-22. Wavves of light passing
p
an iinterface beetween two substances of differentt
denssity.
The refractive inde
ex varies with the waveleng
gth of light and
d the temperature because bboth alter the
enerrgy of interaction. Normally, these values are identified when a refrac
ctive index is l isted; for
197
Dr. Murtadha Alshareifi e-Library
exam
mple, nD20 sign
nifies the refra
active index ussing the D-line
e emission of sodium,
s
at 5899 nm, at a
temp
perature of 20°C. Pressure must also be h
held constant in measuring the refractivee index of gase
es.
As discussed in Chapter 2, temperature and p
pressure are related.
r
The re
efractive indexx can be used to
identtify a substancce, to measure
e its purity, an
nd to determin
ne the concenttration of one ssubstance
disso
olved in anoth
her. Typically, a refractomete
er is used to determine
d
refra
active index. R
Refractometerrs
can b
be an importa
ant tool for stud
dying pharmacceutical comp
pounds that do
o not undergo extensive UV
V–Vis
abso
orption due to their electroniic structure.
The molar refractio
on, Rm, is rela
ated to both the
e refractive index and the molecular
m
propperties of a
comp
pound being tested.
t
It is expressed as
wherre M is the mo
olecular weigh
ht and ρ is the density of the
e compound. The
T Rm value oof a compound
can o
often be prediicted from the structural fea
atures of the molecule,
m
and one
o should noote the analogous
equa
ation to those utilized with dielectrics. Eacch constituent atom or group
p contributes a portion to th
he
final Rm value, as discussed earrlier in connecction with additive–constitutive properties (see Table 4--1).
For e
example, acettone has an Rm produced fro
rom three carb
bons (Rm = 7.2
254), six hydroogens (6.6), and a
3
carbonyl oxygen (2.21) to give a total Rm of 1 6.1 cm /mole.. Because Rm is independennt of the physical
state
e of the molecule, this value
e can often be used to distin
nguish between structurally different
comp
pounds, such as keto and enol
e
tautomerss.
Lightt incident on a molecule ind
duces vibrating
g dipoles due to energy abs
sorption at the interface, where
the g
greater the reffractive index at
a a particularr wavelength, the
t greater the
e dipolar inducction. Simply
state
ed, the interacction of light ph
hotons with po
olarizable elec
ctrons of a dielectric molecu le causes a
reduction in the ve
elocity of light. The dielectricc constant, a measure
m
of po
olarizability, is greatest when the
resulting dipolar in
nteractions witth light are pro
oportionally larrge. The refrac
ctive index forr light of long
wave
elengths, n∞, is related to th
he dielectric co
onstant for a nonpolar
n
molec
cule, ε, by thee expression
P.10
03
Fig.. 4-23. The polarization
n of a dispeersed light beam
b
by a Nicol
N
prism.
Mola
ar polarization, Pi [equation (4-16)], can b e considered roughly equivalent to molarr refraction, Rm,
and can be written
n as
198
Dr. Murtadha Alshareifi e-Library
From
m this equation
n, the polariza
ability, αp, of a nonpolar molecule can be obtained
o
from
m a measureme
ent
of re
efractive index. For practical purposes, the
e refractive ind
dex at a finite wavelength iss used. This
introduces only a relatively
r
small error, appro
oximately 5%, in the calculattion.
Op
ptical Rottation
Electromagnetic ra
adiation comp
prises two sepa
arate, sinusoidal-like wave motions that aare perpendicular
to on
ne another, an
n electric wave
e and a magne
etic wave (Fig
g. 4-4). Both waves
w
have eq ual energy. A
sourrce will producce multiple wav
ves of oscillat ing electromagnetic radiatio
on at any givenn time, so that
multiiple electric an
nd magnetic waves
w
are emi tted. These ellectromagnetic
c radiation waaves travel in
multiiple orientation
ns that are ran
ndomly disperrsed in the res
sulting circular beam of radiaation. Discuss
sions
of intteractions with
h light sources
s are predomin
nantly focused
d on the electrric componentt of these wav
ves,
to wh
hich we restricct ourselves fo
or the remaind
der of this secttion and in the
e Optical Rotat
atory Dispersio
on
sectiion and Circular Dichroism section.
s
Passsing light throu
ugh a polarizin
ng prism such as a Nicol priism sorts the randomly
r
distrributed vibratio
ons
of ele
ectric radiation
n so that only those vibratio
ons occurring in
i a single plane are passedd (Fig. 4-23). The
T
veloccity of this plane-polarized light can beco me slower or faster as it passes through a sample, in a
manner similar to that discussed
d for refraction
n. This change
e in velocity re
esults in refracction of the
polarrized light in a particular dire
ection for an o
optically active
e substance. A clockwise rootation in the planar
p
light,, as observed looking into th
he beam of po
olarized light, defines
d
a subs
stance as dexxtrorotatory. When
W
the p
plane of light iss rotated by th
he sample in a counterclock
kwise manner,, the sample iss defined as
a levvorotatory substance. A dex
xtrorotatory su bstance, whic
ch may be thou
ught of as rotaating the beam
m to
the rright, producess an angle of rotation,
r
α, tha
at is defined as positive, whereas the levoorotatory
subsstance, which rotates the be
eam to the left , has an α tha
at is defined as
s negative. Moolecules that have
h
an asymmetric cen
nter (chiral) an
nd therefore la
ack symmetry about a single
e plane are opptically active,
wherreas symmetric molecules (achiral)
(
are o ptically inactiv
ve and conseq
quently do not rotate the pla
ane of
polarrized light. Op
ptical activity can
c be conside
ered as the intteraction of a plane-polarizeed radiation with
electtrons in a mole
ecule to produ
uce electronic polarization. This
T
interactio
on rotates the direction of
vibra
ation of the rad
diation by altering the electrric field. A pola
arimeter is use
ed to measuree optical
activvity. Figure 4-2
24 illustrates how
h
a polarime
eter can be us
sed to measurre these phenoomena.
Opticcal rotation, α, depends on the density off an optically active
a
substance because eaach molecule
provides an equal but small con
ntribution to the
e rotation. The
especific rotattion, {αtλ} at a sspecified
temp
perature t and wavelength λ (usually the D line of sodiu
um), is charactteristic for a puure, optically
activve substance. It is given by the
t equation
wherre l is the leng
gth in decimete
ers (dm)* of th
he light path th
hrough the sam
mple and g is the number of
gram
ms of optically active substance in v millilitter of volume. If the substan
nce is dissolveed in a solution,
the ssolvent as well as the conce
entration shou ld be reported
d with the spec
cific rotation. T
The specific
rotattion of a moleccule is indicative of its moleccular structure
e
P.10
04
and as such shoulld not change when measurred under the same conditio
ons in any labb. Organic chemists
use sspecific rotatio
on as a tool to
o help confirm the identity off a synthesized compound w
whose molecu
ular
prop
perties are alre
eady defined. Other measurrements are offten performed
d, but this is a quick test to
checck the identity and purity of a reaction pro
oduct. The spe
ecific rotations of some druggs are given
in Ta
able 4-9. The subscript
s
D on
n [α] indicates that the meas
surement of specific rotationn is made at a
wave
elength, λ, of 589
5 nm for sodium light. Wh
hen the conce
entration is nott specified, as in Table 4-9, the
conccentration is offten assumed to be 1 g per milliliter of solvent. Howeve
er, [α] values sshould be repo
orted
at sp
pecific tempera
atures and concentration in cluding solven
nt. Data missing this informaation should be
b
treatted with cautio
on because the
e specific con ditions are no
ot known. The specific rotatioon of steroids,
199
Dr. Murtadha Alshareifi e-Library
carbohydrates, am
mino acids, and other compo
ounds of biolo
ogic importanc
ce are given inn the CRC
Hand
dbook.35
Fig.. 4-24. Scheematic view
w of a polariimeter.
Table 4--9 Specific Rotations
R
Dru
ug
[α]D (deg)
Temperatu
ure (°C)Sollvent
A
Ampicillin
+2283
20
W
Water
A
Aureomycinn
+2296
23
W
Water
B
Benzylpeniccillin
+3305
25
W
Water
C
Camphor
+441 to +43
25
E
Ethanol
C
Colchicine
–1121
17
C
Chloroform
m
C
Cyanocobalamin
–660
23
W
Water
E
Ergonovine
–116
20
PPyridine
N
Nicotine
–1162
20
PPure liquid
200
Dr. Murtadha Alshareifi e-Library
Propoxyphene
+67
25
Chloroform
Quinidine
+230
15
Chloroform
Reserpine
–120
25
Chloroform
Tetracycline hydrochloride
–253
24
Methanol
d-Tubocurarine chloride
+190
22
Water
Yohimbine
+51 to +62
20
Ethanol
Optical Rotatory Dispersion
Optical rotation changes as a function of the wavelength of light, and optical rotatory dispersion (ORD) is
the measure of the angle of rotation as a function of the wavelength. Recall that this is light in the UV–
Vis range and that different wavelengths of light have different energies, which may result in changes in
absorption patterns of molecules due to their
P.105
electronic structure. Varying the wavelength of light changes the specific rotation for an optically active
substance because of the electronic structure of the molecule. A graph of specific rotation versus
wavelength shows an inflection and then passes through zero at the wavelength of maximum absorption
of polarized light as shown in Figure 4-25. This change in specific rotation is known as the Cotton effect.
By convention, compounds whose specific rotations show a maximum beforepassing through zero as
the wavelength of polarized light becomes smaller are said to show a positive Cotton effect, whereas if
{α} shows a maximum after passing through zero (under the same conditions of approaching shorter
wavelengths), the compound shows a negative Cotton effect. Enantiomers can be characterized by the
Cotton effect, as shown in Figure 4-26. In addition, ORD is often useful for the structural examination of
organic compounds. Detailed discussions of ORD are given by Campbell and Dwek12 and Crabbe.36
201
Dr. Murtadha Alshareifi e-Library
Fig.. 4-25. The Cotton effeect. Variatioon of the ang
gle of rotatiion (solid linne) in the
vicinity of an absorption
a
band
b
of polaarized light (dashed linee). (Modifieed from W. S.
Breyy, Physical Chemistry and Its Biollogical App
plications, Academic
A
Prress, New
Yorrk, 1978, p. 330.)
Cirrcular Dic
chroism
Plane-polarized lig
ght is describe
ed as the vecto
or sum of two circularly pola
arized componnents. Circularly
polarrized light hass an electric ve
ector that spira
als around the
e direction of propagation.
p
Inn plane-polariz
zed
light,, there can be two such vec
ctors spiraling in the oppositte direction, a right-handed and a left-han
nded
comp
ponent. Handedness is use
ed as a conven
ntion to conce
eptualize circularly polarizedd light. To visualize
hand
dedness, take both hands and
a hold them with the finge
ers pointed stra
aight up and eeach thumb
pointted toward your nose. Now curl the fingerrs around in a loose fist, holding the thum
mbs steady. Th
he
thum
mb on each ha
and indicates the light travel ing toward your nose, and the
t curling of tthe fingers sho
ows
how the light vecto
ors are spiralin
ng into two se
eparate vectors
s, the left and right circular ddirections. For an
opticcally active sub
bstance, the values
v
of the in
ndex of refrac
ction, n, of the two vectors caannot be the
same
e.
Beca
ause of chiralitty, one of the two vectors m
may be absorb
bed differently at one waveleength of light,
wherreas the conve
erse may be true at anotherr wavelength. This differenc
ce changes thee relative rate
e at
whicch the polarize
ed light spirals about its direcction of propa
agation. Likewiise, the speedds of the two
comp
ponents of polarized light be
ecome unequ al as they pas
ss through an optically activve substance that
t
is ca
apable of abso
orbing light ove
er a selected w
wavelength ra
ange. This is th
he same as saaying that the two
comp
ponents of polarized light ha
ave different a
absorptivities at
a a particular wavelength oof light.
Concceptually, it is important to realize
r
that at the source, th
he circularly po
olarized light iss traveling at the
t
same
e speed in a circle.
c
When one componen
nt of light, say the left-hande
ed componentt, has energy
abso
orbed, it will slow down in co
ontrast to the rright-hand com
mponent, strettching the circcle into an ellip
pse.
The converse is trrue as well. Th
his can also be
e illustrated by
y using the hand conventionn. When circullarly
polarrized light is caused
c
to beco
ome ellipticallyy polarized, it is termed circular dichroism
m (CD).12,37
202
Dr. Murtadha Alshareifi e-Library
Fig.. 4-26. Optiical rotary dispersion
d
cuurves for (aa) cis-10-meethyl-2-decaalone and
(b) ttrans-10-meethyl-2-decalone. Notee the positiv
ve effect in (a)
( and the nnegative efffect
in (bb). (From C.
C Djerassi, Optical Rottatory Dispeersion, McG
Graw-Hill, N
New York,
1960. With perrmission.)
The Cotton effect is the name given to the un
nequal absorpttion (elliptic efffect) of the tw
wo components
s of
circu
ularly polarized
d light by a mo
olecule in the w
wavelength re
egion near an absorption baand. Circular
dichrroism spectra are plots of molar
m
ellipticityy, [θ], which is proportional to the differencce in absorptiv
vities
betw
ween the two components
c
off circularly pollarized light, against
a
the wav
velength of ligght. By traditio
onal
convvention, the molar ellipticity is given by
wherre [ψ] is the sp
pecific ellipticitty, analogous to the specific
c rotation, M is
s the moleculaar weight, and
d
εεL a
and εεR are the
e molar absorp
ptivities for the
e left and rightt components
P.10
06
of cirrcularly polarizzed light at a selected
s
wave
elength. Perrin
n and Hart38 review the appplications of CD to
interractions betwe
een small mole
ecules and ma
acromolecules
s, and the theo
ory is describeed by Campbe
ell
and Dwek.12 Circular dichroism
m has its greate
est application
n in the charac
cterization of cchiral
macrromolecules, in
i particular proteins that arre utilized in biotechnology-b
based therapeeutics. Second
dary
203
Dr. Murtadha Alshareifi e-Library
structure in proteins can easily be defined by close investigation of CD spectra.37,39,40,41The
magnitude of the absorption of the components of circularly polarized light is distinctly different for αhelices, β-sheets, and β-turns. By measuring the CD spectrum of proteins whose secondary structure is
known, one can compile database and use it to deconvolve the secondary structural composition of an
unknown protein. The CD spectra for a protein can also serve as standard and be used to determine
whether effects during production or formulation are altering the proteins' folding patterns, which could
dramatically decrease its activity.
Circular dichroism has also been applied to the determination of the activity of penicillin, as described by
Rasmussen and Higuchi.42 The activity was measured as the change in the CD spectra of penicillin
after addition of penicillinase, which enzymatically cleaves the β-lactam ring to form the penicillate ion,
as shown for benzylpenicillin in Figure 4-27. Typical CD spectra for benzylpenicillin and its hydrolysis
product are shown in Figure 4-27. The direct determination of penicillins by CD and the distinction of
penicillins from cephalosporins by their CD spectra has been described by Purdie and Swallows.43 The
penicillins all have single positive CD bands with maxima at 230 nm, whereas the cephalosporins have
two CD bands, a positive one with a maximum at 260 nm and a negative one with maximum at 230 nm
(wavelengths for maxima are given to within ±2 nm). This permits easy differentiation between
penicillins and cephalosporins by CD spectropolarimetry.
Electron and Neutron Scattering and Emission Spectroscopy
Electrons can flow across media in response to charge separation, and do so at certain frequencies and
wavelengths depending on the magnitude of the charge separation. Neutrons can also be generated
and flow in a similar manner. The exact mathematical and theoretical description of this flow is well
beyond the scope of this text. However, the concept that they can flow is important to the discussion of
the following subjects. It must also be noted that whereas electron and neutron waves are not
traditionally thought of as electromagnetic radiation, they can be used to determine important molecular
properties and are included in this text. The main distinction between electrons and neutrons versus
electromagnetic radiation is that electrons and neutrons have a finite resting mass, in contrast to
electromagnetic radiation, which has a zero finite resting mass.12
204
Dr. Murtadha Alshareifi e-Library
Fig.. 4-27. Circuular dichroiism spectra of benzylpeenicillin and
d its penicillloic acid
deriivative. (Froom C. E. Raassmussen aand T. Higu
uchi, J. Pharrm. Sci. 60, 1616, 1971
1.
Witth permissioon.)
Seve
eral technique
es that are pha
armaceuticallyy relevant also
o employ electron or neutronn beams, inclu
uding
smalll-angle neutro
on scattering and
a scanning and transmiss
sion electron microscopy
m
(S
SEM and TEM,
respectively). Scatttering of enerrgy waves ressults when a lo
oss of energy occurs
o
throug h interaction with
w a
medium, as occurs in refraction. Scattering iss usually meas
sured at an an
ngle θ from thee incident beam of
enerrgy and does not
n measure transmittance o
of the energy wave. Both ellectron and neeutron waves play
an im
mportant role in
i identifying some
s
of the m
molecular prope
erties of comp
pounds and exxcipients. They
y are
discu
ussed briefly here.
h
Scan
nning and tran
nsmission elec
ctron microsco
opy are two techniques that are routinely employed in
pharrmaceutical an
nd biologic sciences to prod
duce high-reso
olution images
s of surfaces. LLet us conside
er
SEM
M. In general, the
t image gen
nerated with S
SEM is produce
ed by electron
n wave emissioon from a filam
ment
and bombardmentt of the sample
e, where the e
electrons unde
ergo refraction
n after interacttion with the
samp
ple surface. As
A the electron
ns pass throug
gh the sample surface, they continuously undergo refra
action
until they are refra
acted back outt of the sample
e surface at a lower energy and are deteccted. In the ca
ase of
refra
action, the incident electron beam interactts with electrons in the sample's atomic eelectron cloud,
whicch results in altteration of its path after coll ision. For hea
avier atoms, th
he electron clooud is dense, and
a
the in
ncident electro
ons from the beam
b
undergo
o greater refra
action. In SEM techniques, tthe sample is often
o
sputtter coated (sp
prayed) with go
old to provide a greater elec
ctron density at
a the surface and prevent the
beam
m from passing through the sample. In TE
EM, the electro
ons are transm
mitted throughh the sample. The
refra
acted electronss
P.10
07
205
Dr. Murtadha Alshareifi e-Library
gene
erate a high-re
esolution snap
pshot of the sa
ample surface and provide information abbout its contou
ur and
size;; however, it does
d
not offer molecular info
ormation.
Mole
ecular (atomic) information can
c be obtaine
ed from monitoring the enerrgy, which is inn the form of xx
rays emitted when
n the electrons
s interact with the molecules
s. In general, upon
u
an inelasstic collision th
he
incid
dental electron
n beam can als
so knock out g
ground-state electrons
e
in the molecules. T
This is known as
Auge
er electron em
mission. The outer-shell elecctrons in each atom then relax to the grouund state, whic
ch
results in the emisssion of energy
y in the form o
of x-ray fluores
scence. For each atom, thee atomic structture
is diffferent, and the relaxation energies chang
ge on the basiis of the individual atoms. T
The emission of
o
these
e x-rays can be
b captured us
sing technique
es such as energy-dispersiv
ve x-ray analyssis (EDAX) an
nd
wave
elength-disperrsive x-ray ana
alysis. These techniques prrovide informa
ation about thee molecular
comp
position by de
etecting the diffferent x-ray frrequencies and wavelengths emitted duri ng an SEM ru
un.
Enerrgy-dispersive
e x-ray analysis captures the
e frequency pa
atterns of the emitted x-ray waves, where
eas
wave
elength-disperrsive x-ray ana
alysis measurres the x-ray wavelengths
w
frrom the atomss present. The
ese
emisssions can be measured acrross a linear re
region of the sample or in the whole sampple. They are
qualiitative but can
n be quantitativ
ve in nature, d
depending on the type of an
nalysis perform
med.
A go
ood example of
o the utility of this technique
e is the examin
nation of a coa
ated tablet thaat contains an ironbase
ed drug in its core
c
(Fig. 4-28
8). If an SEM a
and a linear EDAX analysis are performeed across the
surfa
ace of a split ta
ablet, EDAX demonstrates
d
that the iron is
s only presentt in the center of the tablet and
a
not in
n the coated region.
r
SEM provides
p
a pictture of the surfface of the split tablet that ccan be used to
o look
at formulation conditions. If a sim
milar tablet is then placed in
n water for 1 hr
h and subjecteed to the same
analyysis, one migh
ht observe a lo
ower amount o
of iron in the core,
c
and iron present in thee coating, with
h an
imag
ge that shows the difference
e in the tablet'ss core structure upon waterr exposure. Thhis could provide
very valuable inforrmation to a fo
ormulator, who
o may want to
o compare sev
veral different ccoatings to co
ontrol
the rrelease of the iron-based drug or look at h
how different polymorphs
p
(C
Chapter 2) of tthe compound
d
migh
ht behave in th
he same formu
ulation. In factt, this type of analysis
a
is rou
utinely perform
med for many types
t
of formulations inccluding micro- and nanosph eres. The info
ormation about physical propperties of state
(Cha
apter 2) and molecular
m
properties is very iimportant. The
ere are also many
m
other tecchniques
employing similar methods that are being wid
dely incorporatted into the ind
dustrial characcterization and
d
formulation develo
opment of drug
gs.
Fig.. 4-28. A cooated iron-containing taablet that was
w split, sho
owing initiaally a smootth
surfface (left) annd then, afteer exposuree to water fo
or 1 hr, erosiion and a roougher surfaace
(rigght). The linne through th
he tablets shhows how a hypotheticcal scanningg electron
206
Dr. Murtadha Alshareifi e-Library
microscope image combined with an energy-dispersive x-ray analysis (EDAX) linear
profile focused on the iron x-ray fluorescence emission frequency is performed across
the tablet. In the first tablet, the iron is uniformly contained in the center and the
magnitude of the deviation of the line implies a higher iron concentration in the core.
The image shows a smooth surface. After exposure to water for 1 hr, the iron and
some of the excipients have migrated from the core, leaving a rough contour and a
lower EDAX level. The coating has a higher level of iron by this scan, which could be
due to dissolution and the effect of diffusion rates, as discussed in later chapters.
Small-angle neutron scattering is another technique that can be employed to illustrate the physical
properties of a molecule. It is related to techniques like small-angle x-ray scattering and small-angle light
scattering. Collectively, these techniques can be used to give information about the size (even molecular
weights for large polymeric and protein molecules), shape, and even orientation of components in a
sample.44 Small-angle neutron scattering is primarily used for the characterization of polymers, in
particular dendrimers, and colloids.
Chapter Summary
The goal of this chapter was to provide a foundation for understanding the physical properties
of molecules including methods to make those determinations. Students should understand
the nature of intra- and intermolecular forces, molecular energetics of these forces, and their
relevance to different molecules. Another important aspect that was covered was the
determination of these physical properties using a variety of atomic and molecular
spectroscopic techniques. The student should also appreciate the differences in the
techniques with respect to the identification and detection of pharmaceutical agents.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. Y. Bentor, Chemical Element.com—Carbon. Available at
http://www.chemicalelements.com/elements/c.html. Accessed August 28, 2008.
2. J. S. Richardson, Adv. Protein Chem. 34, 167, 1981.
3. C. K. Mathews and K. E. van Holde, Biochemistry, Benjamin Cummings, Redwood City, CA, 1990,
Chapter 5.
4. W. G. Hol, Adv. Biophys. 19, 133, 1985.
5. P. R. Bergethon and E. R. Simons, Biophysical Chemistry: Molecules to Membranes, SpringerVerlag, New York, 1990, Chapter 10.
6. W. G. Gorman and G. D. Hall, J. Pharm. Sci. 53, 1017, 1964.
7. S. N. Pagay, R. I. Poust, and J. L. Colaizzi, J. Pharm. Sci. 63, 44, 1974.
P.108
8. P. A. Kollman, in M. E. Wolff (Ed.), The Basis of Medicinal Chemistry, Burger's Medicinal Chemistry,
Part I, Wiley, New York, 1980, Chapter 7.
9. V. I. Minkin, O. A. Osipov, and Y. A. Zhadanov, Dipole Moments in Organic Chemistry, Plenum Press,
New York, 1970.
10. D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, University Science
Books, Sausalito, CA, 1997.
11. W. S. Brey, Physical Chemistry and Its Biological Applications, Academic Press, New York, 1978,
Chapter 9.
12. I. D. Campbell and R. A. Dwek, Biological Spectroscopy, Benjamin/ Cummings, Menlo Park, CA,
1984.
13. S. G. Jivani and V. J. Stella, J. Pharm. Sci. 74, 1274, 1985.
207
Dr. Murtadha Alshareifi e-Library
14. A. L. Gutman, V. Ribon, and J. P. Leblanc, Anal. Chem. 57, 2344, 1985.
15. S. G. Schulman and B. S. Vogt, in J. W. Munson (Ed.), Pharmaceutical Analysis: Modern Methods,
Part B, Marcel Dekker, New York, 1984, Chapter 8.
16. A. H. Zewail, J. Phys. Chem. A 104, 5660, 2000.
17. B. Valeur, Molecular Fluorescence Principles and Applications, Wiley-VCH Verlag, Weinheim,
Germany, 2002, Chapter 11.
18. S. G. Schulman and R. J. Sturgeon, in J. W. Munson (Ed.), Pharmaceutical Analysis: Modern
Methods, Part A, Marcel Dekker, New York, 1984, Chapter 4.
19. D. I. Chapman and M. S. Moss, in E. G. C. Clarke (Ed.), Isolation and Identification of Drugs,
Pharmaceutical Press, London, 1969, pp. 103–122.
20. J. R. Durig (Ed.), Chemical, Biological, and Industrial Applications of Infrared Spectroscopy, Wiley,
New York, 1985; R. A. Hoult, Development of the Model 1600 FT-IR Spectrophotometer, Norwalk,
Conn, 1983; R. J. Bell, Introductory Fourier Transform Spectroscopy, Academic Press, New York, 1972;
P. R. Griffith,Chemical Infrared Fourier Transform Spectroscopy, Wiley, New York, 1975; R. J.
Markovich and C. Pidgeon, Pharm. Res. 8, 663, 1991.
21. A. L. Smith, Applied Infrared Spectroscopy, Wiley, New York, 1979.
22. H. H. Willard, L. L. Merritt, J. A. Dean, and F. A. Settle, Instrumental Methods of Analysis, 6th Ed.,
Van Nostrand, New York, 1981, Chapter 7.
23. C. D. Bugay and A. William, in H. G. Brittain (Ed.), Physical Characterization of Pharmaceutical
Solid, Marcel Dekker, New York, 1995, pp. 65–79.
24. H. M. Swartz, J. R. Bolton, and D. C. Borg (Eds.), Biological Applications of Electron Spin
Resonance, Wiley, New York, 1972.
25. P. M. Plonka and M. Elas, Curr. Top. Biophys. 26, 175–189, 2002.
26. R. Galimzyanovich, L. Ivanovna, T. I. Vakul'skaya, and M. G. Voronkov, Electron Paramagnetic
Resonance in Biochemistry and Medicine, Kluwer, Dordrecht, Netherlands, 2001.
27. H. Friebolin, Basic One- and Two-Dimensional NMR Spectroscopy, 2nd Ed., VCH, Weinheim,
Germany, 1993.
28. H. Gunther, NMR Spectroscopy, 2nd Ed., Wiley, New York, 1998.
29. T. C. Farrar, Anal. Chem. 59, 749A, 1987.
30. A. S. V. Burgen and J. C. Metcalfe, J. Pharm. Pharmacol. 22, 153, 1970.
31. D. K. Lawrence and E. W. Gill, Mol. Pharmacol. 11, 595, 1975.
32. J. Tamir and D. Lichtenberg, J. Pharm. Sci. 72, 458, 1983.
33. D. M. Rackham, Talanta 23, 269, 1976.
34. K. Wüthrich, NMR of Proteins and Nucleic Acids, Wiley, New York, 1986.
35. D. Lide (Ed.), CRC Handbook of Chemistry and Physics, 85th Ed., CRC Press, Boca Raton, Fl.,
2004.
36. P. Crabbe, ORD and CD in Chemistry and Biochemistry, Academic Press, New York, 1972.
37. N. Berova, K. Nakanishi, and R. Woody, Circular Dichroism: Principle and Applications, VCH, New
York, 2000.
38. J. H. Perrin and P. A. Hart, J. Pharm. Sci. 59, 431, 1970.
39. M. C. Manning, J. Pharm. Biomed. Anal. 7, 1103, 1989.
40. A. Perczel, K. Park, and G. D. Fasman, Anal. Biochem. 203, 83, 1992.
41. G. D. Fasman (Ed.), Circular Dichroism and the Conformational Analysis of Biomolecules, Plenum
Press, New York, 1996.
42. C. E. Rasmussen and T. Higuchi, J. Pharm. Sci. 60, 1608, 1971.
43. N. Purdie and K. A. Swallows, Anal. Chem. 59, 1349, 1987.
44. S. M. King, in R. A. Pethrick and J. V. Dawkins (Eds.), Modern Techniques for Polymer
Characterisation, Wiley, New York, 1999, Chapter 7.
Recommended Readings
C. R. Cantor and P. R. Schimmel, Biophysical Chemistry Part II: Techniques for the Study of Biological
Structure and Function, W H Freeman, New York, 1980.
208
Dr. Murtadha Alshareifi e-Library
J. B. Lambert, H. F. Shurvell, D. Lightner, and R. G. Cooks, Organic Structural Spectroscopy, Prentice
Hall, Upper Saddle River, NJ, 1998.
S. G. Schulman, Fluorescence and Phosphorescence Spectroscopy, Physicochemical Principles and
Practice, Pergamon Press, Oxford, 1977.
R. M. Silverstein, F. X. Webster, and D. J. Kiemle, Spectrometric Identification of Organic Compounds,
7th Ed., John Wiley and Sons, Hoboken, NJ, 2005.
D. W. Turner, A. D. Baker, C. Baker, and C. R. Brundle, Molecular Photoelectron Spectroscopy, WileyInterscience, New York, 1970.
Chapter Legacy
Fifth Edition: published as Chapter 4. Updated by Gregory Knipp, Dea Herrera-Ruiz, and
Hugo Morales-Rojas.
Sixth Edition: published as Chapter 4. Updated by Patrick Sinko.
209
Dr. Murtadha Alshareifi e-Library
5N
Nonelec
ctrolyte
es
Cha
apter Objec
ctives
At th
he conclusion of this chapter the s
student sho
ould be able to:
1.
2.
3.
4.
5.
6.
7.
Identify and describe
e the four co lligative prop
perties of non
nelectrolytess in solution.
Understand the vario
ous types of pharmaceuttical solutions
s.
Calculatte molarity, normality,
n
mo
olality, mole fraction,
f
and percentage expressions
s.
Calculatte equivalentt weights.
Define id
deal and real solutions ussing Raoult's
s and Henry's laws.
Use Rao
oult's law to calculate
c
parrtial and total vapor press
sure.
Calculatte vapor pres
ssure lowerin
ng, boiling po
oint elevation
n, freezing pooint lowering
g,
and presssure for solutions of non
nelectrolytes.
8. Use colligative prope
erties to dete
ermine molec
cular weight.
Intrroduction
In this chapter, the
e student will begin
b
to learn about pharma
aceutical syste
ems. In the phharmaceutical
scien
nces, a system
m is generally considered to
o be a bounde
ed space or an
n exact quantitty of a materia
al
subsstance. Material substances
s can be mixed
d together to form
f
a variety of pharmaceuutical mixtures
s (or
dispe
ersions) such as true solutio
ons, colloidal d
dispersions, and
a coarse dis
spersions. A di
dispersion cons
sists
of at least two pha
ases with one or more dispe
ersed (internall) phases conttained in a sin gle continuous
(exte
ernal) phase. The
T term phas
se is defined a
as a distinct ho
omogeneous part of a systeem separated by
defin
nite boundaries from other parts
p
of the sysstem. Each ph
hase may be consolidated
c
i nto a contiguo
ous
masss or region, su
uch as a single
e tea leaf floatting in water. A true solution
n is defined ass a mixture of two
or more componen
nts that form a homogeneou
us molecular dispersion,
d
in other words, a one-phase
syste
em. In a true solution,
s
the suspended parrticles complettely dissolve and
a are not larrge enough to
scattter light but arre small enoug
gh to be evenl y dispersed re
esulting in a homogeneous appearance. The
T
diam
meter of particles in coarse dispersions
d
is greater than ~500
~
nm (0.5 µm). Two com
mmon
pharrmaceutical co
oarse dispersio
ons are emulssions (liquid–liquid dispersio
ons) and suspeensions (solid
d–
liquid
d dispersions)). A colloidal dispersion reprresents a systtem having a particle
p
size inntermediate
betw
ween that of a true solution and
a a coarse d
dispersion, rou
ughly 1 to 500
0 nm. A colloiddal dispersion may
be co
onsidered as a two-phase (heterogeneou
(
us) system under some circumstances. H
However, it ma
ay
also be considered
d as a one-phase system (h
homogeneous
s) under other circumstancees. For example,
lipossomes or micro
ospheres in an aqueous de
elivery vehicle are considere
ed to be heteroogeneous colloidal
dispe
ersions becau
use they consist of distinct p
particles consttituting a separate phase. O
On the other ha
and,
a collloidal dispersion of acacia or
o sodium carrboxymethylce
ellulose in wate
er is homogenneous since it does
not d
differ significan
ntly from a sollution of sucro
ose. Therefore
e, it may be co
onsidered as a single-phase
e
syste
em or true solution.1 Anothe
er example off a homogeneo
ous colloidal dispersion
d
thatt is considered
d to
be a true solution is drug-polymer conjugatess since they ca
an completely dissolve in waater.
Key Co
oncept
Pha
armaceutica
al Dispersio
ons
Whe
en two materrials are mixe
ed, one beco
omes dispers
sed in the oth
her. To classsify a
pharrmaceutical dispersion,
d
only
o
the size of the dispersed phase and
a not its coomposition is
s
conssidered. The
e two compon
nents may be
ecome dispe
ersed at the molecular
m
levvel forming a true
solu
ution. In other words, the dispersed ph
hase comple
etely dissolve
es, cannot sccatter light, and
cann
not be visuallized using microscopy.
m
I f the dispers
sed phase is in the size raange of 1 to 500
nm, it is conside
ered to be a colloidal
c
disp
persion. Com
mmon examples of colloiddal dispersion
ns
inclu
ude blood, lip
posomes, an
nd zinc oxide
e paste. If the
e particle size
e is greater thhan 500 nm (or
0.5 µ
µm), it is con
nsidered to be
b a coarse d
dispersion. Two common examples oof coarse
disp
persions are emulsions
e
an
nd suspensio
ons.
210
Dr. Murtadha Alshareifi e-Library
This chapter focuses on molecular dispersions, which are also known as true solutions. A solution
composed of only two substances is known as a binary solution, and the components or constituents
are referred to as the solvent and the solute. Commonly, the terms component and constituent are used
interchangeably to represent the pure chemical substances that make up a solution. The number of
components has a definite significance in the phase rule. The constituent present in the greater amount
in a binary solution is arbitrarily designated as the solvent and the constituent in the lesser amount as
the solute. When a solid is dissolved in a liquid, however, the liquid is usually taken as the solvent and
the solid as the solute, irrespective of the relative amounts of the constituents. When water is one of the
constituents of a liquid mixture, it is usually considered the solvent. When dealing with mixtures of
liquids that are miscible in all
P.110
proportions, such as alcohol and water, it is less meaningful to classify the constituents as solute and
solvent.
Physical Properties of Substances
The physical properties of substances can be classified as colligative, additive, and constitutive. Some
of the constitutive and additive properties of molecules were considered in Chapter 4. In the field of
thermodynamics, physical properties of systems are classified as extensive properties, which depend on
the quantity of the matter in the system (e.g., mass and volume), and intensive properties, which are
independent of the amount of the substances in the system (e.g., temperature, pressure, density,
surface tension, and viscosity of a pure liquid).
Colligative properties depend mainly on the number of particles in a solution. The colligative properties
of solutions are osmotic pressure, vapor pressure lowering, freezing point depression, and boiling point
elevation. The values of the colligative properties are approximately the same for equal concentrations
of different nonelectrolytes in solution regardless of the species or chemical nature of the constituents.
In considering the colligative properties of solid-in-liquid solutions, it is assumed that the solute is
nonvolatile and that the pressure of the vapor above the solution is provided entirely by the solvent.
Additive properties depend on the total contribution of the atoms in the molecule or on the sum of the
properties of the constituents in a solution. An example of an additive property of a compound is the
molecular weight, that is, the sum of the masses of the constituent atoms. The masses of the
components of a solution are also additive, the total mass of the solution being the sum of the masses of
the individual components.
Constitutive properties depend on the arrangement and to a lesser extent on the number and kind of
atoms within a molecule. These properties give clues to the constitution of individual compounds and
groups of molecules in a system. Many physical properties may be partly additive and partly constitutive.
The refraction of light, electric properties, surface and interfacial characteristics, and the solubility of
drugs are at least in part constitutive and in part additive properties; these are considered in other
sections of the book.
Types of Solutions
A solution can be classified according to the states in which the solute and solvent occur, and because
three states of matter (gas, liquid, and crystalline solid) exist, nine types of homogeneous mixtures of
solute and solvent are possible. These types, together with some examples, are given in Table 5-1.
When solids or liquids dissolve in a gas to form a gaseous solution, the molecules of the solute can be
treated thermodynamically like a gas; similarly, when gases or solids dissolve in liquids, the gases and
the solids can be considered to exist in the liquid state. In the formation of solid solutions, the atoms of
the gas or liquid take up positions in the crystal lattice and behave like atoms or molecules of solids.
Table 5-1 Types of Solutions
211
Dr. Murtadha Alshareifi e-Library
Solute
Solvent
Example
Gas
Gas
Air
Liquid
Gas
Water in oxygen
Solid
Gas
Iodine vapor in air
Gas
Liquid
Carbonated water
Liquid
Liquid
Alcohol in water
Solid
Liquid
Aqueous sodium chloride solution
Gas
Solid
Hydrogen in palladium
Liquid
Solid
Mineral oil in paraffin
Solid
Solid
Gold—silver mixture, mixture of alums
The solutes (whether gases, liquids, or solids) are divided into two main
classes: nonelectrolytes and electrolytes. Nonelectrolytes are substances that do not ionize when
dissolved in water and therefore do not conduct an electric current through the solution. Examples of
nonelectrolytes are sucrose, glycerin, naphthalene, and urea. The colligative properties of solutions of
nonelectrolytes are fairly regular. A 0.1-molar (M) solution of a nonelectrolyte produces approximately
the same colligative effect as any other nonelectrolytic solution of equal concentration. Electrolytes are
substances that form ions in solution, conduct electric current, and show apparent “anomalous”
colligative properties; that is, they produce a considerably greater freezing point depression and boiling
point elevation than do nonelectrolytes of the same concentration. Examples of electrolytes are
hydrochloric acid, sodium sulfate, ephedrine, and phenobarbital.
Electrolytes may be subdivided further into strong electrolytes and weak electrolytes depending on
whether the substance is completely or only partly ionized in water. Hydrochloric acid and sodium
sulfate are strong electrolytes, whereas ephedrine and phenobarbital are weak electrolytes. The
classification of electrolytes according to Arrhenius and the discussion of the modern theories of
electrolytes are given later in the book.
Concentration Expressions
The concentration of a solution can be expressed either in terms of the quantity of solute in a
definite volume of solution or as the quantity of solute in a definite mass of solvent or solution. The
various expressions are summarized in Table 5-2.
Molarity and Normality
Molarity and normality are the expressions commonly used in analytical work.2 All solutions of the same
molarity
P.111
212
Dr. Murtadha Alshareifi e-Library
contain the same number of solute molecules in a definite volume of solution. When a solution contains
more than one solute, it may have different molar concentrations with respect to the various solutes. For
example, a solution can be 0.001 M with respect to phenobarbital and 0.1 M with respect to sodium
chloride. One liter of such a solution is prepared by adding 0.001 mole of phenobarbital (0.001 mole ×
232.32 g/mole = 0.2323 g) and 0.1 mole of sodium chloride (0.1 mole × 58.45 g/mole = 5.845 g) to
enough water to make 1000 mL of solution.
Table 5-2 Concentration Expressions
Expression
Symbol Definition
Molarity
M,c
Moles (gram molecular weights) of solute in 1
liter of solution
Normality
N
Gram equivalent weights of solute in 1 liter of
solution
Molality
m
Moles of solute in 1000 g of solvent
Mole fraction
X,N
Ratio of the moles of one constituent (e.g., the
solute) of a solution to the total moles of all
constituents (solute and solvent)
Mole percent
Moles of one constituent in 100 moles of the
solution; mole percent is obtained by
multiplying mole fraction by 100
Percent by
weight
%
w/w
Grams of solute in 100 g of solution
Percent by
volume
%
v/v
Milliliters of solute in 100 mL of solution
Percent
weight-involume
%
w/v
Grams of solute in 100 mL of solution
Milligram
percent
—
Milligrams of solute in 100 mL of solution
Difficulties are sometimes encountered when one desires to express the molarity of an ion or radical in a
solution. A molar solution of sodium chloride is 1 M with respect to both the sodium and the chloride ion,
whereas a molar solution of Na2CO3 is 1 M with respect to the carbonate ion and 2 M with respect to the
213
Dr. Murtadha Alshareifi e-Library
sodiu
um ion becausse each mole of this salt co
ontains 2 moles of sodium io
ons. A molar ssolution of sodium
chlorride is also 1 normal
n
(1 N) with
w respect to
o both its ions; however, a molar
m
solution of sodium
carbonate is 2 N with
w respect to
o both the sodiium and the carbonate ion.
Mola
ar and normal solutions are popular in che
emistry becau
use they can be brought to a convenient
volum
me; a volume aliquot of the solution, reprresenting a known weight off solute, is eassily obtained by
b the
use o
of the burette or pipette.
Both
h molarity and normality hav
ve the disadva
antage of chan
nging value witth temperaturee because of the
expa
ansion or contraction of liquids and should
d not be used when one wis
shes to study tthe properties
s of
soluttions at variou
us temperature
es. Another diffficulty arises in the use of molar
m
and norrmal solutions for
the sstudy of prope
erties such as vapor pressurre and osmotic
c pressure, wh
hich are relateed to the
conccentration of th
he solvent. The volume of th
he solvent in a molar or a no
ormal solutionn is not usually
y
know
wn, and it varie
es for different solutions of tthe same concentration, depending uponn the solute an
nd
solve
ent involved.
Mo
olality
A mo
olal solution iss prepared in terms
t
of weigh
ht units and do
oes not have the
t disadvantaages just
discu
ussed; therefo
ore, molal conc
centration app
pears more fre
equently than molarity and nnormality in
theoretical studiess. It is possible
e to convert m olality into mo
olarity or normality if the finaal volume of th
he
soluttion is observe
ed or if the density is determ
mined. In aque
eous solutions
s more dilute thhan 0.1 M, it
usua
ally may be asssumed for pra
actical purpose
es that molalitty and molarity
y are equivaleent. For examp
ple, a
1% ssolution by we
eight of sodium
m chloride with
h a specific gra
avity of 1.0053
3 is 0.170 M aand 0.173 molal
(0.17
73 m). The following differen
nce between m
molar and molal solutions should also bee noted. If anotther
solutte, containing neither sodium
m nor chloride
e ions, is adde
ed to a certain volume of a m
molal solution of
sodiu
um chloride, th
he solution remains 1 m in ssodium chlorid
de, although the total volum
me and the weight
of the solution incrrease. Molarity
y, of course, d
decreases whe
en another solute is added because of the
incre
ease in volume
e of the solutio
on.
Mola
al solutions are
e prepared by
y adding the prroper weight of
o solvent to a carefully weigghed quantity of
the ssolute. The volume of the so
olvent can be calculated from the specific
c gravity, and tthe solvent ca
an
then be measured
d from a burettte rather than weighed.
Mo
ole Fractiion
Mole
e fraction is ussed frequently in experimenttation involvin
ng theoretical considerations
c
s because it gives
a me
easure of the relative
r
proportion of moles of each consttituent in a sollution. It is exppressed as
for a system of two
o constituents
s. Here X1 is th
he mole fractio
on of constitue
ent 1 (the subsscript 1 is
ordin
narily used as the designatio
on for the solvvent), X2 is the
e mole fraction
n of constituennt 2 (usually th
he
solutte), and n1 and
d n2 are the nu
umbers of mo
oles of the resp
pective constittuents in the ssolution. The sum
s
of the mole fractions of solute and solvent mu
ust equal
P.112
unityy. Mole fraction
n is also expre
essed in perce
entage terms by multiplying X1 or X2 by 1 00. In a solution
conta
aining 0.01 mole of solute and
a 0.04 mole
e of solvent, the mole fraction of the solutee is X2 = 0.01//(0.04
+ 0.0
01) = 0.20. Be
ecause the mo
ole fractions off the two constituents must equal 1, the m
mole fraction of
o the
solve
ent is 0.8. The
e mole percent of the solute
e is 20%; the mole
m
percent of
o the solvent iis 80%.
The manner in wh
hich mole fracttion is defined allows one to
o express the relationship
r
beetween the
number of solute and
a solvent molecules in a ssimple, direct way. In the ex
xample just givven, it is readily
seen
n that 2 of every 10 molecules in the soluttion are solute
e molecules, and
a it will be obbserved later that
many of the prope
erties of solute
es and solventts are directly related to their mole fractionn in the solutio
on.
For e
example, the partial
p
vapor pressure
p
abovve a solution brought about by the presencce of a volatile
e
solutte is equal to the
t vapor pres
ssure of the pu
ure solute multiplied by the mole fraction of the solute in
i the
soluttion.
214
Dr. Murtadha Alshareifi e-Library
Perrcentage
e Express
sions
The percentage method
m
of exprressing the co ncentration off pharmaceutic
cal solutions iss quite commo
on.
Perccentage by we
eight signifies the
t number off grams of solu
ute per 100 g of solution. A 10% by weigh
ht (%
w/w)) aqueous solu
ution of glycerrin contains 10
0 g of glycerin dissolved in enough
e
water (90 g) to mak
ke
100 g of solution. Percentage by
y volume is exxpressed as th
he volume of solute
s
in millilitters contained
d in
100 mL of the solu
ution. Alcohol (United Statess Pharmacope
eia) contains 92.3%
9
by weigght and 94.9%
% by
volum
me of C2H5OH
H at 15.56°C; that is, it conta
ains 92.3 g of C2H5OH in 10
00 g of solutioon or 94.9 mL of
C2H5OH in 100 mL
L of solution.
Calculation
ns Involviing Conc
centration
n Expres
ssions
The calculations in
nvolving the va
arious concen
ntration expres
ssions are illus
strated in the ffollowing exam
mple.
Exa
ample 5-1
Solu
utions of Fe
errous Sulffate
An a
aqueous solu
ution of exsic
ccated ferrou
us sulfate wa
as prepared by
b adding 41 .50 g of
FeS
SO4 to enough water to make
m
1000 m L of solution at 18°C. The density of tthe solution is
1.03
375 and the molecular
m
we
eight of FeSO
O4 is 151.9. Calculate
C
(a)) the molarityy; (b) the
mola
ality; (c) the mole fraction
n of FeSO4, tthe mole frac
ction of water, and the moole percent of
o
the ttwo constitue
ents; and (d)) the percenta
age by weight of FeSO4.
215
Dr. Murtadha Alshareifi e-Library
One can use the table of conversion equation
ns, Table 5-3, to convert a concentration
c
eexpression, sa
ay
mola
ality, into its va
alue in molarity
y or mole fracction. Alternativ
vely, knowing the weight, w 1, of a solventt, the
weig
ght, w2, of the solute,
s
and the molecular w
weight, M2, of the
t solute, one
e can calculatee the molarity
y, c,
or the molality, m, of the solution. As an exerccise, the reade
er should deriv
ve an expresssion
relating X1 to X2 to
o the weights w1 and w2 and
d the solute's molecular
m
weight, M2. The ddata inExamplle 51 are
e useful for de
etermining whe
ether your derrived equation
n is correct.
Equivalent Weights
One gram atom off hydrogen we
eighs 1.008 g a
and consists of
o 6.02 × 1023 atoms (Avogaadro's numberr) of
hydrrogen. This gra
am atomic weight of hydrog
gen combines with 6.02 × 10
023 atoms of flluorine and with
23
3
half o
of 6.02 × 10 atoms of oxygen. One gram
m atom of fluo
orine weighs 19 g, and 1 g aatom of oxyge
en
weig
ghs 16 g. Therrefore, 1.008 g of hydrogen combines with
h 19 g of fluorine and with hhalf of 16 or 8 g of
P.113
oxyg
gen. The quan
ntities of fluorin
ne and oxygen
n combining with
w 1.008 g of hydrogen aree referred to as
s the
equivvalent weight of the combin
ning atoms. On
ne equivalent (Eq) of fluorine (19 g) combbines with 1.00
08 g
of hyydrogen. One equivalent of oxygen (8 g) a
also combines
s with 1.008 g of hydrogen.
216
Dr. Murtadha Alshareifi e-Library
Tab
ble 5-3 Con
nversion Eq
quations forr Concentrration Term
ms
It is o
observed that 1 equivalent weight
w
(19 g) of fluorine is identical with itts atomic weigght. Not so witth
oxyg
gen; its gram equivalent
e
weight (8 g) is eq
qual to half its atomic weightt. Stated otherrwise, the atomic
weig
ght of fluorine contains
c
1 Eq of fluorine, w hereas the ato
omic weight off oxygen contaains 2 Eq. The
e
equa
ation relating these
t
atomic quantities
q
is ass follows (the equation for molecules
m
is quuite similar to that
for a
atoms, as seen
n in the next paragraph):
p
217
Dr. Murtadha Alshareifi e-Library
The number of equivalents per atomic
a
weightt, namely, 1 fo
or fluorine and 2 for oxygen,, is the
comm
mon valence of
o these eleme
ents. Many ele
ements may have
h
more tha
an one valencee and hence
seve
eral equivalentt weights, dep
pending on the
e reaction under consideration. Magnesiu m will combine
with two atoms of fluorine, and each
e
fluorine ccan combine with
w one atom
m of hydrogen. Therefore, the
valen
nce of magnesium is 2, and
d its equivalen t weight, acco
ording to equation (5-3), is oone half of its
atom
mic weight (24//2 = 12 g/Eq). Aluminum willl combine with three atoms
s of fluorine; thhe valence of
alum
minum is thereffore 3 and its equivalent we
eight is one thiird of its atomiic weight, or 227/3 = 9 g/Eq.
The concept of eq
quivalent weigh
hts not only ap
pplies to atom
ms but also extends to moleccules. The
equivvalent weight of sodium chloride is identiccal to its mole
ecular weight, 58.5 g/Eq; thaat is, the equiv
valent
weig
ght of sodium chloride
c
is the
e sum of the eq
quivalent weig
ghts of sodium
m (23 g) and chhlorine (35.5 g),
g or
58.5 g/Eq. The equivalent weigh
ht of sodium cchloride is iden
ntical to its mo
olecular weightt, 58.5 g, beca
ause
the vvalence of sod
dium and chlorrine is each 1 in the compou
und. The equivalent weight of Na2CO3 is
2numerically half off its molecularr weight. The vvalence of the
e carbonate ion
n, CO3 , is 2, and its equiva
alent
weig
ght is 60/2 = 30
0 g/Eq. Althou
ugh the valencce of sodium is
s 1, two atoms
s are present in Na2CO3,
providing a weightt of 2 × 23 g = 46 g; its equi valent weight is one half of this, or 23 g/E
Eq. The equiva
alent
weig
ght of Na2CO3 is therefore 30 + 23 = 53 g , which is one half the molecular weight. T
The relationsh
hip of
equivvalent weight to molecular weight
w
for mollecules such as
a NaCl and Na
N 2CO3 is [com
mpare equatio
on (53) fo
or atoms]
Exa
ample 5-2
Calc
culation of Equivalentt Weight
(a) W
What is the number
n
of eq
quivalents pe
er mole of K3PO4, and wh
hat is the equuivalent weig
ght of
this salt? (b) Wh
hat is the equ
uivalent weig ht of KNO3? (c) What is the
t number oof equivalentts
per mole of Ca3(PO
( 4)2, and what
w
is the e quivalent we
eight of this salt?
s
a. K3PO4 re
epresents 3 Eq/mole, an d its equivale
ent weight is
s numericallyy equal to one
e
third of its molecularr weight, nam
mely, (212 g/m
mole) ÷ (3 Eq/mole) = 700.7 g/Eq.
b. The equ
uivalent weight of KNO3 iss also equal to its molecu
ular weight, oor 101 g/Eq.
c. The num
mber of equiv
valents per m
mole for Ca3(PO
( 4)2 is 6 (i.e., three calccium ions ea
ach
with a va
alence of 2 or
o two phosp hate ions ea
ach with a valence of 3). T
The equivale
ent
weight of
o Ca3(PO4)2 is therefore one sixth of its molecular weight, or 3310/6 = 51.7
g/Eq.
P.114
For a complex saltt such as mon
nobasic potasssium phospha
ate (potassium
m acid phosphaate),
KH2P
PO4 (molecula
ar weight, 136 g), the equiva
alent weight depends on ho
ow the compouund is used. Iff it is
used
d for its potasssium content, the
t equivalentt weight is identical to its mo
olecular weighht, or 136 g. When
W
it is u
used as a bufffer for its hydrogen content,, the equivalen
nt weight is on
ne half of the m
molecular weig
ght,
136/2 = 68 g, beca
ause two hydrrogen atoms a
are present. When
W
used for its phosphatee content, the
equivvalent weight of KH2PO4 is one third of th
he molecular weight,
w
136/3 = 45.3 g, becaause the valen
nce
of ph
hosphate is 3.
As defined in Tablle 5-2, the norrmality of a so lution is the eq
quivalent weig
ght of the solutte in 1 liter of
soluttion. For NaF, KNO3, and HCl, the numbe
er of equivalen
nt weights equ
uals the numbber of molecula
ar
weig
ghts, and norm
mality is identic
cal with molariity. For H3PO4, the equivale
ent weight is onne third of the
e
mole
ecular weight, 98 g/3 = 32.67 g/Eq, assum
ming complete
e reaction, and
d a 1 N solutioon of H3PO4 is
218
Dr. Murtadha Alshareifi e-Library
prep
pared by weigh
hing 32.67 g of
o H3PO4 and b
bringing it to a volume of 1 liter with waterr. For a 1 N
soluttion of sodium
m bisulfate (sod
dium acid sulfa
fate), NaHSO4 (molecular weight
w
120 g), tthe weight of salt
s
need
ded depends on
o the species
s for which the
e salt is used. If used for sod
dium or hydroogen, the
equivvalent weight would equal the molecular weight, or 120
0 g/Eq. If the solution
s
were uused for its su
ulfate
conte
ent, 120/2 = 60
6 g of NaHSO
O4 would be w
weighed out an
nd sufficient wa
ater added to make 1 liter of
o
soluttion.
In ele
ectrolyte repla
acement thera
apy, solutions ccontaining varrious electroly
ytes are injecteed into a patie
ent to
corre
ect serious ele
ectrolyte imbalances. The co
oncentrations are usually ex
xpressed as eequivalents pe
er liter
or milliequivalents per liter. For example, the normal plasma concentratio
on of sodium iions in human
ns is
abou
ut 142 mEq/lite
er; the normal plasma conce
entration of biicarbonate ion
n, HCO3-, is 277 mEq/liter.
Equa
ation (5-4) is useful
u
for calcu
ulating the qua
antity of salts needed to pre
epare electrolyyte solutions in
n
hosp
pital practice. The
T moles in the
t numeratorr and denomin
nator of equation (5-4) may be replaced with,
w
say, liters to give
or
o molecular we
eight) is expre
essed in g/Eq, or what amouunts to the sam
me
Equivalent weight (analogous to
unitss, mg/mEq.
Exa
ample 5-3
Ca2++ in Human
n Plasma
Hum
man plasma contains
c
abo
out 5 mEq/lite
er of calcium
m ions. How many
m
milligraams of calciu
um
chlo
oride dihydratte, CaCl2 • 2H2O (molecu
ular weight 147 g/mole), are required to prepare 750
7
2
mL o
of a solution equal in Ca2+
to human plasma? The
e equivalent weight of thee dihydrate salt
s
CaC
Cl2 • 2H2O is half of its mo
olecular weig
ght, 147/2 = 73.5
7
g/Eq, or 73.5 mg/mE
Eq. Using
equa
ation (5-6), we
w obtain
Exa
ample 5-4
Equ
uivalent We
eight and Mo
olecular We
eight
Calcculate the nu
umber of equivalents per liter of potas
ssium chloride, molecularr weight 74.5
55
g/mo
ole, present in a 1.15% w/v
w solution o
of KCl.
Usin
ng equation (5-5)
(
and notting that the equivalent weight
w
of KCl is identical tto its molecu
ular
weig
ght, we obtain
Exa
ample 5-5
Sod
dium Content
Wha
at is the Na+ content in mEq/liter
m
of a solution con
ntaining 5.00 g of NaCl peer liter of
solu
ution? The molecular weig
ght and there
efore the equ
uivalent weig
ght of NaCl iss 58.5 g/Eq or
o
58.5
5 mg/mEq.
219
Dr. Murtadha Alshareifi e-Library
Ide
eal and Real
R
Soluttions
As stated earlier, the
t colligative properties of nonelectrolyte
es are ordinarily regular; on the other han
nd,
soluttions of electro
olytes show apparent devia
ations. The rem
mainder of this
s chapter relattes to solutions of
none
electrolytes, exxcept where comparison
c
wi th an electroly
yte system is desirable
d
for cclarity. Solutions of
electtrolytes are de
ealt with in Chapter 6.
An id
deal gas is deffined in Chaptter 2 as one in
n which there is no attraction
n between thee molecules, and
a it
is fou
und desirable to establish an
a ideal gas eq
quation to which the propertties of real gasses tend as th
he
pressure approach
hes zero. Con
nsequently, the
e ideal gas law
w is referred to
o as a limitingg law. It is
convvenient to defin
ne an ideal so
olution as one in which there
e is no change
e in the properrties of the
comp
ponents, other than dilution, when they a
are mixed to fo
orm the solutio
on. No heat is evolved or
abso
orbed during th
he mixing proc
cess, and the final volume of
o the solution represents ann additive property
of the individual co
onstituents. Sttated another way, no shrinkage or expan
nsion occurs w
when the
subsstances are mixed. The constitutive prope
erties, for exam
mple, the vapo
or pressure, reefractive index
x,
surfa
ace tension, and viscosity of the solution, are the weigh
hted averages
s of the properrties of the purre
indivvidual constituents.
P.115
Key Co
oncept
Idea
ality
Idea
ality in a gas implies the complete
c
abssence of attrractive forces
s, and idealityy in a solutio
on
mea
ans complete
e uniformity of
o attractive fforces. Because a liquid is a highly coondensed sta
ate,
it ca
annot be expe
ected to be devoid
d
of attrractive forces
s; neverthele
ess, if, in a m
mixture of A and
a
B molecules, the
e forces betw
ween A and A
A, B and B, and
a A and B are all of thee same orderr, the
solu
ution is consid
dered to be ideal
i
accordiing to the de
efinition just given.
g
Mixin
ng substancess with similar properties
p
form
ms ideal solutiions. For exam
mple, when 1000 mL of meth
hanol
is miixed with 100 mL of ethanoll, the final volu
ume of the sollution is 200 mL,
m and no heaat is evolved or
o
abso
orbed. The sollution is nearly
y ideal.
Whe
en 100 mL of sulfuric
s
acid is combined witth 100 mL of water,
w
howeve
er, the volumee of the solution is
abou
ut 180 mL at ro
oom temperatture, and the m
mixing is atten
nded by a cons
siderable evollution of heat; the
soluttion is said to be nonideal, or
o real. As with
h gases, some
e solutions are
e quite ideal inn moderate
conccentrations, wh
hereas others approach ide
eality only under extreme dilution.
Esc
caping Tendency
T
y3
Two bodies are in thermal equilibrium when t heir temperatu
ures are the same. If one boody is heated to a
highe
er temperature than the oth
her, heat will flo
ow “downhill” from the hotte
er to the coldeer body until bo
oth
bodie
es are again in thermal equ
uilibrium. This process is des
scribed in ano
other way by uusing the conc
cept
of esscaping tendency, and say that
t
the heat i n the hotter bo
ody has a greater escaping tendency tha
an
that in the colder one.
o
Temperature is a quan
ntitative measu
ure of the esca
aping tendenccy of heat, and
d at
therm
mal equilibrium
m, when both bodies finally have the sam
me temperature
e, the escapinng tendency off
each
h constituent iss the same in all parts of the
e system.
A qu
uantitative mea
asure of the es
scaping tende
encies of mate
erial substance
es undergoingg physical and
chem
mical transform
mations is free
e energy. For a pure substance, the free energy
e
per moole, or the mollar
free energy, provid
des a measure
e of escaping tendency; forr the constituent of a solutionn, it is the parrtial
mola
ar free energy orchemical po
otential that iss used as an expression
e
of escaping
e
tenddency. Chemic
cal
potential is discussed in Chapte
er 3. The free e
energy of 1 mole
m
of ice is grreater than thaat of liquid water at
1 atm
m above 0°C and
a is spontan
neously conve
erted into wate
er because
At 0°°C, at which te
emperature the system is in
n equilibrium, the
t molar free energies of icce and water are
a
identtical and ∆G = 0. In terms of
o escaping ten
ndencies, the escaping tend
dency of ice iss greater than the
220
Dr. Murtadha Alshareifi e-Library
esca
aping tendencyy of liquid water above 0°C,, whereas at equilibrium,
e
the
e escaping tenndencies of water
w
in bo
oth phases are
e identical.
Ide
eal Solutiions and Raoult's
s Law
The vapor pressurre of a solution
n is a particula
arly important property beca
ause it serves as a quantitative
exprression of esca
aping tendenc
cy. In 1887, Ra
aoult recogniz
zed that, in an ideal solutionn, the partial va
apor
pressure of each volatile
v
constittuent is equal to the vapor pressure
p
of the
e pure constituuent multiplied
d by
its m
mole fraction in
n the solution. Thus, for two constituents A and B,
wherre pA and pB are
a the partial vapor pressurres of the cons
stituents over the solution w
when the mole
e
fraction concentrations are XA and
a XB, respecctively. The va
apor pressures
s of the pure ccomponents
are pA° and pB°, re
espectively. Fo
or example, if the vapor pre
essure of ethylene chloride iin the pure sta
ate is
236 mm Hg at 50°°C, then in a solution
s
consissting of a mole
e fraction of 0.4 ethylene chlloride and 0.6
6
benzzene, the partiial vapor press
sure of ethylen
ne chloride is 40% of 236 mm,
m or 94.4 mm
m. Thus, in an
n
ideall solution, whe
en liquid A is mixed
m
with liqu
uid B, the vapo
or pressure off A is reduced by dilution with B
in a m
manner depen
nding on the mole
m
fractions of A and B prresent in the final solution. T
This will diminish
the e
escaping tendency of each constituent,
c
le
eading to a red
duction in the rate of escapee of the molec
cules
of A and B from th
he surface of the liquid.
Exa
ample 5-6
Parttial Vapor Pressure
P
Wha
at is the partiial vapor pressure of ben
nzene and off ethylene chloride in a soolution at a mole
m
fracttion of benze
ene of 0.6? The
T vapor pre
essure of pure benzene at
a 50°C is 2668 mm, and the
t
corresponding pA° for ethylen
ne chloride iss 236 mm. We
W have
If additional volatile components
s are present in the solution
n, each will pro
oduce a partiaal pressure above
the ssolution, which
h can be calcu
ulated from Ra
aoult's law. Th
he total pressure is the sum of the partial
pressures of all the constituents
s. In Example 5-6, the total vapor
v
pressure P is calculatted as follows:
The vapor pressurre–compositio
on curve for the
e binary syste
em benzene and ethylene chhloride at 50°C
C is
show
wn in Figure 5-1. The three lines represen
nt the partial pressure
p
of eth
hylene chloridee, the partial
pressure of benze
ene, and the to
otal
P.116
pressure of the so
olution as a fun
nction of the m
mole fraction of
o the constitue
ents.
221
Dr. Murtadha Alshareifi e-Library
Fig.. 5-1. Vaporr pressure–ccompositionn curve for an ideal bin
nary system
m.
Aerosols an
nd Raoullt's Law
Aero
osol dispenserrs have been used
u
to packa
age some drug
gs since the ea
arly 1950s. Ann aerosol conttains
the d
drug concentra
ated in a solve
ent or carrier l iquid and a prropellant mixtu
ure of the propper vapor
charracteristics. Ch
hlorofluorocarb
bons (CFCs) w
were very pop
pular propellan
nts in aerosolss until about 1989.
Since that time the
ey have been replaced in ne
early every co
ountry because
e of the negattive effects tha
at
CFC
Cs have on the
e Earth's ozone layer. Todayy, two volatile hydrocarbons
s are commonnly used as
prop
pellants in metered dose inhalers for treat ing asthma, hydrofluoroalka
ane 134a
(1,1,1,2,tetrafluoro
oethane) or hy
ydrofluoroalka ne 227 (1,1,1,,2,3,3,3-hepta
afluoropropanee) or combinattions
of the two. Early metered
m
dose inhalers comm
monly used tric
chloromonoflu
uoromethane ((CFC propella
ant
11) a
and dichlorodifluoromethane
e (CFC prope llant 12) as prropellants. CFC 11 and CFC
C 12 were use
ed in
vario
ous proportion
ns to yield the proper vapor pressure and density at roo
om temperaturre. Although still
used
d with drugs, th
hese halogena
ated hydrocarrbons are no lo
onger used in cosmetic aero
rosols and hav
ve
been
n replaced by nitrogen and unsubstituted
u
hydrocarbons
s. Since prope
ellants represeent the vast
majo
ority (>99%) off the dose delivered by a m etered dose in
nhaler, it mustt be nontoxic tto the patient.
Exa
ample 5-7
Aerrosol Vaporr Pressure
The vapor presssure of pure CFC
C
11 (mollecular weigh
ht 137.4) at 21°C
2
is p11° = 13.4
lb/in
n2 (psi) and that of CFC 12
1 (molecula
ar weight 120
0.9) is p12° = 84.9 psi. A 550:50 mixture
e by
gram
m weight of the two prope
ellants consissts of 50 ÷ 137.4 g/mole = 0.364 molle of CFC 11 and
50 g ÷ 120.9 g/m
mole = 0.414 mole of CFC
C 12. What is
s the partial pressure of C
CFCs 11 and
d 12
in th
he 50:50 mixtture, and what is the tota
al vapor pressure of this mixture?
m
Wee write
222
Dr. Murtadha Alshareifi e-Library
The total vapor pressure
p
of the mixture iss
To cconvert to ga
auge pressure (psig), one
e subtracts th
he atmosphe
eric pressure of 14.7 psi:
The psi values ju
ust given are
e measured w
with respect to zero press
sure rather thhan with respect
to th
he atmosphe
ere and are sometimes w ritten psia to
o signify abso
olute pressurre.
Real Solutio
ons
Ideality in solutions presupposes complete un
niformity of atttractive forces. Many exampples of solution
pairss are known, however,
h
in wh
hich the “cohe
esive” attractio
on of A for A ex
xceeds the “addhesive” attra
action
existting between A and B. Simillarly, the attracctive forces be
etween A and B may be greeater than thos
se
betw
ween A and A or B and B. Th
his may occurr even though the liquids are
e miscible in aall proportions.
Such
h mixtures are
e real or nonid
deal; that is, the
ey do not adh
here to Raoult's law throughoout the entire
range of composittion. Two types of deviation from Raoult's
s law are recog
gnized, negatiive
devia
ation and posiitive deviation.
Whe
en the “adhesivve” attractions
s between mo lecules of diffe
erent species exceed the “ccohesive”
attra
actions betwee
en like molecules, the vaporr pressure of the solution is less than thatt expected from
m
Raou
ult's ideal solu
ution law, and negative deviiation occurs. If the deviation is sufficientlyy great, the to
otal
vapo
or pressure cu
urve shows a minimum,
m
as o
observed in Figure 5-2, whe
ere A is chlorofform and B is
aceto
one.
The dilution of con
nstituent A by addition of B n
normally would be expected
d to reduce thee partial vapor
pressure of A; thiss is the simple dilution effectt embodied in Raoult's law. In the case off liquid pairs th
hat
show
w negative devviation from th
he law, howeve
er, the additio
on of B to A ten
nds to reduce the vapor
pressure of A to a greater exten
nt than can be accounted for by the simple dilution effecct. Chloroform
m and
aceto
one manifest such an attrac
ction for one a
another throug
gh the formatio
on of a hydroggen bond, thus
s
further reducing th
he escaping te
endency of eacch constituentt. This pair forms a weak coompound,
that can be isolate
ed and identifie
ed. Reactionss between dipo
olar molecules
s, or between a dipolar and a
nonp
polar molecule
e, may also lea
ad to negative
e deviations. The
T interaction
n in these casees, however, is
usua
ally so weak th
hat no definite compound ca
an be isolated.
Whe
en the interactiion between A and B moleccules is less th
han that betwe
een moleculess of the pure
consstituents,
P.117
the p
presence of B molecules red
duces the inte
eraction of the A molecules, and A molecuules
corre
espondingly re
educe the B—
—B interaction. Accordingly, the dissimilarity of polaritiess or internal
pressures of the constituents
c
results in a grea
ater escaping tendency of both
b
the A andd the B molecu
ules.
The partial vapor pressure
p
of the constituentss is greater tha
an that expectted from Raouult's law, and the
syste
em is said to exhibit
e
positive
e deviation. Th
he total vapor pressure often shows a maaximum at one
e
particular composition if the dev
viation is sufficciently large. An
A example off positive deviaation is shown
n
in Fig
gure 5-3. Liqu
uid pairs that demonstrate
d
p
positive deviatiion are benzene and ethyl aalcohol, carbon
disullfide and aceto
one, and chlorroform and eth
hyl alcohol.
223
Dr. Murtadha Alshareifi e-Library
Fig.. 5-2. Vaporr pressure of
o a system sshowing neegative deviation from R
Raoult's law
w.
Raou
ult's law does not apply ove
er the entire co
oncentration ra
ange in a nonideal solution.. It describes the
t
beha
avior of either component off a real liquid p
pair only when
n that substance is present in high
conccentration and thus is consid
dered to be th e solvent. Rao
oult's law can be expressedd as
in su
uch a situation
n, and it is valid
d only for the ssolvent of a nonideal solutio
on that is suffi ciently dilute with
w
respect to the solu
ute. It cannot hold
h
for the co
omponent in lo
ow concentration, that is, thee solute, in a dilute
d
nonid
deal solution.
Thesse statements will become clearer
c
when o
one observes,, in Figure 5-2, that the actuual vapor pressure
curve
e of chloroform
m (componentt A) approache
es the ideal cu
urve defined by
b Raoult's law
w as the solutiion
comp
position appro
oaches pure chloroform. Ra
aoult's law can
n be used to de
escribe the beehavior of
chlorroform when itt is present in high concentrration (i.e., wh
hen it is the so
olvent). The ideeal equation is
s not
appliicable to aceto
one (compone
ent B), howeve
er, which is prresent in low concentration
c
iin this region of
o
the d
diagram, beca
ause the actua
al curve for ace
etone does no
ot coincide with the ideal linee. When one
studiies the left sid
de of Figure 5--2, one observves that the co
onditions are reversed: Acettone is considered
to be
e the solvent here,
h
and its vapor pressure
e curve tends to
t coincide with the ideal cuurve. Chloroform is
the ssolute in this ra
ange, and its curve
c
does no
ot approach th
he ideal line. Similar
S
consideerations apply
to Figure 5-3.
224
Dr. Murtadha Alshareifi e-Library
Fig.. 5-3. Vaporr pressure of
o a system sshowing po
ositive deviaation from R
Raoult's law
w.
Henry's Law
w
The vapor pressurre curves for both
b
acetone a
and chloroform
m as solutes are
a observed to lie considera
ably
below
w the vapor pressure of an ideal mixture of this pair. Th
he molecules of solute, beinng in relatively
y
smalll number in th
he two regions
s of the diagra
am, are comple
etely surround
ded by molecuules of solventt and
so re
eside in a unifo
orm environment. Therefore
e, the partial pressure
p
or escaping tendenncy of chlorofo
orm
at low
w concentratio
on is in some way proportio
onal to its mole
e fraction, but, as observed in Figure 5-2, the
prop
portionality con
nstant is not eq
qual to the va por pressure of
o the pure substance. The vapor pressure–
comp
position relatio
onship of the solute
s
cannot be expressed
d by Raoult's la
aw but insteadd by an equatiion
know
wn as Henry's law:
wherre k for chloroform is less th
han p°CHCL3. H
Henry's law app
plies to the so
olute and Raouult's law applie
es to
the ssolvent in dilutte solutions off real liquid pa irs. Of course, Raoult's law also applies oover the entire
e
conccentration rang
ge (to both solvent and solu
ute) when the constituents are
a sufficientlyy similar to form
m an
ideall solution. Und
der any circum
mstance, when
n the partial va
apor pressures
s of both of th e constituents
s are
direcctly
P.118
225
Dr. Murtadha Alshareifi e-Library
prop
portional to the
e mole fraction
ns over the enttire range, the
e solution is sa
aid to be ideal;; Henry's law
beco
omes identicall with Raoult's law, and k be
ecomes equal to p°. Henry's
s law is used ffor the study of
o gas
solub
bilities discusssed later in the
e book.
Dis
stillation of Binary
y Mixture
es
The relationship between vaporr pressure (and
d hence boilin
ng point) and composition
c
off binary liquid
phasses is the unde
erlying princip
ple in distillatio
on. In the case
e of miscible liq
quids, insteadd of plotting va
apor
pressure versus composition, it is more usefu
ul to plot the bo
oiling points of
o the various m
mixtures,
determined at atm
mospheric pres
ssure, against composition.
The higher the vap
por pressure of
o a liquid—tha
at is, the more
e volatile it is—
—the lower is tthe boiling point.
Beca
ause the vapo
or of a binary mixture
m
is alwa
ays richer in th
he more volatile constituent,, the process of
distillation can be used to separrate the more volatile from the
t less volatile constituent. Figure 5-4 sh
hows
a mixxture of a high
h-boiling liquid
d A and a low--boiling liquid B.
B A mixture of
o these substaances having the
comp
position a is distilled
d
at the boiling point b
b. The compos
sition of the va
apor v1 in equiilibrium with th
he
liquid
d at this tempe
erature is c; th
his is also the composition of
o the distillate
e when it is conndensed. The
e
vapo
or is therefore richer in B tha
an the liquid frrom which it was
w distilled. If a fractionatingg column is us
sed,
A an
nd B can be co
ompletely sepa
arated. The va
apor rising in the
t column is met by the coondensed vapo
or or
down
nward-flowing
g liquid. As the
e rising vapor iis cooled by co
ontact with the
e liquid, somee of the lowerboilin
ng fraction con
ndenses, and the vapor con
ntains more off the volatile co
omponent thaan it did when it left
the rretort. Therefo
ore, as the vap
por proceeds u
up the fraction
nating column, it becomes pprogressively richer
r
in the
e more volatile
e component B, and the liqu
uid returning to
t the distilling retort becom es richer in the
less volatile component A.
Fig.. 5-4. Boilinng point diaagram of an ideal binary
y mixture.
226
Dr. Murtadha Alshareifi e-Library
Figure 5-4 shows the situation for
f a pair of miiscible liquids exhibiting ideal behavior. B
Because vaporr
pressure curves can show maxiima and minim
ma (see Figs. 5-2and 5-3), itt follows that bboiling point curves
will sshow correspo
onding minima
a and maxima , respectively.. With these mixtures,
m
distill ation produce
es
eithe
er pure A or pu
ure B plus a mixture
m
of consstant composittion and consttant boiling pooint. This latterr is
know
wn as an azeo
otrope (Greek: “boil unchang
ged”) or azeottropic mixture. It is not posssible to separa
ate
such
h a mixture com
mpletely into two
t
pure comp
ponents by sim
mple fractiona
ation. If the vappor pressure
curve
es show a min
nimum (i.e., ne
egative deviattion from Raou
ult's law), the azeotrope
a
hass the highest
boilin
ng point of all the mixtures possible;
p
it is ttherefore least volatile and remains in thee flask, wherea
as
eithe
er pure A or pu
ure B is distille
ed off. If the va
apor pressure curve exhibits
s a maximum (showing a
posittive deviation from Raoult's law), the azeo
otrope has the
e lowest boiling point and foorms the distillate.
Eithe
er pure A or pu
ure B then rem
mains in the fla
ask.
Whe
en a mixture off HCl and wate
er is distilled a
at atmospheric
c pressure, an
n azeotrope is obtained thatt
conta
ains 20.22% by
b weight of HCl
H and that bo
oils at 108.58°°C. The compo
osition of this mixture is
accu
urate and repro
oducible enou
ugh that the so
olution can be used as a sta
andard in analyytic chemistry
y.
Mixtu
ures of water and acetic aciid and of chlorroform and ac
cetone yield az
zeotropic mixtuures with max
xima
in the
eir boiling poin
nt curves and minima in the
eir vapor press
sure curves. Mixtures
M
of ethhanol and wate
er
and of methanol and
a benzene both
b
show the reverse behavior, namely, minima in the boiling point
curve
es and maxim
ma in the vapor pressure currves.
Whe
en a mixture off two practically immiscible liquids is heatted while being agitated to eexpose the
surfa
aces of both liq
quids to the va
apor phase, e
each constituent independen
ntly exerts its oown vapor
pressure as a funcction of tempe
erature as thou
ugh the other constituent we
ere not presennt. Boiling beg
gins,
and distillation ma
ay be effected when the sum
m of the partial pressures of the two immisscible liquids just
j
exce
eeds the atmospheric pressure. This princciple is applied
d in steam distillation, whereeby many organic
comp
pounds insolu
uble in water can
c be purified
d at a tempera
ature well below the point at which
deco
omposition occcurs. Thus, bromobenzene alone boils att 156.2°C, whe
ereas water booils at 100°C at
a a
pressure of 760 mm
m Hg. A mixtu
ure of the two , however, in any proportion
n, boils at 95°C
C. Bromobenz
zene
can tthus be distille
ed at a temperrature 61°C be
elow its norma
al boiling pointt. Steam distilllation is
particularly useful for obtaining volatile
v
oils fro
om plant tissues without dec
composing thee oils.
Co
olligative Propertie
es
Whe
en a nonvolatille solute is com
mbined with a volatile solve
ent, the vapor above the soluution is provid
ded
solelly by the solve
ent. The solute
e reduces the escaping tend
dency of the solvent,
s
and, oon the basis off
Raou
ult's law, the vapor
v
pressure
e of a solution
n
P.119
conta
aining a nonvo
olatile solute is lowered pro
oportional to th
he relative num
mber.
Key Co
oncept
Colligative Pro
operties
The freezing point, boiling po
oint, and osm
motic pressurre of a solutio
on also depeend on the
relattive proportio
on of the molecules of the
e solute and the solvent. These are ccalled colliga
ative
prop
perties (Gree
ek: “collected
d together”) b
because they
y depend chiefly on the nnumber rathe
er
than
n on the natu
ure of the con
nstituents.
Low
wering of
o the Vap
por Press
sure
Acco
ording to Raou
ult's law, the vapor pressure
e, p1, of a solv
vent over a dilu
ute solution is equal to the vapor
v
pressure of the pu
ure solvent, p1°, times the m
mole fraction off solvent in the
e solution, X1. Because the
solutte under discu
ussion here is considered to
o be nonvolatile, the vapor pressure
p
of thee solvent, p1, is
i
identtical to the tota
al pressure off the solution, p
p.
It is m
more convenie
ent to express
s the vapor pre
essure of the solution
s
in term
ms of the conccentration of the
t
solutte rather than the mole fracttion of the solvvent, and this may be accom
mplished in thhe following wa
ay.
The sum of the mo
ole fractions of
o the constitue
ents in a solution is unity:
227
Dr. Murtadha Alshareifi e-Library
Therrefore,
wherre X1 is the mo
ole fraction of the solvent an
nd X2 is the mole
m
fraction off the solute. R
Raoult's equation
can b
be modified by substituting equation (5-1 2) for X1 to giv
ve
In eq
quation (5-15), ∆p = p1° - p is the lowering
g of the vapor pressure and ∆p/p1° is the relative vaporr
presssure lowering. The relative vapor pressurre lowering de
epends only on
n the mole frac
action of the
solutte, X2, that is, on the numbe
er of solute pa
articles in a definite volume of
o solution. Thherefore, the
relative vapor presssure lowering
g is a colligativve property.
Exa
ample 5-8
Rela
ative Vaporr Pressure Lowering
L
o
of a Solution
n
Calcculate the rellative vapor pressure
p
low
wering at 20°C for a solution containinng 171.2 g off
sucrrose (w2) in 100
1 g (w1) off water. The m
molecular we
eight of sucro
ose (M2) is 3342.3 and the
e
mole
ecular weigh
ht of water (M
M1) is 18.02 g
g/mole. We have
h
Key Concept
Osm
molality
In m
most solution
ns, changes in concentra
ration are ac
ccompanied by linear annd proportion
nal
changes in the cardinal colligative prop
perties of the
e solvent—v
vapor pressuure, freezing
g
poin
nt, and boilin
ng point. Measuring anyy of these prroperties pro
ovides an inddirect indication
of osmolality, bu
ut among them, only va
apor pressure can be de
etermined paassively with
hout
a forced change
e in the sam
mple's physiccal state.
Noticce that in Exam
mple 5-8, the relative vaporr pressure low
wering is a dimensionless nuumber, as wou
uld be
expe
ected from its definition. The
e result can alsso be stated as
a a percentag
ge; the vapor ppressure of th
he
soluttion has been lowered 0.89% by the 0.5 m
mole of sucros
se.
The mole fraction, n2/(n1 + n2), is nearly equa
al to, and may be replaced by,
b the mole raatio n2/n1 in a dilute
d
soluttion such as th
his one. Then,, the relative vvapor pressure
e lowering can
n be expresseed in terms of molal
m
conccentration of th
he solute by se
etting the weig
ght of solvent w1 equal to 10
000 g. For an aqueous solu
ution,
Exa
ample 5-9
Calc
culation of the Vapor Pressure
P
Calcculate the va
apor pressure
e when 0.5 m
mole of sucro
ose is added to 1000 g off water at 20°C.
The vapor presssure of water at 20°C is 1 7.54 mm Hg
g. The vapor pressure low
wering of the
solu
ution is
The final vapor pressure
p
is
Ele
evation of
o the Boiiling Poin
nt
228
Dr. Murtadha Alshareifi e-Library
The normal boiling
g point is the temperature att which the va
apor pressure of the liquid bbecomes equa
al to
an external pressu
ure of
P.12
20
760 mm Hg. A solution will boil at a higher tem
mperature tha
an will the pure
e solvent. Thiss is the colliga
ative
prop
perty called boiling point elev
vation. As sho
own in Figure 5-5, the more of the solute tthat is dissolved,
the g
greater is the effect.
e
The boiling point of a solution of a nonvolatile so
olute is higherr than that of th
he
pure
e solvent owing
g to the fact th
hat the solute lowers the vap
por pressure of
o the solvent.. This may be seen
by re
eferring to the curves in Figu
ure 5-6. The vvapor pressure
e curve for the
e solution lies below that of the
pure
e solvent, and the temperatu
ure of the solu tion must be elevated
e
to a value
v
above thhat of the solv
vent
in orrder to reach th
he normal boiling point. The
e elevation of the boiling point is shown inn the figure as
sTTo = ∆Tb. The ratiio of the eleva
ation of the boiiling point, ∆T
Tb, to the vaporr pressure low
wering, ∆p = p° - p,
at 10
00°C is approxximately a con
nstant at this te
emperature; itt is written as
Fig.. 5-5. Theorretical plot of
o the norm
mal boiling point
p
for waater (solventt) as a functtion
of m
molality in solutions
s
co
ontaining suucrose (a non
nvolatile solute) in incrreasing
conccentrations.. Note that the
t normal bboiling poin
nt of water increases
i
ass the
conccentration of
o sucrose in
ncreases. Thhis is known
n as boiling
g point elevaation.
or
More
eover, becausse p° is a cons
stant, the boilin
ng point eleva
ation may be considered prooportional to ∆p/p°,
∆
the rrelative lowerin
ng of vapor prressure. By Ra
aoult's law, ho
owever, the relative vapor prressure lowering is
equa
al to the mole fraction of the
e solute; thereffore,
229
Dr. Murtadha Alshareifi e-Library
Fig.. 5-6. Boilinng point elevation of thhe solvent due to addition of a soluute (not to
scalle).
Tab
ble 5-4 Ebu
ullioscopic (Kb) and C
Cryoscopic (Kf) Consta
ants for Vaarious Solveents
Sub
bstance
Boiling Point
(°C))
Kb
Freezin
ng Point (°C
C) Kf
A
Acetic acid
11
18.0
2.93
16.7
3.9
A
Acetone
56
6.0
1.71
-94.82*
2.40*
B
Benzene
80
0.1
2.53
5.5
5.12
C
Camphor
20
08.3
5.95
178.4
4
37.7
C
Chloroform
61
1.2
3.54
-63.5
4.96
E
Ethyl alcohool
78
8.4
1.22
-114.49*
3*
E
Ethyl ether
34
4.6
2.02
-116.3
1.79*
P
Phenol
18
81.4
3.56
42.0
7.27
230
Dr. Murtadha Alshareifi e-Library
W
Water
10
00.0
0.51
0.00
1.86
*F
From G. Koortum and J.
J O'M. Bocckris, Textbo
ook of Electtrochemistryy, Vol. II,
E
Elsevier, New
w York, 1951, pp. 618 , 620.
Beca
ause the boilin
ng point elevattion depends o
only on the mole fraction off the solute, it is a colligative
e
prop
perty.
In dillute solutions,, X2 is equal approximately tto m/(1000/M1) [equation (5
5-16)], and equuation (5-19) can
c
be w
written as
or
wherre ∆Tb is know
wn as the boiliing point eleva
ation and Kb is
s called the mo
olal elevation cconstant or
the e
ebullioscopic constant.
c
Kb has a characte
eristic value forr each solventt, as seen in T
Table 5-4. It may
be co
onsidered as the
t boiling point elevation fo
or an ideal 1 m solution. Sta
ated another w
way, Kb is the ratio
of the boiling pointt elevation to the
t molal conccentration in an
a extremely dilute
d
solution in which the
syste
em is approxim
mately ideal.
The preceding discussion consttitutes a plaussible argumentt leading to the
e equation forr boiling point
eleva
ation. A more satisfactory derivation of eq
quation (5-21)), however, inv
volves the appplication of the
e
Clap
peyron equatio
on, which is wrritten as
wherre Vv and V1 are
a the molar volume
v
of the gas and the molar
m
volume of the liquid, re
respectively, Tb is
the b
boiling point off the solvent, and
a ∆Hv is the
e molar heat of
o vaporization. Because V1 is negligible
comp
pared to Vv, th
he equation be
ecomes
P.12
21
and Vv, the volume
e of 1 mole off gas, is replacced by RTb/p° to give
or
m equation (5--16), ∆p/p1° = X2, and equattion (5-25) can
n be written as
s
From
231
Dr. Murtadha Alshareifi e-Library
whicch provides a more
m
exact eq
quation with w hich to calcula
ate ∆Tb.
Repllacing the rela
ative vapor pre
essure lowerin
ng ∆p/p1° by m/(1000/M
m
ccording to thee approximate
e
1) ac
exprression (5-16),, in which w2/M
M2 = m and w1 = 1000 s, we
e obtain the formula
Equa
ation (5-27) prrovides a less exact expresssion with whic
ch to calculate ∆Tb.
For w
water at 100°C
C, we have Tb = 373.2 K, ∆H
Hv = 9720 cal//mole, M1 = 18
8.02 g/mole, aand R = 1.987
cal/m
mole deg.
Exa
ample 5-10
Calc
culation of the Elevation Constan
nt
A 0.200 m aqueo
ous solution of a drug ga
ave a boiling point elevation of 0.103°°C. Calculate
e the
apprroximate mo
olal elevation constant forr the solvent,, water. Subs
stituting into equation (5-21)
yield
ds
The proportionalityy between ∆T
Tb and the mola
ality is exact only
o
at infinite dilution, at whhich the properties
of re
eal and ideal solutions coinc
cide. The ebulllioscopic cons
stant, Kb, of a solvent
s
can bee obtained
expe
erimentally by measuring ∆T
Tb at various m
molal concentrrations and ex
xtrapolating to infinite dilutio
on
(m = 0), as seen in
nFigure 5-7.
Dep
pression
n of the Freezing
F
P
Point
The normal freezin
ng point or me
elting point of a pure compo
ound is the tem
mperature at w
which the solid
d and
the liiquid phases are
a in equilibriium under a p
pressure of 1 atm.
a
Equilibrium here meanss that the tend
dency
for th
he solid to passs into the liqu
uid state is the
e same as the tendency for the
t reverse prrocess to occu
ur,
beca
ause both the liquid and the solid have the
e same escap
ping tendency.. The value T00, shown in Fig
gure
5-8, for water satu
urated with air at this pressu
ure is arbitrarily
y assigned a temperature
t
oof 0°C. The trip
ple
pointt of air-free wa
ater, at which solid, liquid, a
and vapor are in equilibrium, lies at a presssure of 4.58
mm Hg and a temperature of 0.0098°C. It is n
not identical with
w the ordinary freezing pooint of water att
atmo
ospheric presssure but is rath
her the freezin
ng point of watter under the pressure of itss own vapor. We
W
shalll use the triple
e point in the fo
ollowing argum
ment because
e the depressio
on ∆Tf here dooes not differ
signiificantly from ∆T
∆ f at a pressure of 1 atm. T
The two freezing point deprressions referrred to are
illusttrated in Figure 5-7. The ∆T
Tb of Figure 5-6
6 is also show
wn in the diagram.
232
Dr. Murtadha Alshareifi e-Library
Fig.. 5-7. The innfluence of concentratiion on the ebullioscopi
e
c constant.
If a ssolute is dissolved in the liqu
uid at the triple
e point, the es
scaping tendency or vapor ppressure of the
e
liquid
d solvent is low
wered below that
t
of the purre solid solven
nt. The temperrature must dro
rop to reestabllish
equillibrium betwee
en the liquid and
a the solid. B
Because of this fact, the freezing point off a solution is
alwa
ays lower than that of the pu
ure solvent. It iis assumed th
hat the solventt freezes out inn the pure state
rathe
er than as a so
olid solution co
ontaining som
me of the solute. When such a complicatioon does arise,
speccial calculation
ns, not conside
ered here, mu
ust be used.
The more concenttrated the solu
ution, the farth
her apart are th
he solvent and
d the solution curves in the
diagrram (see Fig. 5-8) and the greater
g
is the ffreezing pointt depression. Accordingly,
A
a situation exis
sts
analo
ogous to that described
P.12
22
for th
he boiling poin
nt elevation, and the freezin g point depres
ssion is propo
ortional to the m
molal
conccentration of th
he solute. The
e equation is
233
Dr. Murtadha Alshareifi e-Library
Fig.. 5-8. Depreession of thee freezing ppoint of the solvent, waater, by a sollute (not to
scalle).
Fig.. 5-9. The innfluence of concentratiion on the cryoscopic
c
constant
c
for water.
or
234
Dr. Murtadha Alshareifi e-Library
∆Tf is the freezing
g point depress
sion, and Kf iss the molal dep
pression constant or the cryyoscopic constant,
whicch depends on
n the physical and chemical properties of the solvent.
The freezing pointt depression of
o a solvent is a function only
y of the number of particless in the solution,
and ffor this reason
n it is referred to as a colliga
ative property. The depression of the freeezing point, like the
boilin
ng point eleva
ation, is a direc
ct result of the
e lowering of th
he vapor press
sure of the soolvent. The value
of Kf for water is 1.86. It can be determined exxperimentally by measuring
g ∆Tf/m at sevveral molal
conccentrations and extrapolating to zero conccentration. As seen in Figurre 5-9, Kfapprooaches the value
of 1.86 for water solutions
s
of sucrose and glyccerin as the concentrations tend toward zzero, and
equa
ation (5-28) is valid only in very
v
dilute solu
utions. The ap
pparent cryosc
copic constantt for higher
conccentrations can
n be obtained from Figure 5
5-9. For work in
i pharmacy and
a biology, thhe Kf value of 1.86
can b
be rounded offf to 1.9, which
h is good apprroximation for practical use with aqueouss solutions, wh
here
conccentrations are
e usually lower than 0.1 M. T
The value of Kf for the solve
ent in a solutioon of citric acid
d is
obse
erved not to ap
pproach 1.86. This abnorma
al behavior is to be expected when dealinng with solutio
ons of
electtrolytes. Their irrationality will
w be explaine
ed in Chapter 6, and proper steps will be ttaken to corre
ect
the d
difficulty.
Kf ca
an also be derrived from Rao
oult's law and the Clapeyron
n equation. Fo
or water at its ffreezing point, Tf =
273.2 K, ∆Hf is 14
437 cal/mole, and
a
The cryoscopic co
onstants, togetther with the e
ebullioscopic constants,
c
for some solventss at infinite dilution
are g
given in Table 5-4.
Exa
ample 5-11
Calc
culation of Freezing Point
Wha
at is the freezzing point of a solution co
ontaining 3.4
42 g of sucrose and 500 g of water? The
T
mole
ecular weigh
ht of sucrose is 342. In th is relatively dilute
d
solution, Kf is approoximately eq
qual
to 1.86. We have
e
Therefore, the frreezing pointt of the aqueo
ous solution is -0.037°C.
Exa
ample 5-12
Free
ezing Pointt Depressio
on
Wha
at is the freezzing point de
epression of a 1.3 m solution of sucro
ose in water??
From
m the graph in Figure 5-8
8, one observves that the cryoscopic
c
constant
c
at thhis concentra
ation
is ab
bout 2.1 rather than 1.86. Thus, the ccalculation be
ecomes
Os
smotic Prressure
If cobalt chloride iss placed in a parchment
p
sacc and suspend
ded in a beake
er of water, th e water gradu
ually
beco
omes red as th
he solute diffuses throughou
ut the vessel. In this process of diffusion, both the solve
ent
and the solute molecules migratte freely. On t he other hand
d, if the solutio
on is confined in a membran
ne
perm
meable only to the solvent molecules,
m
the phenomenon
n known as osm
mosis (Greek:: “a push or
impu
ulse”)4 occurs, and the barrier that permitts only the mo
olecules of one
e of the compoonents (usually
wate
er) to pass thro
ough is known
n as a semiperrmeable mem
mbrane. A thistle tube over thhe wide opening of
whicch is stretched a piece of untreated cellop
phane can be used
u
to demonstrate the priinciple, as sho
own
in Fig
gure 5-10. The tube is partly filled with a concentrated solution of sucrose, and thee apparatus is
s
lowe
ered into a bea
aker of water. The passage of water throu
ugh the semipermeable mem
mbrane into th
he
235
Dr. Murtadha Alshareifi e-Library
soluttion eventuallyy creates enou
ugh pressure tto drive the su
ugar solution up
u the tube unntil the hydrosttatic
pressure of the co
olumn of liquid equals the prressure causin
ng the water to
o pass throughh the membra
ane
and enter the thisttle tube. When
n this occurs, tthe solution ce
eases to rise in the tube. Ossmosis is therrefore
defin
ned as the passsage of the solvent into a ssolution throug
gh a semiperm
meable membrrane. This pro
ocess
tends to equalize the
t escaping tendency
t
of th
he solvent on both
b
sides of the
t membranee. Escaping
tendency can be measured
m
in te
erms of vapor pressure or th
he closely rela
ated colligativee property osm
motic
presssure. It should
d be evident th
hat osmosis ca
an also
P.12
23
take place when a concentrated
d solution is se
eparated from a less concen
ntrated solutioon by a
semiipermeable membrane.
Fig.. 5-10. Apparatus for demonstratin
d
ng osmosis.
Osm
mosis in some cases is believed to involve
e the passage of solvent through the mem
mbrane by a
distillation processs or by dissolv
ving in the matterial of the membrane in which the solutee is insoluble. In
other cases, the membrane
m
may
y act as a sievve, having a pore
p
size suffic
ciently large too allow passag
ge of
solve
ent but not of solute molecu
ules.
In eitther case, the phenomenon
n of osmosis d
depends on the
e fact that the chemical poteential (a
therm
modynamic exxpression of escaping tende
ency) of a solv
vent molecule in solution is lless than exis
sts in
the p
pure solvent. Solvent
S
therefo
ore passes sp
pontaneously into
i
the solutio
on until the chhemical potenttials
of so
olvent and solu
ution are equa
al. The system
m is then at equilibrium. It may be advantaageous for the
e
236
Dr. Murtadha Alshareifi e-Library
stude
ent to conside
er osmosis in terms
t
of the fo
ollowing seque
ence of events
s. (a) The adddition of a
nonvvolatile solute to the solventt forms a soluttion in which th
he vapor pressure of the soolvent is reduc
ced
(see Raoult's law). (b) If pure so
olvent is now p
placed adjacent to the solution but separaated from it by
ya
semiipermeable membrane, solv
vent molecule
es will pass thrrough the mem
mbrane into thhe solution in an
a
attem
mpt to dilute out the solute and
a raise the vvapor pressurre back to its original
o
value ((namely, that of
the o
original solven
nt). (c) The osm
motic pressure
e that is set up
p as a result of
o this passagee of solvent
mole
ecules can be determined either by meassuring the hydrrostatic head appearing
a
in thhe solution orr by
applyying a known pressure that just balancess the osmotic pressure
p
and prevents
p
any nnet movement of
solve
ent molecules into the solution. The latterr is the preferrred technique. The osmotic pressure thus
s
obtained is proporrtional to the re
eduction in va
apor pressure brought aboutt by the conceentration of solute
present. Because this is a function of the mol ecular weight of the solute, osmotic presssure is a
collig
gative propertyy and can be used to determ
mine molecule
e weights.
Fig.. 5-11. Osm
motic pressurre osmometter.
An o
osmotic pressu
ure osmomete
er (Fig. 5-11) iss based on the same princip
ple as the thisstle tube apparatus
show
wn in Figure 5-10. Once equ
uilibrium has b
been attained, the height of the solution inn the capillary tube
on th
he solution sid
de of the membrane is greatter by the amo
ount h than the
e height in
P.12
24
the ccapillary tube on
o the solventt (water) side. The hydrosta
atic head, h, is related to thee osmotic pres
ssure
throu
ugh the expresssion osmotic pressure π (a
atm) = Height h× Solution density ρ × Graavity accelerattion.
The two tubes of large bore are for filling and discharging the liquids from
m the comparttments of the
appa
aratus. The he
eight of liquid in these two la
arge tubes doe
es not enter in
nto the calculaation of osmotiic
pressure. The dete
ermination of osmotic presssure is discuss
sed in some detail in the nex
ext section.
237
Dr. Murtadha Alshareifi e-Library
Me
easureme
ent of Os
smotic Prressure
The osmotic presssure of the suc
crose solution referred to in the last sectio
on is not meassured conveniently
by ob
bserving the height
h
that the solution attai ns in the tube at equilibrium
m. The concenntration of the final
f
soluttion is not kno
own because the passage o
of water into th
he solution dilu
utes it and alteers the
conccentration. A more
m
exact me
easure of the o
osmotic pressure of the und
diluted solutionn is obtained by
b
determining the exxcess pressure on the soluttion side that just prevents the passage of solvent throu
ugh
the m
membrane. Ossmotic pressure is defined a
as the excess pressure, or pressure
p
greaater than that
abovve the pure so
olvent, that mu
ust be applied to the solution
n to prevent th
he passage off the solvent
throu
ugh a perfect semipermeab
s
le membrane.. In this definittion, it is assum
med that a sem
mipermeable sac
conta
aining the solu
ution is immerrsed in the purre solvent.
In 18
877, the botan
nist Wilhelm Pffeffer measure
ed the osmotic
c pressure of sugar solutionns, using a porous
cup iimpregnated with
w a deposit of cupric ferro
ocyanide, Cu2Fe(CN)6, as th
he semipermeeable membra
ane.
The apparatus wa
as provided witth a manometter to measure
e the pressure
e. Although maany improvem
ments
have
e been made through
t
the ye
ears, including
g the attachme
ent of sensitive
e pressure traansducers to th
he
mem
mbrane that ca
an be electronically amplified
d to produce a signal,5 the direct measurrement of osm
motic
pressure remains difficult and in
nconvenient. N
Nevertheless, osmotic press
sure is the collligative property
best suited to the determination of the molecu
ular weight of polymers such as proteins.
van
n't Hoff and
a Mors
se Equatio
ions for Osmotic
O
Pressure
P
e
In 18
886, Jacobus van't
v
Hoff reco
ognized in Pfe
effer's data pro
oportionality between osmot
otic pressure,
conccentration, and
d temperature, suggested a relationship that
t
correspon
nded to the eq uation for an ideal
i
gas. van't Hoff con
ncluded that th
here was an a
apparent analo
ogy between solutions
s
and ggases and tha
at the
osmo
otic pressure in a dilute solu
ution was equ al to the press
sure that the solute
s
would eexert if it were a
gas o
occupying the
e same volume
e. The equatio
on is
wherre π is the osm
motic pressure
e in atm, V is tthe volume of the solution in
n liters, n is thhe number of moles
m
of so
olute, R is the gas constant, equal to 0.08
82 liter atm/mo
ole deg, and T is the absolutte temperature.
The student should be cautioned not to take vvan't Hoff's an
nalogy too literrally, for it leadds to the belie
ef that
the ssolute moleculles “produce” the osmotic p
pressure by ex
xerting pressurre on the mem
mbrane, just as
s gas
mole
ecules create a pressure by striking the w
walls of a vessel. It is more correct,
c
howevver, to conside
er the
osmo
otic pressure as resulting from the relativve escaping tendencies of th
he solvent mollecules on the
e two
sidess of the memb
brane. Actually
y, equation (5--30) is a limitin
ng law applyin
ng to dilute sollutions, and it
simp
plifies into this form from a more
m
exact exp
pression [equation (5-36)] only
o after introoducing a number
of asssumptions tha
at are not valid
d for real solu tions.
Exa
ample 5-13
Calc
culating the
e Osmotic Pressure
P
off a Sucrose
e Solution
One
e gram of succrose, molec
cular weight 3
342, is dissolved in 100 mL
m of solutioon at 25°C. What
W
is th
he osmotic prressure of the solution? W
We have
Equa
ation (5-30), th
he van't Hoff equation,
e
can be expressed as
wherre c is the con
ncentration of the
t solute in m
moles/liter (mo
olarity). Morse and others haave shown tha
at
when
n the concentrration is expre
essed in molallity rather than
n in molarity, th
he results com
mpare more ne
early
with the experimen
ntal findings. The
T Morse eq uation is
The
ermodyn
namics off Osmotiic Pressu
ure and Vapor
V
Pre
essure
Low
wering
238
Dr. Murtadha Alshareifi e-Library
Osm
motic pressure and the lowerring of vapor p
pressure, both
h colligative properties, are iinextricably
related, and this re
elationship can be obtained from certain thermodynam
t
ic consideratioons.
We b
begin by considering a sucrrose solution in
n the right-han
nd compartme
ent of the appaaratus shown
in Fig
gure 5-12 and
d the pure solv
vent—water—
—in the left-han
nd compartment. A semiperm
meable memb
brane
throu
ugh which watter molecules,, but not sucro
ose molecules
s, can pass separates the tw
wo compartme
ents.
It is a
assumed that the gate in the air space co
onnecting the solutions can be shut duringg osmosis. Th
he
exterrnal pressure, say 1 atm, ab
bove the pure solvent is Po and the press
sure on the so lution, provide
ed by
the p
piston in Figurre 5-12 and ne
eeded to main tain equilibrium, is P. The difference
d
betw
ween the two
pressures at equilibrium, P - Po, or
P.12
25
the e
excess pressu
ure on the solu
ution just requ ired to preven
nt passage of water
w
into the solution is the
e
osmo
otic pressure π.
Fig.. 5-12. Apparatus for demonstratin
d
ng the relatiionship betw
ween osmottic pressure and
vapoor pressure lowering.
Let u
us now consid
der the alternative transport of water throu
ugh the air spa
ace above thee liquids. Shou
uld
the m
membrane be closed off and
d the gate in t he air space opened,
o
waterr molecules woould pass from
m the
pure
e solvent to the
e solution by way
w of the vap
por state by a distillation pro
ocess. The spaace above the
e
liquid
ds actually serrves as a “sem
mipermeable m
membrane,” ju
ust as does the
e real membraane at the low
wer
part of the apparattus. The vapor pressure p° of water in the
e pure solventt under the inffluence of the
atmo
ospheric presssure Po is grea
ater than the vvapor pressure
e p of water in
n the solution bby an amount p° p = ∆p. To bring about
a
equilibrium, a pressurre P must be exerted
e
by the
e piston on thee solution to
incre
ease the vapor pressure of the
t solution un
ntil it is equal to
t that of the pure
p
solvent, pp°. The excess
pressure that musst be applied, P - Po, is again
n the osmotic pressure π. The
T operation of such an
appa
aratus thus de
emonstrates th
he relationship
p between osm
motic pressure
e and vapor prressure lowering.
239
Dr. Murtadha Alshareifi e-Library
By fo
ollowing this analysis
a
further, it should be
e possible to obtain an equation relating oosmotic pressu
ure
and vvapor pressurre. Observe th
hat both the ossmosis and the distillation process are baased on the
princciple that the escaping
e
tende
ency of water in the pure so
olvent is greater than that inn the solution. By
appliication of an excess
e
pressure, P - Po = π , on the solutio
on side of the apparatus, it is possible to
make
e the escaping
g tendencies of
o water in the
e solvent and solution
s
identical. A state off equilibrium is
s
produced; thus, th
he free energy of solvent on both sides of the membran
ne or on both ssides of the air
spacce is made equ
ual, and ∆G = 0.
To re
elate vapor pressure lowerin
ng and osmot ic pressure, we
w must obtain
n the free enerrgy changes
invollved in (a) tran
nsferring 1 mo
ole of solvent ffrom solvent to
o solution by a distillation prrocess through the
vapo
or phase and (b)
( transferring
g 1 mole of so
olvent from sollvent to solutio
on by osmosiss. We have
as th
he increase in free energy at
a constant tem
mperature for the
t passage of
o 1 mole of waater to the solution
throu
ugh the vapor phase, and
as th
he increase in free energy at
a a definite tem
mperature for the passage of
o 1 mole of w
water into the
soluttion by osmossis. In equation
n (5-34), V1 is the volume off 1 mole of solvent, or, moree correctly, it is
the p
partial molar volume,
v
that is, the change i n volume of th
he solution on the addition oof 1 mole of
solve
ent to a large quantity of solution.
Setting equations (5-33) and (5--34) equal give
es
e minus sign by
b inverting th
he logarithmic term yields
and eliminating the
Equa
ation (5-36) is a more exactt expression fo
or osmotic pre
essure than are equations (55-31) and (5-3
32),
and it applies to co
oncentrated as well as dilute
e solutions, prrovided that th
he vapor follow
ws the ideal ga
as
lawss.
The simpler equattion (5-32) for osmotic presssure can be ob
btained from equation
e
(5-366), assuming that
the ssolution obeyss Raoult's law,
Equa
ation (5-36) ca
an thus be written
and ln(1 - X2) can be expanded into a series,
Whe
en X2 is small, that is, when the solution iss dilute, all terms in the expa
ansion beyondd the first may
y be
negle
ected, and
so th
hat
For a dilute solutio
on, X2 equals approximately
a
y the mole ratio n2/n1, and equation (5-42)) becomes
wherre n1V1, the nu
umber of mole
es of solvent m
multiplied by th
he volume of 1 mole, is equ al to the total
volum
me of solvent V in liters. For a dilute aque
eous solution, the equation becomes
whicch is Morse's expression,
e
eq
quation (5-32)..
P.12
26
240
Dr. Murtadha Alshareifi e-Library
Exa
ample 5-14
Com
mpute π
Com
mpute π for a 1 m aqueou
us solution o
of sucrose us
sing both equ
uation (5-32) and the morre
exacct thermodyn
namic equation (5-36). Th
he vapor pre
essure of the solution is 331.207 mm Hg
H
and the vapor prressure of wa
ater is 31.82 4 mm Hg at 30.0°C. The molar volum
me of water at
a
this temperature
e is 18.1 cm3/mole,
/
or 0.0
0181 liter/mole.
a. By the Morse
M
equation,
b. By the th
hermodynam
mic equation,,
The experimenta
al value for th
he osmotic p
pressure of a 1 m solution
n of sucrose at 30°C is 27.2
atm.
Mo
olecular Weight
W
Determina
ation
The four colligative
e properties th
hat have been
n discussed in this chapter—
—vapor pressuure lowering,
freezzing point lowe
ering, boiling point
p
elevation
n, and osmotic
c pressure—can be used too calculate the
mole
ecular weightss of nonelectro
olytes present as solutes. Th
hus, the lowerring of the vappor pressure of
o a
soluttion containing
g a nonvolatile
e solute depen
nds only on the mole fraction of the solutee. This allows the
mole
ecular weight of
o the solute to
o be calculate
ed in the follow
wing manner.
Beca
ause the mole
e fraction of so
olvent, n1 = w1//M1, and the mole
m
fraction of
o solute, n2 = w2/M2, in
whicch w1 and w2 are
a the weights
s of solvent an
nd solute of molecular
m
weights M1 and M22 respectively,,
equa
ation (5-15) ca
an be expresse
ed as
In dillute solutions in which w2/M
M2 is negligible
e compared wiith w1/M1, the former term m
may be omitted
d
from the denomina
ator, and the equation
e
simp
plifies to
ained by rearra
anging equatio
on (5-46) to
The molecular weight of the solute M2 is obta
b determined from the boiliing point eleva
ation
The molecular weight of a nonvolatile solute ccan similarly be
nowing Kb, the molal elevatio
on constant, fo
or the solvent and determinning Tb, the boiling
of the solution. Kn
ne can calcula
ate the molecu
ular weight of a nonelectrolyte. Because 11000w2/w1 is the
pointt elevation, on
weig
ght of solute pe
er kilogram of solvent, mola
ality (moles/kilo
ogram of solve
ent) can be exxpressed as
and
Then
n,
241
Dr. Murtadha Alshareifi e-Library
or
Exa
ample 5-15
Dete
ermination of the Mole
ecular Weig
ght of Sucro
ose by Boiling Point E
Elevation
A so
olution containing 10.0 g of sucrose d
dissolved in 100
1 g of wate
er has a boiliing point of
100.149°C. Wha
at is the mole
ecular weightt of sucrose?
? We write
As shown in Figurre 5-8, the low
wering of vaporr pressure aris
sing from the addition
a
of a nnonvolatile solute
to a solvent resultss in a depress
sion of the free
ezing point. By
y rearranging equation (5-299), we obtain
wherre w2 is the nu
umber of gram
ms of solute disssolved in w1 grams of solve
ent. It is thus ppossible to
calcu
ulate the mole
ecular weight of
o the solute frrom cryoscopiic data of this type.
Exa
ample 5-16
Calc
culating Mo
olecular We
eight Using Freezing Point
P
Depression
The freezing point depressio
on of a solutio
on of 2.000 g of 1,3-dinitrrobenzene inn 100.0 g of
benzzene was de
etermined by the equilibri um method and
a was found to be 0.60095°C. Calcu
ulate
the m
molecular we
eight of 1,3-d
dinitrobenzen
ne. We write
e
The van't Hoff and
d Morse equattions can be u
used to calcula
ate the molecu
ular weight of ssolutes from
osmo
otic pressure data, provided
d the solution is sufficiently dilute and ideal. The manneer in which osmotic
pressure is used to calculate the
e molecular w
weight of colloidal materials is
i discussed inn Chapter 17.
P.12
27
Exa
ample 5-17
Dete
ermining Molecular
M
Weight
W
by Os
smotic Pres
ssure
Fifte
een grams off a new drug dissolved in
n water to yie
eld 1000 mL of
o solution att 25°C was
foun
nd to produce
e an osmotic
c pressure off 0.6 atm. Wh
hat is the mo
olecular weig ht of the solu
ute?
We write
whe
ere cg is in g/lliter of solutio
on. Thus,
or
Ch
hoice of Colligativ
C
ve Properrties
Each
h of the colliga
ative properties seems to ha
ave certain advantages and disadvantagees for the
determination of molecular
m
weig
ghts. The boilin
ng point method can be use
ed only when tthe solute is
nonvvolatile and wh
hen the substa
ance is not de
ecomposed at boiling tempe
eratures. The ffreezing point
meth
hod is satisfacctory for solutio
ons containing
g volatile soluttes, such as alcohol, becausse the freezing
pointt of a solution depends on the vapor presssure of the so
olvent alone. The
T freezing ppoint method is
s
easilly executed an
nd yields results of high acccuracy for solu
utions of small molecules. It is sometimes
s
incon
nvenient to usse freezing point or boiling p
point methods, however, because they muust be carried out
242
Dr. Murtadha Alshareifi e-Library
at de
efinite tempera
atures. Osmottic pressure m
measurements do not have this disadvantaage, and yet the
difficculties inheren
nt in this metho
od preclude itss wide use. In summary, it may
m be said thhat the cryosco
opic
and newer vapor pressure
p
techn
niques are the
e methods of choice,
c
exceptt for high polym
mers, in which
h
insta
ance the osmo
otic pressure method
m
is used
d.
Beca
ause the collig
gative properties are interrellated, it should
d be possible to determine tthe value of one
prop
perty from know
wledge of any
y other. The re
elationship bettween vapor pressure
p
lowerring and osmo
otic
pressure has alrea
ady been show
wn. Freezing p
point depressiion and osmottic pressure caan be related
apprroximately as follows.
f
The molality
m
from th
he equation m = ∆Tf/Kf is su
ubstituted in thhe osmotic
pressure equation
n, π = RTm, to give, at 0°C,
or
Lewiis6 suggested
d the equation
whicch gives accurate results.
Tab
ble 5-5 App
proximate Expression
E
s for the Colligative Properties
P
Exa
ample 5-18
Osm
motic Press
sure of Hum
man Blood S
Serum
A sa
ample of hum
man blood se
erum has a frreezing pointt of -0.53°C. What is the approximate
e
osm
motic pressure
e of this sam
mple at 0°C? What is its more
m
accuratte value as ggiven by the
Lew
wis equation?
? We write
Table 5-5 presentss the equation
ns and their co
onstants in sum
mmary form. All
A equations aare approxima
ate
and are useful only for dilute solutions in whicch the volume occupied by the
t solute is nnegligible with
respect to that of the
t solvent.
Cha
apter Summ
mary
Thiss chapter focused on an important ph armaceutica
al mixture kno
own as a moolecular
disp
persion or true solution. Nine
N
types off solutions, cllassified acco
ording to thee states in wh
hich
the ssolute and so
olvent occur, were define
ed. The conc
centration of a solution waas expressed in
two ways: eitherr in terms of the
t quantity o
of solute in a definite volume of soluttion or as the
e
quan
ntity of solute
e in a definite
e mass of so
olvent or solu
ution. You should be ablee to calculate
e
243
Dr. Murtadha Alshareifi e-Library
molarity, normality, molality, mole fraction, and percentage expressions. Ideal and real
solutions were described using Raoult's and Henry's laws. Finally, the colligative properties of
solutions (osmotic pressure, vapor pressure lowering, freezing point depression, and boiling
point elevation) were described. Colligative properties depend mainly on the number of
particles in a solution and are approximately the same for equal concentrations of different
nonelectrolytes in solution regardless of the species or chemical nature of the constituents.
P.128
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. H. R. Kruyt, Colloid Science, Vol. II, Elsevier, New York, 1949, Chapter 1.
2. W. H. Chapin and L. E. Steiner, Second Year College Chemistry, Wiley, New York, 1943, Chapter 13;
M. J. Sienko, Stoichiometry and Structure, Benjamin, New York, 1964.
3. G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, New York, 1923, Chapter 16, J. M.
Johlin, J. Biol. Chem. 91, 551, 1931.
4. M. P. Tombs and A. R. Peacock, The Osmotic Pressure of Biological Macromolecules, Clarendon,
Oxford, 1974, p. 1.
5. W. K. Barlow and P. G. Schneider, Colloid Osmometry, Wescor, Inc., Logan, Utah, 1978.
6. G. N. Lewis, J. Am. Chem. Soc. 30, 668, 1908.
Recommended Readings
P. W. Atkins and J. De Paula, Atkins' Physical Chemistry, 8th Ed., W. H. Freeman, New York, 2006.
J. H. Hildebrand and R. L. Scott, The Solubility of Nonelectrolytes, 3rd Ed., Dover Publications, New
York, 1964.
Chapter Legacy
Fifth Edition: published as Chapter 5. Updated by Patrick Sinko.
Sixth Edition: published as Chapter 5. Updated by Patrick Sinko.
244
Dr. Murtadha Alshareifi e-Library
6 Electrolyte Solutions
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Understand the important properties of solutions of electrolytes.
2. Understand and apply Faraday's law and electrolytic conductance.
3. Calculate the conductance of solutions, the equivalent conductance, and the
equivalent conductance of electrolytes.
4. Compare and contrast the colligative properties of electrolytic solutions and
concentrated solutions of nonelectrolytes.
5. Apply the Arrhenius theory of electrolytic dissociation.
6. Apply the theory of strong electrolytes; for example, calculate degree of dissociation,
activity coefficients, and so on.
7. Calculate ionic strength.
8. Calculate osmotic coefficients, osmolality, and osmolarity.
9. Understand the differences between osmolality and osmolarity.
The first satisfactory theory of ionic solutions was that proposed by Arrhenius in 1887. The theory was
based largely on studies of electric conductance by Kohlrausch, colligative properties by van't Hoff, and
chemical properties such as heats of neutralization by Thomsen. Arrhenius1 was able to bring together
the results of these diverse investigations into a broad generalization known as the theory of electrolytic
dissociation.
Although the theory proved quite useful for describing weak electrolytes, it was soon found
unsatisfactory for strong and moderately strong electrolytes. Accordingly, many attempts were made to
modify or replace Arrhenius's ideas with better ones, and finally, in 1923, Debye and Hückel put forth a
new theory. It is based on the principles that strong electrolytes are completely dissociated into ions in
solutions of moderate concentration and that any deviation from complete dissociation is due to
interionic attractions. Debye and Hückel expressed the deviations in terms of activities, activity
coefficients, and ionic strengths of electrolytic solutions. These quantities, which had been introduced
earlier by Lewis, are discussed in this chapter together with the theory of interionic attraction. Other
aspects of modern ionic theory and the relationships between electricity and chemical phenomena are
considered in following chapters.
This chapter begins with a discussion of some of the properties of ionic solutions that led to the
Arrhenius theory of electrolytic dissociation.
Properties of Solutions of Electrolytes
Electrolysis
When, under a potential of several volts, a direct electric current (dc) flows through an electrolytic cell
(Fig. 6-1), a chemical reaction occurs. The process is known aselectrolysis. Electrons enter the cell from
the battery or generator at the cathode (road down); they combine with positive ions or cations in the
solution, and the cations are accordingly reduced. The negative ions, or anions, carry electrons through
the solution and discharge them at the anode (road up), and the anions are accordingly
oxidized. Reduction is the addition of electrons to a chemical species, and oxidation is removal of
electrons from a species. The current in a solution consists of a flow of positive and negative ions
toward the electrodes, whereas the current in a metallic conductor consists of a flow of free electrons
migrating through a crystal lattice of fixed positive ions. Reduction occurs at the cathode, where
electrons enter from the external circuit and are added to a chemical species in solution. Oxidation
occurs at the anode, where the electrons are removed from a chemical species in solution and go into
the external circuit.
In the electrolysis of a solution of ferric sulfate in a cell containing platinum electrodes, a ferric ion
migrates to the cathode, where it picks up an electron and is reduced:
245
Dr. Murtadha Alshareifi e-Library
The sulfate ion carries the curre
ent through the
e solution to th
he anode, but it is not easilyy oxidized;
there
efore, hydroxyyl ions of the water
w
are convverted into molecular oxygen
n, which escaapes at the ano
ode,
and sulfuric acid iss found in the solution aroun
nd the electrode. The oxidation reaction aat the anode is
s
es are used he
ere because t hey do not pa
ass into solutio
on to any extennt.
Platinum electrode
en attackable metals,
m
such as
a copper or zzinc, are used as the anode, their atoms ttend to lose
Whe
electtrons, and the metal passes
s into solution as the positiv
vely charged io
on.
In the electrolysis of cupric chloride between platinum electtrodes, the rea
action at the ccathode is
30
P.13
wherreas at the anode, chloride and hydroxyl ions are conv
verted, respecttively, into gasseous molecules of
chlorrine and oxyge
en, which then
n escape. In e
each of these two
t
examples, the net resullt is the transfe
er of
one electron from the cathode to
o the anode.
Fig.. 6-1. Electrrolysis in an
n electrolytiic cell.
Tra
ansferenc
ce Numb
bers
It sho
ould be noted that the flow of electrons th
hrough the sollution from right to left in Figgure 6-1 is
acco
omplished by the
t movementt of cations to the right as well
w as anions to the left. Thee fraction of to
otal
curre
ent carried by the cations orr by the anionss is known as the transport or transferencce number t+ or
o t-:
246
Dr. Murtadha Alshareifi e-Library
The sum of the tw
wo transference
e numbers is o
obviously equal to unity:
The transference numbers
n
are related
r
to the vvelocities of th
he ions, the fa
aster-moving ioon carrying the
grea
ater fraction of current. The velocities
v
of th
he ions in turn depend on hy
ydration as weell as ion size and
charrge. Hence, the speed and the
t transferencce numbers are
a not necess
sarily the samee for positive and
a
for negative ions. For example, the transferen
nce number off the sodium io
on in a 0.10 M solution of NaCl
N
is 0.3
385. Because it is greatly hy
ydrated, the litthium ion in a 0.10 M solutio
on of LiCl movves more slow
wly
than the sodium io
on and hence has a lower trransference nu
umber, 0.317.
Key Co
oncept
Fara
aday's Law
ws
In 18
833 and 183
34, Michael Faraday
F
anno
ounced his fa
amous laws of electricity,, which may be
sum
mmarized in the statement, the passag
ge of 96,500
0 coulombs of
o electricity th
through a
cond
ductivity cell produces a chemical cha
ange of 1 g equivalent
e
weight
w
of any substance. The
T
quan
ntity 96,500 is known as the faraday, F. The best estimate of the value todday is 9.6484
456
× 10
04 coulombs//gram equiva
alent.
Ele
ectrical Units
U
Acco
ording to Ohm
m's law, the stre
ength of an el ectric current I in amperes flowing
f
througgh a metallic
cond
ductor is relate
ed to the differrence in applie
ed potential orr voltageE and
d the resistancce R in ohms, as
follow
ws:
The current streng
gth I is the rate
e of flow of cu rrent or the qu
uantity Q of ele
ectricity (electtronic charge) in
coulo
ombs flowing per unit time:
and
The quantity of ele
ectric charge is expressed in
n coulombs (1
1 coulomb = 3 × 109 electrosstatic units of
charrge, or esu), th
he current in amperes, and tthe electric po
otential in volts
s.
Electric energy consists of an in
ntensity factor,, electromotive
e force or volta
age, and a quuantity factor,
coulo
ombs.
Farraday's Laws
L
A un
nivalent negative ion is an attom to which a valence elec
ctron has been
n added; a un ivalent positiv
ve ion
is an
n atom from which an electro
on has been rremoved. Each gram equiva
alent of ions of any electroly
yte
carries Avogadro'ss number (6.02 × 1023) of po
ositive or nega
ative charges. Hence, from Faraday's law
ws,
the p
passage of 96,500 coulomb
bs of electricityy results in the
e transport of 6.02
6
× 1023 eleectrons in the cell.
One faraday is an Avogadro's number of elecctrons, corresp
ponding to the
e mole, which iis an Avogadrro's
number of molecu
ules. The pass
sage of 1 farad
day of electriciity causes the electrolytic deeposition of th
he
+
+
2+
2+
3+
follow
wing number of
o gram atoms
s or “moles” off various ions: 1 Ag , 1 Cu , ½ Cu , ½ Fee , 1/3 Fe .
Thuss, the number of positive charges carried by 1 g equiva
alent of Fe3+ is
s 6.02 × 1023, bbut the numbe
er of
23
posittive charges carried
c
by 1 g atom
a
or 1 molle of ferric ions
s is 3 × 6.02 × 10 .
P.13
31
247
Dr. Murtadha Alshareifi e-Library
Fara
aday's laws ca
an be used to compute
c
the ccharge on an electron
e
in the
e following wayy. Because 6.02 ×
23
10 electrons are associated with
w 96,500 cou
ulombs of elec
ctricity, each electron
e
has a charge e of
Ele
ectrolytic
c Conduc
ctance
The resistance, R, in ohms of any uniform me
etallic or electrrolytic conductor is directly pproportional to
o its
2
lengtth, l, in cm and
d inversely pro
oportional to itts cross-sectio
onal area, A, in cm ,
wherre ρ is the resistance betwe
een opposite fa
aces of a 1-cm
m cube of the conductor andd is known as
the sspecific resista
ance.
The conductance, C, is the recip
procal of resisstance,
a a measure o
of the ease wiith which curre
ent can pass tthrough the
and hence can be considered as
cond
ductor. It is exp
pressed in rec
ciprocal ohms or mhos. From
m equation (6-13):
The specific condu
uctance κ is th
he reciprocal o
of specific resiistance and is expressed in mhos/cm:
It is tthe conductan
nce of a solutio
on confined in
n a cube 1 cm on an edge as seen in Figuure 6-2. The
relationship betwe
een specific co
onductance an
nd conductanc
ce or resistanc
ce is obtained by combining
g
equa
ations (6-15) and
a (6-16):
Fig.. 6-2. Relatiionship betw
ween speciffic conductaance and equ
uivalent connductance.
248
Dr. Murtadha Alshareifi e-Library
Fig.. 6-3. Wheaatstone bridg
ge for conduuctance measurements.
Me
easuring the Cond
ductance
e of Soluttions
The Wheatstone bridge
b
assemb
bly for measurring the condu
uctance of a so
olution is show
wn in Figure 6-3.
The solution of unknown resista
ance Rx is placced in the cell and connecte
ed in the circuiit. The contactt
pointt is moved alo
ong the slide wire
w bc until at some point, say
s d, no curre
ent from the soource of
alterrnating currentt (oscillator) flo
ows through th
he detector (e
earphones or oscilloscope).
o
When the brid
dge
is ba
alanced, the po
otential at a is
s equal to that at d, the soun
nd in the earph
hones or the ooscillating patttern
on th
he oscilloscope is at a minim
mum, and the resistances Rs, R1, and R2 are read. In thhe balanced state,
the rresistance of the solution Rx is obtained frrom the equattion
The variable condenser across resistance Rs is used to pro
oduce a sharp
per balance. Soome conductivity
bridg
ges are calibra
ated in conduc
ctance as welll as resistance
e values. The electrodes in tthe cell are
platin
nized with plattinum black by
y electrolytic d
deposition so that
t
catalysis of the reactionn will occur at the
platin
num surfaces and formation
n of a noncond
ducting gaseo
ous film will not occur on thee electrodes.
Wate
er that is careffully purified by
b redistillation
n in the presen
nce of a little permanganate
p
e is used to
prep
pare the solutio
ons. Conductiv
vity water, as it is called, ha
as a specific co
onductance off about 0.05 × 106
mh
ho/cm at 18°C,, whereas ordinary distilled water has a value
v
somewhat greater thann 1 × 10-6 mho
o/cm.
For m
most conductivity studies, “e
equilibrium wa
ater” containin
ng CO2 from th
he atmospheree is satisfactory. It
has a specific conductance of about 0.8 × 10--6 mho/cm.
The specific condu
uctance, κ, is computed from
m the resistan
nce, Rx, or con
nductance, C, by use of
equa
ation (6-17). The
T quantity l/A
A, the ratio of d
distance betw
ween electrode
es to the area of the electrod
de,
has a definite valu
ue for each conductance
P.13
32
cell; it is known ass the cell constant, K. Equattion (6-17) thu
us can be written as
e subscript x iss no longer needed on R an
nd is therefore dropped.) It would
w
be difficcult to
(The
measure l and A, but
b it is a simp
ple matter to d
determine the cell constant experimentally
e
y. The specific
c
cond
ductance of se
everal standard solutions ha
as been determ
mined in careffully calibratedd cells. For
exam
mple, a solutio
on containing 7.45263
7
g of p
potassium chlo
oride in 1000 g of water hass a specific
cond
ductance of 0.012856 mho/c
cm at 25°C. A solution of this concentratio
on contains 0..1 mole of saltt per
249
Dr. Murtadha Alshareifi e-Library
cubicc decimeter (1
100 cm3) of wa
ater and is kno
own as a 0.1-d
demal solution
n. When such a solution is
place
ed in a cell an
nd the resistan
nce is measure
ed, the cell constant can be determined bby use of
equa
ation (6-19).
Exa
ample 6-1
Calc
culating K
A 0.1-demal solu
ution of KCl was
w placed i n a cell whos
se constant K was desireed. The
resisstance R was found to be
e 34.69 ohm s at 25°C. Thus,
Exa
ample 6-2
Calc
culating Sp
pecific Cond
ductance
Whe
en the cell de
escribed in Example
E
6-1 was filled with a 0.01 N Na
N 2SO4 soluttion, it had a
resisstance of 397
7 ohms. Wha
at is the speccific conducttance? We write
w
Equivalent Conducttance
To study the disso
ociation of mollecules into io ns, independe
ent of the conc
centration of thhe electrolyte,, it is
convvenient to use equivalent co
onductance ratther than spec
cific conductan
nce. All solutees of equal
norm
mality produce the same num
mber of ions w
when complete
ely dissociated
d, and equivallent conductan
nce
measures the currrent-carrying capacity
c
of thiss given numbe
er of ions. Spe
ecific conductaance, on the other
o
hand
d, measures th
he current-carrrying capacityy of all ions in a unit volume of solution annd accordingly
y
varie
es with concen
ntration.
Equiivalent conducctance Λ is defined as the cconductance of
o a solution off sufficient voluume to contain
n1g
equivvalent of the solute
s
when measured
m
in a ccell in which the electrodes are spaced 1 cm apart. The
equivvalent conducctance Λc at a concentration
n of c gram equivalents per liter is calculat
ated from the
product of the spe
ecific conducta
ance κ and the
e volume V in cm3 that conta
ains 1 g equivvalent of solute
e.
The cell may be im
magined as ha
aving electrode
es 1 cm apartt and to be of sufficient
s
areaa so that it can
n
conta
ain the solutio
on. The cell is shown in Figu
ure 6-2. We ha
ave
The equivalent conductance is obtained
o
when
n κ, the condu
uctance per cm
m3 of solution (i.e., the spec
cific
3
cond
ductance), is multiplied
m
by V,
V the volume iin cm that contains 1 g equ
uivalent weighht of solute. He
ence,
the e
equivalent con
nductance, Λc, expressed in
n units of mho cm2/Eq, is giv
ven by the exppression
If the
e solution is 0..1 N in concen
ntration, then tthe volume co
ontaining 1 g equivalent
e
of thhe solute will be
b
3
10,000 cm , and, according
a
to equation
e
(6-21 ), the equivale
ent conductan
nce will be 10,0000 times as great
he specific con
nductance. Th
his is seen in th
he following example.
e
as th
Exa
ample 6-3
Spe
ecific and Equivalent
E
Conductanc
C
ce
The measured conductance
c
of a 0.1 N so
olution of a drug
d
is 0.056
63 ohm at 255°C. The cell
-1
consstant at 25°C
C is 0.520 cm
m . What is th
he specific conductance
c
and what is the equivale
ent
cond
ductance of the
t solution at
a this conce
entration? We
e write
Equivalent Conducttance of S
Strong and Weak
k Electrollytes
As th
he solution of a strong electtrolyte is dilute
ed, the specific
c conductance
e κ decreasess because the
number of ions pe
er unit volume of solution is reduced. It so
ometimes goes
s through a maaximum beforre
250
Dr. Murtadha Alshareifi e-Library
decrreasing. Conve
ersely, the equ
uivalent condu
uctance Λ of a solution of a strong electroolyte
stead
dily increasess on dilution. The
T increase in
n Λ with dilutio
on is explained
d as follows. T
The quantity of
electtrolyte remains constant at 1 g equivalentt according to the definition of equivalent conductance;;
howe
ever, the ions are hindered less by their n
neighbors in th
he more dilute
e solution and hence can mo
ove
faste
er. The equiva
alent conducta
ance of a weakk electrolyte also increases on dilution, buut not as rapid
dly at
first.
Kohlrausch was one of the first investigators tto study this phenomenon.
p
He found thatt the equivalen
nt
cond
ductance was a linear function of the squa
are root of the
e concentration
n for strong el ectrolytes in dilute
d
soluttions, as illustrrated in Figure
e 6-4. The exp
pression for Λc, the equivale
ent conductancce at a
conccentration c (E
Eq/L), is
wherre Λ0 is the inttercept on the vertical axis a
and is known as
a the equivallent conductannce at infinite
dilutiion. The consttant
P.13
33
b is tthe slope of th
he line for the strong electro
olytes shown in
n Figure 6-4.
Fig.. 6-4. Equivvalent condu
uctance of sstrong and weak
w
electro
olytes.
Whe
en the equivale
ent conductan
nce of a weak electrolyte is plotted
p
agains
st the square rroot of the
conccentration, as shown for ace
etic acid in Fig
gure 6-4, the curve
c
cannot be
b extrapolatedd to a limiting
value
e, and Λ0 musst be obtained by a method such as is des
scribed in the following paraagraph. The
steep
ply rising curvve for acetic ac
cid results from
m the fact thatt the dissociation of weak ellectrolytes
incre
eases on dilutiion, with a larg
ge increase in the number of
o ions capable
e of carrying thhe current.
Kohlrausch conclu
uded that the ions of all elecctrolytes begin
n to migrate independently aas the solution
n is
dilute
ed; the ions in
n dilute solutions are so far a
apart that they
y do not intera
act in any way.. Under these
251
Dr. Murtadha Alshareifi e-Library
cond
ditions, Λ0 is th
he sum of the equivalent co
onductances of the cations lco and the anioons lao at infinite
dilutiion
Base
ed on this law,, the known Λ0 values for ce
ertain electroly
ytes can be ad
dded and subttracted to yield
d
Λ0 fo
or the desired weak electroly
yte. The meth
hod is illustrate
ed in the follow
wing example.
Exa
ample 6-4
Equ
uivalent Con
nductance of Phenoba
arbital
Wha
at is the equivalent condu
uctance at in
nfinite dilution
n of the weak
k acid phenoobarbital? The
Λ0 o
of the strong electrolytes HCl, sodium
m phenobarbittal (NaP), an
nd NaCl are oobtained from
m
the e
experimenta
al results shown in Figure
e 6-4. The va
alues are Λ0,H
HCl = 426.2, Λ 0,NaP = 73.5, and
Λ0,NaaCl = 126.5 mho
m cm2/Eq.
Now
w, by Kohlrau
usch's law of the indepen
ndent migration of ions,
and
whicch, on simpliffying the righ
ht-hand side of the equattion, become
es
Therefore,
and
Co
olligative Propertie
es of Ele
ectrolytic Solution
ns and
Co
oncentratted Soluttions of N
Nonelectrrolytes
As stated in the prrevious chapte
er, van't Hoff o
observed that the osmotic pressure, π, off dilute solutions of
none
electrolytes, su
uch as sucros
se and urea, co
ould be expressed satisfacttorily by the eqquation π
= RT
Tc [equation (5
5-34)], where R is the gas co
onstant, T is the absolute te
emperature, annd c is the
conccentration in moles/liter.
m
van
n't Hoff found, however, thatt solutions of electrolytes
e
gaave osmotic
pressures approximately two, th
hree, and more
e times largerr than expected from this eqquation, depen
nding
on th
he electrolyte investigated. Introducing
I
a ccorrection fac
ctor i to accoun
nt for the irratioonal behaviorr of
ionicc solutions, he wrote
By th
he use of this equation, van't Hoff was ab
ble to obtain ca
alculated value
es that compaared favorably
y with
the e
experimental results
r
of osmotic pressure. Van't Hoff rec
cognized that i approached the number of
o
ions into which the
e molecule dis
ssociated as th
he solution wa
as made increasingly dilute..
The i factor is plottted against the molal conce
entration of bo
oth electrolytes
s and nonelect
ctrolytes in Figure
6-5. For nonelectrolytes, it is seen to approacch unity, and fo
or strong
P.13
34
electtrolytes, it tend
ds toward a va
alue equal to tthe number off ions formed upon
u
dissociat
ation. For
exam
mple, i approaches the value
e of 2 for solu tes such as NaCl
N
and CaSO
O4, 3 for K2SO
O4 and CaCl2, and
4 forr K3Fe(C)6 and
d FeCl3.
252
Dr. Murtadha Alshareifi e-Library
Fig.. 6-5. van't Hoff
H i facto
or of represeentative com
mpounds.
The van't Hoff facttor can also be
e expressed a
as the ratio of any colligative
e property of a real solution to
that of an ideal solution of a non
nelectrolyte be
ecause i repre
esents the num
mber of times ggreater that th
he
collig
gative effect iss for a real solution (electrol yte or nonelec
ctrolyte) than for
f an ideal noonelectrolyte.
The colligative pro
operties in dilu
ute solutions o
of electrolytes are expressed
d on the molall scale by the
equa
ations
Equa
ation (6-25) ap
pplies only to aqueous
a
soluttions, whereas
s (6-26) throug
gh (6-28) are independent of
o the
solve
ent used.
Exa
ample 6-5
Osm
motic Press
sure of Sodium Chlorid
de
Wha
at is the osmotic pressure
e of a 2.0 m solution of sodium chloride at 20°C?
The i factor for a 2.0 m solution of sodium
m chloride as
s observed in
n Figure 6-5 is about 1.9..
Thus,
The
eory of Electrolyt
E
tic Dissociation
Durin
ng the period in which van't Hoff was devveloping the so
olution laws, th
he Swedish chhemist Svante
e
Arrhe
enius was pre
eparing his doc
ctoral thesis o
on the properties of electroly
ytes at the Uniiversity of Upp
psala
in Sw
weden. In 188
87, he publishe
ed the results of his investig
gations and proposed the noow classic the
eory
of disssociation.1 The
T new theory
y resolved ma
any of the anomalies encountered in the eearlier
interrpretations of electrolytic
e
sollutions. Althou
ugh the theory
y was viewed with
w disfavor bby some influe
ential
253
Dr. Murtadha Alshareifi e-Library
scien
ntists of the nineteenth centtury, Arrheniuss's basic princ
ciples of electrrolytic dissociaation were
gradually accepted
d and are still considered va
alid today. The
e theory of the
e existence off ions in solutio
ons
of ele
ectrolytes eve
en at ordinary temperatures remains intac
ct, aside from some modificaations and
elabo
orations that have
h
been ma
ade through th e years to brin
ng it into line with
w certain stuubborn
expe
erimental factss.
Key Co
oncept
van't Hoff Facttor i
The factor i mayy also be considered to exxpress the departure of concentrated
c
d solutions off
none
electrolytes from
f
the laws
s of ideal sollutions. The deviations off concentrateed solutions of
none
electrolytes can
c be expla
ained on the same basis as deviations
s of real soluutions from
Rao
oult's law, con
nsidered in th
he preceding
g chapter. Th
hey included differences of internal
presssures of the
e solute and solvent,
s
pola
arity, compou
und formation
n or complexxation, and
asso
ociation of either the solu
ute or the sol vent. The de
eparture of electrolytic soolutions from the
colligative effects in ideal sollutions of non
nelectrolytes
s may be attrributed—in aaddition to the
e
facto
ors just enum
merated—to dissociation of weak elec
ctrolytes and
d to interactioon of the ions
s of
stron
ng electrolyte
es. Hence, th
he van't Hofff factor i acco
ounts for the deviations oof real solutio
ons
of no
onelectrolyte
es and electrolytes, regarrdless of the reason for th
he discrepanncies.
The original Arrhenius theory, to
ogether with th
he alterations that have com
me about as a result of the
inten
nsive research
h on electrolyte
es, is summarrized as follow
ws. When electtrolytes are disssolved in wa
ater,
the ssolute exists in
n the form of io
ons in the solu
ution, as seen
n in the followin
ng equations:
The solid form of sodium
s
chlorid
de is marked w
with plus and minus
m
signs in
n reaction (6-229) to indicate that
um chloride exxists as ions even
e
in the cryystalline state. If electrodes are connectedd to a source of
sodiu
curre
ent and are pla
aced in a mas
ss of fused sod
dium chloride,, the molten co
ompound will conduct the
electtric current because the crystal lattice of tthe pure salt consists
c
of ions. The additioon of water to the
t
solid
d dissolves the
e crystal and separates
s
the ions in solutio
on.
Hydrrogen chloride
e exists essentially as neutra
al molecules rather
r
than as ions in the puure form and does
d
not cconduct electricity. When it reacts with wa
ater, however,, it ionizes acc
cording to reacction [equation
n (6+
30)]. H3O is the modern
m
repres
sentation of the
e hydrogen ion in water and
d is known as the hydronium
m or
nium ion. In ad
ddition to H3O+, other hydratted species off the proton prrobably exist inn solution, butt they
oxon
need
d not be considered here.2
Sodium chloride and
a hydrochlorric acid are strrong electrolyttes because th
hey exist almoost completely
y in
the io
onic
P.13
35
form in moderatelyy concentrated
d aqueous sollutions. Inorga
anic acids such as HCl, HNO
O3, H2SO4, an
nd HI;
inorg
ganic bases ass NaOH and KOH
K
of the alkkali metal family and Ba(OH
H)2 and Ca(OH
H)2 of the alkalline
earth
h group; and most
m
inorganic
c and organic salts are highly ionized and
d belong to thee class of stron
ng
electtrolytes.
254
Dr. Murtadha Alshareifi e-Library
Acettic acid is a we
eak electrolyte
e, the opposite
ely directed arrrows in equation (6-31) indiicating that
equillibrium betwee
en the molecu
ules and the io
ons is establish
hed. Most organic acids andd bases and some
s
inorg
ganic compounds, such as H3BO3, H2CO3 , and NH4OH
H, belong to the class of weaak electrolytes
s.
+
2+
Even
n some salts (lead
(
acetate, HgCl2, HgI, an
nd HgBr) and the complex ions Hg(NH3)22 , Cu(NH3)4 , and
Fe(C
CN)63- are wea
ak electrolytes.
Fara
aday applied th
he term ion (G
Greek: “wande
erer”) to these species of ele
ectrolytes and recognized th
hat
the ccations (positivvely charged ions) and anio
ons (negatively
y charged ions
s) were responnsible for
cond
ducting the ele
ectric current. Before the tim
me of Arrhenius's publication
ns, it was belieeved that a so
olute
was not spontaneo
ously decomp
posed in waterr but rather dis
ssociated apprreciably into ioons only when
n an
electtric current wa
as passed thro
ough the solut ion.
Dru
ugs and Ionizatio
I
n
Som
me drugs, such
h as anionic an
nd cationic anttibacterial and
d antiprotozoal agents, are m
more active when
w
in the
e ionic state. Other
O
compou
unds, such as the hydroxybe
enzoate esters (parabens) aand many gen
neral
anessthetics, bring about their biologic effects as nonelectro
olytes. Still other compoundss, such as the
e
sulfo
onamides, are thought to ex
xert their drug action both as
s ions and as neutral moleccules.3
Degree of Dissociat
D
tion
Arrhe
enius did not originally
o
cons
sider strong ellectrolytes to be
b ionized com
mpletely exceppt in extremely
y
dilute
e solutions. He differentiated between strrong and weak
k electrolytes by the fractionn of the molec
cules
ionizzed: the degre
ee of dissociatiion, α. A stron
ng electrolyte was
w one that dissociated
d
intto ions to a hig
gh
degrree and a wea
ak electrolyte was
w one that d
dissociated intto ions to a low
w degree.
Arrhe
enius determined the degre
ee of dissociattion directly fro
om conductance measurem
ments. He
recognized that the equivalent conductance
c
a
at infinite dilutiion Λ0 was a measure
m
of thee complete
disso
ociation of the
e solute into its
s ions and thatt Λc representted the numbe
er of solute paarticles present as
ions at a concentration c. Hence
e, the fraction of solute mole
ecules ionized
d, or the degreee of dissociattion,
was expressed byy the equation4
4
wherre Λc/Λ0 is kno
own as the con
nductance rattio.
Exa
ample 6-6
Deg
gree of Diss
sociation off Acetic Aciid
The equivalent conductance
c
of acetic ac id at 25°C an
nd at infinite dilution is 3990.7 ohm
cm2//Eq. The equ
uivalent cond
ductance of a 5.9 × 10-3 M solution off acetic acid iis 14.4 ohm
cm2//Eq. What is the degree of
o dissociatio
on of acetic acid
a
at this concentrationn? We write
The van't Hoff facttor, i, can be connected
c
with
h the degree of
o dissociation
n, α, in the folloowing way.
The i factor equalss unity for an ideal solution o
of a nonelectrrolyte; howeve
er, a term musst be added to
acco
ount for the pa
articles produc
ced when a mo
olecule of an electrolyte
e
diss
sociates. For 1 mole of calc
cium
chlorride, which yie
elds three ions
s per molecule
e, the van't Ho
off factor is giv
ven by
or, in
n general, for an
a electrolyte yielding v ionss,
from which we obttain an expres
ssion for the de
egree of disso
ociation,
The cryoscopic me
ethod is used to determine i from the exp
pression
or
Exa
ample 6-7
255
Dr. Murtadha Alshareifi e-Library
Deg
gree of Ionizzation of Ac
cetic Acid
The freezing point of a 0.10 m solution off acetic acid is -0.188°C. Calculate thhe degree of
ionizzation of ace
etic acid at this concentra
ation. Acetic acid
a
dissocia
ates into two ions, that is,, v =
2. W
We write
In other words, acccording to the
e result of Exa mple 6-7, the fraction of ace
etic acid preseent as free ion
ns in
a 0.1
10 m solution is 0.011. State
ed in percenta
age terms, ace
etic acid in 0.1 m concentrattion is ionized to
the e
extent of abou
ut 1%.
The
eory of Strong
S
Electrolyte
es
Arrhe
enius used α to
t express the
e degree of disssociation of both
b
strong an
nd weak electrrolytes, and va
an't
Hoff introduced the factor i to ac
ccount for the deviation of strong
s
and wea
ak
P.13
36
electtrolytes and no
onelectrolytes
s from the idea
al laws of the colligative
c
properties, regarddless of the na
ature
of these discrepan
ncies. According to the earlyy ionic theory, the degree off dissociation oof ammonium
m
chlorride, a strong electrolyte, wa
as calculated in the same manner
m
as thatt of a weak eleectrolyte.
Exa
ample 6-8
Deg
gree of Diss
sociation
The freezing point depressio
on for a 0.01 m solution of
o ammonium
m chloride is 00.0367°C.
Calcculate the “de
egree of diss
sociation” of this electroly
yte. We write
e
The Arrhenius theory is now acc
cepted for desscribing the be
ehavior only of weak electroolytes. The degree
of disssociation of a weak electro
olyte can be ca
alculated satis
sfactorily from the conductaance ratio Λc/Λ
Λ0 or
obtained from the van't Hoff i factor.
Many inconsistenccies arise, how
wever, when a
an attempt is made
m
to apply the theory to solutions of strong
electtrolytes. In dilu
ute and moderately concenttrated solution
ns, they dissoc
ciate almost coompletely into
o
ions,, and it is not satisfactory
s
to
o write an equi librium expres
ssion relating the concentraation of the ion
ns
and the minute am
mount of undis
ssociated mole
ecules, as is done
d
for weak electrolytes (C
Chapter 7).
More
eover, a discre
epancy exists between α ca
alculated from the i value an
nd αcalculatedd from the
cond
ductivity ratio for
f strong elec
ctrolytes in aqu
ueous solution
ns having concentrations grreater than about
0.5 M
M.
For tthese reasonss, one does no
ot account for the deviation of strong electrolyte from iddeal nonelectro
olyte
beha
avior by calcullating a degree
e of dissociati on. It is more convenient to
o consider a sttrong electroly
yte as
comp
pletely ionized
d and to introd
duce a factor t hat expresses
s the deviation
n of the solute from 100%
ionizzation. The acttivityand osmo
otic coefficientt, discussed in
n subsequent paragraphs, aare used for th
his
purp
pose.
Activity and
d Activity
y Coeffic
cients
An a
approach that conforms welll to the facts a
and that has ev
volved from a large numberr of studies on
n
soluttions of strong
g electrolytes ascribes
a
the b
behavior of stro
ong electrolyte
es to an electrrostatic attracttion
betw
ween the ions.
Key Co
oncept
Wea
ak and Stro
ong Electrolytes
The large numbe
er of oppositely charged ions in solutions of electrolytes influeence one ano
other
through interioniic attractive forces.
f
Altho ugh this interference is negligible
n
in ddilute solutions, it
256
Dr. Murtadha Alshareifi e-Library
beco
omes apprecciable at mod
derate conce
entrations. In solutions off weak electro
rolytes,
rega
ardless of concentration, the number of ions is sm
mall and the interionic attrraction
correspondingly insignificantt. Hence, the
e Arrhenius th
heory and the concept off the degree of
dissociation are valid for solu
utions of wea
ak electrolyte
es but not forr strong electtrolytes.
Not o
only are the io
ons interfered with in their m
movement by the “atmosphe
ere” of opposite
tely charged io
ons
surro
ounding them,, they can also
o associate at high concentration into gro
oups known ass ion pairs (e.g
g.,
+ + +
Na C
Cl ) and ion trip
plets (Na Cl Na
N ). Associatiions of still hig
gher orders ma
ay exist in solvvents of low
diele
ectric constantt, in which the force of attracction of oppos
sitely charged ions is large.
Beca
ause of the ele
ectrostatic attrraction and ion
n association in moderately concentratedd solutions of
stron
ng electrolytess, the values of
o the freezing point depress
sion and the other
o
colligativee properties are
a
less than expected
d for solutions
s of unhindered
d ions. Conse
equently, a stro
ong electrolytee may
be co
ompletely ioniized, yet incom
mpletely disso
ociated into fre
ee ions.
One may think of the
t solution as
s having an “e
effective conce
entration” or, as
a it is called, an activity. Th
he
activvity, in generall, is less than the
t actual or sstoichiometric concentration
n of the solute,, not because the
stron
ng electrolyte is only partly ionized, but ra
ather because some of the ions are effecttively “taken out of
play”” by the electrostatic forces of interaction..
At infinite dilution, in which the ions are so wid
dely separated that they do not interact w
with one anoth
her,
the a
activity a of an
n ion is equal to
t its concentrration, express
sed as molality
y or molarity. IIt is written on
na
mola
al basis at infin
nite dilution as
s
or
he concentratiion of the solu
ution is increassed, the ratio becomes
b
less than unity bec
ecause the effe
ective
As th
conccentration or activity
a
of the io
ons becomes less than the stoichiometric
c or molal con centration. Th
his
ratio is known as the
t practical activity
a
coefficiient, γm, on the
e molal scale, and the formuula is written, for a
particular ionic spe
ecies, as
or
On the molarity sccale, another practical
p
activiity coefficient, γc, is defined as
P.13
37
and on the mole frraction scale, a rational actiivity coefficient is defined as
s
One sees from equations (6-41)) through (6-4 3) that these coefficients
c
arre proportionallity constants
relating activity to molality, mola
arity, and mole
e fraction, resp
pectively, for an
a ion. The acctivity coefficie
ents
take on a value off unity and are thus identica l in infinitely dilute solutions
s. The three cooefficients usu
ually
decrrease and assume different values as the
e concentration
n is increased; however, thee differences
among the three activity
a
coefficiients may be d
disregarded in
n dilute solutio
ons, in which c [congruent] m <
0.01. The conceptts of activity an
nd activity coe
efficient were first
f
introduced
d by Lewis annd Randall5 an
nd
can b
be applied to solutions of no
onelectrolytess and weak ele
ectrolytes as well
w as to the i ons of strong
electtrolytes.
A ca
ation and an an
nion in an aqu
ueous solution
n may each ha
ave a different ionic activity. This is recogn
nized
by ussing the symb
bol a+ when sp
peaking of the activity of a cation
c
and the symbol a- wheen speaking of
o the
activvity of an anion
n. An electroly
yte in solution contains each
h of these ions
s, however, soo it is convenie
ent to
defin
ne a relationsh
hip between th
he activity of th
he electrolyte a± and the ac
ctivities of the individual ions
s.
The activity of an electrolyte
e
is defined
d
by its m
mean ionic ac
ctivity, which is
s given by thee relation
257
Dr. Murtadha Alshareifi e-Library
wherre the exponents m and n give
g
the stoich iometric numb
bers of given ions that are inn solution. Thu
us,
an N
NaCl solution has
h a mean ionic activity of
wherreas an FeCl3 solution has a mean ionic a
activity of
The ionic activitiess of equation (6-44)
(
can be expressed in terms of concentrations usi ng any of
equa
ations (6-41) to
o (6-43). Using equation (6--42), one obta
ains from equa
ation (6-44) thee expression
or
The mean ionic acctivity coefficie
ent for the elecctrolyte can be
e defined by
and
Subsstitution of equ
uation (6-47) into equation ((6-46) yields
In ussing equation (6-49), it shou
uld be noted th
hat the concen
ntration of the electrolyte c iss related to th
he
conccentration of its ions by
and
Exa
ample 6-9
Mea
an Ionic Acttivity of FeC
Cl3
Wha
at is the mea
an ionic activity of a 0.01 M solution of
o FeCl3? We write
It is p
possible to ob
btain the mean
n ionic activity coefficient, γ±
±, of an electro
olyte by severa
ral experimenttal
meth
hods as well as
a by a theorettical approach
h. The experim
mental methods include distrribution coeffic
cient
studiies, electromo
otive force mea
asurements, ccolligative prop
perty methods
s, and solubilitty determinatio
ons.
Thesse results can then be used to obtain app
proximate activ
vity coefficientts for individuaal ions, where this
is de
esired.6
Debyye and Hückel developed a theoretical m ethod by whic
ch it is possible
e to calculate the activity
coeffficient of a sin
ngle ion as well as the mean
n ionic activity coefficient of a solute withoout recourse to
o
expe
erimental data. Although the
e theoretical eq
quation agree
es with experim
mental findingss only in dilute
e
soluttions (so dilute
e, in fact, that some chemissts have referred jokingly to such solutionss as “slightly
conta
aminated wate
er”), it has cerrtain practical value in solutiion calculation
ns. Furthermorre, the DebyeHückkel equation provides
p
a rem
markable confirrmation of modern solution theory.
The mean ionic acctivity coefficie
ents of a numb
ber of strong electrolytes
e
are
e given in Tabble 6-1. The re
esults
of va
arious investig
gators vary in the
t third decim
mal place; therrefore, most of the entries inn the table are
e
recorded only to tw
wo places, pro
oviding sufficie
ent precision for
f the calculations in this boook. Although the
value
es in the table
e are given at various
v
molaliities, we can accept
a
these activity
a
coefficiients for problems
invollving molar co
oncentrations (in
( which m $0
0.1) because, in dilute solutions, the differrence between
mola
ality and molarrity is not grea
at.
The mean values of Table 6-1 fo
or NaCl, CaCll2, and ZnSO4 are plotted in
n Figure 6-6 aggainst the square
root of the molalityy. The reason for plotting the
e square root of the concen
ntration is due to the form th
hat
the D
Debye-Hückel equation take
es. The activityy coefficient approaches
a
un
nity with increaasing dilution. As
the cconcentrationss of some of th
he electrolytess are increase
ed, their curves
s pass throughh minima and rise
again
n to values greater than uniity. Although t he curves for different electtrolytes of the same ionic class
258
Dr. Murtadha Alshareifi e-Library
coincide at lower concentrations, they differ widely at higher values. The initial decrease in the activity
coefficient with increasing concentration is due to the interionic attraction, which causes the
P.138
activity to be less than the stoichiometric concentration. The rise in the activity coefficient following the
minimum in the curve of an electrolyte, such as HCl and CaCl2, can be attributed to the attraction of the
water molecules for the ions in concentrated aqueous solution. This solvation reduces the interionic
attractions and increases the activity coefficient of the solute. It is the same effect that results in the
salting out of nonelectrolytes from aqueous solutions to which electrolytes have been added.
Table 6-1 Mean Ionic Activity Coefficients of Some Strong Electrolytes at 25°C
On the Molal Scale
Molalit
y (m)
HCl
NaCl
0.0
00
1.
00
1.
00
1.
00
1.
00
1.
00
1.
00
1.
00
1.0
0
1.0
0
0.0
05
0.
93
0.
93
0.
93
–
0.
79
0.
64
0.
78
0.5
3
0.4
8
0.0
1
0.
91
0.
90
0.
90
0.
90
0.
72
0.
55
0.
72
0.4
0
0.3
9
0.0
5
0.
83
0.
82
0.
82
0.
81
0.
58
0.
34
0.
51
0.2
1
0.2
0
0.1
0
0.
80
0.
79
0.
77
0.
76
0.
52
0.
27
0.
44
0.1
5
0.1
5
0.5
0
0.
77
0.
68
0.
65
0.
68
0.
51
0.
16
0.
27
0.0
67
0.0
63
1.0
0
0.
81
0.
66
0.
61
0.
67
0.
73
0.
13
0.
21
0.0
42
0.0
44
2.0
0
1.
01
0.
67
0.
58
0.
69
1.
55
0.
13
0.
15
–
0.0
35
4.0
0
1.
74
0.
79
0.
58
0.
90
2.
93
0.
17
0.
14
–
–
KCl NaOH CaCl2 H2SO2
Activity of the Solvent
259
Na2SO
2
CuSO2 ZnSO2
Dr. Murtadha Alshareifi e-Library
Thuss far, the discu
ussion of activ
vity and activityy coefficients has centered on the solute and particularrly on
electtrolytes. It is customary
c
to define
d
the activvity of the solv
vent on the mo
ole fraction scaale. When a
soluttion is made in
nfinitely dilute,, it can be con
nsidered to con
nsist essentially of pure solvvent. Thereforre,
X1 [ccongruent] 1, and
a the solven
nt behaves ide
eally in conformity with Raoult's law. Undeer this conditio
on,
the m
mole fraction can
c be set equ
ual to the activvity of the solv
vent, or
Fig.. 6-6. Meann ionic activity coefficieents of representative electrolytes
e
plotted agaainst
the square root of concentrration.
As th
he solution becomes more concentrated
c
iin solute, the activity
a
of the solvent ordinaarily becomes less
than the mole fracction concentra
ation, and the ratio can be given,
g
as for th
he solute, by tthe rational ac
ctivity
coeffficient,
or
The activity of a vo
olatile solvent can be determ
mined rather simply.
s
The ratio of the vapoor pressure, p1, of
the ssolvent in a so
olution to the vapor
v
pressure
e of pure solve
ent,p1°, is approximately eq ual to
the a
activity of the solvent
s
at ordiinary pressure
es: a1 = p1/p°.
Exa
ample 6-10
260
Dr. Murtadha Alshareifi e-Library
Calc
culating Es
scaping Ten
ndency
The vapor presssure of water in a solution
n containing 0.5 mole of sucrose
s
in 10000 g of wate
er is
17.3
38 mm and th
he vapor pre
essure of pure
re water at 20
0°C is 17.54 mm. What iss the activity (or
esca
aping tenden
ncy) of water in the solutio
on? We write
e
Reference State
S
The assignment of activities to the
t componen
nts of solutions
s provides a measure
m
of thee extent of
depa
arture from ide
eal solution be
ehavior. For th
his purpose, a reference state must be est
stablished in which
w
each
h component behaves
b
ideally. The referen
nce state can be defined as
s the solution iin which the
conccentration (mo
ole fraction, mo
olal or molar) o
of the compon
nent is equal to the activity,
Activvity = Concenttration
P.13
39
or, w
what amounts to the same th
hing, the activvity coefficient is unity,
The reference statte for a solven
nt on the mole
e fraction scale
e was shown in equation (6--52) to be the pure
ent.
solve
The reference statte for the solute can be cho
osen from one of several pos
ssibilities. If a liquid solute is
misccible with the solvent
s
(e.g., in a solution off alcohol in wa
ater), the conc
centration can be expressed
d in
mole
e fraction, and the pure liquid can be take
en as the referrence state, as
s was done forr the solvent. For a
liquid
d or solid solute having a lim
mited solubilityy in the solven
nt, the referenc
ce state is orddinarily taken as
a
the in
nfinitely dilute solution in wh
hich the conce
entration of the
e solute and the ionic strenggth (see the
follow
wing) of the so
olution are sm
mall. Under the
ese conditions, the activity is
s equal to the concentration
n and
the a
activity coefficient is unity.
Sta
andard State
S
The activities ordin
narily used in chemistry are
e relative activities. It is not possible
p
to knoow the absolu
ute
value
e of the activitty of a compon
nent; therefore
e, a standard must be estab
blished just ass was done
in Ch
hapter 1 for th
he fundamenta
al measurable properties.
The standard state
e of a compon
nent in a solutiion is the state
e of the compo
onent at unit aactivity. The
relative activity in any solution is
s then the ratio
o of the activitty in that state
e to the value iin the standard
state
e. When define
ed in these terrms, activity iss a dimensionlless number.
The pure liquid at 1 atm and at a definite tem perature is ch
hosen as the standard state of a solvent or
o of
a liqu
uid solute miscible with the solvent becau
use, for the pu
ure liquid, a = 1. Because thhe mole fractio
on of
a pure solvent is also
a
unity, mole fraction is e
equal to activity
y, and the refe
erence state iss identical with
h the
stand
dard state.
The standard state
e of the solven
nt in a solid so
olution is the pure
p
solid at 1 atm and at a definite
temp
perature. The assignment of a = 1 to pure
e liquids and pure
p
solids willl be found to bbe convenient in
laterr discussions on
o equilibria and electromottive force.
The standard state
e for a solute of
o limited solu
ubility is more difficult to define. The activiity of the solutte in
an in
nfinitely dilute solution, altho
ough equal to the concentra
ation, is not unity, and the sttandard state is
thus not the same as the reference state. The
e standard sta
ate of the solutte is defined aas a hypothetic
cal
soluttion of unit con
ncentration (m
mole fraction, m
molal or molarr) having, at th
he same time, the characterristics
of an
n infinitely dilute or ideal solution. For com
mplete understtanding, this definition
d
requiires careful
deve
elopment, as carried
c
out by Klotz and Rossenberg.7
Ion
nic Streng
gth
In dillute solutions of nonelectrollytes, activitiess and concenttrations are co
onsidered to bbe practically
identtical because electrostatic forces
f
do not b
bring about de
eviations from ideal behavioor in these
soluttions. Likewise
e, for weak ele
ectrolytes thatt are present alone
a
in solutio
on, the differeences between
n the
261
Dr. Murtadha Alshareifi e-Library
ionicc concentration
n terms and activities are ussually disregarded in ordina
ary calculationss because the
e
number of ions pre
esent is small and the elect rostatic forces
s are negligible
e.
How
wever, for stron
ng electrolytes
s and for solut ions of weak electrolytes
e
together with saalts and other
electtrolytes, such as exist in bufffer systems, iit is important to use activities instead of cconcentrations.
The activity coefficcient, and hen
nce the activityy, can be obtained by using one of the forrms of the Deb
byeHückkel equation (cconsidered latter) if one know
ws the ionic strength of the solution. Lew
wis and
Rand
dall8 introduce
ed the concep
pt of ionic stren
ngth, µ, to rela
ate interionic attractions
a
andd activity
coeffficients. The io
onic strength is
i defined on tthe molar scalle as
or, in
n abbreviated notation,
wherre the summation symbol in
ndicates that th
he product of cz2 terms for all
a the ionic sppecies in the
soluttion, from the first one to the
e jth species, is to be added
d together. The term ci is thee concentratio
on in
mole
es/liter of any of
o the ions and
d zi is its valen
nce. Ionic stre
engths represe
ent the contribbution to the
electtrostatic forcess of the ions of
o all types. It d
depends on th
he total numbe
er of ionic charrges and not on
o
the sspecific properrties of the salts present in tthe solution. Itt was found th
hat bivalent ionns are equivalent
not to two, but to four
f
univalent ions; hence, b
by introducing the square off the valence, one gives pro
oper
weig
ght to the ions of higher charge. The sum is divided by 2 because positive ion-negaative ion pairs
s
contrribute to the to
otal electrostatic interaction , whereas we are interested
d in the effect of each ion
sepa
arately.
Exa
ample 6-11
Calc
culating Ion
nic Strength
h
Wha
at is the ionicc strength of (a) 0.010 M KCl, (b) 0.01
10 M BaSO4, and (c) 0.0 10 M Na2SO
O4,
and (d) what is the ionic strength of a sollution contain
ning all three
e electrolytess together witth
saliccylic acid in 0.010
0
M conc
centration in aqueous so
olution?
(a) K
KCl:
(b) B
BaSO4:
(c) N
Na2 SO4:
(d) T
The ionic stre
ength of a 0.010 M soluti on of salicyliic acid is 0.003 as calculaated from a
know
wledge of the
e ionization of
o the acid att this concen
ntration (using the equatioon [H3O+] =
√Kacc). Unionized
d salicylic acid does not ccontribute to the ionic stre
ength.
The ionic strength of the mix
xture of electrrolytes is the
e sum of the ionic strengtth of the
indivvidual salts. Thus,
T
P.14
40
Exa
ample 6-12
Ioniic Strength of a Solutio
on
262
Dr. Murtadha Alshareifi e-Library
A bu
uffer containss 0.3 mole off K2HPO4 an
nd 0.1 mole of
o KH2PO4 pe
er liter of soluution. Calculate
the iionic strength of the solution.
The concentratio
ons of the ion
ns of K2HPO
O4 are [K+] = 0.3
0 × 2 and [HPO42-] = 0. 3. The value
es for
+
KH2PO4 are [K ] = 0.1 and [H
H2PO4 ] = 0.1
1. Any contrib
butions to µ by further disssociation off
[HPO
O42-] and [H2PO4-] are ne
eglected. Thu
us,
It will be observed
d in Example 6-11
6
that the io
onic strength of
o a 1:1 electro
olyte such as KCl is the sam
me as
the m
molar concenttration; µ of a 1:2 electrolyte
e such as Na2SO
S 4 is three times the conce
centration;
and µ for a 2:2 ele
ectrolyte is four times the co
oncentration.
The mean ionic acctivity coefficie
ents of electro lytes should be
b expressed at
a various ioniic strengths
inste
ead of concenttrations. Lewis
s showed the uniformity in activity
a
coefficients when theey are related to
ionicc strength:
a.
b.
The activvity coefficient of a strong el ectrolyte is roughly constan
nt in all dilute ssolutions of the
same ion
nic strength, irrrespective of tthe type of sallts that are use
ed to provide tthe additional ionic
strength.
The activvity coefficients
s of all strong electrolytes of
o a single clas
ss, for examplee, all uni-univa
alent
electrolyttes, are approximately the ssame at a definite ionic stren
ngth, providedd the solutions
s are
dilute.
The results in Tab
ble 6-1 illustratte the similarityy of the mean ionic activity coefficients foor 1:1 electroly
ytes
at low
w concentratio
ons (below 0.1
1 m) and the d
differences tha
at become ma
arked at higherr concentratio
ons.
Bull9
9 pointed out the
t importance
e of the princip
ple of ionic strrength in bioch
hemistry. In thhe study of the
e
influe
ence of pH on
n biologic actio
on, the effect o
of the variable
e salt concentrration in the buuffer may obsc
cure
the rresults unless the buffer is adjusted
a
to a cconstant ionic strength in ea
ach experimennt. If the
bioch
hemical action
n is affected by the specific salts used, ho
owever, even this precautionn may fail to yield
y
satissfactory resultss. Further use will be made of ionic streng
gth in the chap
pters on ionic equilibria,
solub
bility, and kine
etics.
The
e Debye--Hückel Theory
T
Debyye and Hückel derived an equation
e
based
d on the princiiples that stron
ng electrolytess are complete
ely
ionizzed in dilute so
olution and tha
at the deviatio
ons of electroly
ytic solutions from
f
ideal behhavior are due to
the e
electrostatic efffects of the oppositely charrged ions. The
e equation rela
ates the activitty coefficient of
o a
particular ion or th
he mean ionic activity coefficcient of an ele
ectrolyte to the
e valence of thhe ions, the ion
nic
stren
ngth of the solution, and the
e characteristiccs of the solve
ent. The mathe
ematical derivvation of the
equa
ation is not attempted here but
b can be fou
und in Lewis and
a Randall's Thermodynam
mics as revised
d by
Pitze
er and Brewerr.10 The equattion can be ussed to calculatte the activity coefficients off drugs whose
e
value
es have not be
een obtained experimentallyy and are not available in th
he literature.
Acco
ording to the th
heory of Deby
ye and Hückel , the activity coefficient,
c
γi, of
o an ion of vaalence zi is giv
ven
by th
he expression
Equa
ation (6-57) yields a satisfac
ctory measure
e of the activity
y coefficient off an ion speciees up to an ionic
stren
ngth, µ, of abo
out 0.02. For water
w
at 25°C, A, a factor tha
at depends on
nly on the tem
mperature and the
diele
ectric constantt of the medium
m, is approxim
mately equal to
o 0.51. The va
alues of A for vvarious solven
nts of
pharrmaceutical im
mportance are found in Table
e 6-2.
The form of the De
ebye-Hückel equation
e
for a binary electro
olyte consisting
g of ions with valences
of z+ and z- and prresent in a dilu
ute solution (µ
µ < 0.02) is
The symbols z+ an
nd z- stand forr the valences or charges, ig
gnoring algebrraic signs, on the ions of the
e
electtrolyte whose mean ionic ac
ctivity coefficie
ent is sought. The
T coefficien
nt in equation ((6-58) should
263
Dr. Murtadha Alshareifi e-Library
actua
ally be γx, the rational activity coefficient ((i.e., γ± on the
e mole fraction
n scale), but inn dilute solutio
ons
for w
which the Debyye-Hückel equ
uation is appliccable, γx can be
b assumed without
w
seriouss error to be equal
e
also to the practica
al coefficients, γm and Λc, o n the molal an
nd molar scale
es.
Table 6-2 Valuess of A For Solvents
S
at 25°C*
Soolvent
Dielectricc Constant, ε
Acalc
Acetone
20.70
3.766
Ethanol
24.30
2.966
Water
78.54
0.5009
*Acalc = (1..824 × 106)//(ε × T)3/2, w
where ε is th
he dielectricc constant aand T is the
absolute teemperature on
o the Kelvvin scale.
P.14
41
Exa
ample 6-13
Mea
an Ionic Acttivity Coefficient
Calcculate the me
ean ionic acttivity coefficie
ent for 0.005
5 M atropine sulfate
s
(1:2 eelectrolyte) in
n an
aque
eous solution
n containing 0.01 M NaC
Cl at 25°C. Be
ecause the drug
d
is a uni--bivalent
elecctrolyte, z1z2 = 1 × 2 = 2. For water at 25°C, A is 0.51.
0
We hav
ve
With
h the presentt-day access
sibility of the handheld ca
alculator, the intermediatee step in this
calcculation (need
ded only whe
en log tabless are used) can
c be delete
ed.
Thuss, one observe
es that the acttivity coefficien
nt of a strong electrolyte in dilute
d
solution depends on the
t
total ionic strength
h of the solutio
on, the valence
e of the ions of
o the drug involved, the natture of the solv
vent,
and the temperatu
ure of the solution. Notice th
hat although th
he ionic streng
gth term resultts from the
contrribution of all ionic
i
species in solution, the
e z1z2 terms apply
a
only to th
he drug whosee activity
coeffficient is being
g determined.
Exttension of
o the De
ebye-Hüc
ckel Equa
ation to Higher
H
Co
oncentrattions
The limiting expressions (6-57) and (6-58) are
e not satisfacttory above an ionic strengthh of about 0.02
2,
and equation (6-58
8) is not comp
pletely satisfacctory for use in
nExample 6-13. A formula thhat applies up
p to
onic strength of
o perhaps 0.1 is
an io
264
Dr. Murtadha Alshareifi e-Library
The term ai is the mean distance of approach
h of the ions an
nd is called the mean effecttive ionic
diam
meter or the ion
n size parame
eter. Its exact ssignificance is
s not known; however,
h
it is ssomewhat
analo
ogous to the b term in the van
v der Waalss gas equation
n. The term B, like A, is a coonstant influenced
only by the nature of the solvent and the temp
perature. The values of ai fo
or several elecctrolytes at 25°C
are g
given in Table 6-3 and the values
v
of B an d A for water at various tem
mperatures aree shown in Table
6-4. The values off A for various solvents, as p
previously me
entioned, are listed in Table 6-2.
-8
Beca
ause ai for mo
ost electrolytes
s equals 3 to 4 × 10 and B for water at 25°C
2
equals 0..33 × 108, the
product of ai and B is approxima
ately unity. Eq
quation (6-59) then simplifies to
Tab
ble 6-3 Meaan Effectivee Ionic Diaameter for Some
S
Electtrolytes at 225°C
Electrolyte
ai (cm)
HCl
5.3 × 10-8
NaCl
4.4 × 10-8
KCl
4.1 × 10-8
Methapyrilene HCl
3.9 × 10-8
MgSO4
3.4 × 10-8
K2SO4
3.0 × 10-8
AgNO3
2.3 × 10-8
Sodium phhenobarbitall
2.0 × 10-8
Exa
ample 6-14
Com
mparing Ac
ctivity Coeffficients
Calcculate the acctivity coefficient of a 0.00
04 M aqueou
us solution off sodium pheenobarbital at
a
25°C
C that has be
een brought to an ionic sttrength of 0.09 by the addition of soddium chloride
e.
Use equations (6
6-58) through (6-60) and compare the
e results.
Thesse results can be compared
d with the expe
erimental values for some uni-univalent eelectrolytes
in Ta
able 6-1 at a molal
m
concentrration of aboutt 0.1.
265
Dr. Murtadha Alshareifi e-Library
For sstill higher con
ncentrations, that is, at ionicc strengths above 0.1, the observed activiity coefficients
s for
some
e electrolytes pass through minima and t hen increase with concentra
ation; in somee cases, they
beco
ome greater th
han unity, as seen
s
in Figure 6-6. To accou
unt for the increase in γ± at higher
conccentrations, an
n empirical term Cµ can be added to the Debye-Hücke
D
l equation, ressulting in the
exprression
es satisfactory
y results in solu
utions of conc
centrations as high as 1 M. T
The mean ionic
This equation give
activvity coefficient
P.14
42
obtained from equ
uation (6-61) is
s γx; however, it is not signifficantly different from γm andd γc even at th
his
conccentration. Zog
grafi et al.11 used
u
the exten
nded Debye-H
Hückel equation [(equation (66-61)] in a stu
udy of
the in
nteraction bettween the dye orange II and
d quarternary ammonium
a
sa
alts.
Tab
ble 6-4 Valu
ues of A an
nd B for Waater at Varrious Temperatures
Temperaturre (°C)
A
B
0
0.4
488
0.3 25 × 108
15
0.5
500
0.3 28 × 108
25
0.5
509
0.3 30 × 108
40
0.5
524
0.3 33 × 108
70
0.5
560
0.3 39 × 108
100
0.6
606
0.3 48 × 108
Invesstigations havve resulted in equations
e
thatt extend the co
oncentration to about 5 molles/liter.12
Co
oefficients
s for Exp
pressing Colligative Prope
erties
The
e L Value
e
The van't Hoff exp
pression ∆Tf = iKfm probablyy provides the
e best single equation for coomputing the
collig
gative propertiies of nonelec
ctrolytes, weakk electrolytes, and strong electrolytes. It ccan be modifie
ed
sligh
htly for conven
nience in dilute
e solutions by substituting molar
m
concentrration c and byy writing iKf as
s L,
so th
hat
Goya
an et al.13 com
mputed L from
m experimenta
al data for a nu
umber of drugs. It varies witth the
conccentration of th
he solution. Att a concentratiion of drug tha
at is isotonic with
w body fluidss, L = iKf is
desig
gnated here as
a Liso. It has a value equal tto about 1.9 (a
actually 1.86) for nonelectroolytes, 2.0 for weak
w
electtrolytes, 3.4 fo
or uni-univalen
nt electrolytes,, and larger va
alues for electrolytes of highh valences. A plot
of iK
Kf against the concentration
c
of some drugss is presented
d in Figure 6-7, where each curve is
repre
esented as a band
b
to show the variability of the L value
es within each ionic class. T
The
apprroximate Liso fo
or each of the ionic classes can be obtain
ned from the dashed
d
line runnning through
h the
figurre.
266
Dr. Murtadha Alshareifi e-Library
Fig.. 6-7. Liso vaalues of varrious ionic cclasses.
Key Co
oncept
Activities
Altho
ough activitie
es can be us
sed to bring tthe colligative
e properties of strong eleectrolytes into
o
line with experim
mental results
s, the equatio
ons are complicated and are not treaated in this bo
ook.
Activvities are mo
ore valuable in connectio n with equilib
brium expres
ssions and ellectrochemic
cal
calcculations. The
e use of activ
vities for calcculating the colligative
c
pro
operties of w
weak electrolytes
is pa
articularly incconvenient because it alsso requires knowledge
k
off the degree of dissociatio
on.
Os
smotic Co
oefficientt
Othe
er methods of correcting for the deviationss of electrolyte
es from ideal colligative
c
behhavior have be
een
sugg
gested. One off these is base
ed on the fact that as the so
olution becomes more dilutee, i approache
es ν,
the n
number of ionss into which an electrolyte d
dissociates, an
nd at infinite dilution, i = ν, oor i/ν = 1.
Procceeding in the direction of more
m
concentra
ated solutions, i/ν becomes less (and som
metimes greater)
than unity.
The ratio i/ν is dessignated as g and
a is known as the practic
cal osmotic coe
efficient whenn expressed on
na
mola
al basis. In the
e case of a weak electrolyte , it provides a measure of th
he degree of ddissociation. For
F
stron
ng electrolytess, g is equal to
o unity for com
mplete dissocia
ation, and the departure of g from unity, th
hat
is, 1 - g, in modera
ately concentrrated solutionss is an indicatiion of the interrionic attractioon. Osmotic
coeffficients, g, for electrolytes and
a nonelectro
olytes are plottted against ion
nic concentrattion, νm, in Fig
gure
6-8. Because g = 1/ν or i = gν in
n a dilute soluttion, the cryos
scopic equation can be writtten as
267
Dr. Murtadha Alshareifi e-Library
The molal osmoticc coefficients of
o some salts a
are listed in Table 6-5.
Exa
ample 6-15
Molality and Molarity
M
The osmotic coe
efficient of LiB
Br at 0.2 m iss 0.944 and the Liso value
e is 3.4. Com
mpute ∆Tf forr this
com
mpound using
g g and Liso. Disregard
D
the
e difference between mo
olality and moolarity. We have
Os
smolality
Altho
ough osmotic pressure classically is given
n in atmosphe
eres, in clinical practice it is expressed in
terms of osmols (O
Osm) or millios
smols (mOsm
m). A solution containing
c
1 mole
m
(1 g moleecular weight) of a
nonio
onizable subsstance in 1 kg of water (a 1 m solution) is referred to as
s a 1-osmolal ssolution. It
conta
ains 1 osmol (Osm)
(
or 1000
0 milliosmols ((mOsm) of sollute per kilogra
am of solvent.. Osmolality
measures the tota
al number of pa
articles dissolvved in 1 kg of water, that is,
P.14
43
the o
osmols per kilo
ogram of wate
er, and depend
ds on the elec
ctrolytic nature
e of the solute.. An ionic species
disso
olved in waterr will dissociate
e to form ions or “particles.”” These ions te
end to associaate somewhatt,
howe
ever, owing to
o their ionic intteractions. The
e apparent nu
umber of “particles” in solutioon, as measurred
by ossmometry or one
o of the othe
er colligative m
methods, will depend
d
on the
e extent of theese interaction
ns. An
un-io
onized materia
al (i.e., a none
electrolyte) is u
used as the re
eference solute
e for osmolalitty measureme
ents,
ionicc interactions being
b
insignific
cant for a non electrolyte. Fo
or an electroly
yte that dissocciates into ions
s in a
dilute
e solution, osm
molality or milliosmolality ca
an be calculate
ed from
Fig.. 6-8. Osmootic coefficient, g, for s ome comm
mon solutes. (From G. SScatchard, W.
W
268
Dr. Murtadha Alshareifi e-Library
Ham
mer, and S. Wood, J. Am.
A Chem. S
Soc.60, 306
61, 1938. With permissiion.)
wherre i is approxim
mately the num
mber of ions fo
formed per mo
olecule and mm
m is the millim
molal
conccentration. If no ionic
P.14
44
interractions occurrred in a solutio
on of sodium cchloride, i wou
uld equal 2.0. In a typical caase, for a 1:1
electtrolyte in dilute
e solution, i is approximatelyy 1.86 rather than
t
2.0, owing to ionic inteeraction between
the p
positively and negatively cha
arged ions.
Tablee 6-5 Osmootic Coefficcient, g, at 25°C*
2
N
NaCl
KCl
H2SO4
Sucrose
Urea
Glycerin
n
0.1
0.9342
0.9264
0
0.6784
1.0073
0.9959
1.0014
0.2
0.9255
0.9131
0
0.6675
1.0151
0.9918
1.0028
0.4
0.9217
0.9023
0
0.6723
1.0319
0.9841
1.0055
0.6
0.9242
0.8987
0
0.6824
1.0497
0.9768
1.0081
0.8
0.9295
0.8980
0
0.6980
1.0684
0.9698
1.0105
1.0
0.9363
0.8985
0
0.7176
1.0878
0.9631
1.0128
1.6
0.9589
0.9024
0
0.7888
1.1484
0.9496
1.0192
2.0
0.9786
0.9081
0
0.8431
1.1884
0.9346
1.0230
3.0
1.0421
0.9330
0
0.9922
1.2817
0.9087
1.0316
4.0
1.1168
0.9635
0
1.1606
1.3691
0.8877
1.0393
5.0
1.2000
0.9900
0
–
1.4477
0.8700
1.0462
m
*From G. Scatchard, W.
W G. Ham
mer, and S. E.
E Wood, J. Am. Chem . Soc. 60,
3061, 19388. With perm
mission.
Exa
ample 6-16
269
Dr. Murtadha Alshareifi e-Library
Calc
culating Milliosmolalitty
Wha
at is the millio
osmolality off a 0.120 m ssolution of po
otassium bromide? Whatt is its osmotic
presssure in atmo
ospheres?
A 1--osmolal solu
ution raises the boiling po
oint 0.52°C, lowers
l
the freezing pointt 1.86°C, and
d
prod
duces an osm
motic pressure of 24.4 at m at 25°C. Therefore,
T
a 0.223-Osm/kkg solution yields
an o
osmotic presssure of 24.4 × 0.223 = 5..44 atm.
Stren
ng et al.14 and Murty et al.1
15 discuss the
e use of osmolality and osmolarity in cliniccal pharmacy..
Mola
arity (moles off solute per lite
er of solution) is used in clin
nical practice more
m
frequentlly than molalitty
(moles of solute per kilogram off solvent). In a
addition, osmo
olarity is used more
m
frequenttly than osmollality
in lab
beling parente
eral solutions in
i the hospitall. Yet, osmolarity cannot be measured annd must be
calcu
ulated from the experimenta
ally determine
ed osmolality of
o a solution. As
A shown by M
Murty et al.,15 the
convversion is mad
de using the re
elation
Acco
ording to Stren
ng et al.,14 os
smolality is con
nverted to osm
molarity using the equation
wherre d1° is the de
ensity of the solvent
s
and [v with bar abov
ve]°2 is the parttial molal volu me of the solu
ute at
infiniite dilution.
Exa
ample 6-17
Con
nverting Os
smolality to Osmolarity
y
A 30
0-g/liter soluttion of sodium
m bicarbonatte contains 0.030
0
g/mL of
o anhydrouss sodium
bica
arbonate. The
e density of this
t
solution was found to
o be 1.0192 g/mL
g
at 20°C
C and its
mea
asured milliossmolality was
s 614.9 mOssm/kg. Conve
ert milliosmo
olality to millioosmolarity. We
W
have
e
Exa
ample 6-18
Calc
culating Milliosmolaritty
A 0.154 m sodiu
um chloride solution
s
has a milliosmola
ality of 286.4 mOsm/kg (ssee Example
e 619). Calculate th
he milliosmolarity, mOsm /liter solution
n, using equa
ation (6-66). The density of
the ssolvent—water—at 25°C
C is d1° = 0.99
971 g/cm3 an
nd the partial molal volum
me of the
solu
ute—sodium chloride—is [v with bar a
above]°2 = 16
6.63 mL/mole
e.
As noted, osmolarrity differs from
m osmolality b
by only 1% or 2%.
2 However, in more conccentrated soluttions
of po
olyvalent electtrolytes togeth
her with bufferss, preservative
es, and other ions, the diffeerence may
beco
ome significan
nt. For accurac
cy in the prepa
aration and lab
beling of paren
nteral solutionns, osmolality
shou
uld be measurred carefully with
w a vapor prressure or free
ezing point osm
mometer (rathher than
calcu
ulated) and the results conv
verted to osmo
olarity using equation (6-65)) or (6-66).
Who
ole blood, plasma, and serum
m are complexx liquids consisting of proteins, glucose, nnonprotein
nitrogenous materrials, sodium, potassium, ca
alcium, magne
esium, chloride
e, and bicarboonate ions. The
serum electrolytess, constituting less than 1% of the blood's
s weight, deterrmine the osm
molality of the
blood
d. Sodium chloride contribu
utes a milliosm
molality of 275,, whereas gluc
cose and the oother constitue
ents
together provide about
a
10 mOsm
m/kg to the blo
ood.
270
Dr. Murtadha Alshareifi e-Library
Colligative propertties such as frreezing point d
depression are
e related to os
smolality throuugh equations (627) a
and (6-63):
wherre i = gν and im
m = gνm is os
smolality.
Exa
ample 6-19
Calc
culate Free
ezing Point Depression
n
Calcculate the fre
eezing point depression
d
o
of (a) 0.154 m solution off NaCl and (bb) a
0.15
54 m solution
n of glucose. What are th e milliosmola
alities of thes
se two solutioons?
(a) F
From Table 6-5,
6 g for NaCl at 25°C iss about 0.93, and becaus
se NaCl ionizzes into two
ionss, i = νg = 2 × 0.93 = 1.86
6. From equa
ation (6-64), the
t osmolalitty of a 0.154 m solution
is im
m = 1.86 × 0.154 = 0.2864
4. The milliossmolality of this
t
solution is therefore 2286.4 mOsm
m/kg.
Usin
ng equation (6-67),
(
with Kf also equal to 1.86, we obtain for the freezing pooint depression
of a 0.154 m solution—or its equivalent, a 0.2864-Os
sm/kg solution—of NaCl:
(b) G
Glucose is a nonelectroly
yte, producin g only one particle
p
for ea
ach of its mollecules in
solu
ution, and forr a nonelectro
olyte, i = ν = 1 and g = i/ν
ν = 1. Therefore, the freez
ezing point
deprression of a 0.154 m solu
ution of gluco
ose is approx
ximately
whicch is nearly one
o half of th
he freezing po
oint depress
sion provided
d by sodium cchloride, a 1:1
elecctrolyte that provides
p
two particles rat her than one
e particle in solution.
s
The osmolality of
o a nonelectrolyte such a
as glucose is
s identical to its molal conncentration
beca
ause osmola
ality = i × mollality, and i fo
or a nonelecttrolyte is 1.00
0. The milliossmolality of a
solu
ution is 1000 times its osm
molality or, in
n this case, 154
1 mOsm/kg
g.
P.14
45
Ohw
waki et al.16 sttudied the effe
ect of osmolalitty on the nasa
al absorption of
o secretin, a hhormone used
d in
the treatment of du
uodenal ulcers
s. They found that maximum
m absorption through
t
the naasal mucosa
occu
urred at a sodium chloride milliosmolarity
m
of about 860 mOsm/liter (0.462 M), posssibly owing to
strucctural changess in the epithelial cells of the
e nasal mucos
sa at this high mOsm/liter vaalue.
Altho
ough the osmo
olality of blood
d and other bo
ody fluids is co
ontributed mainly by the conntent of sodium
m
chlorride, the osmo
olality and milliosmolality of these comple
ex solutions by
y convention aare calculated on
the b
basis of i for nonelectrolytes
s, that is, i is ta
aken as unity, and osmolalitty becomes eqqual to molalitty.
This principle is be
est described by an exampl e.
Exa
ample 6-20
Milliosmolarity
y of Blood
Free
ezing points were
w
determ
mined using th
he blood of 20
2 normal ind
dividuals andd were avera
aged
to -0
0.5712°C. Th
his value of course
c
is equ
uivalent to a freezing
f
poin
nt depressionn of +0.5712°C
belo
ow the freezin
ng point of water
w
becausse the freezin
ng point of wa
ater is taken as 0.000°C at
atmo
ospheric pre
essure. What is the avera
age milliosmo
olality, x, of th
he blood of tthese subjects?
Usin
ng equation (6-67)
(
with th
he arbitrary cchoice of i = 1 for body flu
uids, we obtaain
It is n
noted in Exam
mple 6-20 that although the o
osmolality of blood
b
and its freezing
f
point depression arre
contrributed mainlyy by NaCl, an i value of 1 wa
as used for blo
ood rather tha
an gν = 1.86 foor an NaCl
soluttion.
271
Dr. Murtadha Alshareifi e-Library
The milliosmolality for blood obtained by various workers using osmometry, vapor pressure, and
freezing point depression apparatus ranges from about 250 to 350 mOsm/kg.17 The normal osmolality
of body fluids is given in medical handbooks18 as 275 to 295 mOsm/kg, but normal values are likely to
fall in an even narrower range of 286 ± 4 mOsm/kg.19 Freezing point and vapor pressure osmometers
are now used routinely in the hospital. A difference of 50 mOsm/kg or more from the accepted values of
a body fluid suggests an abnormality such as liver failure, hemorrhagic shock, uremia, or other toxic
manifestations. Body water and electrolyte balance are also monitored by measurement of
milliosmolality.
Chapter Summary
In this chapter, solutions of electrolytes were introduced and discussed. Understanding the
properties of electrolytes in solution is still very important today in the pharmaceutical
sciences. You should be able to understand Faraday's law and calculate the conductance in
solutions. The rationale for using the concept of activity rather than concentrations is
important in many chemical calculations since solutions that contain ionic solutes do not
behave ideally even at very low concentrations. The information provided establishes a
framework for calculating ionic strength, osmolality, and osmolarity. The successful student
will not only be able to do these calculations but will understand the differences between
osmolality and osmolarity.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. S. Arrhenius, Z. Physik. Chem. 1, 631, 1887.
2. H. L. Clever, J. Chem. Educ. 40, 637, 1963; P. A. Giguère, J. Chem. Educ. 56, 571, 1979; R. P.
Bell, The Proton in Chemistry, 2nd Ed., Chapman and Hall, London, 1973, pp. 13–25.
3. A. Albert, Selective Toxicity, Chapman and Hall, London, 1979, pp. 344, 373–384.
4. S. Arrhenius, J. Am. Chem. Soc. 34, 361, 1912.
5. G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, New York, 1923, p. 255.
6. S. Glasstone, An Introduction to Electrochemistry, Van Nostrand, New York, 1942, pp. 137–139.
7. I. M. Klotz and R. M. Rosenberg, Chemical Thermodynamics, 3rd Ed., W. A. Benjamin, Menlo Park,
Calif., 1973, Chapter 19.
8. G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, New York, 1923, p. 373.
9. H. B. Bull, Physical Biochemistry, 2nd Ed., Wiley, New York, 1951, p. 79.
10. G. N. Lewis and M. Randall, Thermodynamics, Rev. K. S. Pitzer and L. Brewer, 2nd Ed., McGrawHill, New York, 1961, Chapter 23.
11. G. Zografi, P. Patel, and N. Weiner, J. Pharm. Sci. 53, 545, 1964.
12. R. H. Stokes and R. A. Robinson, J. Am. Chem. Soc. 70, 1870, 1948; M. Eigen and E. Wicke, J.
Phys. Chem. 58, 702, 1954.
13. F. M. Goyan, J. M. Enright, and J. M. Wells, J. Am. Pharm. Assoc. Sci. Ed. 33, 74, 1944.
14. W. H. Streng, H. E. Huber, and J. T. Carstensen, J. Pharm. Sci. 67, 384, 1978; H. E. Huber, W. H.
Streng, and H. G. H. Tan, J. Pharm. Sci. 68, 1028, 1979.
15. B. S. R. Murty, J. N. Kapoor, and P. O. DeLuca, Am. J. Hosp. Pharm. 33, 546, 1976.
16. T. Ohwaki, H. Ando, F. Kakimoto, K. Uesugi, S. Watanabe, Y. Miyake, and M. Kayano, J. Pharm.
Sci. 76, 695, 1987.
17. C. Waymouth, In Vitro 6, 109, 1970.
18. M. A. Kaupp, et al. (Eds.), Current Medical Diagnosis and Treatment, Lange, Los Altos, Calif., 1986,
pp. 27, 1113.
19. L. Glasser, D. D. Sternalanz, J. Combie, and A. Robinson, Am. J. Clin. Pathol. 60, 695–699, 1973.
Recommended Readings
R. A. Robinson and R. H. Stokes, Electrolyte Solutions: Second Revised Edition, Dover Publications,
Mineola NY, 2002.
Chapter Legacy
272
Dr. Murtadha Alshareifi e-Library
Fifth Edition: published as Chapter 6 (Electrolyte Solutions). Updated by Patrick Sinko.
Sixth Edition: published as Chapter 6 (Electrolyte Solutions). Updated by Patrick Sinko.
273
Dr. Murtadha Alshareifi e-Library
7 Ionic Equilibria
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Describe the Brönsted–Lowry and Lewis electronic theories.
2. Identify and define the four classifications of solvents.
3. Understand the concepts of acid–base equilibria and the ionization of weak acids and
weak bases.
4. Calculate dissociation constants Ka and Kb and understand the relationship
between Ka and Kb.
5. Understand the concepts of pH, pK, and pOH and the relationship between hydrogen
ion concentration and pH.
6. Calculate pH.
7. Define strong acid and strong base.
8. Define and calculate acidity constants.
Introduction
Arrhenius defined an acid as a substance that liberates hydrogen ions and a base as a substance that
supplies hydroxyl ions on dissociation. Because of a need for a broader concept, Brönsted in
Copenhagen and Lowry in London independently proposed parallel theories in 1923.1 The Brönsted–
Lowry theory, as it has come to be known, is more useful than the Arrhenius theory for the
representation of ionization in both aqueous and nonaqueous systems.
Brönsted–Lowry Theory
According to the Brönsted–Lowry theory, an acid is a substance, charged or uncharged, that is capable
of donating a proton, and a base is a substance, charged or uncharged, that is capable of accepting a
proton from an acid. The relative strengths of acids and bases are measured by the tendencies of these
substances to give up and take on protons. Hydrochloric acid is a strong acid in water because it gives
up its proton readily, whereas acetic acid is a weak acid because it gives up its proton only to a small
extent. The strength of an acid or a base varies with the solvent. Hydrochloric acid is a weak acid in
glacial acetic acid and acetic acid is a strong acid in liquid ammonia. Consequently, the strength of an
acid depends not only on its ability to give up a proton but also on the ability of the solvent to accept the
proton from the acid. This is called the basic strength of the solvent.
Key Concept
Classification of Solvents
Solvents can be classified as protophilic, protogenic, amphiprotic, and aprotic. A protophilic or
basic solvent is one that is capable of accepting protons from the solute. Such solvents as
acetone, ether, and liquid ammonia fall into this group. A protogenic solvent is a protondonating compound and is represented by acids such as formic acid, acetic acid, sulfuric acid,
liquid HCl, and liquid HF. Amphiprotic solvents act as both proton acceptors and proton
donors, and this class includes water and the alcohols. Aprotic solvents, such as the
hydrocarbons, neither accept nor donate protons, and, being neutral in this sense, they are
useful for studying the reactions of acids and bases free of solvent effects.
-
-
In the Brönsted–Lowry classification, acids and bases may be anions such as HSO4 and CH3COO ,
+
+
cations such as NH4 and H3O , or neutral molecules such as HCl and NH3. Water can act as either an
acid or a base and thus is amphiprotic. Acid–base reactions occur when an acid reacts with a base to
form a new acid and a new base. Because the reactions involve a transfer of a proton, they are known
as protolytic reactions or protolysis.
In the reaction between HCl and water, HCl is the acid and water the base:
274
Dr. Murtadha Alshareifi e-Library
Acid1 and Base1 stand for an acid–base pair or conjugate pair, as do Acid2 and Base2. Because the bare
+
proton, H , is practically nonexistent in aqueous solution, what is normally referred to as the hydrogen
+
ion consists of the hydrated proton, H3O , known as the hydronium ion. Higher solvated forms can also
exist in solution.* In an ethanolic solution, the “hydrogen ion” is the proton attached to a molecule of
+
solvent, represented as C2H5OH2 . In equation (7-1), hydrogen chloride, the acid, has donated a proton
+
-
to water, the base, to form the corresponding acid, H3O , and the base, Cl .
The reaction of HCl with water is one of ionization. Neutralization and hydrolysis are also considered as
acid–base reactions or proteolysis following the broad definitions of the Brönsted–Lowry concept.
Several examples illustrate these types of reactions, as shown in Table 7-1. The displacement reaction,
a special type of neutralization, involves the displacement of a weaker acid, such as acetic acid, from its
salt as in the reaction shown later.
P.147
Table 7-1 Examples of Acid–Base Reactions
Acid1
Base2
OH-
NH4
Neutralizati
on
H3O
+
OH-
=
H2O
+
H2
O
Neutralizati
on
HCl
+
NH3
=
NH4+
+
Cl-
Hydrolysis
H2O
+
CH3CO
O-
=
CH3COO
H
+
OH
NH4
+
H2O
=
H3O+
+
+
+
HCl
H2O
+
NH
3
+
Displaceme
nt
=
Base1
Neutralizati
on
Hydrolysis
+
Acid2
-
NH
3
+
CH3CO
O-
=
CH3COO
H
+
Cl-
Lewis Electronic Theory
Other theories have been suggested for describing acid–base reactions, the most familiar of which is
the electronic theory of Lewis.3
According to the Lewis theory, an acid is a molecule or an ion that accepts an electron pair to form a
covalent bond. A base is a substance that provides the pair of unshared electrons by which the base
coordinates with an acid. Certain compounds, such as boron trifluoride and aluminum chloride, although
not containing hydrogen and consequently not serving as proton donors, are nevertheless acids in this
scheme. Many substances that do not contain hydroxyl ions, including amines, ethers, and carboxylic
acid anhydrides, are classified as bases according to the Lewis definition. Two Lewis acid–base
reactions are
275
Dr. Murtadha Alshareifi e-Library
The Lewis system is probably too broad for convenient application to ordinary acid–base reactions, and
those processes that are most conveniently expressed in terms of this electronic classification should be
referred to simply as a form of electron sharing rather than as acid–base reactions.4 The Lewis theory is
finding increasing use for describing the mechanism of many organic and inorganic reactions. It will be
mentioned again in the chapters on solubility and complexation. The Brönsted–Lowry nomenclature is
particularly useful for describing ionic equilibria and is used extensively in this chapter.
Key Concept
Equilibrium
Equilibrium can be defined as a balance between two opposing forces or actions. This
statement does not imply cessation of the opposing reactions, suggesting rather a dynamic
equality between the velocities of the two. Chemical equilibrium maintains the concentrations
of the reactants and products constant. Most chemical reactions proceed in both a forward
and a reverse direction if the products of the reaction are not removed as they form. Some
reactions, however, proceed nearly to completion and, for practical purposes, may be
regarded as irreversible. The topic of chemical equilibria is concerned with truly reversible
systems and includes reactions such as the ionization of weak electrolytes.
Acid–Base Equilibria
The ionization or proteolysis of a weak electrolyte, acetic acid, in water can be written in the Brönsted–
Lowry manner as
The arrows pointing in the forward and reverse directions indicate that the reaction is proceeding to the
right and left simultaneously. According to the law of mass action, the velocity or rate of the forward
reaction, Rf, is proportional to the concentration of the reactants:
The speed of the reaction is usually expressed in terms of the decrease in the concentration of either
the reactants per unit time. The terms rate, speed, and velocity have the same meaning here. The
reverse reaction
expresses the rate, Rr, of re-formation of un-ionized acetic acid. Because only 1 mole of each
constituent appears in the
P.148
reaction, each term is raised to the first power, and the exponents need not appear in subsequent
expressions for the dissociation of acetic acid and similar acids and bases. The symbols k1 and k2 are
proportionality constants commonly known as specific reaction rates for the forward and the reverse
reactions, respectively, and the brackets indicate concentrations. A better representation of the facts
would be had by replacing concentrations with activities, but for the present discussion, the approximate
equations are adequate.
Ionization of Weak Acids
276
Dr. Murtadha Alshareifi e-Library
According to the concept of equilibrium, the rate of the forward reaction decreases with time as acetic
acid is depleted, whereas the rate of the reverse reaction begins at zero and increases as larger
quantities of hydrogen ions and acetate ions are formed. Finally, a balance is attained when the two
rates are equal, that is, when
The concentrations of products and reactants are not necessarily equal at equilibrium; the speeds of the
forward and reverse reactions are what are the same. Because equation (7-7) applies at equilibrium,
equations (7-5) and (7-6) may be set equal:
and solving for the ratio k1/k1, one obtains
In dilute solutions of acetic acid, water is in sufficient excess to be regarded as constant at about 55.3
moles/liter (1 liter H2O at 25°C weights 997.07 g, and 997.07/18.02 = 55.3). It is thus combined
with k1/k2 to yield a new constant Ka, the ionization constant or the dissociation constant of acetic acid.
Equation (7-10) is the equilibrium expression for the dissociation of acetic acid, and the dissociation
constant Ka is an equilibrium constant in which the essentially constant concentration of the solvent is
incorporated. In the discussion of equilibria involving charged as well as uncharged acids, according to
the Brönsted–Lowry nomenclature, the term ionization constant, Ka, is not satisfactory and is replaced
by the name acidity constant. Similarly, for charged and uncharged bases, the termbasicity constant is
now often used for Kb, to be discussed in the next section.
In general, the acidity constant for an uncharged weak acid HB can be expressed by
Equation (7-10) can be presented in a more general form, using the symbol c to represent the initial
+
molar concentration of acetic acid and x to represent the concentration [H3O ]. The latter quantity is also
-
equal to [Ac ] because both ions are formed in equimolar concentration. The concentration of acetic acid
remaining at equilibrium [HAc] can be expressed as c - x. The reaction [equation (7-4)] is
and the equilibrium expression (7-10) becomes
where c is large in comparison with x. The term c - x can be replaced by c without appreciable error,
giving the equation
which can be rearranged as follows for the calculation of the hydrogen ion concentration of weak acids:
Example 7-1
In a liter of a 0.1 M solution, acetic acid was found by conductivity analysis to dissociate into
-3
1.32 × 10 g ions (“moles”) each of hydrogen and acetate ion at 25°C. What is the acidity or
dissociation constant Ka for acetic acid?
According to equation (7-4), at equilibrium, 1 mole of acetic acid has dissociated into 1 mole
each of hydrogen ion and acetate ion. The concentration of ions is expressed as moles/liter
and less frequently as molality. A solution containing 1.0078 g of hydrogen ions in 1 liter
represents 1 g ion or 1 mole of hydrogen ions. The molar concentration of each of these ions
277
Dr. Murtadha Alshareifi e-Library
is expressed as x. If the original amount of acetic acid was 0.1 mole/liter, then at equilibrium
the undissociated acid would equal 0.1 - x because x is the amount of acid that has
dissociated. The calculation according to equation (7-12) is
-3
It is of little significance to retain the small number, 1.32 × 10 , in the denominator, and the
calculations give
The value of Ka in Example 7-1 means that, at equilibrium, the ratio of the product of the ionic
-5
concentrations to that of the undissociated acid is 1.74 × 10 ; that is, the dissociation of acetic acid into
its ions is small, and acetic acid may be considered as a weak electrolyte.
P.149
When a salt formed from a strong acid and a weak base, such as ammonium chloride, is dissolved in
water, it dissociates completely as follows:
-
-
The Cl is the conjugate base of a strong acid, HCl, which is 100% ionized in water. Thus, the Cl cannot
+
react any further. In the Brönsted–Lowry system, NH4 is considered to be a cationic acid, which can
form its conjugate base, NH3, by donating a proton to water as follows:
+
In general, for charged acids BH , the reaction is written as
and the acidity constant is
Ionization of Weak Bases
Nonionized weak bases B, exemplified by NH3, react with water as follows:
which, by a procedure like that used to obtain equation (7-16), leads to
Example 7-2
-7
The basicity or ionization constant Kb for morphine base is 7.4 × 10 at 25°C. What is the
hydroxyl ion concentration of a 0.0005 M aqueous solution of morphine? We have
Salts of strong bases and weak acids, such as sodium acetate, dissociate completely in aqueous
solution to given ions:
The sodium ion cannot react with water, because it would form NaOH, which is a strong electrolyte and
would dissociate completely into its ions. The acetate anion is a Brönsted–Lowry weak base, and
278
Dr. Murtadha Alshareifi e-Library
-
In general, for an anionic base B ,
The Ionization of Water
The concentration of hydrogen or hydroxyl ions in solutions of acids or bases may be expressed as
gram ions/liter or as moles/liter. A solution containing 17.008 g of hydroxyl ions or 1.008 g of hydrogen
ions per liter is said to contain 1 g ion or 1 mole of hydroxyl or hydrogen ions per liter. Owing to the
ionization of water, it is possible to establish a quantitative relationship between the hydrogen and
hydroxyl ion concentrations of any aqueous solution.
The concentration of either the hydrogen or the hydroxyl ion in acidic, neutral, or basic solutions is
usually expressed in terms of the hydrogen ion concentration or, more conveniently, in pH units.
In a manner corresponding to the dissociation of weak acids and bases, water ionizes slightly to yield
hydrogen and hydroxyl ions. As previously observed, a weak electrolyte requires the presence of water
or some other polar solvent for ionization. Accordingly, one molecule of water can be thought of as a
weak electrolytic solute that reacts with another molecule of water as the solvent.
This autoprotolytic reaction is represented as
The law of mass action is then applied to give the equilibrium expression
The term for molecular water in the denominator is squared because the reactant is raised to a power
equal to the number of molecules appearing in the equation, as required by the law of mass action.
Because molecular water exists in great excess relative to the concentrations of hydrogen and hydroxyl
ions, [H2O]2 is considered as a constant and is combined with k to give a new constant, Kw, known as
the dissociation constant, the autoprotolysis constant, or the ion product of water:
-14
The value of the ion product is approximately 1 × 10 at 25°C; it depends strongly on temperature, as
shown in Table 7-2. In any calculations involving the ion product, one must be certain to use the proper
value of Kw for the temperature at which the data are obtained.
P.150
Table 7-2 Ion Product of Water at Various Temperatures*
279
Dr. Murtadha Alshareifi e-Library
Temperature (°C)
Kw × 1014
pKw
0
0.1139
14.944
10
0.2920
14.535
20
0.6809
14.167
24
1.000
14.000
25
1.008
13.997
30
1.469
13.833
37
2.57
13.59
40
2.919
13.535
50
5.474
13.262
60
9.614
13.017
70
15.1
12.82
80
23.4
12.63
90
35.5
12.45
100
51.3
12.29
300
400
11.40
*From H. S. Harned and R. A. Robinson, Trans. Faraday Soc. 36, 973, 1940.
Substituting equation (7-30) into (7-29) gives the common expression for the ionization of water:
280
Dr. Murtadha Alshareifi e-Library
In pure water, the hydrogen and hydroxyl ion concentrations are equal, and each has the value of
-7
approximately 1 × 10 mole/liter at 25°C.*
When an acid is added to pure water, some hydroxyl ions, provided by the ionization of water, must
always remain. The increase in hydrogen ions is offset by a decrease in the hydroxyl ions so
-14
that Kw remains constant at about 1 × 10 at 25°C.
Example 7-3
Calculate [OH-]
-3
A quantity of HCl (1.5 × 10 M) is added to water at 25°C to increase the hydrogen ion
-7
-3
concentration from 1 × 10 to 1.5 × 10 mole/liter. What is the new hydroxyl ion
concentration?
From equation (7-31),
Relationship Between Ka and Kb
A simple relationship exists between the dissociation constant of a weak acid HB and that of its
+
conjugate base B , or between BH and B, when the solvent is amphiprotic. This can be obtained by
multiplying equation (7-12) by equation (7-27):
and
or
Example 7-4
Calculate Ka
-5
+
Ammonia has a Kb of 1.74 ×10 at 25°. Calculate Ka for its conjugate acid, NH4 . We have
Ionization of Polyprotic Electrolytes
Acids that donate a single proton and bases that accept a single proton are called monoprotic
electrolytes. A polyprotic (polybasic) acid is one that is capable of donating two or more protons, and a
polyprotic base is capable of accepting two or more protons. A diprotic (dibasic) acid, such as carbonic
acid, ionizes in two stages, and a triprotic (tribasic) acid, such as phosphoric acid, ionizes in three
stages. The equilibria involved in the protolysis or ionization of phosphoric acid, together with the
equilibrium expressions, are
281
Dr. Murtadha Alshareifi e-Library
In any polyprotic electrolyte, the primary protolysis is greatest, and succeeding stages become less
complete at any given acid concentration.
2The negative charges on the ion HPO4 make it difficult for water to remove the proton from the
phosphate ion, as reflected in the small value of K3. Thus, phosphoric acid is weak in the third stage of
3-
ionization, and a solution of this acid contains practically no PO 4 ions.
P.151
Each of the species formed by the ionization of a polyprotic acid can also act as a base. Thus, for the
phosphoric acid system,
In general, for a polyprotic acid system for which the parent acid is H nA, there are n + 1 possible species
in solution:
where j represents the number of protons dissociated from the parent acid and goes from 0 to n. The
total concentration of all species must be equal to Ca, or
Each of the species pairs in which j differs by 1 constitutes a conjugate acid–base pair, and in general
where Kj represents the various acidity constants for the system. Thus, for the phosphoric acid system
described by equations (7-37) to (7-45),
Ampholytes
In the preceding section, equations (7-37), (7-38), (7-41), and (7-43) demonstrated that in the
-
2-
phosphoric acid system, the species H2PO4 and HPO4 can function either as an acid or a base. A
species that can function either as an acid or as a base is called an ampholyte and is said to
be amphoteric in nature. In general, for a polyprotic acid system, all the species, with the exception of
nHnA and A , are amphoteric.
Amino acids and proteins are ampholytes of particular interest in pharmacy. If glycine hydrochloride is
dissolved in water, it ionizes as follows:
+
-
The species NH3CH2COO is amphoteric in that, in addition to reacting as an acid as shown in
equation (7-51), it can react as a base with water as follows:
282
Dr. Murtadha Alshareifi e-Library
+
-
The amphoteric species NH3CH2COO is called a zwitterion and differs from the amphoteric species
formed from phosphoric acid in that it carries both a positive and a negative charge, and the whole
molecule is electrically neutral. The pH at which the zwitterion concentration is a maximum is known as
the isoelectric point. At the isoelectric point the net movement of the solute molecules in an electric field
is negligible.
Sörensen's pH
The hydrogen ion concentration of a solution varies from approximately 1 in a 1 M solution of a strong
-14
acid to about 1 × 10 in a 1 M solution of a strong base, and the calculations often become unwieldy.
To alleviate this difficulty, Sörensen5 suggested a simplified method of expressing hydrogen ion
+
concentration. He established the term pH, which was originally written as pH , to represent the
hydrogen ion potential, and he defined it as the common logarithm of the reciprocal of the hydrogen ion
concentration:
According to the rules of logarithms, this equation can be written as
and because the logarithm of 1 is zero,
Equations (7-53) and (7-55) are identical; they are acceptable for approximate calculations involving pH.
The pH of a solution can be considered in terms of a numeric scale having values from 0 to 14, which
expresses in a quantitative way the degree of acidity (7 to 0) and alkalinity (7-14). The value 7 at which
the hydrogen and hydroxyl ion concentrations are about equal at room temperature is referred to as
the neutral point, or neutrality. The neutral pH at 0°C is 7.47, and at 100°C it is 6.15 (Table 7-2). The
scale relating pH to the hydrogen and hydroxyl ion concentrations of a solution is given in Table 7-3.
Conversion of Hydrogen Ion Concentration to pH
The student should practice converting from hydrogen ion concentration to pH and vice versa until he or
she is proficient in these logarithmic operations. The following examples are given to afford a review of
the mathematical operations
P.152
involving logarithms. Equation (7-55) is more convenient for these calculations than equation (7-53).
Table 7-3 The pH Scale and Corresponding Hydrogen and Hydroxyl Ion
Concentrations
pH
[H3O+] (moles/liter)[OH- 1] (moles/liter)
0
100 = 1
10-14
1
10-1
10-13
2
10-2
10-12
3
10-3
10-11
4
10-4
10-10
Acidic
283
Dr. Murtadha Alshareifi e-Library
5
10-5
10-9
6
10-6
10-8
7
10-7
10-7
8
10-8
10-6
9
10-9
10-5
10
10-10
10-4
11
10-11
10-3
12
10-12
10-2
13
10-13
10-1
14
10-14
100 = 1
Neutral
Basic
Example 7-5
pH Calculation
The hydronium ion concentration of a 0.05 M solution of HCl is 0.05 M. What is the pH of this
solution? We write
A handheld calculator permits one to obtain pH simply by use of the log function followed by a
change of sign.
A better definition of pH involves the activity rather than the concentration of the ions:
and because the activity of an ion is equal to the activity coefficient multiplied by the molal or molar
concentration [equation (7-42)],
the pH may be computed more accurately from the formula
Example 7-6
Solution pH
The mean molar ionic activity coefficient of a 0.05 M solution of HCl is 0.83 at 25°C. What is
the pH of the solution? We write
284
Dr. Murtadha Alshareifi e-Library
If sufficient NaCl is added to the HCl solution to produce a total ionic strength of 0.5 for this
mixture of uni-univalent electrolytes, the activity coefficient is 0.77. What is the pH of this
solution? We write
Hence, the addition of a neutral salt affects the hydrogen ion activity of a solution, and activity
coefficients should be used for the accurate calculation of pH.
Example 7-7
Solution pH
The hydronium ion concentration of a 0.1 M solution of barbituric acid was found to be 3.24 ×
-3
10 M. What is the pH of the solution? We write
For practical purposes, activities and concentrations are equal in solutions of weak electrolytes to which
no salts are added, because the ionic strength is small.
Conversion of pH to Hydrogen Ion Concentration
+
The following example illustrates the method of converting pH to [H 3O ].
Example 7-8
Hydronium Ion Concentration
If the pH of a solution is 4.72, what is the hydronium ion concentration? We have
The use of a handheld calculator bypasses this two-step procedure. One simply enters -4.72
x
+
into the calculator and presses the key for antilog or 10 to obtain [H3O ].
pK and pOH
The use of pH to designate the negative logarithm of hydronium ion concentration has proved to be so
convenient that expressing numbers less than unity in “p” notation has become a standard procedure.
The mathematician would say that “p” is a mathematical operator that acts on the quantity
+
[H ], Ka, Kb, Kw, and so on to convert the value into the negative of its common logarithm. In other
words, the term “p” is used to express the negative logarithm of the term following the “p.” For example,
pOH expresses -log[OH ], pKa is used for -log Ka, and pKw is -log Kw. Thus, equations (7-31) and (733) can be expressed as
where pK is often called the dissociation exponent.
P.153
The pK values of weak acidic and basic drugs are ordinarily determined by ultraviolet spectrophometry
, ,
(95) and potentiometric titration (202). They can also be obtained by solubility analysis6 7 8 (254) and by
a partition coefficient method.8
Species Concentration as a Function of pH
As shown in the preceding sections, polyprotic acids, HnA, can ionize in successive stages to yield n + 1
possible species in solution. In many studies of pharmaceutical interest, it is important to be able to
calculate the concentration of all acidic and basic species in solution.
The concentrations of all species involved in successive acid–base equilibria change with pH and can
be represented solely in terms of equilibrium constants and the hydronium ion concentration. These
285
Dr. Murtadha Alshareifi e-Library
relationships can be obtained by defining all species in solution as fractions α of total acid, Ca, added to
the system [see equation (7-47) for Ca]:
and in general
and
where j represents the number of protons that have ionized from the parent acid. Thus, dividing
equation (7-47) by Ca and using equations (7-60) to (7-63) gives
+
All of the α values can be defined in terms of equilibrium constants α0 and [H3O ] as follows:
Therefore,
or
and, in general,
Inserting the appropriate forms of equation (7-69) into equation (7-64) gives
Solving for α0 yields
or
where D represents the denominator of equation (7-71). Thus, the concentration of HnA as a function of
+
[H3O ] can be obtained by substituting equation (7-60) into equation (7-72) to give
Substituting equation (7-61) into equation (7-66) and the resulting equation into equation (7-72) gives
In general,
and
Although these equations appear complicated, they are in reality quite simple. The term D in
+
equations (7-72) to (7-76) is a power series in [H3O ], each term multiplied by equilibrium constants. The
+
series starts with [H3O ] raised to the power representing n, the total number of dissociable hydrogens in
the parent acid, HnA. The last term is the product of all the acidity constants. The intermediate terms can
286
Dr. Murtadha Alshareifi e-Library
+
be obtained from the last term by substituting [H3O ] for Kn to obtain the next to last term, then
+
substituting [H3O ] for Kn-1 to obtain the next term, and so on, until the first term is reached. The
following equations show the denominators D to be used in equations (7-72) to (7-76) for various types
of polyprotic acids:
P.154
In all instances, for a species in which j protons have ionized, the numerator in equations (7-72) to (7+
76) is Ca multiplied by the term from the denominator D that has [H3O ] raised to the n - j power. Thus,
for the parent acid H2A, the appropriate equation for D is equation (7-79). The molar concentrations of
-
2-
the species HnA (j = 0), HA (j = 1), and A (j = 2) can be given as
These equations can be used directly to solve for molar concentrations. It should be obvious, however,
that lengthy calculations are needed for substances such as citric acid or ethylenediaminetetraacetic
acid, requiring the use of a digital computer to obtain solutions in a reasonable time. Graphic methods
have been used to simplify the procedure.9
Calculation of pH
Proton Balance Equations
According to the Brönsted–Lowry theory, every proton donated by an acid must be accepted by a base.
Thus, an equation accounting for the total proton transfers occurring in a system is of fundamental
importance in describing any acid–base equilibria in that system. This can be accomplished by
establishing a proton balance equation (PBE) for each system. In the PBE, the sum of the concentration
terms for species that form by proton consumption is equated to the sum of the concentration terms for
species that are formed by the release of a proton.
+
+
For example, when HCl is added to water, it dissociates completely into H 3O and Cl ions. The H3O is
-
a species that is formed by the consumption of a proton (by water acting as a base), and the Cl is
+
formed by the release of a proton from HCl. In all aqueous solutions, H 3O and OH result from the
dissociation of two water molecules according to equation (7-28). Thus, OH is a species formed from
the release of a proton. The PBE for the system of HCl in water is
+
Although H3O is formed from two reactions, it is included only once in the PBE. The same would be
-
true for OH if it came from more than one source.
The general method for obtaining the PBE is as follows:
a.
b.
Always start with the species added to water.
On the left side of the equation, place all species that can form when protons are consumed by
c.
the starting species.
On the right side of the equation, place all species that can form when protons are released
from the starting species.
287
Dr. Murtadha Alshareifi e-Library
d.
Each species in the PBE should be multiplied by the number of protons lost or gained when it
e.
is formed from the starting species.
+
Add [H3O ] to the left side of the equation and [OH ] to the right side of the equation. These
result from the interaction of two molecules of water, as shown previously.
Example 7-9
Proton Balance Equations
What is the PBE when H3PO4 is added to water?
The species H2PO4 forms with the release of one proton.
2The species HPO4 forms with the release of two protons.
3The species PO4 forms with the release of three protons. We thus have
Example 7-10
Proton Balance Equations
What is the PBE when Na2HPO4 is added to water?
2+
+
The salt dissociates into two Na and one HPO4 ; Na is neglected in the PBE because it is
2not formed from the release or consumption of a proton; HPO4 , however, does react with
water and is considered to be the starting species.
The species H2PO4 results with the consumption of one proton.
The species of H3PO4 can form with the consumption of two protons.
3The species PO4 can form with the release of one proton.
Thus, we have
Example 7-11
Proton Balance Equations
What is the PBE when sodium acetate is added to water?
+
3
The salt dissociates into one Na and one CH3COO ion. The CH COO is considered to be
the starting species. The CH3COOH can form when CH3COO consumes one proton. Thus,
The PBE allows the pH of any solution to be calculated readily, as follows:
a. Obtain the PBE for the solution in question.
b. Express the concentration of all species as a function of equilibrium constants and
+
[H3O ] using equations (7-73) to (7-76).
+
c. Solve the resulting expression for [H3O ] using any assumptions that appear valid for
the system.
d. Check all assumptions.
+
e. If all assumptions prove valid, convert [H3O ] to pH.
If the solution contains a base, it is sometimes more convenient to solve the expression
obtained in part (b) for [OH ], then convert this to pOH, and finally to pH by use of equation (758).
Solutions of Strong Acids and Bases
-2
Strong acids and bases are those that have acidity or basicity constants greater than about 10 . Thus,
they are considered
P.155
to ionize 100% when placed in water. When HCl is placed in water, the PBE for the system is given by
288
Dr. Murtadha Alshareifi e-Library
which can be rearranged to give
where Ca is the total acid concentration. This is a quadratic equation of the general form
which has the solution
Thus, equation (7-85) becomes
+
where only the positive root is used because [H3O ] can never be negative.
-6
When the concentration of acid is 1 × 10 M or greater, [Cl ] becomes much greater than* [OH ] in
equation (7-84) and Ca2 becomes much greater than 4Kw in equation (7-88). Thus, both equations
simplify to
A similar treatment for a solution of a strong base such as NaOH gives
and
6
if the concentration of base is 1 × 10 M or greater.
Conjugate Acid–Base Pairs
Use of the PBE enables us to develop one master equation that can be used to solve for the pH of
solutions composed of weak acids, weak bases, or a mixture of a conjugate acid–base pair. To do this,
-
consider a solution made by dissolving both a weak acid, HB, and a salt of its conjugate base, B , in
water. The acid–base equilibria involved are
The PBE for this system is
The concentrations of the acid and the conjugate base can be expressed as
Equation (7-96) contains Cb (concentration of base added as the salt) rather than Ca because in terms
-
of the PBE, the species HB was generated from the species B added in the form of the salt.
Equation (7-97) contains Ca (concentration of HB added) because the species B in the PBE came from
the HB added. Inserting equations (7-96) and (7-97) into equation (7-95) gives
which can be rearranged to yield
This equation is exact and was developed using no assumptions.* It is, however, quite difficult to solve.
Fortunately, for real systems, the equation can be simplified.
289
Dr. Murtadha Alshareifi e-Library
Solutions Containing Only a Weak Acid
+
-
If the solution contains only a weak acid, Cb is zero, and [H3O ] is generally much greater than [OH ].
Thus, equation (7–99) simplifies to
which is a quadratic equation with the solution
+
In many instances, Ca is much greater than [H3O ], and equation (7-100) simplifies to
Example 7-12
Calculate pH
-3
Calculate the pH of a 0.01 M solution of salicylic acid, which has a Ka = 1.06 × 10 at 25°C.
a. Using equation (7-102), we find
b. Using equation (7-101), we find
P.156
The example just given illustrates the importance of checking the validity of all assumptions made in
+
deriving the equation used for calculating [H3O ]. The simplified equation (7-102) gives an answer for
+
[H3O ] with a relative error of 18% as compared with the correct answer given by equation (7-101).
Example 7-13
Calculate pH
Calculate the pH of a 1-g/100 mL solution of ephedrine sulfate. The molecular weight of the
-5
salt is 428.5, and Kb for ephedrine base is 2.3 × 10 .
+
+
a. The ephedrine sulfate, (BH )2SO4, dissociates completely into two BH cations and
2one SO4 anion. Thus, the concentration of the weak acid (ephedrine cation) is twice
the concentration, Cs, of the salt added.
b.
c.
290
Dr. Murtadha Alshareifi e-Library
All assumptions are valid. We have
Solutions Containing Only a Weak Base
-
+
If the solution contains only a weak base, Ca is zero, and [OH ] is generally much greater than [H3O ].
Thus, equation (7-99) simplifies to
+
-
+
This equation can be solved for either [H3O ] or [OH ]. Solving for [H3O ] using the left and rightmost
parts of equation (7-103) gives
which has the solution
+
If Ka is much greater than [H3O ], which is generally true for solutions of weak bases, equation (7100) gives
-
Equation (7-103) can be solved for [OH ] by using the left and middle portions and converting Ka to Kb to
give
-
and if Cb is much greater than [OH ], which generally obtains for solutions of weak bases,
A good exercise for the student would be to prove that equation (7-106) is equal to equation (7-108).
The applicability of both these equations will be shown in the following examples.
Example 7-14
Calculate pH
What is the pH of a 0.0033 M solution of cocaine base, which has a basicity constant of 2.6 ×
-6
10 ? We have
All assumptions are valid. Thus,
Example 7-15
Calculate pH
Calculate the pH of a 0.165 M solution of sodium sulfathiazole. The acidity constant for
-8
sulfathiazole is 7.6 × 10 .
+ +
(a) The salt Na B dissociates into one Na and one B as described by equations (7-24) to (727). Thus, Cb = Cs = 0.165 M. Because Ka for a weak acid such as sulfathiazole is usually
given rather than Kb for its conjugate base, equation (7-106) is preferred over equation (7108):
All assumptions are valid. Thus,
Solutions Containing a Single Conjugate Acid–Base Pair
291
Dr. Murtadha Alshareifi e-Library
In a solution composed of a weak acid and a salt of that acid (e.g., acetic acid and sodium acetate) or a
weak base and a salt of that base (e.g., ephedrine and ephedrine hydrochloride), Ca and Cb are
+
generally much greater than either [H3O ] or [OH ]. Thus, equation (7-99) simplifies to
P.157
Example 7-16
Calculate pH
What is the pH of a solution containing acetic acid 0.3 M and sodium acetate 0.05 M? We
write
All assumptions are valid. Thus,
Example 7-17
Calculate pH
What is the pH of a solution containing ephedrine 0.1 M and ephedrine hydrochloride 0.01 M?
-5
Ephedrine has a basicity constant of 2.3 × 10 ; thus, the acidity constant for its conjugate
-10
acid is 4.35 × 10 .
All assumptions are valid. Thus,
Solutions made by dissolving in water both an acid and its conjugate base, or a base and its conjugate
acid, are examples of buffer solutions. These solutions are of great importance in pharmacy and are
covered in greater detail in the next two chapters.
Two Conjugate Acid–Base Pairs
The Brönsted–Lowry theory and the PBE enable a single equation to be developed that is valid for
solutions containing an ampholyte, which forms a part of two dependent acid–base pairs. An amphoteric
species can be added directly to water or it can be formed by the reaction of a diprotic weak acid, H 2A,
2-
or a diprotic weak base, A . Thus, it is convenient to consider a solution containing a diprotic weak acid,
2H2A, a salt of its ampholyte, HA , and a salt of its diprotic base, A , in concentrations Ca, Cab, and Cb,
respectively. The total PBE for this system is
where the subscripts refer to the source of the species in the PBE, that is, [H2A]ab refers to H2A
generated from the ampholyte and [H2A]b refers to the H2A generated from the diprotic base. Replacing
+
these species concentrations as a function of [H3O ] gives
+
Multiplying through by [H3O ] and D, which is given by equation (7-79), gives
292
Dr. Murtadha Alshareifi e-Library
This is a general equation that has been developed using no assumptions and that can be used for
solutions made by adding a diprotic acid to water, adding an ampholyte to water, adding a diprotic base
to water, and by adding combinations of these substances to water. It is also useful for tri- and
quadriprotic acid systems because K3 and K4 are much smaller than K1 and K2 for all acids of
pharmaceutical interest. Thus, these polyprotic acid systems can be handled in the same manner as a
diprotic acid system.
Solutions Containing Only a Diprotic Acid
If a solution is made by adding a diprotic acid, H2A, to water to give a concentration Ca, the terms
Cab and Cb in equation (7-112) are zero. In almost all instances, the terms containing Kw can be
+
dropped, and after dividing through by [H3O ], we obtain from equation (7-112)
If Ca ≫ K2, as is usually true,
+
+
If [H3O ] is much greater than 2K2, the term 2K1K2Ca can be dropped, and dividing through by [H3O ]
yields the quadratic equation
+
The assumptions Ca is much greater than K2 and [H3O ] is much greater than 2K2 will be valid
whenever K2 is much less than K1. Equation (7-115) is identical to equation (7-100), which was obtained
+
for a solution containing a monoprotic weak acid. Thus, if Ca is much greater than [H3O ], equation (7115) simplifies to equation (7-100).
Example 7-18
Calculate pH
-3
-5
Calculate the pH of a 1.0 × 10 M solution of succinic acid. K1 = 6.4 × 10 and K2 = 2.3 × 10
6
.
a. Use equation (7-102) because K1 is approximately 30 times K2:
+
The assumption that Ca is much greater than [H3O ] is not valid.
b. Use the quadratic equation (7-115):
+
Note that Ca is much greater than K2, and [H3O ] is much greater than 2K2. Thus, we have
P.158
293
-
Dr. Murtadha Alshareifi e-Library
Solutions Containing Only an Ampholyte
If an ampholyte, HA , is dissolved in water to give a solution with concentration Cab, the
-
terms Ca and Cb in equation (7-112) are zero. For most systems of practical importance, the first, third,
and fifth terms of equation (7-112) are negligible when compared with the second and fourth terms, and
the equation becomes
The term K2Cab is generally much greater than Kw, and
If the solution is concentrated enough that Cab is much greater than K1,
Example 7-19
Calculate pH
-3
Calculate the pH of a 5.0 × 10 M solution of sodium bicarbonate at 25°C. The acidity
-7
-11
constants for carbonic acid are K1 = 4.3 × 10 and K2 = 4.7 × 10 .
-14
Because K2Cab (23.5 × 10 ) is much greater than Kw and Cab is much greater than K1,
equation (7-118) can be used. We have
Solutions Containing Only a Diacidic Base
-
In general, the calculations for solutions containing weak bases are easier to handle by solving for [OH ]
+
+
rather than [H3O ]. Any equation in terms of [H3O ] and acidity constants can be converted into terms of
+
[OH ] and basicity constants by substituting [OH ] for [H3O ], Kb1 for K1, Kb2 for K2, and Cb for Ca. These
substitutions are made into equation (7-112). Furthermore, for a solution containing only a diacidic
base, Ca and Cab are zero; all terms containing Kw can be dropped; Cb is much greater than Kb2; and
-
[OH ] is much greater than 2Kb2. The following expression results:
-
If Cb is much greater than [OH ], the equation simplifies to
Example 7-20
Calculate pH
-3
Calculate the pH of a 1.0 × 10 M solution of Na2CO3. The acidity constants for carbonic acid
-7
-11
are K1 = 4.31 × 10 and K2 = 4.7 × 10 .
a. Using equation (7-48), we obtain
b. Because Kb2 is much greater than Kb2, one uses equation (7-120):
-
The assumption that Cb is much greater than [OH ] is not valid, and equation (7-119) must be
used. [See equations (7-86) and (7-87) for the solution of a quadratic equation.] We obtain
294
Dr. Murtadha Alshareifi e-Library
-
Use of the simplified equation (7-120) gives an answer for [OH ] that has a relative error of 24% as
compared with the correct answer given by equation (7-119). It is absolutely essential that all
+
-
assumptions made in the calculation of [H3O ] or [OH ] be verified!
Two Independent Acid–Base Pairs
Consider a solution containing two independent acid–base pairs:
A general equation for calculating the pH of this type of solution can be developed by considering a
solution made by adding to water the acids HB1 and HB2 in concentrations Ca1
P.159
-
-
and Ca2 and the bases B1 and B2 in concentrations Cb1 and Cb2. The PBE for this system is
where the subscripts refer to the sources of the species in the PBE. Replacing these species
+
concentrations as a function of [H3O ] gives
which can be rearranged to
Although this equation is extremely complex, it simplifies readily when applied to specific systems.
Solutions Containing Two Weak Acids
In systems containing two weak acids, Cb1 and Cb2 are zero, and all terms in Kw can be ignored in
equation (7-125). For all systems of practical importance, Ca1and Ca2 are much greater than K1 and K2,
so the equation simplifies to
+
If Ca1 and Ca2 are both greater than [H3O ], the equation simplifies to
Example 7-21
Calculate pH
295
Dr. Murtadha Alshareifi e-Library
What is the pH of a solution containing acetic acid, 0.01 mole/liter, and formic acid, 0.001
mole/liter? We have
Solutions Containing a Salt of a Weak Acid and a Weak Base
The salt of a weak acid and a weak base, such as ammonium acetate, dissociates almost completely in
+
+
aqueous solution to yield NH4 and Ac , the NH4 is an acid and can be designated as HB1, and the base
Ac can be designated as B2 in equations (7-121) and (7-122). Because only a single acid, HB1, and a
-
single base, B2 , were added to water in concentrations Ca1 and Cb2, respectively, all other stoichiometric
concentration terms in equation (7-115) are zero. In addition, all terms containing Kw are negligibly small
and may be dropped, simplifying the equation to
In solutions containing a salt such as ammonium acetate, Ca1 = Cb2 = Cs, where Cs is the concentration
of salt added. In all systems of practical importance, Cs is much greater than K1 or K2, and equation (7128) simplifies to
which is a quadratic equation that can be solved in the usual manner. In most instances, however, Cs is
+
much greater than [H3O ], and the quadratic equation reduces to
Equations (7-121) and (7-122) illustrate the fact that K1 and K2 are not the successive acidity constants
for a single diprotic acid system, and equation (7-130) is not the same as equation (7-118);
instead, K1 is the acidity constant for HB1 (Acid1) and K2 is the acidity constant for the conjugate acid,
-
HB2 (Acid2), of the base B2 . The determination of Acid1 and Acid2 can be illustrated using ammonium
acetate and considering the acid and base added to the system interacting as follows:
Thus, for this system, K1 is the acidity constant for the ammonium ion and K2 is the acidity constant for
acetic acid.
Example 7-22
Calculate pH
Calculate the pH of a 0.01 M solution of ammonium acetate. The acidity constant for acetic
-5
-5
acid is K2 = Ka = 1.75 × 10 , and the basicity constant for ammonia is Kb = 1.74 × 10 .
(a) K1 can be found by dividing Kb for ammonia into Kw:
Note that all of the assumptions are valid. We have
+
When ammonium succinate is dissolved in water, it dissociates to yield two NH 4 cations and one
2succinate (S ) anion. These ions can enter into the following acid–base equilibrium:
P.160
296
Dr. Murtadha Alshareifi e-Library
In this system, Cb2 = Cs and Ca1 = 2Cs, the concentration of salt added. If Cs is much greater than
either K1 or K2, equation (7-125) simplifies to
+
and if 2K2 is much greater than [H3O ],
In this example, equation (7-132) shows that K1 is the acidity constant for the ammonium cation and K2,
-
referring to Acid2, must be the acidity constant for the bisuccinate species HS or the second acidity
constant for succinic acid.
In general, when Acid2 comes from a polyprotic acid HnA, equation (7-128) simplifies to
and
using the same assumptions that were used in developing equations (7-132) and (7-133).
It should be pointed out that in deriving equations (7-132) to (7-136), the base was assumed to be
monoprotic. Thus, it would appear that these equations should not be valid for salts such as ammonium
succinate or ammonium phosphate. For all systems of practical importance, however, the solution to
these equations yields a pH value above the final pKa for the system. Therefore, the concentrations of
all species formed by the addition of more than one proton to a polyacidic base will be negligibly small,
and the assumption of only a one-proton addition becomes quite valid.
Example 7-23
Calculate pH
Calculate the pH of a 0.01 M solution of ammonium succinate. As shown in equation (7132), K1 is the acidity constant for the ammonium cation, which was found in the previous
-10
example to be 5.75 × 10 , and K2 refers to the acid succinate (HS ) or the second acidity
-6
constant for the succinic acid system. Thus, K2 = 2.3 × 10 . We have
Solutions Containing a Weak Acid and a Weak Base
In the preceding section, the acid and base were added in the form of a single salt. They can be added
as two separate salts or an acid and a salt, however, forming buffer solutions whose pH is given by
equation (7-130). For example, consider a solution made by dissolving equimolar amounts of sodium
acid phosphate, NaH2PO4, and disodium citrate, Na2HC6H5O7, in water. Both salts dissociate to give the
-
2-
amphoteric species H2PO4 and HC6H5O7 , causing a problem in deciding which species to designate
as HB1 and which to designate as B2 in equations (7-121) and (7-122). This problem can be resolved by
-
considering the acidity constants for the two species in question. The acidity constant for H 2PO4 is 7.2
2and that for the species HC6H5O7 is 6.4. The citrate species, being more acidic, acts as the acid in the
following equilibrium:
Thus, K1 in equation (7-130) is K3 for the citric acid system, and K2 in equation (7-130) is K1 for the
phosphoric acid system.
Example 7-24
Calculate pH
What is the pH of a solution containing NaH2PO4 and disodium citrate (disodium hydrogen
citrate) Na2HC6H5O7, both in a concentration of 0.01 M? The third acidity constant for
2-7
-3
HC6H5O7 is 4.0 × 10 , whereas the first acidity constant for phosphoric acid is 7.5 × 10 . We
have
All assumptions are valid. We find
297
Dr. Murtadha Alshareifi e-Library
The equilibrium shown in equation (7-137) illustrates the fact that the system made by dissolving
NaH2PO4 and Na2HC6H5O7 in water is identical to that made by dissolving H3PO4 and Na3C6H5O7 in
water. In the latter case, H3PO4 is HB1 and the tricitrate is B2 , and if the two substances are dissolved in
equimolar amounts, equation (7-130) is valid for the system.
A slightly different situation arises for equimolar combinations of substances such as succinic acid,
H2C4H4O4, and tribasic sodium phosphate, Na3PO4. In this case, it is obvious that succinic acid is the
2acid, which can protonate the base to yield the species HC4H4O4 and HPO4 . The acid succinate
2(pKa 5.63) is a stronger acid than HPO4 (pKa 12.0), however, and an equilibrium cannot be established
2-
between these species and the species originally added to water. Instead, the HPO 4 is protonated by
2-
-
the acid succinate to give C4H4O4 and H2PO4 . This is illustrated in the following:
Thus, K1 in equation (7-139) is K2 for the succinic acid system, and K2 in equation (7-130) is
actually K2 from the phosphoric acid system.
P.161
Example 7-25
Calculate pH
Calculate the pH of a solution containing succinic acid and tribasic sodium phosphate, each at
a concentration of 0.01 M. The second acidity constant for the succinic acid system is 2.3 ×
-6
-8
10 . The second acidity constant for the phosphoric acid system is 6.2 ×10 . Write
a.
All assumptions are valid. We have
b. Equation (7-130) can also be solved by taking logarithms of both sides to yield
Equations (7-138) and (7-139) illustrate the fact that solutions made by dissolving equimolar amounts of
H2C4H4O4 and Na3PO4, NaHC4H4O4 and Na2HPO4, or Na2C4H4O4 and NaH2PO4 in water all equilibrate
to the same pH and are identical.
Acidity Constants
One of the most important properties of a drug molecule is its acidity constant, which for many drugs
,
,
can be related to physiologic and pharmacologic activity,10 11 12 solubility, rate of solution,13 extent of
binding,14 and rate of absorption.15
Effect of Ionic Strength on Acidity Constants
In the preceding sections, the solutions were considered dilute enough that the effect of ionic strength
on the acid–base equilibria could be ignored. A more exact treatment for the ionization of a weak acid,
for example, would be
298
Dr. Murtadha Alshareifi e-Library
where K is the thermodynamic acidity constant, and the charges on the species have been omitted to
make the equations more general. Equation (7-141) illustrates the fact that in solving equations involving
acidity constants, both the concentration and the activity coefficient of each species must be considered.
One way to simplify the problem would be to define the acidity constant as an apparent constant in
terms of the hydronium ion activity and species concentrations and activity coefficients, as follows:
and
The following form of the Debye–Hückel equation16 can be used for ionic strengths up to about 0.3 M:
where Zi is the charge on the species i. The value of the constant αB can be taken to be approximately 1
at 25°C, and Ks is a “salting-out” constant. At moderate ionic strengths, Ks can be assumed to be
approximately the same for both the acid and its conjugate base.16 Thus, for an acid with
charge Z going to a base with chargeZ - 1,
Example 7-26
Calculate pK′2
Calculate pK′2 for citric acid at an ionic strength of 0.01 M. Assume that pK2 = 4.78. The
charge on the acidic species is -1. We have
If either the acid or its conjugate base is a zwitterion, it will have a large dipole moment, and the
expression for its activity coefficient must contain a term Kr, the “salting-in” constant.17 Thus, for the
zwitterion [+ -],
The first ionization of an amino acid such as glycine hydrochloride involves an acid with a charge of +1
going to the zwitterion, [+ -]. Combining equations (7-146) and (7-144) with equation (7-143) gives
The second ionization step involves the zwitterion going to a species with a charge of -1. Thus, using
equations (7-146), (7-144), and (7-143) gives
The “salting-in” constant, Kr, is approximately 0.32 for alpha-amino acids in water and approximately 0.6
for dipeptides.17 Use of these values for Kr enables equations (7-147) and (7-148) to be used for
solutions with ionic strengths up to about 0.3 M.
The procedure to be used in solving pH problems in which the ionic strength of the solution must be
considered is as follows:
a.
Convert all pK values needed for the problem into pK′ values.
b.
Solve the appropriate equation in the usual manner.
299
Dr. Murtadha Alshareifi e-Library
P.162
Example 7-27
Calculate pH
Calculate the pH of a 0.01 M solution of acetic acid to which enough KCl had been added to
give an ionic strength of 0.01 M at 25°C. The pKa for acetic acid is 4.76.
a.
b. Taking logarithms of equation (7-99) gives
in which we now write pKa as pK′a:
Example 7-28
Calculate pH
-3
Calculate the pH of a 10 M solution of glycine at an ionic strength of 0.10 at 25°C. The
pKa values for glycine are pK1 = 2.35 and pK2 = 9.78.
Chapter Summary
In this chapter, the student is introduced to ionic equilibria in the pharmaceutical sciences.
The Brönsted–Lowry and Lewis electronic theories are introduced. The four classes of
solvents (protophilic, protogenic, amphiprotic, and aprotic) are described as well. The student
should understand the concepts of acid–base equilibria and the ionization of weak acids and
weak bases. Further, you should be able to use this theory in your practice. In other words,
you should be able to calculate dissociation constants Ka and Kb and understand the
relationship between Ka and Kb. It is also very important to understand the concepts of pH,
pK, and pOH and the relationship between hydrogen ion concentration and pH. Of course,
you should be able to calculate pH. Finally, strong acids and strong bases were defined and
described. You should strive for a working understanding of acidity constants.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. J. N. Brönsted, Rec. Trav. Chim. 42, 718, 1923; Chem. Rev. 5, 231, 1928; T. M. Lowry, J. Chem.
Soc. 123, 848, 1923.
2. A. W. Castleman Jr, J. Chem. Phys. 94, 3268, 1991; Chem. Eng. News 69, 47, 1991.
300
Dr. Murtadha Alshareifi e-Library
3. W. F. Luder and S. Zuffanti, Electronic Theory of Acids and Bases, Wiley, New York, 1947; G. N.
Lewis, Valency and the Structure of Atoms and Molecules, Reinhold, New York, 1923.
4. R. P. Bell, Acids and Bases, Methuen, London, 1952, Chapter 7.
5. S. P. L. Sörensen, Biochem. Z. 21, 201, 1909.
6. S. F. Kramer and G. L. Flynn, J. Pharm. Sci. 61, 1896, 1972.
7. P. A. Schwartz, C. T. Rhodes and J. W. Cooper Jr, J. Pharm. Sci. 66, 994, 1977.
8. J. Blanchard, J. O. Boyle and S. Van Wagenen, J. Pharm. Sci. 77, 548, 1988.
9. T. S. Lee and L. Gunnar Sillen, Chemical Equilibrium in Analytical Chemistry, Interscience, New York,
1959; J. N. Butler, Solubility and pH Calculations, Addison–Wesley, Reading, Mass., 1964; A. J.
Bard, Chemical Equilibrium, Harper & Row, New York, 1966.
10. P. B. Marshall, Br. J. Pharmacol. 10, 270, 1955.
11. P. Bell and R. O. Roblin, J. Am. Chem. Soc. 64, 2905, 1942.
12. I. M. Klotz, J. Am. Chem. Soc. 66, 459, 1944.
13. W. E. Hamlin and W. I. Higuchi, J. Pharm. Sci. 55, 205, 1966.
14. M. C. Meyer and D. E. Guttman, J. Pharm. Sci. 57, 245, 1968.
15. B. B. Brodie, in T. B. Binns (Ed.), Absorption and Distribution of Drugs, Williams & Wilkins,
Baltimore, 1964, pp. 16–48.
16. J. T. Edsall and J. Wyman, Biophysical Chemistry, Vol. 1, Academic Press, New York, 1958, p. 442.
17. J. T. Edsall and J. Wyman, Biophysical Chemistry, Vol. 1, Academic Press, New York, 1958, p. 443.
Recommended Reading
J. N. Butler, Ionic Equilibrium: Solubility and pH Calculations, Wiley Interscience, Hoboken, NJ, 1998.
*Reports have appeared in the literature2 describing the discovery of a polymer of the hydrogen ion
consisting of 21 molecules of water surrounding one hydrogen ion, namely,
*Under laboratory conditions, distilled water in equilibrium with air contains about 0.03% by volume of
-6
CO2, corresponding to a hydrogen ion concentration of about 2 × 10 (pH [congruent] 5.7).
*To adopt a definite and consistent method of making approximations throughout this chapter, the
expression “much greater than” means that the larger term is at least 20 times greater than the smaller
term.
*Except that, in this and all subsequent developments for pH equations, it is assumed that concentration
may be used in place of activity.
Chapter Legacy
Fifth Edition: published as Chapter 7 (Ionic Equilibria). Updated by Patrick Sinko.
Sixth Edition: published as Chapter 7 (Ionic Equilibria). Updated by Patrick Sinko.
301
Dr. Murtadha Alshareifi e-Library
8 Buffered and Isotonic Solutions
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Understand the common ion effect.
2. Understand the relationship between pH, pKa, and ionization for weak acids and
weak bases.
3. Apply the buffer equation, also known as the Henderson–Hasselbalch equation, for a
weak acid or base and its salt.
4. Understand the relationship between activity coefficients and the buffer equation.
5. Discuss the factors influencing the pH of buffer solutions.
6. Understand the concept and be able to calculate buffer capacity.
7. Describe the influence of concentration on buffer capacity.
8. Discuss the relationship between buffer capacity and pH on tissue irritation.
9. Describe the relationship between pH and solubility.
10. Describe the concept of tonicity and its importance in pharmaceutical systems.
11. Calculate solution tonicity and tonicity adjustments.
Buffers are compounds or mixtures of compounds that, by their presence in solution, resist changes in
pH upon the addition of small quantities of acid or alkali. The resistance to a change in pH is known
as buffer action. According to Roos and Borm,1 Koppel and Spiro published the first paper on buffer
action in 1914 and suggested a number of applications, which were later elaborated by Van Slyke.2
If a small amount of a strong acid or base is added to water or a solution of sodium chloride, the pH is
altered considerably; such systems have no buffer action.
The Buffer Equation
Common Ion Effect and the Buffer Equation for a Weak Acid and
Its Salt
The pH of a buffer solution and the change in pH upon the addition of an acid or base can be calculated
by use of the buffer equation. This expression is developed by considering the effect of a salt on the
ionization of a weak acid when the salt and the acid have an ion in common.
Key Concept
What is a Buffer?
A combination of a weak acid and its conjugate base (i.e., its salt) or a weak base and its
conjugate acid acts as a buffer. If 1 mL of a 0.1 N HCl solution is added to 100 mL of pure
water, the pH is reduced from 7 to 3. If the strong acid is added to a 0.01 M solution
containing equal quantities of acetic acid and sodium acetate, the pH is changed only 0.09 pH
units because the base Ac ties up the hydrogen ions according to the reaction
If a strong base, sodium hydroxide, is added to the buffer mixture, acetic acid neutralizes the
hydroxyl ions as follows:
For example, when sodium acetate is added to acetic acid, the dissociation constant for the weak acid,
-
is momentarily disturbed because the acetate ion supplied by the salt increases the [Ac ] term in the
-5
+
numerator. To reestablish the constant Ka at 1.75 × 10 , the hydrogen ion term in the numerator [H3O ]
is instantaneously decreased, with a corresponding increase in [HAc]. Therefore, the
constant Ka remains unaltered, and the equilibrium is shifted in the direction of the reactants.
Consequently, the ionization of acetic acid,
302
Dr. Murtadha Alshareifi e-Library
-
is repressed upon the addition of the common ion, Ac . This is an example of the common ion effect.
The pH of the final solution is obtained by rearranging the equilibrium expression for acetic acid:
If the acid is weak and ionizes only slightly, the expression [HAc] may be considered to represent the
total concentration of acid, and it is written simply as [Acid]. In the slightly ionized acidic solution, the
acetate concentration [Ac ] can be considered as having come entirely from the salt, sodium acetate.
-
Because 1 mole of sodium acetate yields 1 mole of acetate ion, [Ac ] is equal to the total salt
concentration and is replaced by the term [Salt]. Hence, equation (8-5) is written as
P.164
Equation (8-6) can be expressed in logarithmic form, with the signs reversed, as
from which is obtained an expression, known as the buffer equation or the Henderson–Hasselbalch
equation, for a weak acid and its salt:
The ratio [Acid]/[Salt] in equation (8-6) has been inverted by undertaking the logarithmic operations in
equation (8-7), and it appears in equation (8-8) as [Salt]/[Acid]. The term pKa, the negative logarithm
of Ka, is called the dissociation exponent.
The buffer equation is important in the preparation of buffered pharmaceutical solutions; it is satisfactory
for calculations within the pH range of 4 to 10.
Example 8-1
pH Calculation
What is the pH of 0.1 M acetic acid solution, pKa = 4.76? What is the pH after enough sodium
acetate has been added to make the solution 0.1 M with respect to this salt?
The pH of the acetic acid solution is calculated by use of the logarithmic form of equation (7102):
The pH of the buffer solution containing acetic acid and sodium acetate is determined by use
of the buffer equation (8-8):
It is seen from Example 8-1 that the pH of the acetic acid solution has been increased almost
2 pH units; that is, the acidity has been reduced to about 1/100 of its original value by the
addition of an equal concentration of a salt with a common ion. This example bears out the
statement regarding the repression of ionization upon the addition of a common ion.
Sometimes it is desired to know the ratio of salt to acid in order to prepare a buffer of a
definite pH. The following example demonstrates the calculation involved in such a problem.
Example 8-2
pH and [Salt]/[Acid] Ratio
What is the molar ratio, [Salt]/[Acid], required to prepare an acetate buffer of pH 5.0? Also
express the result in mole percent.
303
Dr. Murtadha Alshareifi e-Library
Therefore, the mole ratio of salt to acid is 1.74/1. Mole percent is mole fraction multiplied by
100. The mole fraction of salt in the salt–acid mixture is 1.74/(1 + 1.74) = 0.635, and in mole
percent, the result is 63.5%.
The Buffer Equation for a Weak Base and Its Salt
Buffer solutions are not ordinarily prepared from weak bases and their salts because of the volatility and
instability of the bases and because of the dependence of their pH on pKw, which is often affected by
temperature changes. Pharmaceutical solutions—for example, a solution of ephedrine base and
ephedrine hydrochloride—however, often contain combinations of weak bases and their salts.
The buffer equation for solutions of weak bases and the corresponding salts can be derived in a manner
analogous to that for the weak acid buffers. Accordingly,
-
+
and using the relationship [OH ] = Kw/[H3O ], the buffer equation is obtained
Example 8-3
Using the Buffer Equation
What is the pH of a solution containing 0.10 mole of ephedrine and 0.01 mole of ephedrine
hydrochloride per liter of solution? Since the pKb of ephedrine is 4.64,
Activity Coefficients and the Buffer Equation
A more exact treatment of buffers begins with the replacement of concentrations by activities in the
equilibrium of a weak acid:
The activity of each species is written as the activity coefficient multiplied by the molar concentration.
The activity coefficient of the undissociated acid, γHAc, is essentially 1 and may be dropped. Solving for
+
the hydrogen ion activity and pH, defined as -log aH3O , yields the equations
From the Debye–Hückel expression for an aqueous solution of a univalent ion at 25°C having an ionic
strength not greater than about 0.1 or 0.2, we write
P.165
and equation (8-13) then becomes
304
Dr. Murtadha Alshareifi e-Library
The general equation for buffers of polybasic acids is
where n is the stage of the ionization.
Example 8-4
Activity Coefficients and Buffers
A buffer contains 0.05 mole/liter of formic acid and 0.10 mole/liter of sodium formate. The
pKa of formic acid is 3.75. The ionic strength of the solution is 0.10. Compute the pH (a) with
and (b) without consideration of the activity coefficient correction.
Some Factors Influencing the pH of Buffer Solutions
The addition of neutral salts to buffers changes the pH of the solution by altering the ionic strength, as
shown in equation (8-13). Changes in ionic strength and hence in the pH of a buffer solution can also be
brought about by dilution. The addition of water in moderate amounts, although not changing the pH,
may cause a small positive or negative deviation because it alters activity coefficients and because
water itself can act as a weak acid or base. Bates3 expressed this quantitatively in terms of a dilution
value, which is the change in pH on diluting the buffer solution to one half of its original strength. Some
dilution values for National Bureau of Standards buffers are given in Table 8-1. A positive dilution value
signifies that the pH rises with dilution and a negative value signifies that the pH decreases with dilution
of the buffer.
Table 8-1 Buffer Capacity of Solutions Containing Equimolar Amounts (0.1 M)
of Acetic Acid And Sodium Acetate
Moles of NaOH AddedpH of SolutionBuffer Capacity, β
0
4.76
0.01
4.85
0.11
0.02
4.94
0.11
0.03
5.03
0.11
0.04
5.13
0.10
0.05
5.24
0.09
0.06
5.36
0.08
305
Dr. Murtadha Alshareifi e-Library
Temperature also influences buffers. Kolthoff and Tekelenburg4 determined the temperature coefficient
of pH, that is, the change in pH with temperature, for a large number of buffers. The pH of acetate
buffers was found to increase with temperature, whereas the pH of boric acid–sodium borate buffers
decreased with temperature. Although the temperature coefficient of acid buffers was relatively small,
the pH of most basic buffers was found to change more markedly with temperature, owing to Kw, which
appears in the equation of basic buffers and changes significantly with temperature. Bates3 referred to
several basic buffers that show only a small change of pH with temperature and can be used in the pH
range of 7 to 9. The temperature coefficients for the calomel electrode are given in the study by Bates.
Drugs as Buffers
It is important to recognize that solutions of drugs that are weak electrolytes also manifest buffer action.
Salicylic acid solution in a soft glass bottle is influenced by the alkalinity of the glass. It might be thought
at first that the reaction would result in an appreciable increase in pH; however, the sodium ions of the
soft glass combine with the salicylate ions to form sodium salicylate. Thus, there arises a solution of
salicylic acid and sodium salicylate—a buffer solution that resists the change in pH. Similarly, a solution
of ephedrine base manifests a natural buffer protection against reductions in pH. Should hydrochloric
acid be added to the solution, ephedrine hydrochloride is formed, and the buffer system of ephedrine
plus ephedrine hydrochloride will resist large changes in pH until the ephedrine is depleted by reaction
with the acid. Therefore, a drug in solution may often act as its own buffer over a definite pH range.
Such buffer action, however, is often too weak to counteract pH changes brought about by the carbon
dioxide of the air and the alkalinity of the bottle. Additional buffers are therefore frequently added to drug
solutions to maintain the system within a certain pH range. A quantitative measure of the efficiency or
capacity of a buffer to resist pH changes will be discussed in a later section.
pH Indicators
Indicators may be considered as weak acids or weak bases that act like buffers and also exhibit color
changes as their degree of dissociation varies with pH. For example, methyl red shows its full alkaline
color, yellow, at a pH of about 6 and its full acid color, red, at about pH 4.
The dissociation of an acid indicator is given in simplified form as
The equilibrium expression is
P.166
Table 8-2 Color, pH, and Indicator Constant, pKIn, of Some Common Indicators
306
Dr. Murtadha Alshareifi e-Library
Indicator
Color
Acid
Base
pH Range
pKIn
Thymol blue (acid range)
Red
Yellow
1.2–2.8
1.5
Methyl violet
Blue
Violet
1.5–3.2
–
Methyl orange
Red
Yellow
3.1–4.4
3.7
Bromcresol green
Yellow
Blue
3.8–5.4
4.7
Methyl red
Red
Yellow
4.2–6.2
5.1
Bromcresol purple
Yellow
Purple
5.2–6.8
6.3
Bromthymol blue
Yellow
Blue
6.0–7.6
7.0
Phenol red
Yellow
Red
6.8–8.4
7.9
Cresol red
Yellow
Red
7.2–8.8
8.3
Thymol blue (alkaline
range)
Yellow
Blue
8.0–9.6
8.9
Phenolphthalein
Colorless
Red
8.3–10.0
9.4
Alizarin yellow
Yellow
Lilac
10.0–
12.0
–
Indigo carmine
Blue
Yellow
11.6–14
–
-
HIn is the un-ionized form of the indicator, which gives the acid color, and In is the ionized form, which
produces the basic color. KIn is referred to as the indicator constant. If an acid is added to a solution of
the indicator, the hydrogen ion concentration term on the right-hand side of equation (8-16) is increased,
and the ionization is repressed by the common ion effect. The indicator is then predominantly in the form
+
of HIn, the acid color. If base is added, [H3O ] is reduced by reaction of the acid with the base,
reaction (8-16) proceeds to the right, yielding more ionized indicator In , and the base color
predominates. Thus, the color of an indicator is a function of the pH of the solution. A number of
indicators with their useful pH ranges are listed in Table 8-2.
307
Dr. Murtadha Alshareifi e-Library
The equilibrium expression (8-16) can be treated in a manner similar to that for a buffer consisting of a
weak acid and its salt or conjugate base. Hence
-
and because [HIn] represents the acid color of the indicator and the conjugate base [In ] represents the
basic color, these terms can be replaced by the concentration expressions [Acid] and [Base]. The
formula for pH as derived from equation (8-18) becomes
Example 8-5
Calculate pH
-
3
An indicator, methyl red, is present in its ionic form In , in a concentration of 3.20 × 10 M and
3
in its molecular form, HIn, in an aqueous solution at 25°C in a concentration of 6.78 × 10 M.
From Table 8-2 a pKIn of 5.1 is observed for methyl red. What is the pH of this solution? We
have
Just as a buffer shows its greatest efficiency when pH = pKa, an indicator exhibits its middle tint when
[Base]/[Acid] = 1 and pH = pKIn. The most efficient indicator range, corresponding to the effective buffer
interval, is about 2 pH units, that is, pKIn ± 1. The reason for the width of this color range can be
explained as follows. It is known from experience that one cannot discern a change from the acid color
to the salt or conjugate base color until the ratio of [Base] to [Acid] is about 1 to 10. That is, there must
be at least 1 part of the basic color to 10 parts of the acid color before the eye can discern a change in
color from acid to alkaline. The pH value at which this change is perceived is given by the equation
Conversely, the eye cannot discern a change from the alkaline to the acid color until the ratio of [Base]
to [Acid] is about 10 to 1, or
Therefore, when base is added to a solution of a buffer in its acid form, the eye first visualizes a change
in color at pKIn - 1, and the color ceases to change any further at pKIn + 1. The effective range of the
indicator between its full acid and full basic color can thus be expressed as
Chemical indicators are typically compounds with chromophores that can be detected in the visible
range and change color in response to a solution's pH. Most chemicals used as indicators respond only
to a narrow pH range. Several indicators can be combined to yield so-called universal indicators just as
buffers can be mixed to cover a wide pH range. A universal indicator is a pH indicator that displays
different colors as the pH transitions from pH 1 to 12. A typical universal indicator will display a color
range from red to purple8
P.167
For example, a strong acid (pH 0–3) may display as red in color, an acid (pH 3–6) as orange–yellow,
neutral pH (pH 7) as green, alkaline pH (pH 8–11) as blue, and purple for strong alkaline pH (pH 11–14).
The colorimetric method for the determination of pH is probably less accurate and less convenient but is
also less expensive than electrometric methods and it can be used in the determination of the pH of
aqueous solutions that are not colored or turbid. This is particularly useful for the study of acid–base
reactions in nonaqueous solutions. A note of caution should be added regarding the colorimetric
method. Because indicators themselves are acids (or bases), their addition to unbuffered solutions
whose pH is to be determined will change the pH of the solution. The colorimetric method is therefore
not applicable to the determination of the pH of sodium chloride solution or similar unbuffered
pharmaceutical preparations unless special precautions are taken in the measurement. Some medicinal
308
Dr. Murtadha Alshareifi e-Library
solutions and pharmaceutical vehicles, however, to which no buffers have been added are buffered by
the presence of the drug itself and can withstand the addition of an indicator without a significant change
in pH. Errors in the result can also be introduced by the presence of salts and proteins, and these errors
must be determined for each indicator over the range involved.
Recently, Kong et al.5 reported on a rapid method for determining pKa based on spectrophotometric
titration using a universal pH indicator. Historically, potentiometric titration, which typically uses pH
electrodes, has been the most commonly used method for determining pK a values. This method takes
time and requires the daily calibration of the pH electrode. Spectrophotometric titration has the
advantage that less sample is required, it is not affected by CO2 interference, and it can provide
multiwavelength absorbance information. The method can be applied only to compounds with
chromophores placed close to the titratable groups. The indicator spectra can then be used to calculate
the pH value of a solution from the pKa values, concentration, and molar extinction coefficients of the
indicator species. In contrast to pH electrodes, chemical indicators respond rapidly and do not require
frequent calibration.
Buffer Capacity
Thus far it has been stated that a buffer counteracts the change in pH of a solution upon the addition of
a strong acid, a strong base, or other agents that tend to alter the hydrogen ion concentration.
Furthermore, it has been shown in a rather qualitative manner how combinations of weak acids and
weak bases together with their salts manifest this buffer action. The resistance to changes of pH now
remains to be discussed in a more quantitative way.
The magnitude of the resistance of a buffer to pH changes is referred to as the buffer capacity, β. It is
also known as buffer efficiency, buffer index, and buffer value. Koppel and Spiro1 and Van
Slyke2 introduced the concept of buffer capacity and defined it as the ratio of the increment of strong
base (or acid) to the small change in pH brought about by this addition. For the present discussion, the
approximate formula
can be used, in which delta, Δ, has its usual meaning, a finite change, and ΔB is the small increment in
gram equivalents (g Eq)/liter of strong base added to the buffer solution to produce a pH change of Δ
pH. According to equation (8-23), the buffer capacity of a solution has a value of 1 when the addition of
1 g Eq of strong base (or acid) to 1 liter of the buffer solution results in a change of 1 pH unit. The
significance of this index will be appreciated better when it is applied to the calculation of the capacity of
a buffer solution.
Approximate Calculation of Buffer Capacity
Consider an acetate buffer containing 0.1 mole each of acetic acid and sodium acetate in 1 liter of
solution. To this are added 0.01-mole portions of sodium hydroxide. When the first increment of sodium
hydroxide is added, the concentration of sodium acetate, the [Salt] term in the buffer equation, increases
by 0.01 mole/liter and the acetic acid concentration, [Acid], decreases proportionately because each
increment of base converts 0.01 mole of acetic acid into 0.01 mole of sodium acetate according to the
reaction
The changes in concentration of the salt and the acid by the addition of a base are represented in the
buffer equation (8-8) by using the modified form
Before the addition of the first portion of sodium hydroxide, the pH of the buffer solution is
The results of the continual addition of sodium hydroxide are shown in Table 8-1. The student should
verify the pH values and buffer capacities by the use of equations (8-25) and (8-23), respectively.
309
Dr. Murtadha Alshareifi e-Library
As can be seen from Table 8-1, the buffer capacity is not a fixed value for a given buffer system but
instead depends on the amount of base added. The buffer capacity changes as the ratio
log([Salt]/[Acid]) increases with added base. With the addition of more sodium hydroxide, the buffer
capacity decreases rapidly, and, when sufficient base has been added to convert the acid completely
into sodium ions and acetate ions, the solution no longer possesses an acid reserve. The buffer has its
greatest capacity before any base is added, where [Salt]/[Acid] = 1, and, therefore, according to
equation (8-8), pH = pKa. The buffer capacity is also influenced
P.168
by an increase in the total concentration of the buffer constituents because, obviously, a great
concentration of salt and acid provides a greater alkaline and acid reserve. The influence of
concentration on buffer capacity is treated following the discussion of Van Slyke's equation.
A More Exact Equation for Buffer Capacity
The buffer capacity calculated from equation (8-23) is only approximate. It gives the average buffer
capacity over the increment of base added. Koppel and Spiro1 and Van Slyke2 developed a more exact
equation,
where C is the total buffer concentration, that is, the sum of the molar concentrations of the acid and the
salt. Equation (8-27) permits one to compute the buffer capacity at any hydrogen ion concentration—for
example, at the point where no acid or base has been added to the buffer.
Example 8-6
Calculating Buffer Capacity
-5
At a hydrogen ion concentration of 1.75 × 10 (pH = 4.76), what is the capacity of a buffer
containing 0.10 mole each of acetic acid and sodium acetate per liter of solution? The total
concentration, C = [Acid] + [Salt], is 0.20 mole/liter, and the dissociation constant is 1.75 × 10
5
. We have
Example 8-7
Buffer Capacity and pH
Prepare a buffer solution of pH 5.00 having a capacity of 0.02. The steps in the solution of the
problem are as follows:
a. Choose a weak acid having a pKa close to the pH desired. Acetic acid, pKa = 4.76, is
suitable in this case.
b. The ratio of salt and acid required to produce a pH of 5.00 was found in Example 82 to be [Salt]/[Acid] = 1.74/1.
c. Use the buffer capacity equation (8-27) to obtain the total buffer concentration, C =
[Salt] + [Acid]:
d. Finally from (b), [Salt] = 1.74 × [Acid], and from (c),
310
Dr. Murtadha Alshareifi e-Library
Therefore,
and
The Influence of Concentration on Buffer Capacity
The buffer capacity is affected not only by the [Salt]/[Acid] ratio but also by the total concentrations of
acid and salt. As shown in Table 8-1, when 0.01 mole of base is added to a 0.1 molar acetate buffer, the
pH increases from 4.76 to 4.85, for a ΔpH of 0.09.
If the concentration of acetic acid and sodium acetate is raised to 1 M, the pH of the original buffer
solution remains at about 4.76, but now, upon the addition of 0.01 mole of base, it becomes 4.77, for a
ΔpH of only 0.01. The calculation, disregarding activity coefficients, is
Therefore, an increase in the concentration of the buffer components results in a greater buffer capacity
or efficiency. This conclusion is also evident in equation (8-27), where an increase in the total buffer
concentration, C = [Salt] + [Acid], obviously results in a greater value of β.
Maximum Buffer Capacity
An equation expressing the maximum buffer capacity can be derived from the buffer capacity formula of
Koppel and Spiro1 and Van Slyke,2 equation (8-27). The maximum buffer capacity occurs where pH =
+
+
pKa, or, in equivalent terms, where [H3O ] = Ka. Substituting [H3O ] for Ka in both the numerator and the
denominator of equation (8-27) gives
where C is the total buffer concentration.
Example 8-8
Maximum Buffer Capacity
What is the maximum buffer capacity of an acetate buffer with a total concentration of 0.020
mole/liter? We have
Key Concept
Buffer Capacity
The buffer capacity depends on (a) the value of the ratio [Salt]/[Acid], increasing as the ratio
approaches unity, and (b) the magnitude of the individual concentrations of the buffer
components, the buffer becoming more efficient as the salt and acid concentrations are
increased.
P.169
Neutralization Curves and Buffer Capacity
A further understanding of buffer capacity can be obtained by considering the titration curves of strong
and weak acids when they are mixed with increasing quantities of alkali. The reaction of an equivalent of
an acid with an equivalent of a base is called neutralization; it can be expressed according to the
method of Brönsted and Lowry. The neutralization of a strong acid by a strong base and a weak acid by
a strong base is written in the form
311
Dr. Murtadha Alshareifi e-Library
+
-
where (H3O )(Cl ) is the hydrated form of HCl in water. The neutralization of a strong acid by a strong
base simply involves a reaction between hydronium and hydroxyl ions and is usually written as
-
+
Because (Cl ) and (Na ) appear on both sides of the reaction equation just given, they may be
disregarded without influencing the result. The reaction between the strong acid and the strong base
proceeds almost to completion; however, the weak acid–strong base reaction is incomplete because Ac
reacts in part with water, that is, it hydrolyzes to regenerate the free acid.
The neutralization of 10 mL of 0.1 N HCl (curve I) and 10 mL of 0.1 N acetic acid (curve II) by 0.1 N
NaOH is shown in Figure 8-1. The plot of pH versus milliliters of NaOH added produces the titration
curve. It is computed as follows for HCl. Before the first increment of NaOH is added, the hydrogen ion
-1
concentration of the 0.1 N solution of HCl is 10 mole/liter, and the pH is 1, disregarding activities and
assuming HCl to be completely ionized. The addition of 5 mL of 0.1 N NaOH neutralizes 5 mL of 0.1 N
+
-
HCl, leaving 5 mL of the original HCl in 10 + 5 = 15 mL of solution, or [H 3O ] = 5/15 × 0.1 = 3.3 × 10
2
mole/liter, and the pH is 1.48. When 10 mL of base has been added, all the HCl is converted to NaCl,
and the pH, disregarding the difference between activity and concentration resulting from the ionic
strength of the NaCl solution, is 7. This is known as the equivalence point of the titration. Curve I
in Figure 8-1 results from plotting such data. It is seen that the pH does not change markedly until nearly
all the HCl is neutralized. Hence, a solution of a strong acid has a high buffer capacity below a pH of 2.
Likewise, a strong base has a high buffer capacity above a pH of 12.
Fig. 8-1. Neutralization of a strong acid and a
weak acid by a strong base.
The buffer capacity equations considered thus far have pertained exclusively to mixtures of weak
electrolytes and their salts. The buffer capacity of a solution of a strong acid was shown by Van Slyke to
be directly proportional to the hydrogen ion concentration, or
The buffer capacity of a solution of a strong base is similarly proportional to the hydroxyl ion
concentration,
The total buffer capacity of a water solution of a strong acid or base at any pH is the sum of the separate
capacities just given, equations (8-31) and (8-32), or
Example 8-9
Calculate Buffer Capacity
312
Dr. Murtadha Alshareifi e-Library
What is the buffer capacity of a solution of hydrochloric acid having a hydrogen ion
-2
concentration of 10 mole/liter?
-12
The hydroxyl ion concentration of such a solution is 10 , and the total buffer capacity is
-
The OH concentration is obviously so low in this case that it may be neglected in the
calculation.
Three equations are normally used to obtain the data for the titration curve of a weak acid (curve II
of Fig. 8-1), although a single equation that is somewhat complicated can be used. Suppose that
increments of 0.1 N NaOH are added to 10 mL of a 0.1 N HAc solution.
a.
The pH of the solution before any NaOH has been added is obtained from the equation for a
weak acid,
b.
At the equivalence point, where the acid has been converted completely into sodium ions and
acetate ions, the
P.170
pH is computed from the equation for a salt of a weak acid and strong base in log form:
The concentration of the acid is given in the last term of this equation as 0.05 because the
solution has been reduced to half its original value by mixing it with an equal volume of base at
the equivalence point.
c.
Between these points on the neutralization curve, the increments of NaOH convert some of the
acid to its conjugate base Ac to form a buffer mixture, and the pH of the system is calculated
from the buffer equation. When 5 mL of base is added, the equivalent of 5 mL of 0.1 N acid
remains and 5 mL of 0.1 N Ac is formed, and using the Henderson–Hasselbalch equation, we
obtain
The slope of the curve is a minimum and the buffer capacity is greatest at this point, where the
solution shows the smallest pH change per g Eq of base added. The buffer capacity of a
313
Dr. Murtadha Alshareifi e-Library
solution is the reciprocal of the slope of the curve at a point corresponding to the composition
of the buffer solution. As seen inFigure 8-1, the slope of the line is a minimum, and the buffer
capacity is greatest at half-neutralization, where pH = pKa.
The titration curve for a tribasic acid such as H3PO4 consists of three stages, as shown in Figure 8-2.
These can be considered as being produced by three separate acids (H 3PO4, pK1 = 2.21; H2PO4 , pK2 =
27.21; and HPO4 , pK3 = 12.67) whose strengths are sufficiently different so that their curves do not
overlap. The curves can be plotted by using the buffer equation and their ends joined by smooth lines to
produce the continuous curve of Figure 8-2.
Fig. 8-2. Neutralization of a tribasic acid.
Fig. 8-3. Neutralization curve for a universal
buffer. (From H. T. Britton, Hydrogen Ions, Vol.
I, Van Nostrand, New York, 1956, p. 368.)
A mixture of weak acids whose pKa values are sufficiently alike (differing by no more than about 2 pH
units) so that their buffer regions overlap can be used as auniversal buffer over a wide range of pH
values. A buffer of this type was introduced by Britton and Robinson.6 The three stages of citric acid,
pK1 = 3.15, pK2 = 4.78, and pK3 = 6.40, are sufficiently close to provide overlapping of neutralization
-
curves and efficient buffering over this range. Adding Na 2HPO4, whose conjugate acid, H2PO4 , has a
pK2 of 7.2, diethylbarbituric acid, pK1 = 7.91, and boric acid, pK1 = 9.24, provides a universal buffer that
covers the pH range of about 2.4 to 12. The neutralization curve for the universal buffer mixture is linear
between pH 4 and 8, as seen in Figure 8-3, because the successive dissociation constants differ by only
a small value.
314
Dr. Murtadha Alshareifi e-Library
A titration curve depends on the ratio of the successive dissociation constants. Theoretically, when
one K is equal to or less than 16 times the previous K, that is, when successive pKs do not differ by
greater than 1.2 units, the second ionization begins well before the first is completed, and the titration
curve is a straight line with no inflection points. Actually, the inflection is not noticeable until one K is
about 50 to 100 times that of the previous K value.
The buffer capacity of several acid–salt mixtures is plotted against pH in Figure 8-4. A buffer solution is
useful within a range of about ±1 pH unit about the pKa of its acid, where the buffer capacity is roughly
greater than 0.01 or 0.02, as observed in Figure 8-4. Accordingly, the acetate buffer should be effective
over a pH range of about 3.8 to 5.8, and the borate buffer should be effective over a range of 8.2 to
10.2. In each case, the greatest capacity occurs where [Salt]/[Acid] = 1 and pH = pKa. Because of
interionic effects, buffer capacities do not in general exceed a value of 0.2. The buffer capacity of a
solution of the strong acid HCl becomes marked below a pH of 2, and the buffer capacity of a strong
base NaOH becomes significant above a pH of 12.
P.171
Fig. 8-4. The buffer capacity of several buffer
systems as a function of pH. (Modified from R.
G. Bates, Electrometric pH Determinations,
Wiley, New York, 1954.)
The buffer capacity of a combination of buffers whose pKa values overlap to produce a universal buffer
is plotted in Figure 8-5. It is seen that the total buffer capacity Σβ is the sum of the β values of the
individual buffers. In this figure, it is assumed that the maximum βs of all buffers in the series are
identical.
Buffers in Pharmaceutical and Biologic Systems
In Vivo Biologic Buffer Systems
Blood is maintained at a pH of about 7.4 by the so-called primary buffers in the plasma and the
secondary buffers in the erythrocytes. The plasma contains carbonic acid/bicarbonate and acid/alkali
sodium salts of phosphoric acid as buffers. Plasma proteins, which behave as acids in blood, can
combine with bases and so act as buffers. In the erythrocytes, the two buffer systems consist of
hemoglobin/oxyhemoglobin and acid/alkali potassium salts of phosphoric acid.
The dissociation exponent pK1 for the first ionization stage of carbonic acid in the plasma at body
temperature and an ionic strength of 0.16 is about 6.1. The buffer equation for the carbonic
acid/bicarbonate buffer of the blood is
315
Dr. Murtadha Alshareifi e-Library
Fig. 8-5. The total buffer capacity of a universal
buffer as a function of a pH. (From I. M. Kolthoff
and C. Rosenblum, Acid–Base Indicators,
Macmillan, New York, 1937, p. 29.)
where [H2CO3] represents the concentration of CO2 present as H2CO3 dissolved in the blood. At a pH of
7.4, the ratio of bicarbonate to carbonic acid in normal blood plasma is
or
This result checks with experimental findings because the actual concentrations of bicarbonate and
carbonic acid in the plasma are about 0.025 M and 0.00125 M, respectively.
The buffer capacity of the blood in the physiologic range pH 7.0 to 7.8 is obtained as follows. According
to Peters and Van Slyke,7 the buffer capacity of the blood owing to hemoglobin and other constituents,
exclusive of bicarbonate, is about 0.025 g equivalents per liter per pH unit. The pH of the bicarbonate
buffer in the blood (i.e., pH 7.4) is rather far removed from the pH (6.1) where it exhibits maximum buffer
capacity; therefore, the bicarbonate's buffer action is relatively small with respect to that of the other
blood constituents. According to the calculation just given, the ratio [NaHCO 3]/[H2CO3] is 20:1 at pH 7.4.
-7
Using equation (8-27), we find the buffer capacity for the bicarbonate system (K1 = 4 × 10 ) at a pH of
+
-8
7.4 ([H3O ] = 4 × 10 ) to be roughly 0.003. Therefore, the total buffer capacity of the blood in the
physiologic range, the sum of the capacities of the various constituents, is 0.025 + 0.003 = 0.028.
Salenius8 reported a value of 0.0318 ± 0.0035 for whole blood, whereas Ellison et al.9 obtained a buffer
capacity of about 0.039 g equivalents per liter per pH unit for whole blood, of which 0.031 was
contributed by the cells and 0.008 by the plasma.
It is usually life-threatening for the pH of the blood to go below 6.9 or above 7.8. The pH of the blood in
diabetic coma is as low as about 6.8.
Lacrimal fluid, or tears, have been found to have a great degree of buffer capacity, allowing a dilution of
1:15 with neutral distilled water before an alteration of pH is noticed.10 In the terminology of
Bates,11 this would be referred to today as dilution value rather than buffer capacity. The pH of tears is
about 7.4, with a range of 7 to 8 or slightly higher. It is generally thought that eye drops within a pH
range of 4 to 10 will not harm the cornea.12 However, discomfort and a flow of tears will occur below pH
6.6 and above pH 9.0.12 Pure conjunctival fluid is probably more acidic than the tear fluid commonly
used in pH measurements. This is because pH increases rapidly when the sample is removed for
analysis because of the loss of CO2 from the tear fluid.
Urine
The 24-hr urine collection of a normal adult has a pH averaging about 6.0 units; it may be as low as 4.5
or as high as 7.8. When the pH of the urine is below normal values, hydrogen ions are excreted by the
kidneys. Conversely, when the
P.172
316
Dr. Murtadha Alshareifi e-Library
urine is above pH 7.4, hydrogen ions are retained by action of the kidneys in order to return the pH to its
normal range of values.
Pharmaceutical Buffers
Buffer solutions are used frequently in pharmaceutical practice, particularly in the formulation of
ophthalmic solutions. They also find application in the colorimetric determination of pH and for research
studies in which pH must be held constant.
Gifford13 suggested two stock solutions, one containing boric acid and the other monohydrated sodium
carbonate, which, when mixed in various proportions, yield buffer solutions with pH values from about 5
to 9.
Sörensen14 proposed a mixture of the salts of sodium phosphate for buffer solutions of pH 6 to 8.
Sodium chloride is added to each buffer mixture to make it isotonic with body fluids.
A buffer system suggested by Palitzsch15 and modified by Hind and Goyan16 consists of boric acid,
sodium borate, and sufficient sodium chloride to make the mixtures isotonic. It is used for ophthalmic
solutions in the pH range of 7 to 9.
The buffers of Clark and Lubs,17 based on the original pH scale of Sörensen, have been redetermined
at 25°C by Bower and Bates18 so as to conform to the present definition of pH. Between pH 3 and 11,
the older values were about 0.04 unit lower than the values now assigned, and at the ends of the scale,
the differences were greater. The original values were determined at 20°C, whereas most experiments
today are performed at 25°C.
The Clark–Lubs mixtures and their corresponding pH ranges are as follows:
a.
HCl and KCl, pH 1.2 to 2.2
b.
c.
HCl and potassium hydrogen phthalate, pH 2.2 to 4.0
NaOH and potassium hydrogen phthalate, pH 4.2 to 5.8
d.
e.
NaOH and KH2PO4, pH 5.8 to 8.0
H3BO3, NaOH, and KCl, pH 8.0 to 10.0
With regard to mixture (a), consisting of HCl and KCl and used for the pH range from 1.0 to 2.2, it will be
recalled from the discussion of the neutralization curve I inFigure 8-1 that HCl alone has considerable
buffer efficiency below pH 2. KCl is a neutral salt and is added to adjust the ionic strength of the buffer
+
solutions to a constant value of 0.10; the pH calculated from the equation -log aH = -log (y ± c)
corresponds closely to the experimentally determined pH. The role of the KCl in the Clark–Lubs buffer is
sometimes erroneously interpreted as that of a salt of the buffer acid, HCl, corresponding to the part
played by sodium acetate as the salt of the weak buffer acid, HAc. Potassium chloride is added to (e),
the borate buffer, to produce an ionic strength comparable to that of (d), the phosphate buffer, where the
pH of the two buffer series overlaps.
Key Concept
Phosphate Buffered Saline
There are several variations in the formula for preparing PBS. Two common examples follow:
Formula One: Take 8 g NaCl, 0.2 g KCl, 1.44 g Na2HPO4, and 0.24 g KH2PO4 in 800 mL
distilled water. Adjust pH to 7.4 using HCl. Add sufficient (qs ad) distilled water to achieve 1
liter.
Formula Two: Another variant of PBS. This one is designated as ―10X PBS (0.1 M PBS, pH
7.2)‖ since it is much more concentrated than PBS and the pH is not yet adjusted to pH 7.4.
Take 90 g NaCl, 10.9 g Na2HPO4, and 3.2 g NaH2PO4 in 1000 mL distilled water. Dilute 1:10
using distilled water and adjust pH as necessary.
Many buffers are available today. One of the most common biological buffers is phosphate buffered
saline (PBS). Phosphate buffered saline contains sodium chloride (NaCl) and dibasic sodium phosphate
317
Dr. Murtadha Alshareifi e-Library
(Na2PO4). It may also contain potassium chloride (KCl), monobasic potassium phosphate (KH2PO4),
calcium chloride (CaCl2), and magnesium sulfate (MgSO4).
General Procedures for Preparing Pharmaceutical Buffer
Solutions
The pharmacist may be called upon at times to prepare buffer systems for which the formulas do not
appear in the literature. The following steps should be helpful in the development of a new buffer.
a.
Select a weak acid having a pKa approximately equal to the pH at which the buffer is to be
b.
used. This will ensure maximum buffer capacity.
From the buffer equation, calculate the ratio of salt and weak acid required to obtain the
desired pH. The buffer equation is satisfactory for approximate calculations within the pH range
of 4 to 10.
c.
d.
Consider the individual concentrations of the buffer salt and acid needed to obtain a suitable
buffer capacity. A concentration of 0.05 to 0.5 M is usually sufficient, and a buffer capacity of
0.01 to 0.1 is generally adequate.
Other factors of some importance in the choice of a pharmaceutical buffer include availability of
chemicals, sterility of the final solution, stability of the drug and buffer on aging, cost of
materials, and freedom from toxicity. For example, a borate buffer, because of its toxic effects,
e.
certainly cannot be used to stabilize a solution to be administered orally or parenterally.
Finally, determine the pH and buffer capacity of the completed buffered solution using a
reliable pH meter. In some cases, sufficient accuracy is obtained by the use of
P.173
pH papers. Particularly when the electrolyte concentration is high, it may be found that the pH
calculated by use of the buffer equation is somewhat different from the experimental value.
This is to be expected when activity coefficients are not taken into account, and it emphasizes
the necessity for carrying out the actual determination.
Influence of Buffer Capacity and pH on Tissue Irritation
Friedenwald et al.18 claimed that the pH of solutions for introduction into the eye may vary from 4.5 to
11.5 without marked pain or damage. This statement evidently would be true only if the buffer capacity
were kept low. Martin and Mims19 found that Sörensen's phosphate buffer produced irritation in the
eyes of a number of individuals when used outside the narrow pH range of 6.5 to 8, whereas a boric
acid solution of pH 5 produced no discomfort in the eyes of the same individuals. Martin and Mims
concluded that a pH range of nonirritation cannot be established absolutely but instead depends upon
the buffer employed. In light of the previous discussion, this apparent anomaly can be explained partly in
terms of the low buffer capacity of boric acid as compared with that of the phosphate buffer and partly to
the difference of the physiologic response to various ion species.
Riegelman and Vaughn20 assumed that the acid-neutralizing power of tears when 0.1 mL of a 1%
solution of a drug is instilled into the eye is roughly equivalent to 10 µL of a 0.01 N strong base. They
pointed out that although in a few cases, irritation of the eye may result from the presence of the free
base form of a drug at the physiologic pH, it is more often due to the acidity of the eye solution. For
example, because only one carboxyl group of tartaric acid is neutralized by epinephrine base in
epinephrine bitartrate, a 0.06 M solution of the drug has a pH of about 3.5. The prolonged pain resulting
from instilling two drops of this solution into the eye is presumably due to the unneutralized acid of the
bitartrate, which requires 10 times the amount of tears to restore the normal pH of the eye as compared
with the result following two drops of epinephrine hydrochloride. Solutions of pilocarpine salts also
possess sufficient buffer capacity to cause pain or irritation owing to their acid reaction when instilled
into the eye.
318
Dr. Murtadha Alshareifi e-Library
Parenteral solutions for injection into the blood are usually not buffered, or they are buffered to a low
capacity so that the buffers of the blood may readily bring them within the physiologic pH range. If the
drugs are to be injected only in small quantities and at a slow rate, their solutions can be buffered
weakly to maintain approximate neutrality.
According to Mason,21 following oral administration, aspirin is absorbed more rapidly in systems
buffered at low buffer capacity than in systems containing no buffer or in highly buffered preparations.
Thus, the buffer capacity of the buffer should be optimized to produce rapid absorption and minimal
gastric irritation of orally administered aspirin.
Key Concept
Parenteral Solutions
Solutions to be applied to tissues or administered parenterally are liable to cause irritation if
their pH is greatly different from the normal pH of the relevant body fluid. Consequently, the
pharmacist must consider this point when formulating ophthalmic solutions, parenteral
products, and fluids to be applied to abraded surfaces. Of possible greater significance than
the actual pH of the solution is its buffer capacity and the volume to be used in relation to the
volume of body fluid with which the buffered solution will come in contact. The buffer capacity
of the body fluid should also be considered. Tissue irritation, due to large pH differences
between the solution being administered and the physiologic environment in which it is used,
will be minimal (a) the lower is the buffer capacity of the solution, (b) the smaller is the volume
used for a given concentration, and (c) the larger are the volume and buffer capacity of the
physiologic fluid.
In addition to the adjustment of tonicity and pH for ophthalmic preparations, similar requirements are
demanded for nasal delivery of drugs. Conventionally, the nasal route has been used for delivery of
drugs for treatment of local diseases such as nasal allergy, nasal congestion, and nasal
infections.22 The nasal route can be exploited for the systemic delivery of drugs such as small
molecular weight polar drugs, peptides and proteins that are not easily administered via other routes
than by injection, or where a rapid onset of action is required. Examples include buserelin,
desmopressin, and nafarelin.
Stability versus Optimum Therapeutic Response
For the sake of completeness, some mention must be made at this point of the effect of buffer capacity
and pH on the stability and therapeutic response of the drug being used in solution.
As will be discussed later, the undissociated form of a weakly acidic or basic drug often has a higher
therapeutic activity than that of the dissociated salt form. This is because the former is lipid soluble and
can penetrate body membranes readily, whereas the ionic form, not being lipid soluble, can penetrate
membranes only with greater difficulty. Thus, Swan and White23 and Cogan and Kinsey24 observed an
increase in therapeutic response of weakly basic alkaloids (used as ophthalmic drugs) as the pH of the
solution, and hence concentration of the undissociated base, was increased. At a pH of about 4, these
drugs are predominantly in the ionic form, and penetration is slow or insignificant. When the tears bring
the pH to about 7.4, the drugs may exist to a significant degree in the form of the free base, depending
on the dissociation constant of the drug.
P.174
Example 8-10
Mole Percent of Free Base
The pKb of pilocarpine is 7.15 at 25°C. Compute the mole percent of free base present at
25°C and at a pH of 7.4. We have
319
Dr. Murtadha Alshareifi e-Library
Hind and Goyan25 pointed out that the pH for maximum stability of a drug for ophthalmic use may be far
below that of the optimum physiologic effect. Under such conditions, the solution of the drug can be
buffered at a low buffer capacity and at a pH that is a compromise between that of optimum stability and
the pH for maximum therapeutic action. The buffer is adequate to prevent changes in pH due to the
alkalinity of the glass or acidity of CO2 from dissolved air. Yet, when the solution is instilled in the eye,
the tears participate in the gradual neutralization of the solution; conversion of the drug occurs from the
physiologically inactive form to the undissociated base. The base can then readily penetrate the lipoidal
membrane. As the base is absorbed at the pH of the eye, more of the salt is converted into base to
preserve the constancy of pKb; hence, the alkaloidal drug is gradually absorbed.
pH and Solubility
Since the relationship between pH and the solubility of weak electrolytes is treated elsewhere in the
book, it is only necessary to point out briefly the influence of buffering on the solubility of an alkaloidal
base. At a low pH, a base is predominantly in the ionic form, which is usually very soluble in aqueous
media. As the pH is raised, more undissociated base is formed, as calculated by the method illustrated
in Example 8-10. When the amount of base exceeds the limited water solubility of this form, free base
precipitates from solution. Therefore, the solution should be buffered at a sufficiently low pH so that the
concentration of alkaloidal base in equilibrium with its salt is calculated to be less than the solubility of
the free base at the storage temperature. Stabilization against precipitation can thus be maintained.
Buffered Isotonic Solutions
Reference has already been made to in vivo buffer systems, such as blood and lacrimal fluid, and the
desirability for buffering pharmaceutical solutions under certain conditions. In addition to carrying out pH
adjustment, pharmaceutical solutions that are meant for application to delicate membranes of the body
should also be adjusted to approximately the same osmotic pressure as that of the body fluids. Isotonic
solutions cause no swelling or contraction of the tissues with which they come in contact and produce
no discomfort when instilled in the eye, nasal tract, blood, or other body tissues. Isotonic sodium
chloride is a familiar pharmaceutical example of such a preparation.
The need to achieve isotonic conditions with solutions to be applied to delicate membranes is
dramatically illustrated by mixing a small quantity of blood with aqueous sodium chloride solutions of
varying tonicity. For example, if a small quantity of blood, defibrinated to prevent clotting, is mixed with a
solution containing 0.9 g of NaCl per 100 mL, the cells retain their normal size. The solution has
essentially the same salt concentration and hence the same osmotic pressure as the red blood cell
contents and is said to be isotonic with blood. If the red blood cells are suspended in a 2.0% NaCl
solution, the water within the cells passes through the cell membrane in an attempt to dilute the
surrounding salt solution until the salt concentrations on both sides of the erythrocyte membrane are
320
Dr. Murtadha Alshareifi e-Library
identical. This outward passage of water causes the cells to shrink and become wrinkled or crenated.
The salt solution in this instance is said to be hypertonic with respect to the blood cell contents. Finally, if
the blood is mixed with 0.2% NaCl solution or with distilled water, water enters the blood cells, causing
them to swell and finally burst, with the liberation of hemoglobin. This phenomenon is known
as hemolysis, and the weak salt solution or water is said to be hypotonic with respect to the blood.
The student should appreciate that the red blood cell membrane is not impermeable to all drugs; that is,
it is not a perfect semipermeable membrane. Thus, it will permit the passage of not only water
molecules but also solutes such as urea, ammonium chloride, alcohol, and boric acid.26 A 2.0% solution
of boric acid has the same osmotic pressure as the blood cell contents when determined by the freezing
point method and is therefore said to be isosmotic with blood. The molecules of boric acid pass freely
through the erythrocyte membrane, however, regardless of concentration. As a result, this solution acts
essentially as water when in contact with blood cells. Because it is extremely hypotonic with respect to
the blood, boric acid solution brings about rapid hemolysis. Therefore, a solution containing a quantity of
drug calculated to be isosmotic with blood is isotonic only when
P.175
the blood cells are impermeable to the solute molecules and permeable to the solvent, water. It is
interesting to note that the mucous lining of the eye acts as a true semipermeable membrane to boric
acid in solution. Accordingly, a 2.0% boric acid solution serves as an isotonic ophthalmic preparation.
Key Concept
Tonicity
Osmolality and osmolarity are colligative properties that measure the concentration of the
solutes independently of their ability to cross a cell membrane. Tonicity is the concentration of
only the solutes that cannot cross the membrane since these solutes exert an osmotic
pressure on that membrane. Tonicity is not the difference between the two osmolarities on
opposing sides of the membrane. A solution might be hypertonic, isotonic, or hypotonic
relative to another solution. For example, the relative tonicity of blood is defined in reference
to that of the red blood cell (RBC) cytosol tonicity. As such, a hypertonic solution contains a
higher concentration of impermeable solutes than the cytosol of the RBC; there is a net flow
of fluid out of the RBC and it shrinks (Panel A). The concentration of impermeable solutes in
the solution and cytosol are equal and the RBCs remain unchanged, so there is no net fluid
flow (Panel B). A hypotonic solution contains a lesser concentration of such solutes than the
321
Dr. Murtadha Alshareifi e-Library
RBC cytosol and fluid flows into the cells where they swell and potentially burst (Panel C). In
short, a solution containing a quantity of drug calculated to be isosmotic with blood is
isotonic only when the blood cells are impermeable to the solute (drug) molecules and
permeable to the solvent, water.
To overcome this difficulty, Husa27 suggested that the term isotonic should be restricted to solutions
having equal osmotic pressures with respect to a particular membrane. Goyan and Reck28 felt that,
rather than restricting the use of the term in this manner, a new term should be introduced that is
defined on the basis of the sodium chloride concentration. These workers defined the term isotonicity
value as the concentration of an aqueous NaCl solution having the same colligative properties as the
solution in question. Although all solutions having an isotonicity value of 0.9 g of NaCl per 100 mL of
solution need not necessarily be isotonic with respect to the living membranes concerned, many of them
are roughly isotonic in this sense, and all may be considered isotonic across an ideal membrane.
Accordingly, the term isotonic is used with this meaning throughout the present chapter. Only a few
substances—those that penetrate animal membranes at a sufficient rate—will show exception to this
classification.
The remainder of this chapter is concerned with a discussion of isotonic solutions and the means by
which they can be buffered.
Measurement of Tonicity
The tonicity of solutions can be determined by one of two methods. First, in the hemolytic method, the
effect of various solutions of the drug is observed on the appearance of red blood cells suspended in the
solutions. The various effects produced have been described in the previous section. Husa and his
associates27used this method. In their later work, a quantitative method developed by Hunter29 was
used based on the fact that a hypotonic solution liberates oxyhemoglobin in direct proportion to the
number of cells hemolyzed. By such means, the van't Hoff i factor can be determined and the value
compared with that computed from cryoscopic data, osmotic coefficient, and activity coefficient.30
Husa found that a drug having the proper i value as measured by freezing point depression or computed
from theoretical equations nevertheless may hemolyze human red blood cells; it was on this basis that
he suggested restriction of the term isotonic to solutions having equal osmotic pressures with respect to
a particular membrane.
P.176
Table 8-3 Average Liso Values for Various Ionic Types*
Type
Liso
Examples
Nonelectrolytes
1.9
Sucrose, glycerin, urea, camphor
Weak electrolytes
2.0
Boric acid, cocaine, phenobarbital
Di-divalent
electrolytes
2.0
Magnesium sulfate, zinc sulfate
Uni-univalent
electrolytes
3.4
Sodium chloride, cocaine hydrochloride,
sodium phenobarbital
322
Dr. Murtadha Alshareifi e-Library
Uni-divalent
electrolytes
4.3
Sodium sulfate, atropine sulfate
Di-univalent
electrolytes
4.8
Zinc chloride, calcium bromide
Uni-trivalent
electrolytes
5.2
Sodium citrate, sodium phosphate
Tri-univalent
electrolytes
6.0
Aluminum chloride, ferric iodide
Tetraborate
electrolytes
7.6
Sodium borate, potassium borate
*From J. M. Wells, J. Am. Pharm. Assoc. Pract. Ed. 5, 99, 1944.
The second approach used to measure tonicity is based on any of the methods that determine
colligative properties earlier in the book. Goyan and Reck28investigated various modifications of the
Hill–Baldes technique31 for measuring tonicity. This method is based on a measurement of the slight
temperature differences arising from differences in the vapor pressure of thermally insulated samples
contained in constant-humidity chambers.
One of the first references to the determination of the freezing point of blood and tears (as was
necessary to make solutions isotonic with these fluids) is that of Lumiere and Chevrotier,32 in which the
values of -0.56°C and -0.80°C were given, respectively, for the two fluids. Following work by Pedersen,
Bjergaard and coworkers,33 34 however, it is now well established that -0.52°C is the freezing point of
both human blood and lacrimal fluid. This temperature corresponds to the freezing point of a 0.90%
NaCl solution, which is therefore considered to be isotonic with both blood and lacrimal fluid.
Calculating Tonicity Using Liso Values
Because the freezing point depressions for solutions of electrolytes of both the weak and strong types
are always greater than those calculated from the equation ΔTf= Kfc, a new factor, L = i Kf, is introduced
to overcome this difficulty.35 The equation, already discussed is
The L value can be obtained from the freezing point lowering of solutions of representative compounds
of a given ionic type at a concentration c that is isotonic with body fluids. This specific value of L is
written as Liso.
The Liso value for a 0.90% (0.154 M) solution of sodium chloride, which has a freezing point depression
of 0.52°C and is thus isotonic with body fluids, is 3.4: From
we have
323
Dr. Murtadha Alshareifi e-Library
The interionic attraction in solutions that are not too concentrated is roughly the same for all uniunivalent electrolytes regardless of the chemical nature of the various compounds of this class, and all
have about the same value for Liso, namely 3.4. As a result of this similarity between compounds of a
given ionic type, a table can be arranged listing the L value for each class of electrolytes at a
concentration that is isotonic with body fluids. The Liso values obtained in this way are given in Table 83.
It will be observed that for dilute solutions of nonelectrolytes, Liso is approximately equal to Kf. Table 83 is used to obtain the approximate ΔTf for a solution of a drug if the ionic type can be correctly
ascertained. A plot of i Kf against molar concentration of various types of electrolytes, from which the
values of Liso can be read, is shown in Figure 6-7 (in Chapter 6, ―Electrolytes and Ionic Equilibria‖).
Example 8-11
Freezing Point Lowering
What is the freezing point lowering of a 1% solution of sodium propionate (molecular weight
96)? Because sodium propionate is a uni-univalent electrolyte, its Liso value is 3.4. The molar
concentration of a 1% solution of this compound is 0.104. We have
Although 1 g/100 mL of sodium propionate is not the isotonic concentration, it is still proper to use Liso as
a simple average that agrees with the concentration range expected for the finished solution. The
selection of L values in this concentration region is not sensitive to minor changes in concentration; no
pretense to accuracy greater than about 10% is implied or needed in these calculations.
P.177
The calculation of Example 8-11 can be simplified by expressing molarity c as grams of drug contained
in a definite volume of solution. Thus,
or
where w is the grams of solute, MW is the molecular weight of the solute, and v is the volume of solution
in milliliters. Substituting in equation (8-36) gives
The problem in Example 8-11 can be solved in one operation by the use of equation (8-41) without the
added calculation needed to obtain the molar concentration:
The student is encouraged to derive expressions of this type; certainly equations (8-40) and (841) should not be memorized, for they are not remembered for long. The Liso values can also be used
for calculating sodium chloride equivalents and Sprowls V values, as discussed in subsequent sections
of this chapter.
Methods of Adjusting Tonicity and pH
One of several methods can be used to calculate the quantity of sodium chloride, dextrose, and other
substances that may be added to solutions of drugs to render them isotonic.
For discussion purposes, the methods are divided into two classes. In the class I methods, sodium
chloride or some other substance is added to the solution of the drug to lower the freezing point of the
324
Dr. Murtadha Alshareifi e-Library
solution to -0.52°C and thus make it isotonic with body fluids. Under this class are included
the cryoscopic method and thesodium chloride equivalent method. In the class II methods, water is
added to the drug in a sufficient amount to form an isotonic solution. The preparation is then brought to
its final volume with an isotonic or a buffered isotonic dilution solution. Included in this class are
the White–Vincent method and the Sprowls method.
Class I Methods
Cryoscopic Method
The freezing point depressions of a number of drug solutions, determined experimentally or
theoretically, are given in Table 8-4. According to the previous section, the freezing point depressions of
drug solutions that have not been determined experimentally can be estimated from theoretical
considerations, knowing only the molecular weight of the drug and the Liso value of the ionic class.
The calculations involved in the cryoscopic method are explained best by an example.
Example 8-12
Isotonicity
How much sodium chloride is required to render 100 mL of a 1% solution of apomorphine
hydrochloride isotonic with blood serum?
From Table 8-4 it is found that a 1% solution of the drug has a freezing point lowering of
0.08°C. To make this solution isotonic with blood, sufficient sodium chloride must be added to
reduce the freezing point by an additional 0.44°C (0.52°C - 0.08°C). In the freezing point
table, it is also observed that a 1% solution of sodium chloride has a freezing point lowering of
0.58°C. By the method of proportion,
Thus, 0.76% sodium chloride will lower the freezing point the required 0.44°C and will render
the solution isotonic. The solution is prepared by dissolving 1.0 g of apomorphine
hydrochloride and 0.76 g of sodium chloride in sufficient water to make 100 mL of solution.
Sodium Chloride Equivalent Method
A second method for adjusting the tonicity of pharmaceutical solutions was developed by Mellen and
Seltzer.36 The sodium chloride equivalent or, as referred to by these workers, the ―tonicic equivalent‖ of
a drug is the amount of sodium chloride that is equivalent to (i.e., has the same osmotic effect as) 1 g, or
other weight unit, of the drug. The sodium chloride equivalents E for a number of drugs are listed
in Table 8-4.
When the E value for a new drug is desired for inclusion in Table 8-4, it can be calculated from
the Liso value or freezing point depression of the drug according to formulas derived by Goyan et
al.37 For a solution containing 1 g of drug in 1000 mL of solution, the concentration c expressed in
moles/liter can be written as
and from equation (8-36)
Now, E is the weight of NaCl with the same freezing point depression as 1 g of the drug, and for a NaCl
solution containing E grams of drug per 1000 mL,
where 3.4 is the Liso value for sodium chloride and 58.45 is its molecular weight. Equating these two
values of ΔTf yields
P.178
325
Dr. Murtadha Alshareifi e-Library
Table 8-4 Isotonic Values*, †
Substance
MW
E
ΔTf1%
V
Liso
Alcohol, dehydrated
46.07
0.70
23.3
0.41
1.9
Aminophylline
456.46
0.17
5.7
0.10
4.6
Amphetamine sulfate
368.49
0.22
7.3
0.13
4.8
Antipyrine
188.22
0.17
5.7
0.10
1.9
Apomorphine
hydrochloride
312.79
0.14
4.7
0.08
2.6
Ascorbic acid
176.12
0.18
6.0
0.11
1.9
Atropine sulfate
694.82
0.13
4.3
0.07
5.3
Diphenhydramine
hydrochloride
291.81
0.20
6.6
0.34
3.4
Boric acid
61.84
0.50
16.7
0.29
1.8
Caffeine
194.19
0.08
2.7
0.05
0.9
Dextrose·H2O
198.17
0.16
5.3
0.09
1.9
Ephedrine
hydrochloride
201.69
0.30
10.0
0.18
3.6
Ephedrine sulfate
428.54
0.23
7.7
0.14
5.8
Epinephrine
hydrochloride
219.66
0.29
9.7
0.17
3.7
326
Dr. Murtadha Alshareifi e-Library
Glycerin
92.09
0.34
11.3
0.20
1.8
Lactose
360.31
0.07
2.3
0.04
1.7
Morphine
hydrochloride
375.84
0.15
5.0
0.09
3.3
Morphine sulfate
758.82
0.14
4.8
0.08
6.2
Neomycin sulfate
–
0.11
3.7
0.06
–
Penicillin G
potassium
372.47
0.18
6.0
0.11
3.9
Penicillin G Procaine
588.71
0.10
3.3
0.06
3.5
Phenobarbital sodium
254.22
0.24
8.0
0.14
3.6
Phenol
94.11
0.35
11.7
0.20
1.9
Potassium chloride
74.55
0.76
25.3
0.45
3.3
Procaine
hydrochloride
272.77
0.21
7.0
0.12
3.4
Quinine hydrochloride
396.91
0.14
4.7
0.08
3.3
Sodium chloride
58.45
1.00
33.3
0.58
3.4
Streptomycin sulfate
1457.44
0.07
2.3
0.04
6.0
Sucrose
342.30
0.08
2.7
0.05
1.6
Tetracycline
hydrochloride
480.92
0.14
4.7
0.08
4.0
327
Dr. Murtadha Alshareifi e-Library
Urea
60.06
0.59
19.7
0.35
2.1
Zinc chloride
139.29
0.62
20.3
0.37
5.1
*The values were obtained from the data of E. R. Hammarlund and K.
Pedersen-Bjergaard, J. Am. Pharm. Assoc. Pract. Ed. 19, 39, 1958; J. Am.
Pharm. Assoc. Sci. Ed. 47, 107, 1958; and other sources. The values vary
somewhat with concentration, and those in the table are for 1% to 3%
solutions of the drugs in most instances. A complete table of Eand ΔTf values
is found in the Merck Index, 11th Ed., Merck, Rahway, N. J., 1989, pp.
MISC-79 to MISC-103. For the most recent results of Hammarlund, see J.
Pharm. Sci. 70, 1161, 1981; 78, 519, 1989.
Key: MW = molecular weight of the drug; E = sodium chloride equivalent of
the drug; V = volume in mL of isotonic solution that can be prepared by
adding water to 0.3 g of the drug (the weight of drug in 1 fluid ounce of a 1%
solution); ΔTf1% = freezing point depression of a 1% solution of the
drug; Liso = the molar freezing point depression of the drug at a concentration
approximately isotonic with blood and lacrimal fluid.
†
The full table is available at the book's companion website at
thepoint.lww.com/Sinko6e.
Example 8-13
Sodium Chloride Equivalents
Calculate the approximate E value for a new amphetamine hydrochloride derivative
(molecular weight 187).
Because this drug is a uni-univalent salt, it has an Liso value of 3.4. Its E value is calculated
from equation (8-45):
Calculations for determining the amount of sodium chloride or other inert substance to render a solution
isotonic (across an ideal membrane) simply involve multiplying the quantity of each drug in the
prescription by its sodium chloride equivalent and subtracting this value from the concentration of
sodium chloride that is isotonic with body fluids, namely, 0.9 g/100 mL.
Example 8-14
Tonicity Adjustment
A solution contains 1.0 g of ephedrine sulfate in a volume of 100 mL. What quantity of sodium
chloride must be added to make the solution isotonic? How much dextrose would be required
for this purpose?
The quantity of the drug is multiplied by its sodium chloride equivalent, E, giving the weight of
sodium chloride to which the quantity of drug is equivalent in osmotic pressure:
The ephedrine sulfate has contributed a weight of material osmotically equivalent to 0.23 g of
sodium chloride. Because a total of 0.9 g of sodium chloride is required for isotonicity, 0.67 g
(0.90 - 0.23 g) of NaCl must be added.
328
Dr. Murtadha Alshareifi e-Library
If one desired to use dextrose instead of sodium chloride to adjust the tonicity, the quantity
would be estimated by setting up the following proportion. Because the sodium chloride
equivalent of dextrose is 0.16,
P.179
Other agents than dextrose can of course be used to replace NaCl. It is recognized that thimerosal
becomes less stable in eye drops when a halogen salt is used as an ―isotonic agent‖ (i.e., an agent like
NaCl ordinarily used to adjust the tonicity of a drug solution). Reader38 found that mannitol, propylene
glycol, or glycerin—isotonic agents that did not have a detrimental effect on the stability of thimerosal—
could serve as alternatives to sodium chloride. The concentration of these agents for isotonicity is
readily calculated by use of the equation (see Example 8-14)
where X is the grams of isotonic agent required to adjust the tonicity, Y is the additional amount of NaCl
for isotonicity over and above the osmotic equivalence of NaCl provided by the drugs in the solution,
and E is the sodium chloride equivalence of the isotonic agent.
Example 8-15
Isotonic Solutions
Let us prepare 200 mL of an isotonic aqueous solution of thimerosal, molecular weight 404.84
g/mole. The concentration of this anti-infective drug is 1:5000, or 0.2 g/1000 mL. The Liso for
such a compound, a salt of a weak acid and a strong base (a 1:1 electrolyte), is 3.4, and the
sodium chloride equivalent E is
The quantity of thimerosal, 0.04 g for the 200-mL solution, multiplied by its E value gives the
weight of NaCl to which the drug is osmotically equivalent:
Because the total amount of NaCl needed for isotonicity is 0.9 g/100 mL, or 1.8 g for the 200mL solution, and because an equivalent of 0.0057 g of NaCl has been provided by the
thimerosal, the additional amount of NaCl needed for isotonicity, Y, is
This is the additional amount of NaCl needed for isotonicity. The result, 1.8 g of NaCl, shows
that the concentration of thimerosal is so small that it contributes almost nothing to the
isotonicity of the solution. Thus, a concentration of 0.9% NaCl, or 1.8 g/200 mL, is required.
However, from the work of Reader38 we know that sodium chloride interacts with mercury
compounds such as thimerosal to reduce the stability and effectiveness of this preparation.
Therefore, we replace NaCl with propylene glycol as the isotonic agent.
From equation (8-45) we calculate the E value of propylene glycol, a nonelectrolyte with
an Liso value of 1.9 and a molecular weight of 76.09 g/mole:
Using equation (8-46), X = Y/E, we obtain
where X = 4.3 g is the amount of propylene glycol required to adjust the 200-mL solution of
thimerosal to isotonicity.
329
Dr. Murtadha Alshareifi e-Library
Class II Methods
White–Vincent Method
The class II methods of computing tonicity involve the addition of water to the drugs to make an isotonic
solution, followed by the addition of an isotonic or isotonic-buffered diluting vehicle to bring the solution
to the final volume. Stimulated by the need to adjust the pH in addition to the tonicity of ophthalmic
solutions, White and Vincent39 developed a simplified method for such calculations. The derivation of
the equation is best shown as follows.
Suppose that one wishes to make 30 mL of a 1% solution of procaine hydrochloride isotonic with body
fluid. First, the weight of the drug, w, is multiplied by the sodium chloride equivalent, E:
This is the quantity of sodium chloride osmotically equivalent to 0.3 g of procaine hydrochloride.
Second, it is known that 0.9 g of sodium chloride, when dissolved in enough water to make 100 mL,
yields a solution that is isotonic. The volume, V, of isotonic solution that can be prepared from 0.063 g of
sodium chloride (equivalent to 0.3 g of procaine hydrochloride) is obtained by solving the proportion
In equation (8-49), the quantity 0.063 is equal to the weight of drug, w, multiplied by the sodium chloride
equivalent, E, as seen in equation (8-47). The value of the ratio 100/0.9 is 111.1. Accordingly,
equation (8-49) can be written as
where V is the volume in milliliters of isotonic solution that may be prepared by mixing the drug with
water, w is the weight in grams of the drug given in the problem, and E is the sodium chloride equivalent
obtained from Table 8-4. The constant, 111.1, represents the volume in milliliters of isotonic solution
obtained by dissolving 1 g of sodium chloride in water.
P.180
The problem can be solved in one step using equation (8-51):
To complete the isotonic solution, enough isotonic sodium chloride solution, another isotonic solution, or
an isotonic-buffered diluting solution is added to make 30 mL of the finished product.
When more than one ingredient is contained in an isotonic preparation, the volumes of isotonic solution,
obtained by mixing each drug with water, are additive.
Example 8-16
Isotonic Solutions
Make the following solution isotonic with respect to an ideal membrane:
The drugs are mixed with water to make 18 mL of an isotonic solution, and the preparation is
brought to a volume of 100 mL by adding an isotonic diluting solution.
The Sprowls Method
A further simplification of the method of White and Vincent was introduced by Sprowls.40 He recognized
that equation (8-51) could be used to construct a table of values of V when the weight of the drug, w,
330
Dr. Murtadha Alshareifi e-Library
was arbitrarily fixed. Sprowls chose as the weight of drug 0.3 g, the quantity for 1 fluid ounce of a 1%
solution. The volume,V, of isotonic solution that can be prepared by mixing 0.3 g of a drug with sufficient
water can be computed for drugs commonly used in ophthalmic and parenteral solutions. The method
as described by Sprowls40 is further discussed in several reports by Martin and Sprowls.41 The table
can be found in the United States Pharmacopeia. A modification of the original table was made by
Hammarlund and Pedersen-Bjergaard42 and the values of V are given in column 4 of Table 8-4, where
the volume in milliliters of isotonic solution for 0.3 g of the drug, the quantity for 1 fluid ounce of a 1%
solution, is listed. (The volume of isotonic solution in milliliters for 1 g of the drug can also be listed in
tabular form if desired by multiplying the values in column 4 by 3.3.) The primary quantity of isotonic
solution is finally brought to the specified volume with the desired isotonic or isotonic-buffered diluting
solutions.
Chapter Summary
Buffers are compounds or mixtures of compounds that, by their presence in solution, resist
changes in pH upon the addition of small quantities of acid or alkali. The resistance to a
change in pH is known as buffer action. If a small amount of a strong acid or base is added to
water or a solution of sodium chloride, the pH is altered considerably; such systems have no
buffer action. In this chapter, the theory of buffers was introduced as were several formulas
for making commonly used buffers. Finally, the important concept of tonicity was introduced.
Pharmaceutical buffers must usually be made isotonic so that they cause no swelling or
contraction of biological tissues, which would lead to discomfort in the patient being treated.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. A. Roos and W. F. Borm, Respir. Physiol. 40, 1980; M. Koppel and K. Spiro, Biochem. Z. 65, 409,
1914.
2. D. D. Van Slyke, J. Biol. Chem. 52, 525, 1922.
3. R. G. Bates, Electrometric pH Determinations, Wiley, New York, 1954, pp. 97–104, 116, 338.
4. I. M. Kolthoff and F. Tekelenburg, Rec. Trav. Chim. 46, 33, 1925.
5. X. Kong, T. Zhou, Z. Liu, and R. C. Hider, J. Pharm. Sci. 96, 2777, 2007.
6. H. T. S. Britton and R. A. Robinson, J. Chem. Soc. 458, 1931. DOI:10.1039/JR9310000458.
7. J. P. Peters and D. D. Van Slyke, Quantitative Clinical Chemistry, Vol. 1, Williams & Wilkins,
Baltimore, 1931, Chapter 18.
8. P. Salenius, Scand. J. Clin. Lab. Invest. 9, 160, 1957.
9. G. Ellison et al., Clin. Chem. 4, 453, 1958.
10. G. N. Hosford and A. M. Hicks, Arch. Ophthalmol. 13, 14, 1935; 17, 797, 1937.
11. R. G. Bates, Electrometric pH Determinations, Wiley, New York, 1954, pp. 97–108.
12. D. M. Maurice, Br. J. Ophthalmol. 39, 463–473, 1955.
13. S. R. Gifford, Arch. Ophthalmol. 13, 78, 1935.
14. S. L. P. Sörensen, Biochem. Z. 21, 131, 1909; 22, 352, 1909.
15. S. Palitzsch, Biochem. Z. 70, 333, 1915.
16. H. W. Hind and F. M. Goyan, J. Am. Pharm. Assoc. Sci. Ed. 36, 413, 1947.
17. W. M. Clark and H. A. Lubs, J. Bacteriol. 2, 1, 109, 191, 1917.
18. J. S. Friedenwald, W. F. Hughes, and H. Herrman, Arch. Ophthalmol. 31, 279, 1944; V. E. Bower
and R. G. Bates, J. Res. NBS 55, 197, 1955.
19. F. N. Martin and J. L. Mims, Arch. Ophthalmol. 44, 561, 1950; J. L. Mims, Arch.
Ophthalmol. 46, 644, 1951.
20. S. Riegelman and D. G. Vaughn, J. Am. Pharm. Assoc. Pract. Ed. 19, 474, 1958.
21. W. D. Mason, J. Pharm. Sci. 73, 1258, 1984.
22. L. Illum, J. Control. Release. 87, 187, 2003.
23. K. C. Swan and N. G. White, Am. J. Ophthalmol. 25, 1043, 1942.
24. D. G. Cogan and V. E. Kinsey, Arch. Ophthalmol. 27, 466, 661, 696, 1942.
331
Dr. Murtadha Alshareifi e-Library
25. H. W. Hind and F. M. Goyan, J. Am. Pharm. Assoc. Sci. Ed. 36, 33, 1947.
26. Y. Takeuchi, Y. Yamaoka, Y. Morimoto, I. Kaneko, Y. Fukumori, and T. Fukuda, J. Pharm.
Sci. 78, 3, 1989.
27. T. S. Grosicki and W. J. Husa, J. Am. Pharm. Assoc. Sci. Ed. 43, 632, 1954; W. J. Husa and J. R.
Adams, J. Am. Pharm. Assoc. Sci. Ed. 33, 329, 1944; W. D. Easterly and W. J. Husa, J. Am. Pharm.
Assoc. Sci. Ed. 43, 750, 1954; W. J. Husa and O. A. Rossi, J. Am. Pharm. Assoc. Sci. Ed. 31, 270,
1942.
28. F. M. Goyan and D. Reck, J. Am. Pharm. Assoc. Sci. Ed. 44, 43, 1955.
29. F. R. Hunter, J. Clin. Invest. 19, 691, 1940.
30. C. W. Hartman and W. J. Husa, J. Am. Pharm. Assoc. Sci. Ed. 46, 430, 1957.
P.181
31. A. V. Hill, Proc. R. Soc. Lond. A 127, 9, 1930; E. J. Baldes, J. Sci. Instrum. 11, 223, 1934; A.
Weissberger, Physical Methods of Organic Chemistry, 2nd Ed., Vol. 1, Part 1, Interscience, New York,
1949, p. 546.
32. A. Lumiere and J. Chevrotier, Bull. Sci. Pharmacol. 20, 711, 1913.
33. C. G. Lund, P. Nielsen, and K. Pedersen-Bjergaard, The Preparation of Solutions Isosmotic with
Blood, Tears, and Tissue (Danish Pharmacopeia Commission), Vol. 2, Einar Munksgaard, Copenhagen,
1947.
34. A. Krogh, C. G. Lund, and K. Pedersen-Bjergaard, Acta Physiol. Scand. 10, 88, 1945.
35. F. M. Goyan, J. M. Enright, and J. M. Wells, J. Am. Pharm. Assoc. Sci. Ed. 33, 74, 1944.
36. M. Mellen and L. A. Seltzer, J. Am. Pharm. Assoc. Sci. Ed. 25, 759, 1936.
37. F. M. Goyan, J. M. Enright, and J. M. Wells, J. Am. Pharm. Assoc. Sci. Ed. 33, 78, 1944.
38. M. J. Reader, J. Pharm. Sci. 73, 840, 1984.
39. A. I. White and H. C. Vincent, J. Am. Pharm. Assoc. Pract. Ed. 8, 406, 1947.
40. J. B. Sprowls, J. Am. Pharm. Assoc. Pract. Ed. 10, 348, 1949.
41. A. Martin and J. B. Sprowls, Am. Profess. Pharm. 17, 540, 1951; Pa Pharm. 32, 8, 1951.
42. E. R. Hammarlund and K. Pedersen-Bjergaard, J. Am. Pharm. Assoc. Pract. Ed. 19, 38, 1958.
Recommended Readings
R. J. Benyon, Buffer Solutions (The Basics), BIOS Scientific Publishers, Oxford, UK, 1996.
Chapter Legacy
Fifth Edition: published as Chapter 9 (Buffered and Isotonic Solutions). Updated by Patrick
Sinko.
Sixth Edition: published as Chapter 8 (Buffered and Isotonic Solutions). Updated by Patrick
Sinko.
332
Dr. Murtadha Alshareifi e-Library
9 Solubility and Distribution Phenomena
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1.
2.
3.
4.
5.
6.
7.
8.
Define saturated solution, solubility, and unsaturated solution.
Describe and give examples of polar, nonpolar, and semipolar solvents.
Define complete and partial miscibility.
Understand the factors controlling the solubility of weak electrolytes.
Describe the influence of solvents and surfactants on solubility.
Define thermodynamic, kinetic, and intrinsic solubility.
Measure thermodynamic solubility.
Describe what a distribution coefficient and partition coefficient are and their
importance in pharmaceutical systems.
General Principles
Introduction1,2,3,4
Solubility is defined in quantitative terms as the concentration of solute in a saturated solution at a
certain temperature, and in a qualitative way, it can be defined as the spontaneous interaction of two or
more substances to form a homogeneous molecular dispersion. Solubility is an intrinsic material
property that can be altered only by chemical modification of the molecule.1 In contrast to this,
dissolution is an extrinsic material property that can be influenced by various chemical, physical, or
crystallographic means such as complexation, particle size, surface properties, solid-state modification,
or solubilization enhancing formulation strategies.1Dissolution is discussed in Chapter 13. Generally
speaking, the solubility of a compound depends on the physical and chemical properties of the solute
and the solvent as well as on such factors as temperature, pressure, the pH of the solution, and, to a
lesser extent, the state of subdivision of the solute. Of the nine possible types of mixtures, based on the
three states of matter, only liquids in liquids and solids in liquids are of everyday importance to most
pharmaceutical scientists and will be considered in this chapter.
For the most part, this chapter will deal with the thermodynamic solubility of drugs (Fig. 9-1). The
thermodynamic solubility of a drug in a solvent is the maximum amount of the most stable crystalline
form that remains in solution in a given volume of the solvent at a given temperature and pressure under
equilibrium conditions.4 The equilibrium involves a balance of the energy of three interactions against
each other: (1) solvent with solvent, (2) solute with solute, and (3) solvent and solute. Thermodynamic
equilibrium is achieved when the overall lowest energy state of the system is achieved. This means that
only the equilibrium solubility reflects the balance of forces between the solution and the most stable,
lowest energy crystalline form of the solid. In practical terms, this means that one needs to be careful
when evaluating a drug's solubility. For example, let us say that you want to determine the solubility of a
drug and that you were not aware that it was not in its crystalline form. It is well known that a metastable
solid form of a drug will have a higher apparent solubility. Given enough time, the limiting solubility of the
most stable form will eventually dominate and since the most stable crystal form has the lowest
solubility, this means that there will be excess drug in solution resulting in a precipitate. So, initially you
would record a higher solubility but after a period of time the solubility that you measure would be
significantly lower. As you can imagine, this could lead to serious problems. This was vividly illustrated
by Abbott's antiviral drug ritonavir where the slow precipitation of a new stable polymorph from dosing
solutions required the manufacturer to perform an emergency reformulation to ensure consistent drug
,
release characteristics.2 4
Key Concept
Solutions and Solubility
A saturated solution is one in which the solute in solution is in equilibrium with the solid
phase. Solubility is defined in quantitative terms as the concentration of solute in a saturated
333
Dr. Murtadha Alshareifi e-Library
solution at a certain temperature, and in a qualitative way, it can be defined as the
spontaneous interaction of two or more substances to form a homogeneous molecular
dispersion. An unsaturated or subsaturated solution is one containing the dissolved solute in
a concentration below that necessary for complete saturation at a definite temperature.
A supersaturated solution is one that contains more of the dissolved solute than it would
normally contain at a definite temperature, were the undissolved solute present.
P.183
Fig. 9-1. The intermolecular forces that determine thermodynamic solubility. (a)
Solvent and solute are segregated, each interacts primarily with other molecules of the
same type. (b) To move a solute molecule into solution, the interactions among solute
molecules in the crystal (lattice energy) and among solvent molecules in the space
required to accommodate the solute (cavitation energy) must be broken. The system
entropy increases slightly because the ordered network of hydrogen bonds among
solvent molecules has been disrupted. (c) Once the solute molecule is surrounded by
solvent, new stabilizing interactions between the solute and solvent are formed
(solvation energy), as indicated by the dark purple molecules. The system entropy
increases owing to the mingling of solute and solvent (entropy of mixing) but also
decreases locally owing to the new short-range order introduced by the presence of
the solute, as indicated by the light purple molecules.4 (Adapted from Bhattachar et
al. 2006.4)
Solubility Expressions
The solubility of a drug may be expressed in a number of ways. The United States Pharmacopeia (USP)
describes the solubility of drugs as parts of solvent required for one part solute. Solubility is also
quantitatively expressed in terms of molality, molarity, and percentage. The USP describes solubility
using the seven groups listed in Table 9-1. The European Pharmacopoeia lists six categories (it does
not use the practically insoluble grouping). For exact solubilities of many substances, the reader is
referred to standard reference works such as official compendia (e.g., USP) and the Merck Index.
Solvent–Solute Interactions
The pharmacist knows that water is a good solvent for salts, sugars, and similar compounds, whereas
mineral oil is often a solvent for substances that are normally only slightly soluble in water. These
334
Dr. Murtadha Alshareifi e-Library
empirical findings are summarized in the statement, ―like dissolves like.‖ Such a maxim is satisfying to
most of us, but the inquisitive student may be troubled by this vague idea of ―likeness.‖
Table 9-1 Solubility Definition in the United States Pharmacopeia
Parts of Solvent
Solubility
Description Forms
Required for One Part Range
(Solubility Definition) of Solute
(mg/mL)
Solubility
Assigned
(mg/mL)
Very soluble (VS)
<1
>1000
1000
Freely soluble (FS)
From 1 to 10
100–1000
100
Soluble
From 10 to 30
33–100
33
Sparingly soluble
(SPS)
From 30 to 100
10–33
10
Slightly soluble
(SS)
From 100 to 1000
1–10
1
Very slightly
soluble (VSS)
From 1000 to
10,000
0.1–1
0.1
Practically
insoluble (PI)
>10,000
<0.1
0.01
Polar Solvents
The solubility of a drug is due in large measure to the polarity of the solvent, that is, to its dipole
moment. Polar solvents dissolve ionic solutes and other polar substances. Accordingly, water mixes in
all proportions with alcohol and dissolves sugars and other polyhydroxy compounds. Hildebrand
showed, however, that a consideration of dipole moments alone is not adequate to explain the solubility
of polar substances in water. The ability of the solute to form hydrogen bonds is a far more significant
factor than is the polarity as reflected in a high dipole moment. Water dissolves phenols, alcohols,
aldehydes, ketones, amines, and other oxygen- and nitrogen-containing compounds that can form
hydrogen bonds with water:
P.184
335
Dr. Murtadha Alshareifi e-Library
A difference in acidic and basic character of the constituents in the Lewis electron donor–acceptor
sense also contributes to specific interactions in solutions.
In addition to the factors already enumerated, the solubility of a substance also depends on structural
features such as the ratio of the polar to the nonpolar groups of the molecule. As the length of a
nonpolar chain of an aliphatic alcohol increases, the solubility of the compound in water decreases.
Straight-chain monohydroxy alcohols, aldehydes, ketones, and acids with more than four or five carbons
cannot enter into the hydrogen-bonded structure of water and hence are only slightly soluble. When
additional polar groups are present in the molecule, as found in propylene glycol, glycerin, and tartaric
acid, water solubility increases greatly. Branching of the carbon chain reduces the nonpolar effect and
leads to increased water solubility. Tertiary butyl alcohol is miscible in all proportions with water,
whereas n-butyl alcohol dissolves to the extent of about 8 g/100 mL of water at 20°C.
Nonpolar Solvents
The solvent action of nonpolar liquids, such as the hydrocarbons, differs from that of polar substances.
Nonpolar solvents are unable to reduce the attraction between the ions of strong and weak electrolytes
because of the solvents' low dielectric constants. Nor can the solvents break covalent bonds and ionize
weak electrolytes, because they belong to the group known as aprotic solvents, and they cannot form
hydrogen bridges with nonelectrolytes. Hence, ionic and polar solutes are not soluble or are only slightly
soluble in nonpolar solvents.
Nonpolar compounds, however, can dissolve nonpolar solutes with similar internal pressures through
induced dipole interactions. The solute molecules are kept in solution by the weak van der Waals–
London type of forces. Thus, oils and fats dissolve in carbon tetrachloride, benzene, and mineral oil.
Alkaloidal bases and fatty acids also dissolve in nonpolar solvents.
Key Concept
Solubility
The simple maxim that like dissolves like can be rephrased by stating that the solubility of a
substance can be predicted only in a qualitative way in most cases and only after
considerations of polarity, dielectric constant, association, solvation, internal pressures, acid–
336
Dr. Murtadha Alshareifi e-Library
base reactions, and other factors. In short, solubility depends on chemical, electrical, and
structural effects that lead to mutual interactions between the solute and the solvent.
Semipolar Solvents
Semipolar solvents, such as ketones and alcohols, can induce a certain degree of polarity in nonpolar
solvent molecules, so that, for example, benzene, which is readily polarizable, becomes soluble in
alcohol. In fact, semipolar compounds can act as intermediate solvents to bring about miscibility of polar
and nonpolar liquids. Accordingly, acetone increases the solubility of ether in water. Loran and
Guth5 studied the intermediate solvent action of alcohol on water–castor oil mixtures. Propylene glycol
has been shown to increase the mutual solubility of water and peppermint oil and of water and benzyl
benzoate.6
A number of common solvent types are listed in the order of decreasing ―polarity‖ in Table 9-2, together
with corresponding solute classes. The term polarity is loosely used here to represent not only the
dielectric constants of the solvents and solutes but also the other factors enumerated previously.
Solubility of Liquids in Liquids
Frequently two or more liquids are mixed together in the preparation of pharmaceutical solutions. For
example, alcohol is added to water to form hydroalcoholic solutions of various concentrations; volatile
oils are mixed with water to form dilute solutions known as aromatic waters; volatile oils are added to
alcohol to yield spirits and elixirs; ether and alcohol are combined in collodions; and various fixed oils
are blended into lotions, sprays, and medicated oils. Liquid–liquid systems can be divided into two
categories according to the solubility of the substances in one another: (a) complete miscibility and (b)
partial miscibility. The term miscibilityrefers to the mutual solubilities of the components in liquid–liquid
systems.
Complete Miscibility
Polar and semipolar solvents, such as water and alcohol, glycerin and alcohol, and alcohol and acetone,
are said to be
P.185
completely miscible because they mix in all proportions. Nonpolar solvents such as benzene and carbon
tetrachloride are also completely miscible. Completely miscible liquid mixtures in general create no
solubility problems for the pharmacist and need not be considered further.
Table 9-2 Polarity of some Solvents and the Solutes that Readily Dissolve in each
Class of Solvent
Dielectric
Constant of
Solvent, ε
(Approximatel
y)
Solvent
Solute
Decreasin
g Polarity
80
Water
Inorganic
salts,
organic salts
Decreasin
g Water
Solubility
↓
50
Glycols
Sugars,
tannins
↓
337
Dr. Murtadha Alshareifi e-Library
30
Methyl and
ethyl
alcohols
Caster oil,
waxes
20
Aldehydes,
ketones,
and higher
alcohols,
ethers,
esters, and
oxides
Resins,
volatile oils,
weak
electrolytes
including
barbiturates,
alkaloids,
and phenols
5
Hexane,
benzene,
carbon
tetrachlorid
e, ethyl
ether,
petroleum
ether
Fixed oils,
fats,
petrolatum,
paraffin,
other
hydrocarbo
ns
0
Mineral oil
and fixed
vegetable
oils
Partial Miscibility
When certain amounts of water and ether or water and phenol are mixed, two liquid layers are formed,
each containing some of the other liquid in the dissolved state. The phenol–water system has been
discussed in detail in Chapter 2, and the student at this point should review the section dealing with the
phase rule. It is sufficient here to reiterate the following points. (a) The mutual solubilities of partially
miscible liquids are influenced by temperature. In a system such as phenol and water, the mutual
solubilities of the two conjugate phases increase with temperature until, at the critical solution
temperature (or upper consolute temperature), the compositions become identical. At this temperature,
a homogeneous or single-phase system is formed. (b) From a knowledge of the phase diagram, more
especially the tie lines that cut the binodal curve, it is possible to calculate both the composition of each
component in the two conjugate phases and the amount of one phase relative to the other. Example 91 gives an illustration of such a calculation.
Example 9-1
Component Weights
A mixture of phenol and water at 20°C has a total composition of 50% phenol. The tie line at
this temperature cuts the binodal at points equivalent to 8.4% and 72.2% w/w phenol. What is
the weight of the aqueous layer and of the phenol layer in 500 g of the mixture and how many
grams of phenol are present in each of the two layers?
338
Dr. Murtadha Alshareifi e-Library
Let Z be the weight in grams of the aqueous layer. Therefore, 500 - Z is the weight in grams
of the phenol layer, and the sum of the percentages of phenol in the two layers must equal
the overall composition of 50%, or 500 × 0.50 = 250 g. Thus,
In the case of some liquid pairs, the solubility can increase as the temperature is lowered, and the
system will exhibit a lower consolute temperature, below which the two members are soluble in all
proportions and above which two separate layers form. Another type, involving a few mixtures such as
nicotine and water, shows both an upper and a lower consolute temperature with an intermediate
temperature region in which the two liquids are only partially miscible. A final type exhibits no critical
solution temperature; the pair ethyl ether and water, for example, has neither an upper nor a lower
consolute temperature and shows partial miscibility over the entire temperature range at which the
mixture exists.
Three-Component Systems
The principles underlying systems that can contain one, two, or three partially miscible pairs have been
discussed in detail in Chapter 2. Further examples of three-component systems containing one pair of
partially miscible liquids are water, CCl4, and acetic acid; and water, phenol, and acetone. Loran and
Guth5 studied the three-component system consisting of water, castor oil, and alcohol and determined
the proper proportions for use in certain lotions and hair preparations; a triangular diagram is shown in
their report. A similar titration with water of a mixture containing peppermint oil and polyethylene glycol is
shown in Figure 9-2.6Ternary diagrams have also found use in cosmetic formulations
P.186
involving three liquid phases.7 Gorman and Hall8 determined the ternary-phase diagram of the system
of methyl salicylate, isopropanol, and water (Fig. 9-3).
Fig. 9-2. A triangular diagram showing the solubility of peppermint oil in various
proportions of water and polyethylene glycol.
Solubility of Solids in Liquids
339
Dr. Murtadha Alshareifi e-Library
Systems of solids in liquids include the most frequently encountered and probably the most important
type of pharmaceutical solutions. Many important drugs belong to the class of weak acids and bases.
They react with strong acids and bases and, within definite ranges of pH, exist as ions that are ordinarily
soluble in water.
Although carboxylic acids containing more than five carbons are relatively insoluble in water, they react
with dilute sodium hydroxide, carbonates, and bicarbonates to form soluble salts. The fatty acids
containing more than 10 carbon atoms form soluble soaps with the alkali metals and insoluble soaps
with other metal ions. They are soluble in solvents having low dielectric constants; for example, oleic
acid (C17H33COOH) is insoluble in water but is soluble in alcohol and in ether.
Fig. 9-3. Triangular phase diagram for the three-component system methyl salicylate–
isopropanol–water. (From W. G. Gorman and G. D. Hall, J. Pharm. Sci. 53, 1017,
1964. With permission.)
Hydroxy acids, such as tartaric and citric acids, are quite soluble in water because they are solvated
through their hydroxyl groups. The potassium and ammonium bitartrates are not very soluble in water,
although most alkali metal salts of tartaric acid are soluble. Sodium citrate is used sometimes to dissolve
water-insoluble acetylsalicylic acid because the soluble acetylsalicylate ion is formed in the reaction.
The citric acid that is produced is also soluble in water, but the practice of dissolving aspirin by this
means is questionable because the acetylsalicylate is also hydrolyzed rapidly.
Aromatic acids react with dilute alkalies to form water-soluble salts, but they can be precipitated as the
free acids if stronger acidic substances are added to the solution. They can also be precipitated as
heavy metal salts should heavy metal ions be added to the solution. Benzoic acid is soluble in sodium
hydroxide solution, alcohol, and fixed oils. Salicylic acid is soluble in alkalies and in alcohol. The OH
group of salicyclic acid cannot contribute to the solubility because it is involved in an intramolecular
hydrogen bond.
Phenol is weakly acidic and only slightly soluble in water but is quite soluble in dilute sodium hydroxide
solution,
Phenol is a weaker acid than H2CO3 and is thus displaced and precipitated by CO2 from its dilute alkali
solution. For this reason, carbonates and bicarbonates cannot increase the solubility of phenols in
water.
Many organic compounds containing a basic nitrogen atom in the molecule are important in pharmacy.
These include the alkaloids, sympathomimetic amines, antihistamines, local anesthetics, and others.
340
Dr. Murtadha Alshareifi e-Library
Most of these weak electrolytes are not very soluble in water but are soluble in dilute solutions of acids;
such compounds as atropine sulfate and tetracaine hydrochloride are formed by reacting the basic
compounds with acids. Addition of an alkali to a solution of the salt of these compounds precipitates the
free base from solution if the solubility of the base in water is low.
The aliphatic nitrogen of the sulfonamides is sufficiently negative so that these drugs act as slightly
soluble weak acids rather than as bases. They form water-soluble salts in alkaline solution by the
following mechanism. The oxygens of the sulfonyl (—SO2—) group withdraw electrons, and the resulting
electron deficiency of the sulfur atom results in the electrons of the N:H bond being held more closely to
the nitrogen atom. The hydrogen therefore is bound less firmly, and, in alkaline solution, the soluble
sulfonamide anion is readily formed.
The sodium salts of the sulfonamides are precipitated from solution by the addition of a strong acid or by
a salt of a strong acid and a weak base such as ephedrine hydrochloride.
P.187
The barbiturates, like the sulfonamides, are weak acids because the electronegative oxygen of each
acidic carbonyl group tends to withdraw electrons and to create a positive carbon atom. The carbon in
turn attracts electrons from the nitrogen group and causes the hydrogen to be held less firmly. Thus, in
sodium hydroxide solution, the hydrogen is readily lost, and the molecule exists as a soluble anion of the
weak acid. Butler et al.9 demonstrated that, in highly alkaline solutions, the second hydrogen ionizes.
The pK1 for phenobarbital is 7.41 and the pK2 is 11.77. Although the barbiturates are soluble in alkalies,
they are precipitated as the free acids when a stronger acid is added and the pH of the solution is
lowered.
Calculating the Solubility of Weak Electrolytes as Influenced by
pH
From what has been said about the effects of acids and bases on solutions of weak electrolytes, it
becomes evident that the solubility of weak electrolytes is strongly influenced by the pH of the solution.
For example, a 1% solution of phenobarbital sodium is soluble at pH values high in the alkaline range.
The soluble ionic form is converted into molecular phenobarbital as the pH is lowered, and below 9.3,
the drug begins to precipitate from solution at room temperature. On the other hand, alkaloidal salts
such as atropine sulfate begin to precipitate as the pH is elevated.
To ensure a clear homogeneous solution and maximum therapeutic effectiveness, the preparations
should be adjusted to an optimum pH. The pH below which the salt of a weak acid, sodium
phenobarbital, for example, begins to precipitate from aqueous solution is readily calculated in the
following manner.
Representing the free acid form of phenobarbital as HP and the soluble ionized form as P , we write the
equilibria in a saturated solution of this slightly soluble weak electrolyte as
Because the concentration of the un-ionized form in solution, HPsol, is essentially constant, the
equilibrium constant for the solution equilibrium, equation (9-1), is
where So is molar or intrinsic solubility. The constant for the acid–base equilibrium, equation (9-2), is
or
where the subscript ―sol‖ has been deleted from [HP]sol because no confusion should result from this
omission.
341
Dr. Murtadha Alshareifi e-Library
The total solubility, S, of phenobarbital consists of the concentration of the undissociated acid, [HP], and
-
that of the conjugate base or ionized form, [P ]:
-
Substituting So for [HP] from equation (9-3) and the expression from equation (9-5) for [P ] yields
When the electrolyte is weak and does not dissociate appreciably, the solubility of the acid in water or
acidic solutions is So = [HP], which, for phenobarbital is approximately 0.005 mole/liter, in other words,
0.12%.
The solubility equation can be written in logarithmic form, beginning with equation (9-7). By
rearrangement, we obtain
and finally
where pHp is the pH below which the drug separates from solution as the undissociated acid.
In pharmaceutical practice, a drug such as phenobarbital is usually added to an aqueous solution in the
soluble salt form. Of the initial quantity of salt, sodium phenobarbital, that can be added to a solution of a
-
certain pH, some of it is converted into the free acid, HP, and some remains in the ionized form, P
[equation (9-6)]. The amount of salt that can be added initially before the solubility [HP] is exceeded is
therefore equal to S. As seen from equation (9-9), pHp depends on the initial molar concentration, S, of
salt added, the molar solubility of the undissociated acid, So, also known as the intrinsic solubility, and
the pKa. Equation (9-9) has been used to determine the pKa of sulfonamides and other
,
drugs.10 11 Solubility and pH data can also be used to obtain the pK1 and pK2 values of dibasic acids as
suggested by Zimmerman12 and Blanchard et al.13
Example 9-2
Phenobarbital
Below what pH will free phenobarbital begin to separate from a solution having an initial
concentration of 1 g of sodium phenobarbital per 100 mL at 25°C? The molar solubility, So, of
phenobarbital is 0.0050 and the pKa is 7.41 at 25°C. The secondary dissociation of
phenobarbital, referred to previously, can ordinarily be disregarded. The molecular weight of
sodium phenobarbital is 254.
The molar concentration of salt initially added is
P.188
An analogous derivation can be carried out to obtain the equation for the solubility of a weak base as a
function of the pH of a solution. The expression is
where S is the concentration of the drug initially added as the salt and So is the molar solubility of the
free base in water. Here pHp is the pH above which the drug begins to precipitate from solution as the
free base.
342
Dr. Murtadha Alshareifi e-Library
The Influence of Solvents on the Solubility of Drugs
Weak electrolytes can behave like strong electrolytes or like nonelectrolytes in solution. When the
solution is of such a pH that the drug is entirely in the ionic form, it behaves as a solution of a strong
electrolyte, and solubility does not constitute a serious problem. However, when the pH is adjusted to a
value at which un-ionized molecules are produced in sufficient concentration to exceed the solubility of
this form, precipitation occurs. In this discussion, we are now interested in the solubility of
nonelectrolytes and the undissociated molecules of weak electrolytes. The solubility of undissociated
phenobarbital in various solvents is discussed here because it has been studied to some extent by
pharmaceutical investigators.
Frequently, a solute is more soluble in a mixture of solvents than in one solvent alone. This
phenomenon is known as cosolvency, and the solvents that, in combination, increase the solubility of
the solute are called cosolvents. Approximately 1 g of phenobarbital is soluble in 1000 mL of water, in
10 mL of alcohol, in 40 mL of chloroform, and in 15 mL of ether at 25°C. The solubility of phenobarbital
in water–alcohol–glycerin mixtures is plotted on a semilogarithm grid in Figure 9-4from the data of
Krause and Cross.14
By drawing lines parallel to the abscissa in Figure 9-4 at a height equivalent to the required
phenobarbital concentration, it is a simple matter to obtain the relative amounts of the various
combinations of alcohol, glycerin, and water needed to achieve solution. For example, at 22% alcohol,
40% glycerin, and the remainder water (38%), 1.5% w/v of phenobarbital is dissolved, as seen by
following the vertical and horizontal lines drawn on Figure 9-4.
Key Concept
Solvents and Weak Electrolytes
The solvent affects the solubility of a weak electrolyte in a buffered solution in two ways: (a)
The addition of alcohol to a buffered aqueous solution of a weak electrolyte increases the
solubility of the un-ionized species by adjusting the polarity of the solvent to a more favorable
value. (b) Because it is less polar than water, alcohol decreases the dissociation of a weak
electrolyte, and the solubility of the drug goes down as the dissociation constant is decreased
(pKa is increased).
343
Dr. Murtadha Alshareifi e-Library
Fig. 9-4. The solubility of phenobarbital in a mixture of water, alcohol, and glycerin
at 25°C. The vertical axis is a logarithmic scale representing the solubility of
phenobarbital in g/100 mL. (From G. M. Krause and J. M. Cross, J. Am. Pharm.
Assoc. Sci. Ed. 40, 137, 1951. With permission.)
Combined Effect of pH and Solvents
Stockton and Johnson15 and Higuchi et al.16 studied the effect of an increase of alcohol concentration
on the dissociation constant of sulfathiazole, and Edmonson and Goyan17 investigated the effect of
alcohol on the solubility of phenobarbital.
Schwartz et al.10 determined the solubility of phenytoin as a function of pH and alcohol concentration in
various buffer systems and calculated the apparent dissociation constant. Kramer and
Flynn18 examined the solubility of hydrochloride salts of organic bases as a function of pH, temperature,
and solvent composition. They described the determination of the pKa of the salt from the solubility
profile at various temperatures and in several solvent systems. Chowhan11measured and calculated the
solubility of the organic carboxylic acid naproxen and its sodium, potassium, calcium, and magnesium
salts. The observed solubilities were in excellent agreement with the pH–solubility profiles based on
equation (9-9).
The results of Edmonson and Goyan17 are shown in Figure 9-5, where one observes that the pKa of
phenobarbital, 7.41, is raised to 7.92 in a hydroalcoholic solution containing
P.189
30% by volume of alcohol. Furthermore, as can be seen in Figure 9-4, the solubility, So, of un-ionized
phenobarbital is increased from 0.12 g/100 mL or 0.005 M in water to 0.64% or 0.0276 M in a 30%
alcoholic solution. The calculation of solubility as a function of pH involving these results is illustrated in
the following example.
Fig. 9-5. The influence of alcohol concentration on the dissociation constant of
phenobarbital. (From T. D. Edmonson and J. E. Goyan, J. Am. Pharm. Assoc. Sci.
Ed. 47, 810, 1958. With permission.)
Example 9-3
Minimum pH for Complete Solubility
344
Dr. Murtadha Alshareifi e-Library
What is the minimum pH required for the complete solubility of the drug in a stock solution
containing 6 g of phenobarbital sodium in 100 mL of a 30% by volume alcoholic solution?
From equation (9-9),
For comparison, the minimum pH for complete solubility of phenobarbital in an aqueous
solution containing no alcohol is computed using equation (9-9):
From the calculations of Example 9-3, it is seen that although the addition of alcohol increases the pKa,
it also increases the solubility of the un-ionized form of the drug over that found in water sufficiently so
that the pH can be reduced somewhat before precipitation occurs.
Equations (9-9) and (9-10) can be made more exact if activities are used instead of concentrations to
account for interionic attraction effects. This refinement, however, is seldom required for practical work,
where the values calculated from the approximate equations just given serve as satisfactory estimates.
Influence of Complexation in Multicomponent Systems
Many liquid pharmaceutical preparations consist of more than a single drug in solution. Fritz et
al.19 showed that when several drugs together with pharmaceutical adjuncts interact in solution to form
insoluble complexes, simple solubility profiles of individual drugs cannot be used to predict solubilities in
mixtures of ingredients. Instead, the specific multicomponent systems must be studied to estimate the
complicating effects of species interactions.
Influence of Other Factors on the Solubility of Solids
The size and shape of small particles (those in the micrometer range) also affect solubility. Solubility
increases with decreasing particle size according to the approximate equation
where s is the solubility of the fine particles; so is the solubility of the solid consisting of relatively large
particles; γ is the surface tension of the particles, which, for solids, unfortunately, is extremely difficult to
3
obtain; V is the molar volume (volume in cm per mole of particles); r is the final radius of the particles in
7
cm; R is the gas constant (8.314 × 10 ergs/deg mole); and T is the absolute temperature. The equation
can be used for solid or liquid particles such as those in suspensions or emulsions. The following
example is taken from the book by Hildebrand and Scott.20
Example 9-4
Particle Size and Solubility
A solid is to be comminuted so as to increase its solubility by 10%, that is, s/so is to become
1.10. What must be the final particle size, assuming that the surface tension of the solid is
3
100 dynes/cm and the volume per mole is 50 cm ? The temperature is 27°C.
The configuration of a molecule and the type of arrangement in the crystal also has some influence on
solubility, and a symmetric particle can be less soluble than an unsymmetric one. This is because
solubility depends in part on the work required to separate the particles of the crystalline solute. The
molecules of the amino acid α-alanine form a compact crystal with high lattice energy and consequently
low solubility. The molecules of α-amino-n-butyric acid pack less efficiently in the crystal, partly because
of the projecting side chains, and the crystal energy is reduced. Consequently, α-amino-n-butyric acid
has a solubility of 1.80 moles/liter and α-alanine has a solubility of only 1.66 moles/liter in water at 25°C,
although the hydrocarbon chain is longer in α-amino-n-butyric acid than in the other compound.
P.190
345
Dr. Murtadha Alshareifi e-Library
Key Concept
Poor Aqueous Solubility
―Poor aqueous solubility is caused by two main factors: high lipophilicity and strong
intermolecular interactions, which make the solubilization of the solid energetically costly.
What is meant by good and poorly soluble depends partly on the expected therapeutic dose
and potency of the drug. As a rule of thumb from the delivery perspective, a drug with an
average potency of 1 mg/kg should have a solubility of at least 0.1 g/L to be adequately
soluble. If a drug with the same potency has a solubility of less than 0.01 g/L it can be
considered poorly soluble.‖3
Determining Thermodynamic and “Kinetic” Solubility
The Phase Rule and Solubility
Solubility can be described in a concise manner by the use of the Gibbs phase rule, which is described
using
where F is the number of degrees of freedom, that is, the number of independent variables (usually
temperature, pressure, and concentration) that must be fixed to completely determine the system, C is
the smallest number of components that are adequate to describe the chemical composition of each
phase, and P is the number of phases.
The Phase Rule can be used to determine the thermodynamic solubility of a drug substance. This
method is based on the thermodynamic principles of heterogeneous equilibria that are among the
soundest theoretical concepts in chemistry. It does not depend on any assumptions regarding kinetics or
the structure of matter but is applicable to all drugs. The requirements for an analysis are simple, as the
equipment needed is basic to most laboratories and the quantities of substances are small. Basically,
drug is added in a specific amount of solvent. After equilibrium is achieved, excess drug is removed
(usually by filtering) and then the concentration of the dissolved drug is measured using standard
analysis techniques such as high-performance liquid chromatography.
A phase-solubility diagram for a pure drug substance is shown in Figure 9-6.21 At concentrations below
the saturation concentration there is only one degree of freedom since the studies are performed at
constant temperature and pressure. In other words, only the concentration changes. This is represented
in Figure 9-6 by the segment A–B of the line. Once the saturation concentration is reached, the addition
of more drug to the ―system‖ does not result in higher solution concentrations (segment B–C). Rather,
the drug remains in the solid state and the system becomes a two-phase system. Since the
temperature, pressure, and solution concentration are constant at drug concentrations above the
saturation concentration, the system has zero degree of freedom.
346
Dr. Murtadha Alshareifi e-Library
Fig. 9-6. Phase-solubility diagram for a pure drug substance. The line segment A–B
represents one phase since the concentration of drug substance is below the saturation
concentration. Line segment B–C represents a pure solid in a saturated solution at
equilibrium. (From Remington, The Science and Practice of Pharmacy, 21st Ed.,
Lippincott Williams & Wilkins, 2006, p. 216. With permission.)
The situation in Figure 9-6 is valid only for pure drug substances. What if the drug substance is not
pure? This situation is described in Figure 9-7.22 If the system has one impurity, the solution becomes
saturated with the first component at point B. The situation becomes interesting at this point. In segment
B–C of the line, the solution is saturated with component 1 (which is usually the major component such
as the drug), so the drug would precipitate out of solution at concentrations greater than this. However,
the impurity (the minor component or component 2) does not reach saturation
P.191
until it reaches point C on the line. The concentrations of the two components are saturated beyond
point C (segment C–D) of the line. Once true equilibrium is achieved, one can extrapolate back to
the Y axis (solution concentration) to determine the solubility of the two components. Therefore, the
thermodynamic solubility of the drug would be equal to the distance A–E and the solubility of the
impurity would be equal to the distance represented by E–F. As one can see, this procedure can be
used to measure the exact solubility of the pure drug without having a pure form of the drug to start with.
347
Dr. Murtadha Alshareifi e-Library
Fig. 9-7. Phase-solubility curve when the drug substance contains one impurity. At
point B, the solution becomes saturated with component 1 (the drug). The segment B–
C represents two phases—a solution phase saturated with the drug and some of the
impurity and a solid phase of the drug. Segment C–D represents two phases—a liquid
phase saturated with the drug and impurity and a solid phase containing the drug and
the impurity. (From Remington, The Science and Practice of Pharmacy, 21st Ed.,
Lippincott Williams & Wilkins, 2006, p. 217. With permission.)
The practical aspect of measuring thermodynamic solubility is, on the surface, relatively simple but it can
be quite time-consuming.4 Some methods have been developed in attempt to reduce the time that it
takes to get a result. Starting the experiment with a high purity crystalline form of the substance will give
the best chance that the solubility measured after a reasonable incubation period will represent the true
equilibrium solubility. However, this may still take several hours to several days. Also, there is still a risk
that the incubation period will not be sufficient for metastable crystal forms to convert to the most stable
form. This means that the measured concentration may represent the apparent solubility of a different
crystal form. This risk must be taken into consideration when running a solubility experiment with
material that is not known to be the most stable crystalline form.4
Bhattachar and colleagues4 recently reviewed various aspects of solubility and they are summarized
here. In practice, the stable crystalline form of the compound is not available in sufficient purity during
early discovery and so the labor-intensive measurement of thermodynamic solubility is not commonly
made. The amount of compound required to measure a thermodynamic solubility measurement
depends on the volume of solvent used to make the saturated solution and the solubility of the
compound in that solvent. Recent reports for miniaturized systems list compound requirements ranging
from ~100 mg per measurement for poorly soluble compounds23 to 3 to 10 mg for pharmaceutically
relevant compounds.24 Although early-stage solubility information is crucial to drug discovery teams,
the number of compounds being assessed, the scarcity of compound, and questionable purity and
crystallinity make it nearly impossible to assess thermodynamic solubility.
These challenges have been partially met using a high-throughput kinetic measurement of antisolvent
,
,
,
precipitation commonly referred to ―kinetic solubility.‖25 26 27 28 ―Kinetic solubility is a misnomer, not
because it is not kinetic, but because it measures a precipitation rate rather than solubility. Kinetic
solubility methods are designed to facilitate high throughput measurements, using submilligram
quantities of compound, in a manner that closely mimics the actual solubilization process used in
biological laboratories. Typically, the compound is dissolved in dimethyl sulfoxide (because it is a strong
organic solvent) to make a stock solution of known concentration. This stock is added gradually to the
348
Dr. Murtadha Alshareifi e-Library
aqueous solvent of interest until the anti-solvent properties of the water drive the compound out of
solution. The resulting precipitation is detected optically, and the kinetic solubility is defined as the point
at which the aqueous component can no longer solvate the drug. Solubility results obtained from kinetic
measurements might not match the thermodynamic solubility results perfectly; therefore, caution must
be exercised such that the data from the kinetic solubility measurements are used only for their intended
application. Since kinetic solubility is determined for compounds that have not been purified to a high
degree or crystallized, the impurities and amorphous content in the material lead to a higher solubility
than the thermodynamic solubility. Because kinetic solubility experiments begin with the drug in solution,
there is a significant risk of achieving supersaturation of the aqueous solvent through precipitation of an
amorphous or metastable crystalline form. This supersaturation can lead to a measured value that is
significantly higher than the thermodynamic solubility, masking a solubility problem that will become
apparent as soon as the compound is crystallized. Owing to the nature of kinetic solubility
measurements, there is no time for equilibration of the compound in the aqueous solvent of
measurement. Because the compounds tested are in dimethyl sulfoxide solutions, the energy required
to break the crystal lattice is not factored into the solubility measurements.‖4
Key Concept
Effect of pH on Solubility
Solubility must always be considered in the context of pH and pKa. The relationship between
pH and solubility is shown in Figure 9-8. If the measured solubility falls on the steep portion of
the pH–solubility profile, small changes to the pH can have a marked effect on the solubility.4
Fig. 9-8. pH–solubility profile for a compound with a single, basic pKa value of 5.
The four regions of pH-dependent solubility are the salt plateau, pHmax, ionized
compound, and un-ionized compound. (Adapted from Bhattachar et al. 2006,4 with
permission.)
P.192
Some Limitations of Thermodynamic Solubility3
In a recent review, Faller and Ertl3 have discussed some of the limitations of traditional methods for
determining solubility. For example, if the traditional shake-flask method is used, adsorption to the vial or
to the filter, incomplete phase separation, compound instability, and slow dissolution can affect the
result. When the potentiometric method is used, inaccurate pKa determination, compound degradation
349
Dr. Murtadha Alshareifi e-Library
during the titration, slow dissolution, or incorrect data analysis can affect the data quality. It is very
important to define the experimental conditions well. The intrinsic solubility, So, needs to be
distinguished from the solubility measured at a given pH value in a defined medium. Intrinsic solubility
refers to the solubility of the unionized species. Artursson et al.29 has shown that this parameter is
relatively independent of the nature of the medium used. In contrast, solubility measured at a fixed pH
value may be highly dependent on the nature and concentration of the counter ions present in the
medium.30 This is especially critical for poorly soluble compounds that are strongly ionized at the pH of
the measurement. Finally, it is important to note that single pH measurements cannot distinguish
between soluble monomers and soluble aggregates of drug molecules, which may range from dimers to
micelles unless more sophisticated experiments are performed.30
Computational Approaches
In addition to measuring solubility, computational approaches are widely used and were reviewed
recently by Faller and Ertl.3 Briefly, fragment-based models attempt to predict solubility as a sum of
substructure contributions—such as contributions of atoms, bonds, or larger substructures. This
approach is based on a general assumption that molecule properties are determined completely by
molecular structure and may be approximated by the contributions of fragments in the molecule. The
inverse relation between solubility and lipophilicity has also been recognized for a long time and
empirical relationships between log So and log Phave been reported. Finally, numerous other
approaches for predicting water solubility have been reported. The array of possible molecular
descriptors that can be used is nearly unlimited. The polar surface area, which characterizes molecule
polarity and hydrogen bonding features, is one of the most useful descriptors. Polar surface area,
defined as a sum of surfaces of polar atoms, is conceptually easy to understand and seems to encode
in an optimal way a combination of hydrogen-bonding features and molecular polarity.
Distribution of Solutes between Immiscible Solvents
If an excess of liquid or solid is added to a mixture of two immiscible liquids, it will distribute itself
between the two phases so that each becomes saturated. If the substance is added to the immiscible
solvents in an amount insufficient to saturate the solutions, it will still become distributed between the
two layers in a definite concentration ratio.
Key Concept
Hydrophobic Parameters
Meyer in 189931 and Overton in 190132 showed that the pharmacologic effect of simple
organic compounds was related to their oil/water partition coefficient, P. It later became clear
that the partition coefficient was of little value for rationalizing specific drug activity (i.e.,
binding to a receptor) because specificity also relates to steric and electronic effects.
However, in the early 1950s, Collander33 showed that the rate of penetration of plant cell
membranes by organic compounds was related to P. The partition coefficient, P, is a
commonly used way of defining relative hydrophobicity (also known as lipophilicity) of
compounds. For more about partition coefficients, see the text by Hansch and Leo.34
If C1 and C2 are the equilibrium concentrations of the substance in Solvent 1 and Solvent2, respectively,
the equilibrium expression becomes
The equilibrium constant, K, is known as the distribution ratio, distribution coefficient, or partition
coefficient. Equation (9-13), which is known as the distribution law, is strictly applicable only in dilute
solutions where activity coefficients can be neglected.
Example 9-5
Distribution Coefficient
When boric acid is distributed between water and amyl alcohol at 25°C, the concentration in
water is found to be 0.0510 mole/liter and in amyl alcohol it is found to be 0.0155 mole/liter.
What is the distribution coefficient? We have
350
Dr. Murtadha Alshareifi e-Library
No convention has been established with regard to whether the concentration in the water
phase or that in the organic phase should be placed in the numerator. Therefore, the result
can also be expressed as
One should always specify, which of these two ways the distribution constant is being
expressed.
Knowledge of partition is important to the pharmacist because the principle is involved in several areas
of current pharmaceutical interest. These include preservation of oil–water systems, drug action at
nonspecific sites, and the absorption and distribution of drugs throughout the body. Certain aspects of
these topics are discussed in the following sections.
P.193
Fig. 9-9. Schematic representation of the distribution of benzoic acid between water
and an oil phase. The oil phase is depicted as a magnified oil droplet in an oil-in-water
emulsion.
Effect of Ionic Dissociation and Molecular Association on
Partition
The solute can exist partly or wholly as associated molecules in one of the phases or it may dissociate
into ions in either of the liquid phases. The distribution law applies only to the concentration of the
species common to both phases, namely, the monomer or simple molecules of the solute.
Consider the distribution of benzoic acid between an oil phase and a water phase. When it is neither
associated in the oil nor dissociated into ions in the water, equation (9-13) can be used to compute the
351
Dr. Murtadha Alshareifi e-Library
distribution constant. When association and dissociation occur, however, the situation becomes more
complicated. The general case where benzoic acid associates in the oil phase and dissociates in the
aqueous phase is shown schematically in Figure 9-9.
Two cases will be treated. First, according to Garrett and Woods,35 benzoic acid is considered to be
distributed between the two phases, peanut oil and water. Although benzoic acid undergoes
dimerization (association to form two molecules) in many nonpolar solvents, it does not associate in
peanut oil. It ionizes in water to a degree, however, depending on the pH of the solution. Therefore,
in Figure 9-9 for the case under consideration, Co, the total concentration of benzoic acid in the oil
phase, is equal to [HA]o, the monomer concentration in the oil phase, because association does not
occur in peanut oil.
The species common to both the oil and water phases are the unassociated and undissociated benzoic
acid molecules. The distribution is expressed as
where K is the true distribution coefficient, [HA]o = Co is the molar concentration of the simple benzoic
acid molecules in the oil phase, and [HA]w is the molar concentration of the undissociated acid in the
water phase.
The total acid concentration obtained by analysis of the aqueous phase is
and the experimentally observed or apparent distribution coefficient is
As seen in Figure 9-9, the observed distribution coefficient depends on two equilibria: the distribution of
the undissociated acid between the immiscible phases as expressed in equation (9-14) and the species
+
distribution of the acid in the aqueous phase, which depends on the hydrogen ion concentration [H 3O ]
and the dissociation constant Ka of the acid, where
Association of benzoic acid in peanut oil does not occur, and Kd (the equilibrium constant for
dissociation of associated benzoic acid into monomer in the oil phase) can be neglected in this case.
Given these equations and the fact that the concentration, C, of the acid in the aqueous phase before
distribution, assuming equal volumes of the two phases, is*
one arrives at the combined result:†
Expression (9-19) is a linear equation of the form y = a + bx, and therefore a plot of (Ka +
+
+
[H3O ])/Cw against [H3O ]
P.194
yields a straight line with a slope b = (K + 1)/C and an intercept a = Ka/C. The true distribution
coefficient, K, can thus be obtained over the range of hydrogen ion concentration considered.
Alternatively, the true distribution constant could be obtained according to equation (9-14) by analysis of
the oil phase and of the water phase at a sufficiently low pH (2.0) at which the acid would exist
completely in the un-ionized form. One of the advantages of equation (9-19), however, is that the oil
phase need not be analyzed; only the hydrogen ion concentration and Cw, the total concentration
remaining in the aqueous phase at equilibrium, need be determined.
Example 9-6
+
+
According to Garrett and Woods,35 the plot of (Ka + [H3O ])/Cw against [H3O ] for benzoic
acid distributed between equal volumes of peanut oil and a buffered aqueous solution yields a
-5
-5
slope b = 4.16 and an intercept a = 4.22 × 10 . The Ka of benzoic acid is 6.4 × 10 . Compute
352
Dr. Murtadha Alshareifi e-Library
the true partition coefficient, K, and compare it with the value K = 5.33 obtained by the
authors. We have
or
Because
the expression becomes
and
Second, let us now consider the case in which the solute is associated in the organic phase and exists
as simple molecules in the aqueous phase. If benzoic acid is distributed between benzene and acidified
water, it exists mainly as associated molecules in the benzene layer and as undissociated molecules in
the aqueous layer.
The equilibrium between simple molecules HA and associated molecules (HA) n in
and the equilibrium constant expressing the dissociation of associated molecules into simple molecules
in this solvent is
or
Because benzoic acid exists predominantly in the form of double molecules in benzene, Co can replace
[(HA)2], where Co is the total molar concentration of the solute in the organic layer. Then equation (921) can be written approximately as
In conformity with the distribution law as given in equation (9-14), the true distribution coefficient is
always expressed in terms of simple species common to both phases, that is, in terms of [HA] w and
[HA]o. In the benzene–water system, [HA]o is given by equation (9-22), and the modified distribution
constant becomes
The results for the distribution of benzoic acid between benzene and water, as given by
Glasstone,36 are given in Table 9-3.
Extraction
To determine the efficiency with which one solvent can extract a compound from a second solvent—an
operation commonly employed in analytic chemistry and in organic chemistry—we follow
Glasstone.37 Suppose that w grams of a solute is extracted repeatedly from V1 mL of one solvent with
successive portions of V2 mL of a second solvent, which is immiscible with the first. Let w1 be the weight
of the solute remaining in the original solvent after extracting with the first portion of the other solvent.
Then, the concentration
P.195
of solute remaining in the first solvent is (w1/V1) g/mL and the concentration of the solute in the
extracting solvent is (w - w1)/V2 g/mL. The distribution coefficient is thus
Table 9-3 Distribution of Benzoic Acid between Benzene and Acidified Water at
353
Dr. Murtadha Alshareifi e-Library
6°C*,†
[HA]w
K″ - √Co/HAw
Co
0.00329
0.0156
38.0
0.00579
0.0495
38.2
0.00749
0.0835
38.6
0.0114
0.195
38.8
*The concentrations are expressed in mole/liter.
†
From S. Glasstone, Textbook of Physical Chemistry, Van Nostrand, New
York, 1946, p. 738.
or
The process can be repeated, and after n extractions,37
By use of this equation, it can be shown that most efficient extraction results when n is large and V2 is
small, in other words, when a large number of extractions are carried out with small portions of
extracting liquid. The development just described assumes complete immiscibility of the two liquids.
When ether is used to extract organic compounds from water, this is not true; however, the equations
provide approximate values that are satisfactory for practical purposes. The presence of other solutes,
such as salts, can also affect the results by complexing with the solute or by salting out one of the
phases.
Example 9-7
Distribution Coefficient
The distribution coefficient for iodine between water and carbon tetrachloride at 25°C
is K=CH2O/CCCl4 = 0.012. How many grams of iodine are extracted from a solution in water
containing 0.1 g in 50 mL by one extraction with 10 mL of CCl4? How many grams are
extracted by two 5-mL portions of CCl4? We have
Thus, 0.0011 g of iodine remains in the water phase, and the two portions of CCl4 have
extracted 0.0989 g.
354
Dr. Murtadha Alshareifi e-Library
Solubility and Partition Coefficients
Hansch et al.38 observed a relationship between aqueous solubilities of nonelectrolytes and partitioning.
Yalkowsky and Valvani39 obtained an equation for determining the aqueous solubility of liquid or
crystalline organic compounds:
where S is aqueous solubility in moles/liter, K is the octanol–water partition coefficient, ΔSf is the molar
entropy of fusion, and mp is the melting point of a solid compound on the centigrade scale. For a liquid
compound, mp is assigned a value of 25 so that the second right-hand term of equation (9-27) becomes
zero.
The entropy of fusion and the partition coefficient can be estimated from the chemical structure of the
compound. For rigid molecules, ΔSf = 13.5 entropy units (eu). For molecules with n greater than five
nonhydrogen atoms in a flexible chain,
Leo et al.38 provided partition coefficients for a large number of compounds. When experimental values
,
are not available, group contribution methods38 40 are available for estimating partition coefficients.
Example 9-8
Molar Aqueous Solubility
Estimate the molar aqueous solubility of heptyl p-aminobenzoate, mp 75°C, at 25°C:
It is first necessary to calculate ΔSf and log K.
There are nine nonhydrogens in the flexible chain (C, O, and the seven carbons of CH 3).
Using equation (9-28), we obtain
For the partition coefficient, Leo et al.38 give for log K of benzoic acid a value of 1.87, the
contribution of NH2 is -1.16, and that of CH2 is 0.50, or 7 × 0.50 = 3.50 for the seven carbon
atoms of CH3 in the chain:
We substitute these values into equation (9-27) to obtain
The oil–water partition coefficient is an indication of the lipophilic or hydrophobic character of a drug
molecule. Passage of drugs through lipid membranes and interaction with macromolecules at receptor
sites sometimes correlate well with the octanol–water partition coefficient of the drug. In the last few
sections, the student has been introduced to the distribution of drug molecules between immiscible
solvents together with some important applications of partitioning; a number of useful references are
,
,
,
,
,
available for further study on the subject.41 42 43 44 Three excellent books45 46 47 on solubility in the
pharmaceutical sciences will be of interest to the serious student of the subject.
Chapter Summary
The concept of solubility was presented in this chapter. As described, solubility is defined in
quantitative terms as the concentration of solute in a saturated solution at a certain
temperature, and in a qualitative way, it can be defined as the spontaneous interaction of two
or more substances to form a
P.196
homogeneous molecular dispersion. Solubility is an intrinsic material property that can be
altered only by chemical modification of the molecule. Solubilization was not covered in this
chapter. In order to determine the true solubility of a compound, one must measure the
355
Dr. Murtadha Alshareifi e-Library
thermodynamic solubility. However, given the constraints that were discussed an alternate
method, kinetic solubility determination, was presented that offers a more practical alternative
given the realities of the situation. Distribution phenomena were also discussed in some
detail. The distribution behavior of drug molecules is important to many pharmaceutical
processes including physicochemical (e.g., when formulating drug substances) and biological
(e.g., absorption across a biological membrane) processes.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. S. Stegemann, F. Leveiller, D. Franchi, H. de Jong, and H. Linden, Eur. J. Pharm. Sci. 31, 249, 2007.
2. J. Bauer, S. Spanton, R. Henry, J. Quick, W. Dziki, and W. Porter, J. Morris, Pharm. Res. 18, 859,
2001.
3. B. Faller and P. Ertl, Adv. Drug Deliv. Rev. 59, 535, 2007.
4. S. N. Bhattachar, L. A. Deschenes, and J. A. Wesley, Drug Discov. Today 11, 1012, 2006.
5. M. R. Loran and E. P. Guth, J. Am. Pharm. Assoc. Sci. Ed. 40, 465, 1951.
6. W. J. O'Malley, L. Pennati, and A. Martin, J. Am. Pharm. Assoc. Sci. Ed. 47, 334, 1958.
7. R. J. James and R. L. Goldemberg, J. Soc. Cosm. Chem. 11, 461, 1960.
8. W. G. Gorman and G. D. Hall, J. Pharm. Sci. 53, 1017, 1964.
9. T. C. Butler, J. M. Ruth, and G. F. Tucker, J. Am. Chem. Soc. 77, 1486, 1955.
10. P. A. Schwartz, C. T. Rhodes, and J. W. Cooper, Jr., J. Pharm. Sci. 66, 994, 1977.
11. Z. T. Chowhan, J. Pharm. Sci. 67, 1257, 1978.
12. I. Zimmerman, Int. J. Pharm. 31, 69, 1986.
13. J. Blanchard, J. O. Boyle, and S. Van Wagenen, J. Pharm. Sci. 77, 548, 1988.
14. G. M. Krause and J. M. Cross, J. Am. Pharm. Assoc. Sci. Ed. 40, 137, 1951.
15. J. R. Stockton and C. R. Johnson, J. Am. Chem. Soc. 33, 383, 1944.
16. T. Higuchi, M. Gupta, and L. W. Busse, J. Am. Pharm. Assoc. Sci. Ed. 42, 157, 1953.
17. T. D. Edmonson and J. E. Goyan, J. Am. Pharm. Assoc. Sci. Ed. 47, 810, 1958.
18. S. F. Kramer and G. L. Flynn, J. Pharm. Sci. 61, 1896, 1972.
19. B. Fritz, J. L. Lack, and L. D. Bighley, J. Pharm. Sci. 60, 1617, 1971.
20. J. H. Hildebrand and R. L. Scott, Solubility of Nonelectrolytes, Dover, New York, 1964, p. 417.
21. Remington, The Science and Practice of Pharmacy, 21st Ed., Lippincott Williams & Wilkins,
Baltimore, MD, 2006, p. 216.
22. Remington, The Science and Practice of Pharmacy, 21st Ed., Lippincott Williams & Wilkins,
Baltimore, MD, 2006, p. 217.
23. A. Glomme, J. März, and J. B. Dressman, J. Pharm. Sci. 94, 1, 2005.
24. S. N. Bhattachar, J. A. Wesley, and C. Seadeek, J. Pharm. Biomed. Anal. 41, 152, 2006.
25. A. Avdeef, Absorption and Drug Development Solubility, Permeability and Charge State, John Wiley
& Sons, Hoboken, NJ, 2003.
26. K. A. Dehring, J. Pharm. Biomed. Anal. 36, 447, 2004.
27. T. M. Chen, H. Shen, and C. Zhu, Comb. Chem. High Throughput Screen, 5, 575, 2002.
28. C. A. Lipinski, F. Lombardo, B. W. Dominy, and P. J. Feeney, Adv. Drug Del. Rev. 46, 3, 2001.
29. C. A. S. Bergstrom, K. Luthman, and P. Artursson, Eur. J. Pharm. Sci. 22, 387, 2004.
30. A. Avdeef, D. Voloboy, and A. Foreman, Dissolution and Solubility, in J. B. Taylor and D. J. Triggle
(Eds.), Comprehensive Medicinal Chemistry II, Vol. 5, Elsevier, Oxford, UK, ISBN: 978–0-08–044513-7,
2007, pp. 399–423.
31. H. Meyer, Naunyn Schmiedebergs Arch. Pharmacol. 42, 109, 1899.
32. E. Overton, Studien über die Narkose, Fischer, Jena, Germany, 1901.
33. R. Collander, Annu. Rev. Plant Physiol. 8, 335, 1957.
34. C. Hansch and A. Leo, Substituent Constants for Correlation Analysis in Chemistry and Biology,
Wiley, New York, 1979.
35. E. R. Garrett and O. R. Woods, J. Am. Pharm. Assoc. Sci. Ed. 42, 736, 1953.
356
Dr. Murtadha Alshareifi e-Library
36. S. Glasstone, Textbook of Physical Chemistry, Van Nostrand, New York, 1946, p. 738.
37. S. Glasstone, Textbook of Physical Chemistry, Van Nostrand, New York, 1946, pp. 741–742.
38. C. Hansch, J. E. Quinlan, and G. L. Lawrence, J. Org. Chem. 33, 347, 1968; A. J. Leo, C. Hansch,
and D. Elkin, Chem. Rev. 71, 525, 1971; C. Hansch and A. J. Leo, Substituent Constants for Correlation
Analysis in Chemistry and Biology, Wiley, New York, 1979.
39. S. H. Yalkowsky and S. C. Valvani, J. Pharm. Sci. 69, 912, 1980; G. Amidon, Int. J.
Pharmaceutics, 11, 249, 1982.
40. R. F. Rekker, The Hydrophobic Fragmental Constant, Elsevier, New York, 1977; G. G. Nys and R.
F. Rekker, Eur. J. Med. Chem. 9, 361, 1974.
41. C. Hansch and W. J. Dunn III, J. Pharm. Sci. 61, 1, 1972; C. Hansch and J. M. Clayton, J. Pharm.
Sci. 62, 1, 1973; R. N. Smith, C. Hansch, and M. M. Ames, J. Pharm. Sci. 64, 599, 1975.
42. K. C. Yeh and W. I. Higuchi, J. Pharm. Soc. 65, 80, 1976.
43. R. D. Schoenwald and R. L. Ward, J. Pharm. Sci. 67, 787, 1978.
44. W. J. Dunn III, J. H. Block, and R. S. Pearlman (Eds.), Partition Coefficient, Determination and
Estimation, Pergamon Press, New York, 1986.
45. K. C. James, Solubility and Related Properties, Marcel Dekker, New York, 1986.
46. D. J. W. Grant and T. Higuchi, Solubility Behavior of Organic Compounds, Wiley, New York, 1990.
47. S. H. Yalkowsky and S. Banerjee, Aqueous Solubility, Marcel Dekker, New York, 1992.
Recommended Readings
S. N. Bhattachar, L. A. Deschenes, and J. A. Wesley, Drug Discov. Today 11, 1012–1018, 2006.
B. Faller and P. Ertl, Adv. Drug Del. Rev. 59, 533–545, 2007.
C. A. Lipinski, F. Lombardo, B. W. Dominy, and P. J. Feeney, Adv. Drug Del. Rev. 46, 3–26, 2001.
S. H. Yalkowski, Solubility and Solubilization in Aqueous Media, American Chemical Society,
Washington DC, 1999.
*The meaning of C in equation (9-18) is understood readily by considering a simple illustration. Suppose
one begins with 1 liter of oil and 1 liter of water, and after benzoic acid has been distributed between the
two phases, the concentration Co of benzoic acid in the oil is 0.01 mole/liter and the concentration Cw of
benzoic acid in the aqueous phase is 0.01 mole/liter. Accordingly, there is 0.02 mole/2 liter or 0.01 mole
of benzoic acid per liter of total mixture after distribution equilibrium has been attained. Equation (918) gives
The concentration, C, obviously is not the total concentration of the acid in the mixture at equilibrium but,
rather, twice this value. C is therefore seen to be the concentration of benzoic acid in the water phase
(or the oil phase) before the distribution is carried out.
†Equation (9-19) is obtained as follows. Substituting for [A ]w from equation (9-17) into equation (916) gives
Then [HA]w from equation (9-14) is substituted into (a) to eliminate [HA]o from the equation:
The apparent distribution constant is eliminated by substituting equation (b) into equation (9-16) to give
or
Co is eliminated by substituting equation (c) into equation (9-18):
357
Dr. Murtadha Alshareifi e-Library
Rearranging equation (d) gives the final result:
Chapter Legacy
Fifth Edition: published as Chapter 10 (Solubility and Distribution Phenomena). Updated by
Patrick Sinko.
Sixth Edition: published as Chapter 8 (Solubility and Distribution Phenomena). Updated by
Patrick Sinko.
358
Dr. Murtadha Alshareifi e-Library
10
0 Compllexation
n and P
Protein Binding
g
Cha
apter Objec
ctives
At th
he conclusion of this chapter the s
student sho
ould be able to:
1. Define th
he three clas
sses of comp
plexes (coord
dination compounds) andd identify
pharmacceutically relevant examp
ples.
2. Describe
e chelates, th
heir physicallly properties
s, and what differentiates
d
s them from
organic molecular co
omplexes.
3. Describe
e the types of
o forces thatt hold togethe
er organic molecular com
mplexes and give
example
es.
4. Describe
e the forces involved in p
polymer–drug
g complexes used for druug delivery and
situation
ns where reversible or irre
eversible com
mplexes may
y be advantaageous.
5. Discuss the uses and give exam
mples of cyclo
odextrins in pharmaceutic
p
cal applicatio
ons.
6. Determine the stoich
hiometric ratiio and stability constant for
f complex fformation.
7. Describe
e the method
ds of analysiss of complex
xes and theirr strengths annd weakness
ses.
8. Discuss the ways tha
at protein bin
nding can inffluence drug action.
9. Describe
e the equilibrrium dialysis and ultrafiltrration methods for determ
mining protein
binding.
10. Understand the facto
ors affecting complexatio
on and protein binding.
11. Understand the thermodynamic b
basis for the stability of complexes.
c
Com
mplexes or coo
ordination com
mpounds, acco
ording to the classic definitio
on, result from
m a donor–acce
eptor
mech
hanism or Lew
wis acid–base
e reaction betw
ween two or more
m
different chemical
c
consstituents. Any
nonm
metallic atom or
o ion, whethe
er free or conta
ained in a neu
utral molecule or in an ionic compound, th
hat
can d
donate an electron pair can
n serve as the donor. The ac
cceptor, or con
nstituent that aaccepts a sha
are in
the p
pair of electron
ns, is frequenttly a metallic io
on, although itt can be a neu
utral atom. Coomplexes can be
divid
ded broadly intto two classes
s depending o n whether the
e acceptor com
mponent is a m
metal ion or an
n
orga
anic molecule; these are clas
ssified accord
ding to one pos
ssible arrange
ement in Tablee 10-1. A third
classs, the inclusion
n/occlusion co
ompounds, invvolving the enttrapment of on
ne compound in the molecu
ular
framework of anotther, is also included in the ttable.
Interrmolecular forcces involved in
n the formatio
on of complexe
es are the van der Waals foorces of disperrsion,
dipollar, and induced dipolar types. Hydrogen bonding prov
vides a signific
cant force in soome molecula
ar
comp
plexes, and co
oordinate cova
alence is impo
ortant in metall complexes. Charge
C
transfeer and
hydrrophobic intera
action are intro
oduced later in
n the chapter.
Me
etal Comp
plexes
A sa
atisfactory understanding of metal ion com
mplexation is based
b
upon a familiarity withh atomic struc
cture
and molecular forcces, and the re
eader would d
do well to cons
sult texts on in
norganic and oorganic chemistry
to study those secctions dealing with electronicc structure and hybridization
n before proceeeding.
Ino
organic Complexe
C
es
The ammonia mollecules in hexa
amminecobaltt (III) chloride, as the compo
ound [Co(NH33)6]3+ Cl3- is called,
are kknown as the ligandsand
l
are said to be ccoordinated to the cobalt ion. The coordinaation number of
the ccobalt ion, or number
n
of ammonia groupss coordinated to
t the metal io
ons, is six. Othher complex io
ons
+
43+
belon
nging to the in
norganic group
p include [Ag( NH3)2] , [Fe(C
CN)6] , and [Cr(H2O)6] .
Each
h ligand donattes a pair of ellectrons to forrm a coordinatte covalent link
k between itseelf and the cen
ntral
ion h
having an inco
omplete electro
on shell. For e
example,
Hybrridization plays an important part in coord
dination compo
ounds in which sufficient boonding orbitals
s are
not o
ordinarily availlable in the me
etal ion. The rreader's underrstanding of hy
ybridization wi
will be refreshed by
a brief review of th
he argument advanced
a
for t he quadrivalence of carbon. It will be recaalled that the
ground-state confiiguration of ca
arbon is
359
Dr. Murtadha Alshareifi e-Library
This cannot be the
e bonding configuration of ccarbon, howev
ver, because itt normally hass four rather th
han
two vvalence electrrons. Pauling1 suggested th
he possibility of
o hybridization
n to account foor the
quad
drivalence. Acccording to this
s mixing proce
ess, one of the
e 2s electrons is promoted tto
P.19
98
the a
available 2p orrbital to yield four
f
equivalen
nt bonding orbitals:
Tab
ble 10-1 Cllassification
n of Compllexes*
I.
II.
III.
Metal ion co
M
omplexes
A.
Inorg
ganic type
B.
Chellates
C.
Oleffin type
D.
Arom
matic type
1. Pi (π) coomplexes
2. Sigma (σ
σ) complexes
3. “Sandwiich” compo
ounds
O
Organic
mollecular com
mplexes
A.
Quin
nhydrone tyype
B.
Picriic acid type
C.
Cafffeine and othher drug com
mplexes
D.
Poly
ymer type
Inclusion/occclusion com
mpounds
A.
Chan
nnel lattice ttype
B.
Layeer type
C.
Clath
hrates
D.
Mon
nomolecularr type
E. Macrromolecular
ar type
*This claassification does not prretend to deescribe the mechanism
m
or the type
of chemiical bonds involved
i
in complexatiion. It is meant simply to separate
360
Dr. Murtadha Alshareifi e-Library
out the various
v
typees of compleexes that aree discussed in the literaature. A
highly syystematized
d classificatiion of electrron donor–aacceptor intteractions
is given by R. S. Mu
ulliken, J. P
Phys. Chem
m. 56, 801, 1952.
Thesse are directed
d toward the corners
c
of a te
etrahedron, an
nd the structure
e is known ass an sp3 hybrid
d
beca
ause it involves one s and th
hree p orbitalss. In a double bond, the carb
bon atom is coonsidered to be
b
sp2 h
hybridized, an
nd the bonds are
a directed to
oward the corn
ners of a triang
gle.
Orbittals other than
n the 2s and 2p
2 orbitals can
n become invo
olved in hybridization. The trransition elements,
such
h as iron, copp
per, nickel, cob
balt, and zinc, seem to mak
ke use of their 3d, 4s, and 4pp orbitals in
forming hybrids. These
T
hybrids account for th
he differing geometries often
n found for thee complexes of
o the
transsition metal ion
ns. Table 10-2
2shows some compounds in
n which the ce
entral atom orr metal ion is
hybrridized differen
ntly and the ge
eometry that re
esults.
361
Dr. Murtadha Alshareifi e-Library
L
Ligands suchh as H2Ö H3 , C -, orr l- donatte a pair of electrons
e
in forming a
coomplex witth a metal io
on, and the eelectron paiir enters onee of the unfi
filled orbitalls
on the metal ion. A usefful but not iinviolate rulle to follow in estimatinng the type
of hybridizattion in a meetal ion com
mplex is to select that co
omplex in w
which the
m
metal ion has its 3d leveels filled or that can usee the lower--energy 3d aand 4s
orbitals prim
marily in the hybridizatiion. For exaample, the ground-state
g
e electronic
2+
coonfigurationn of Ni caan be given as
Inn combiningg with 4C - ligands too form [Ni(C
CN)4]2-, the electronic
coonfigurationn of the nick
kel ion mayy become eiither
or
362
Dr. Murtadha Alshareifi e-Library
in wh
hich the electrrons donated by
b the ligand a
are shown as dots. The dsp
p2 or square pplanar structure is
predicted to be the
e complex form
med because it uses the low
wer-energy 3d
d orbital. By thhe preparation and
studyy of a numberr of complexes
s, Werner ded
duced many ye
ears ago that this
t
is indeed the structure of
o
the ccomplex.
Similarly, the trivalent cobalt ion
n, Co(III), has tthe ground-sta
ate electronic configuration
and one may inquire into the po
ossible geome
etry of the com
mplex [Co(NH3)6]3+. The elecctronic
confiiguration of the metal ion leading to filled 3d levels is
363
Dr. Murtadha Alshareifi e-Library
and thus the d2sp3 or octahedra
al structure is p
predicted as th
he structure of this complexx. Chelates (se
ee
follow
wing section) of octahedral structure can be resolved in
nto optical isomers, and in aan elegant stu
udy,
Wern
ner2 used thiss technique to prove that co
obalt complexe
es are octahed
dral.
In the case of diva
alent copper, Cu(II),
C
which h
has the electro
onic configurattion
the fformation of th
he complex [Cu(NH3)4]2+ req
quires the prom
motion of one d electron of C
Cu2+ to a 4p le
evel
to ob
btain a filled 3d
d configuration in the comp lexed metal io
on, and a dsp2
2 or planar struucture is obtained:
364
Dr. Murtadha Alshareifi e-Library
Altho
ough the energy required to
o elevate the d electron to th
he 4p level is considerable,
c
the formation of a
plana
ar complex
P.19
99
havin
ng the 3d leve
els filled entire
ely more than ““pays” for the expended ene
ergy.
365
Dr. Murtadha Alshareifi e-Library
Tab
ble 10-2 Bond Types of
o Represen
ntative Com
mpounds
The metal ion Fe(III) has the gro
ound-state con
nfiguration
366
Dr. Murtadha Alshareifi e-Library
and in forming the
e complex [Fe((CN)6]3-, no ele
ectron promottion takes plac
ce,
beca
ause no stabiliization is gaine
ed over that w
which the d2sp
p3configuration
n already posssesses.
Com
mpounds of thiss type, in whic
ch the ligands lie “above” a partially filled orbital, are terrmed outer-sp
phere
comp
plexes; when the ligands lie
e “below” a pa
artially filled orb
bital, as in the
e previous exaample, the
comp
pound is termed aninner-sp
phere complexx. The presenc
ce of unpaired
d electrons in a metal ion
comp
plex can be de
etected by ele
ectron spin ressonance specttroscopy.
Ch
helates
Chellation places stringent
s
steric
c requirementss on both mettal and ligands
s. Ions such ass Cu(II) and Ni(II),
N
whicch form square
e planar complexes, and Fe
e(III) and Co(III), which form octahedral coomplexes, can
n
existt in either of tw
wo geometric forms.
f
As a co
onsequence of
o this isomeris
sm, only cis-cooordinated
ligan
nds—ligands adjacent
a
on a molecule—wi ll be readily re
eplaced by rea
action with a cchelating agen
nt.
Vitam
min B12 and th
he hemoproteins are incapa
able of reacting
g with chelatin
ng agents becaause their metal is
alrea
ady coordinate
ed in such a way
w that only th
he transcoordination positio
ons of the metaal are available for
comp
plexation. In contrast,
c
the metal
m
ion in cerrtain enzymes
s, such as alco
ohol dehydroggenase, which
conta
ains
P.20
00
zinc,, can undergo chelation, sug
ggesting that tthe metal is bo
ound in such a way as to lea
eave
two ccis positions available
a
for ch
helation.
367
Dr. Murtadha Alshareifi e-Library
Fig.. 10-1. Calccium ions seequestered bby ethylendediaminetettraacetic aciid.
Chlo
orophyll and he
emoglobin, tw
wo extremely im
mportant comp
pounds, are naturally occurrring chelates
invollved in the life processes off plants and an
nimals. Album
min is the main carrier of variious metal ion
ns
and small moleculles in the bloo
od serum. The amino-termin
nal portion of human
h
serum albumin binds
s
Cu(II) and Ni(II) with higher affin
nity than that o
of dog serum albumin.
a
This fact partly expplains why humans
are less susceptib
ble to copper poisoning
p
than
n are dogs. Th
he binding of copper to serum
m albumin is
impo
ortant because
e this metal is possibly invollved in severa
al pathologic conditions.3 Thhe synthetic
chela
ating agent ethylenediamine
etetraacetic accid (Fig. 10-1)) has been use
ed to tie up orr sequester iro
on
and copper ions so that they cannot catalyze the oxidative degradation of
o ascorbic aciid in fruit juice
es
and in drug preparrations. In the process of se
equestration, the
t chelating agent
a
and mettal ion form a
wate
er-soluble com
mpound. Ethyle
enediaminetettraacetic acid is widely used
d to sequesterr and remove
calcium ions from hard water.
Key Co
oncept
Che
elates
A su
ubstance con
ntaining two or
o more dono
or groups ma
ay combine with
w a metal to form a sp
pecial
type
e of complex known as a chelate (Gre
eek: “kelos, claw”).
c
Some
e of the bondds in a chelatte
mayy be ionic or of
o the primarry covalent tyype, whereas
s others are coordinate ccovalent links
s.
Whe
en the ligand
d provides on
ne group for a
attachment to
t the central ion, the cheelate is
calle
ed monodenttate. Pilocarp
pine behavess as a monodentate ligan
nd toward Coo(II), Ni(II), and
Zn(II) to form chelates of pse
eudotetrahed
dral geometry
y.7
The donor atom of the ligand
d is the pyrid ine-type nitro
ogen of the imidazole rin g of pilocarp
pine.
Mole
ecules with tw
wo and three
e donor grou
ups are called
d bidentate and
a tridentatee, respective
ely.
Ethyylenediamine
etetraacetic acid
a
has six p
points for atttachment to the
t metal ionn and is
acco
ordinglyhexa
adentate; how
wever, in som
me complexe
es, only four or five of thee groups are
coorrdinated.
Chellation can be applied
a
to the assay of drug
gs. A calorimetric method fo
or assaying proocainamide in
injecctable solutions is based on the formation
n of a 1:1 comp
plex of procain
namide with ccupric ion at pH
H4
to 4.5. The comple
ex absorbs vis
sible radiation at a maximum
m wavelength of 380 nm.4 T
The many use
es to
whicch metal comp
plexes and che
elating agents can be put arre discussed by
b Martell and Calvin.5
Org
ganic Mo
olecular Complex
C
es
368
Dr. Murtadha Alshareifi e-Library
An o
organic coordin
nation compou
und or molecu
ular complex consists
c
of con
nstituents heldd together by weak
w
force
es of the dono
or–acceptor typ
pe or by hydro
ogen bonds.
The difference bettween comple
exation and the
e formation of organic comp
pounds has beeen shown by
Clap
pp.6 The comp
pounds dimeth
hylaniline and 2,4,6-trinitroa
anisole react in
n the cold to g ive a molecula
ar
comp
plex:
On the other hand
d, these two co
ompounds rea
act at an eleva
ated temperatu
ure to yield a ssalt, the
consstituent molecu
ules of which are
a held togetther by primarry valence bon
nds:
The dotted line in the complex of
o equation (10
0-1) indicates that the two molecules
m
are held togetherr by a
v
force. It is not to be considered as
s a clearly deffined bond butt rather as an
weakk secondary valence
overrall attraction between
b
the tw
wo aromatic m
molecules.
The type of bondin
ng existing in molecular com
mplexes in which hydrogen bonding playss no part is no
ot fully
unde
erstood,
P.20
01
but itt may be conssidered for the
e present as in
nvolving an ele
ectron donor–a
acceptor mechhanism
corre
esponding to that
t
in metal complexes
c
butt ordinarily much weaker.
Many organic com
mplexes are so
o weak that the
ey cannot be separated from
m their solutioons as definite
e
comp
pounds, and they
t
are often difficult to dettect by chemic
cal and physic
cal means. Thee energy of
attra
action between
n the constitue
ents is probab ly less than 5 kcal/mole for most organic complexes.
Beca
ause the bond
d distance betw
ween the com
mponents of the
e complex is usually
u
greateer than 3 Å, a
cova
alent link is nott involved. Ins
stead, one mo lecule polarize
es the other, resulting
r
in a tyype of ionic
interraction or charrge transfer, and these mole
ecular complexes are often referred to ass charge transfer
comp
plexes. For exxample, the po
olar nitro grou ps of trinitrobe
enzene induce
e a dipole in thhe readily
polarrizable benzen
ne molecule, and
a the electro
rostatic interac
ction that results leads to coomplex formation:
369
Dr. Murtadha Alshareifi e-Library
X-rayy diffraction sttudies of comp
plexes formed
d between triniitrobenzene and aniline derrivatives have
show
wn that one off the nitro grou
ups of trinitrobe
enzene lies ov
ver the benzene ring of the aniline molecule,
the in
ntermolecularr distance betw
ween the two m
molecules beiing about 3.3 Å.
Å This result strongly suggests
that the interaction
n involves π bonding betwe en the π electtrons of the be
enzene ring annd the electronacce
epting nitro gro
oup.
A facctor of some im
mportance in the
t formation of molecular complexes
c
is the
t steric requuirement. If the
e
apprroach and closse association
n of the donor and acceptor molecules are
e hindered by steric factors,, the
comp
plex is not like
ely to form. Hy
ydrogen bondi ng and other effects must also
a be considdered, and these
are d
discussed in connection
c
with the specific complexes co
onsidered on the following ppages.
Fig.. 10-2. Resoonance in a donor–acceeptor compllex of trinitrrobenzene ((acceptor, to
op)
and hexamethyylbenzene (d
donor, bottoom). (From F. Y. Bullo
ock, in M. FFlorkin and E.
E
H. S
Stotz (Eds.),Comprehen
nsive Biochhemistry, Elssevier, New
w York, 19667, pp. 82–8
85.
370
Dr. Murtadha Alshareifi e-Library
Witth permissioon.)
The difference bettween a donorr–acceptor an d a charge tra
ansfercomplex
x is that in the latter type,
resonance makes the main contribution to co
omplexation, whereas
w
in the former, Londoon dispersion
force
es and dipole–
–dipole interac
ctions contribu
ute more to the
e stability of th
he complex. A resonance
interraction is show
wn inFigure 10
0-2 as depicted
d by Bullock.8
8 Trinitrobenze
ene is the acceeptor, A, mole
ecule
and hexamethylbe
enzene is the donor,
d
D. On tthe left side off the figure, weak dispersionn and dipolar
force
es contribute to
t the interaction of A and D
D; on the right side of the figure, the interaaction of A and
dD
results from a sign
nificant transfe
er of charge, m
making the ele
ectron accepto
or trinitrobenzeene negatively
y
charrged (A-) and leaving the donor, hexameth
hylbenzene, positively
p
charg
ged (D+). The overall donorr–
acce
eptor complex is shown by the double-hea
aded arrow to resonate betw
ween the unchharged D … A and
the ccharged D+ … A-moieties. Iff, as in the casse of hexametthylbenzene–trinitrobenzenee, the resonan
nce is
fairlyy weak, having
g an intermole
ecular binding energy ∆G off about –4700 calories, the ccomplex is refferred
to ass adonor–acce
eptor complex
x. If, on the oth
her hand, reso
onance betwee
en the charge transfer struc
cture
(D+… A-) and the uncharged
u
species (D … A)) contributes greatly
g
to the binding
b
of the donor and
acce
eptor molecule
e, the complex
x is called a ch
harge transferr complex. Fina
ally, those com
mplexes bound
together by van de
er Waals force
es, dipole–dipo
ole interaction
ns, and hydrog
gen bonding bbut lacking cha
arge
transsfer are known
n simply as mo
olecular comp
plexes. In both
h charge transfer and donor–
r–acceptor
comp
plexes, new absorption
a
ban
nds occur in th
he spectra, as shown later in
n Figure 10-133. In this book
k we
do not attempt to separate
s
the first
f
two classe
es, but rather refer to all inte
eractions that produce
abso
orption bands as charge tran
nsfer or as ele
ectron donor–a
acceptor comp
plexes withoutt distinction. Those
T
comp
plexes that do
o not show new
w bands are ccalled molecular complexes.
Charrge transfer co
omplexes are of importance
e in pharmacy. Iodine forms 1:1 charge traansfer comple
exes
with the drugs disu
ulfiram, chlom
methiazole, and
d tolnaftate. These drugs ha
ave recognizeed pharmacolo
ogic
actio
ons of their ow
wn:
P.20
02
Disulfiram is used against alcoh
hol addiction, cclomethiazole is a sedative–
–hypnotic andd anticonvulsant,
and tolnaftate is an antifungal agent. Each of these drugs possesses
p
a nitrogen–carbo
n
on–sulfur moie
ety
(see the accompanying structurre of tolnaftate
e), and a comp
plex may result from the trannsfer of charge
ee electrons on the nitrogen
n and/or sulfurr atoms of thes
se drugs to thee antibonding
from the pair of fre
odine, molecules containing
g the N—C==S
S moiety inhib
bit
orbital of the iodine atom. Thus,, by tying up io
oid action in th
he body.9
thyro
Dru
ug Comp
plexes
371
Dr. Murtadha Alshareifi e-Library
Higuchi and his associates10 investigated the complexing of caffeine with a number of acidic drugs.
They attributed the interaction between caffeine and a drug such as a sulfonamide or a barbiturate to a
dipole–dipole force or hydrogen bonding between the polarized carbonyl groups of caffeine and the
hydrogen atom of the acid. A secondary interaction probably occurs between the nonpolar parts of the
molecules, and the resultant complex is “squeezed out” of the aqueous phase owing to the great internal
pressure of water. These two effects lead to a high degree of interaction.
The complexation of esters is of particular concern to the pharmacist because many important drugs
belong to this class. The complexes formed between esters and amines, phenols, ethers, and ketones
have been attributed to the hydrogen bonding between a nucleophilic carbonyl oxygen and an active
hydrogen. This, however, does not explain the complexation of esters such as benzocaine, procaine,
and tetracaine with caffeine, as reported by Higuchi et al.11 There are no activated hydrogens on
caffeine; the hydrogen in the number 8 position (formula I) is very weak (Ka = 1 × 10-14) and is not likely
to enter into complexation. It might be suggested that, in the caffeine molecule, a relatively positive
center exists that serves as a likely site of complexation. The caffeine molecule is numbered in formula I
for convenience in the discussion. As observed in formula II, the nitrogen at the 2 position presumably
can become strongly electrophilic or acidic just as it is in an imide, owing to the withdrawal of electrons
by the oxygens at positions 1 and 3. An ester such as benzocaine also becomes polarized (formula III)
in such a way that the carboxyl oxygen is nucleophilic or basic. The complexation can thus occur as a
result of a dipole–dipole interaction between the nucleophilic carboxyl oxygen of benzocaine and the
electrophilic nitrogen of caffeine.
372
Dr. Murtadha Alshareifi e-Library
Caffe
eine forms com
mplexes with organic acid a
anions that are
e more soluble
e than the purre xanthine, bu
ut the
comp
plexes formed
d with organic acids, such a
as gentisic acid
d, are less soluble than cafffeine alone. Su
uch
insolluble complexxes provide caffeine in a form
m that masks its normally bitter taste andd should serve
e as a
suita
able state for chewable
c
table
ets. Higuchi an
nd Pitman12 synthesized
s
1:1 and 1:2 cafffeine–gentisic
c acid
comp
plexes and me
easured their equilibrium so
olubility and ra
ates of dissolution. Both the 1:1 and 1:2
comp
plexes were le
ess soluble in water than ca
affeine, and their dissolution
n rates were allso less than that
t
of ca
affeine. Chewa
able tablets fo
ormulated from
m these complexes should provide
p
an exteended-release
e
form of the drug with
w improved taste.
t
Yorkk and Saleh13 studied the effect of sodium
m salicylate on
n the release of
o benzocainee from topical
vehiccles, it being recognized
r
tha
at salicylates fform molecula
ar complexes with
w benzocainne. Complexa
ation
betw
ween drug and
d complexing agents
a
can im prove or impa
air drug absorp
ption and bioaavailability; the
e
authors found thatt the presence
e of sodium sa
alicylate signifiicantly influenced the releasse of benzoca
aine,
depe
ending on the type of vehicle
e involved. Th
he largest increase in absorption was obsserved for a watermisccible polyethyle
ene glycol bas
se.
P.20
03
T
Table 10-3 Some
S
Moleecular Orgaanic Complexes of Ph
harmaceuticcal Interestt*
373
Dr. Murtadha Alshareifi e-Library
Agent
Compounds That Form Complexes with the
Agent Listed in the First Column
Polyethylene glycols
m-Hydroxybenzoic acid, p-hydroxybenzoic
acid, salicylic acid,o-phthalic acid,
acetylsalicylic acid, resorcinol, catechol,
phenol, phenobarbital, iodine (in I2 · KI
solutions), bromine (in presence of HBr)
Povidone (polyvinylpyrrolidone, PVP)
Benzoic acid, m-hydroxybenzoic acid, phydroxybenzoic acid, salicylic acid, sodium
salicylate,p-aminobenzoic acid, mandelic
acid, sulfathiazole, chloramphenicol,
phenobarbital
Sodium
carboxymethylcellulose
Quinine, benadryl, procaine, pyribenzamine
Oxytetracycline and
tetracycline
N-Methylpyrrolidone,N,Ndimethylacetamide, γ-valerolactone, γbutyrolactone, sodium p-aminobenzoate,
sodium salicylate, sodium phydroxybenzoate, sodium saccharin, caffeine
*Compiled from the results of T. Higuchi et al., J. Am. Pharm. Assoc. Sci.
Ed. 43, 393, 398, 456, 1954; 44, 668, 1955; 45, 157, 1956; 46,458, 587,
1957; and J. L. Lach et al., Drug Stand. 24, 11, 1956. An extensive table of
acceptor and donor molecules that form aromatic molecular complexes was
compiled by L. J. Andrews, Chem. Rev. 54, 713, 1954. Also refer to T.
Higuchi and K. A. Connors, Phase Solubility Techniques. Advances in
Analytical Chemistry and Instrumentation, in C. N. Reilley (Ed.), Wiley,
New York, 1965, pp. 117–212.
Polymer Complexes
Polyethylene glycols, polystyrene, carboxymethylcellulose, and similar polymers containing nucleophilic
oxygens can form complexes with various drugs. The incompatibilities of certain polyethers, such as the
Carbowaxes, Pluronics, and Tweens with tannic acid, salicylic acid, and phenol, can be attributed to
these interactions. Marcus14 reviewed some of the interactions that may occur in suspensions,
emulsions, ointments, and suppositories. The incompatibility may be manifested as a precipitate,
flocculate, delayed biologic absorption, loss of preservative action, or other undesirable physical,
chemical, and pharmacologic effects.
Plaizier-Vercammen and De Nève15 studied the interaction of povidone (PVP) with ionic and neutral
aromatic compounds. Several factors affect the binding to PVP of substituted benzoic acid and nicotine
derivatives. Although ionic strength has no influence, the binding increases in phosphate buffer solutions
and decreases as the temperature is raised.
374
Dr. Murtadha Alshareifi e-Library
Crosspovidone, a cross-linked insoluble PVP, is able to bind drugs owing to its dipolar character and
porous structure. Frömming et al.16 studied the interaction of crosspovidone with acetaminophen,
benzocaine, benzoic acid, caffeine, tannic acid, and papaverine hydrochloride, among other drugs. The
interaction is mainly due to any phenolic groups on the drug. Hexylresorcinol shows exceptionally strong
binding, but the interaction is less than 5% for most drugs studied (32 drugs). Crosspovidone is a
disintegrant in pharmaceutical granules and tablets. It does not interfere with gastrointestinal absorption
because the binding to drugs is reversible.
Solutes in parenteral formulations may migrate from the solution and interact with the wall of a polymeric
container. Hayward et al.17showed that the ability of a polyolefin container to interact with drugs
depends linearly on the octanol–water partition coefficient of the drug. For parabens and drugs that
exhibit fairly significant hydrogen bond donor properties, a correction term related to hydrogen-bond
formation is needed. Polymer–drug container interactions may result in loss of the active component in
liquid dosage forms.
Polymer–drug complexes are used to modify biopharmaceutical parameters of drugs; the dissolution
rate of ajmaline is enhanced by complexation with PVP. The interaction is due to the aromatic ring of
ajmaline and the amide groups of PVP to yield a dipole–dipole-induced complex.18
Some molecular organic complexes of interest to the pharmacist are given in Table 10-3. (Complexes
involving caffeine are listed in Table 10-6.)
Inclusion Compounds
The class of addition compounds known as inclusion or occlusioncompounds results more from the
architecture of molecules than from their chemical affinity. One of the constituents of the complex is
trapped in the open lattice or cagelike crystal structure of the other to yield a stable arrangement.
Channel Lattice Type
The cholic acids (bile acids) can form a group of complexes principally involving deoxycholic acid in
combination with paraffins, organic acids, esters, ketones, and aromatic compounds and with solvents
such as ether, alcohol, and dioxane. The crystals of deoxycholic acid are arranged to form a channel
into which the complexing molecule can fit (Fig. 10-3). Such stereospecificity should permit the
resolution of optical isomers. In fact, camphor has been partially resolved by complexation with
deoxycholic acid, anddl-terpineol has been resolved by the use of digitonin, which
P.204
occludes certain molecules in a manner similar to that of deoxycholic acid:
375
Dr. Murtadha Alshareifi e-Library
Fig.. 10-3. (a) A channel complex form
med with urrea moleculles as the hoost. (b) These
mollecules are packed
p
in an
n orderly m
manner and held
h togetheer by hydroggen bonds
betw
ween nitrogen and oxyg
gen atoms. The hexago
onal channels, approxim
mately 5 Å in
diam
meter, proviide room for guest mollecules such
h as long-ch
hain hydrocaarbons, as
show
wn here. (From J. F. Brown, Jr., S
Sci. Am. 207
7, 82, 1962.. Copyright © 1962 by
Scieentific Ameerican, Inc. All
A rights reeserved.) (c)) A hexagon
nal channell complex
(addduct) of metthyl α-lipoaate and 15 g of urea in methanol
m
prrepared withh gentle
heatting. Needlee crystals off the adductt separated overnight
o
att room tempperature. Th
his
inclusion comppound or adduct beginss to decomp
pose at 63°C
C and melts at 163°C.
Thioourea may also
a be used
d to form thhe channel complex.
c
(Frrom H. Minna and M.
Nishhikawa, J. Pharm.
P
Sci. 53, 931, 19964. With peermission.).. (d) Cycloddextrin
(cyccloamylose,, Schardingeer dextrin). (See Merckk Index, 11th Ed., Mercck, Rahway
y,
N.J.., 1989, p. 425.)
4
376
Dr. Murtadha Alshareifi e-Library
Urea
a and thiourea
a also crystalliz
ze in a channe
el-like structurre permitting enclosure
e
of unnbranched
para
affins, alcoholss, ketones, org
ganic acids, an
nd other comp
pounds, as sho
own in Figure 10-3a andb. The
T
well--known starch–iodine solutio
on is a channe
el-type comple
ex consisting of
o iodine moleecules entrapp
ped
within spirals of the glucose residues.
Form
man and Gradyy19 found that monostearin
n, an interfering substance in
n the assay off dienestrol, co
ould
be extracted easilyy from dermattologic creamss by channel-ttype inclusion in urea. Theyy felt that urea
inclu
usion might be
ecome a generral approach ffor separation of long-chain compounds inn assay metho
ods.
The authors review
wed the earlie
er literature on urea inclusion of straight-chain hydrocarrbons and fatty
y
acidss.
P.20
05
Lay
yer Type
Som
me compoundss, such as the clay montmorrillonite, the prrincipal constittuent of bento nite, can trap
hydrrocarbons, alcohols, and gly
ycols between the layers of their lattices.2
20 Graphite caan also interca
alate
comp
pounds betwe
een its layers.
Cla
athrates2
21
The clathrates cryystallize in the form of a cage
ordinating com
mpound is
elike lattice in which the coo
entra
apped. Chemical bonds are
e not involved in these comp
plexes, and on
nly the molecuular size of the
e
enca
aged compone
ent is of imporrtance. Ketelaa
ar22 observed
d the analogy between the sstability of a
clath
hrate and the confinement
c
of
o a prisoner. T
The stability off a clathrate is
s due to the strrength of the
struccture, that is, to
t the high ene
ergy that mustt be expended
d to decompos
se the compouund, just as a
priso
oner is confine
ed by the bars that prevent e
escape.
Pow
well and Palin23 made a deta
ailed study of clathrate com
mpounds and showed
s
that thhe highly toxic
agen
nt hydroquinon
ne (quinol) cry
ystallizes in a ccagelike hydro
ogen-bonded structure, as sseen inFigure 104. Th
he holes have a diameter off 4.2 Å and pe
ermit the entra
apment of one small molecuule to about ev
very
two q
quinol molecu
ules. Small mo
olecules such a
as methyl alco
ohol, CO2, and
d HCl may be trapped in the
ese
cage
es, but smallerr molecules su
uch as H2and larger molecu
ules such as ethanol cannott be
acco
ommodated. Itt is possible th
hat clathrates m
may be used to
t resolve optiical isomers aand to bring ab
bout
other processes of molecular se
eparation.
377
Dr. Murtadha Alshareifi e-Library
Fig.. 10-4. Cageelike structu
ure formed tthrough hyd
drogen bond
ding of hyddroquinone
mollecules. Smaall moleculees such as m
methanol are trapped in
n the cages tto form the
clathhrate. (Moddified from J. F. Brownn, Jr., Sci. Am.
A 207, 82, 1962. Coppyright © 19
962
by S
Scientific American,
A
In
nc. All rightts reserved.))
One official drug, warfarin
w
sodiu
um, United Sta
ates Pharmaco
opeia, is a clathrate of wateer, isopropyl
alcoh
hol, and sodiu
um warfarin in the form of a white crystalliine powder.
Mo
onomolec
cular Incllusion Co
ompound
ds: Cyclo
odextrins
s
Inclu
usion compoun
nds were revie
ewed by Frankk.24a In addition to channel- and cage-tyype (clathrate)
comp
pounds, Frankk added classes of mono- a
and macromolecular inclusio
on compoundss. Monomolec
cular
inclu
usion compoun
nds involve the
e entrapment of a single gu
uest molecule in the cavity oof one host
mole
ecule. Monomolecular host structures
s
are
e represented by the cyclode
extrins (CD). T
These compou
unds
are ccyclic oligosacccharides conttaining a minim
mum of six D-(+)-glucopyranose units attaached by α-1,4
linka
ages produced
d by the action
n on starch of Bacillus mace
eransamylase.. The natural αα-, β-, and γcyclo
odextrins (α-C
CD, β-CD, and γ-CD, respecctively) consistt of six, seven, and eight unnits of glucose,
respectively.
Their ability to form
m inclusion co
ompounds in a
aqueous solutiion is due to th
he typical arraangement of th
he
gluco
ose units (see
e Fig. 10-3d). As
A observed in
n cross sectio
on in the figure
e, the cyclodexxtrin structure
forms a torus or do
oughnut ring. The molecule
e actually existts as a truncatted cone, whicch is seen
in Fig
gure 10-5a; it can accommo
odate moleculles such as mitomycin C to form inclusionn compounds (Fig.
10-5
5b). The interio
or of the cavity
y is relatively h
hydrophobic because
b
of the
e CH2 groups, whereas the
cavitty entrances are
a hydrophilic
c owing to the presence of th
he primary and secondary hhydroxyl
groups.25,26α-CD
D has the smalllest cavity (intternal diamete
er almost 5 Å). β-CD and γ--CD are the most
usefu
ul for pharmacceutical technology owing to
o their larger cavity
c
size (intternal diameteer almost 6 Å and
a 8
Å, re
espectively). Water
W
inside th
he cavity tendss to be squeez
zed out and to
o be replaced by more
hydrrophobic speciies. Thus, molecules of app
propriate size and
a stereoche
emistry can bee included in th
he
cyclo
odextrin cavityy by
P.20
06
378
Dr. Murtadha Alshareifi e-Library
hydrrophobic intera
actions. Comp
plexation doess not ordinarily
y involve the fo
ormation of coovalent bonds..
Som
me drugs may be
b too large to
o be accommo
odated totally in the cavity. As
A shown in F
Figure 10-5b,
mitom
mycin C intera
acts with γ-CD
D at one side o
of the torus. Thus, the azirid
dine ring
Key Co
oncept
Cyc
clodextrins
Acco
ording to Davis and Brew
wster,24b “Cycclodextrins are
a cyclic olig
gomers of gluucose that ca
an
form
m water-solub
ble inclusion complexes w
with small molecules and
d portions of large
com
mpounds. The
ese biocomp
patible, cyclicc oligosaccha
arides do nott elicit immunne responses
s
and have low toxxicities in animals and hu
umans. Cyclo
odextrins are
e used in phaarmaceutical
appllications for numerous
n
pu
urposes, incl uding improv
ving the bioa
availability off drugs. Of
speccific interest is the use off cyclodextrin
n-containing polymers to provide uniqque capabilitiies
for the delivery of
o nucleic aciids.” Davis a
and Brewsterr24b discuss cyclodextrin-b
c
based
therapeutics and
d possible futture applicattions, and rev
view the use of cyclodexttrin-containin
ng
polyymers in drug
g delivery.
Fig.. 10-5. (a) Representati
R
ion of cycloodextrin as a truncated cone. (b) M
Mitomycin C
parttly enclosedd in cyclodextrin to form
m an inclusion complex. (From O . Beckers, Int.
I
J. Phharm. 52, 240, 247, 1989. With peermission.)
379
Dr. Murtadha Alshareifi e-Library
of mitomycin C is protected from
m degradation in acidic solution.27Bakens
sfield et al.28 studied the
1
inclu
usion of indom
methacin with β-CD
β
using a H-NMR techn
nique. The p-c
chlorobenzoyl part of
indom
methacin (sha
aded part of Fig. 10-6) enterrs the β-CD rin
ng, whereas th
he substitutedd indole moiety
y (the
rema
ainder of the molecule)
m
is to
oo large for incclusion and rests against the
e entrance of tthe CD cavity.
Fig.. 10-6. Indoomethacin (IIndocin).
Cyclodextrins are studied as solubilizing and stabilizing age
ents in pharmaceutical dosaage forms. Lach
and associates29 used cyclodextrins to trap, stabilize, and solubilize sulffonamides, tettracyclines,
morp
phine, aspirin, benzocaine, ephedrine, resserpine, and testosterone.
t
The
T aqueous ssolubility of
retinoic acid (0.5 mg/liter),
m
a dru
ug used topica
ally in the treattment of acne,30 is increaseed to 160 mg/liter
by co
omplexation with
w β-CD. Dissolution rate p
plays an important role in bioavailability oof drugs, fast
disso
olution usuallyy favoring abso
orption. Thus,, the dissolutio
on rates of fam
motidine,31 a ppotent drug in the
treattment of gastriic and duoden
nal ulcers, and
d that of tolbuttamide, an ora
al antidiabetic drug, are both
h
incre
eased by comp
plexation with β-cyclodextrin
n.32
Cyclodextrins mayy increase or decrease
d
the rreactivity of th
he guest molec
cule dependinng on the nature of
the rreaction and th
he orientation of the molecu
ule within the CD
C cavity. Thu
us, α-cyclodexxtrin tends to favor
f
380
Dr. Murtadha Alshareifi e-Library
pH-d
dependent hyd
drolysis of indo
omethacin in a
aqueous soluttion, whereas β-cyclodextrinn inhibits
it.28 Unfortunatelyy, the water so
olubility of β-C
CD (1.8 g/100 mL
m at 25°C) is
s often insufficcient to stabiliz
ze
drugs at therapeuttic doses and is also associiated with nep
phrotoxicity wh
hen CD is adm
ministered by
pare
enteral routes.3
33 The relatively low aqueo
ous solubility of
o the cyclodex
xtrins may be due to the
formation of intram
molecular hydrrogen bonds b
between the hydroxyl groups (see Fig. 100-3d), which
prevent their intera
action with wa
ater moleculess.34
Derivvatives of the natural crysta
alline CD have
e been develop
ped to improve
e aqueous sollubility and to
avoid
d toxicity. Parttial methylation (alkylation) of some of the
e OH groups in CD reducess the
interrmolecular hyd
drogen bondin
ng, leaving som
me OH groups
s free to intera
act with water,, thus increasing
the a
aqueous solub
bility of CD.34A
According to M
Müller and Bra
auns,35 a low degree of alkkyl substitution
n is
prefe
erable. Derivatives with a high degree of ssubstitution lo
ower the surfac
ce tension of w
water, and this
s has
been
n correlated with
w the hemoly
ytic activity ob
bserved in som
me CD derivatives. Amorphoous derivatives
s of
β-CD
D and γ-CD arre more effective as solubilizzing agents fo
or sex hormones than the paarent
cyclo
odextrins. Com
mplexes of tes
stosterone with
h amorphous hydroxypropyl β-CD allow aan efficient
transsport of hormo
one into the circulation when
n given sublingually.36 This
s route avoids both metabolism
of the drug in the intestines
i
and rapid first-passs decomposiition in the live
er (seeChapteer 15), thus
imprroving bioavaillability.
In ad
ddition to hydrrophilic derivattives, hydroph
hobic forms of β-CD have be
een found useeful as sustain
nedrelea
ase drug carrie
ers. Thus, the release rate o
of the water-soluble calcium
m antagonist ddiltiazem was
signiificantly decreased by comp
plexation with ethylated β-C
CD. The releas
se rate was coontrolled by miixing
hydrrophobic and hydrophilic
h
derivatives of cyyclodextrins at several ratios
s.37 Ethylatedd β-CD has als
so
been
n used to retarrd the delivery
y of isosorbide
e dinitrate, a va
asodilator.38
Cyclodextrins mayy improve the organoleptic ccharacteristics
s of oral liquid formulations. The bitter tas
ste of
susp
pensions of fem
moxetine, an antidepressan
a
nt, is greatly su
uppressed by complexationn of the drug with
w
β-cycclodextrin.39
P.20
07
Mo
olecular Sieves
S
Macrromolecular in
nclusion comp
pounds, or mollecular sieves
s as they are commonly
c
calleed, include
zeoliites, dextrins, silica gels, an
nd related subsstances. The atoms are arra
anged in threee dimensions to
produce cages and channels. Synthetic
S
zeolittes may be made to a definite pore size sso as to separrate
mole
ecules of differrent dimension
ns, and they a
are also capab
ble of ion exch
hange. See thee review article by
Fran
nk24a for a dettailed discussion of inclusio
on compounds
s.
381
Dr. Murtadha Alshareifi e-Library
Fig.. 10-7. A plot of an add
ditive propeerty against mole fractio
on of one off the speciees in
which complexxation betweeen the speccies has occcurred. The dashed linee is that
expeected if no complex
c
haad formed. ((Refer to C. H. Giles ett al., J. Chem
m. Soc., 195
52,
3799, for similaar figures.)
Me
ethods off Analysis
s40
A de
etermination off the stoichiom
metric ratio of lligand to meta
al or donor to acceptor
a
and a quantitative
exprression of the stability consttant for comple
ex formation are
a important in the study annd application of
coorrdination comp
pounds. A limited number off the more imp
portant methods for obtaininng these quan
ntities
is pre
esented here.
Me
ethod of Continuo
C
ous Varia
ation
Job4
41 suggested the
t use of an additive prope
erty such as th
he spectropho
otometric extinnction coefficie
ent
(diele
ectric constan
nt or the squarre of the refracctive index ma
ay also be use
ed) for the meaasurement of
comp
plexation. If th
he property forr two species is sufficiently different and if no interactioon occurs when the
comp
ponents are mixed,
m
then the
e value of the property is the weighted me
ean of the valuues of the
sepa
arate species in
i the mixture. This means tthat if the add
ditive property, say dielectricc constant, is
plotte
ed against the
e mole fraction
n from 0 to 1 fo
or one of the components
c
of
o a mixture whhere no
comp
plexation occu
urs, a linear re
elationship is o
observed, as shown
s
by the dashed line inn Figure 10-7. If
soluttions of two sp
pecies A and B of equal mo
olar concentrattion (and henc
ce of a fixed tootal concentration
of the species) are
e mixed and iff a complex fo rms between the two specie
es, the value oof the additive
e
prop
perty will pass through a maximum (or min
nimum), as sh
hown by the up
pper curve in F
Figure 10-7. For
F a
consstant total conccentration of A and B, the ccomplex is at itts greatest con
ncentration att a point where
e the
speccies A and B are
a combined in the ratio in w
which they oc
ccur in the com
mplex. The linee therefore shows
a bre
eak or a chang
ge in slope at the mole fracttion correspon
nding to the co
omplex. The cchange in slop
pe
occu
urs at a mole fraction
f
of 0.5 in Figure 10-7
7, indicating a complex of th
he 1:1 type.
Fig.. 10-8. A plot of absorb
bance differrence againsst mole fracction showinng the resullt of
com
mplexation.
Whe
en spectrophottometric absorbance is used
d as the physiical property, the
t observed vvalues obtained at
vario
ous mole fractions when com
mplexation occcurs are usua
ally subtracted from the corre
responding values
that would have be
een expected had no comp
plex resulted. This
T
difference
e, D, is then pllotted against mole
fraction, as shown
n in Figure 10--8. The molar ratio of the co
omplex is readily obtained fro
rom such a curve.
382
Dr. Murtadha Alshareifi e-Library
By m
means of a calculation involv
ving the conce
entration and the
t property being
b
measureed, the stability
y
consstant of the forrmation can be
e determined by a method described
d
by Martell
M
and Caalvin.42 Anoth
her
meth
hod, suggested by Bent and
d French,43 iss given here.
If the
e magnitude of
o the measure
ed property, su
uch as absorb
bance, is propo
ortional only too the concentrration
of the complex MA
An, the molar ratio
r
of ligand A to metal M and the stability constant caan be readily
determined. The equation
e
for co
omplexation ca
an be written as
and the stability co
onstant as
or, in
n logarithmic form,
f
wherre [MAn] is the
e concentration
n of the comp lex, [M] is the
P.20
08
conccentration of th
he uncomplexed metal, [A] iis the concenttration of the uncomplexed
u
lligand, n is the
e
number of moles of
o ligand comb
bined with 1 m
mole of metal ion, and K is th
he equilibrium
m
orsta
ability constan
nt for the comp
plex. The conccentration of a metal ion is held
h
constant w
while the
conccentration of lig
gand is varied
d, and the corrresponding co
oncentration, [M
MAn], of compplex formed is
obtained from the spectrophotometric analysiis.41 Now, acc
cording to equ
uation(10-5), iff log [MAn] is
ed against log
g [A], the slope
e of the line yie
elds the stoich
hiometric ratio
o or the numbeer n of ligand
plotte
mole
ecules coordin
nated to the metal ion, and tthe intercept on
o the vertical axis allows onne to obtain th
he
stabiility constant, K, because [M
M] is a known quantity.
Job rrestricted his method
m
to the formation of a single comp
plex; however, Vosburgh et al.44 modified
d it so
as to
o treat the form
mation of highe
er complexes in solution. Osman
O
and Abu-Eittah45 useed
specctrophotometriic techniques to investigate 1:2 metal–liga
and complexe
es of copper annd barbiturate
es. A
gree
enish-yellow co
omplex is form
med by mixing a blue solutio
on of copper (II) with thiobarrbiturates
(colo
orless). By using the Job me
ethod, an appa
arent stability constant as well
w as the com
mposition of th
he 1:2
comp
plex was obta
ained.
pH
H Titration
n Method
d
This is one of the most reliable methods and can be used whenever
w
the complexation is attended by
b a
chan
nge in pH. The
e chelation of the
t cupric ion by glycine, fo
or example, ca
an be represennted as
Beca
ause two proto
ons are formed in the reacti on of equation
n (10-6), the addition
a
of glyccine to a solution
conta
aining cupric ions
i
should re
esult in a decre
ease in pH.
Titra
ation curves ca
an be obtained
d by adding a strong base to
o a solution off glycine and tto another solu
ution
conta
aining glycine and a copperr salt and plottting the pH ag
gainst the equivalents of basse added. The
e
results of such a potentiometric
p
titration are sshown inFigure
e 10-9. The cu
urve for the meetal–glycine
mixtu
ure is well below that for the
e glycine alone
e, and the dec
crease in pH shows
s
that com
mplexation is
occu
urring througho
out most of the
e neutralizatio
on range. Simiilar results are
e obtained withh other zwitterrions
and weak acids (o
or bases), such
h as N,N′-diaccetylethylened
diamine diacetic acid, which has been studied
for its complexing action with co
opper and calccium ions.
383
Dr. Murtadha Alshareifi e-Library
Fig.. 10-9. Titraation of glyccine and of glycine in the
t presencee of cupric iions. The
diffe
ference in pH
H for a giveen quantity oof base add
ded indicatess the occurrrence of a
com
mplex.
The results can be
e treated quan
ntitatively in th e following ma
anner to obtain stability connstants for the
comp
plex. The two successive or stepwise equ
uilibria betwee
en the copper ion or metal, M
M, and glycine
e or
the liigand, A, can be written in general
g
as
and the overall rea
action, (10-7) and (10-8), is
nstants, and th
he equilibrium constant, β, fo
for the overall
Bjerrrum46 called K1 and K2 the formation con
reaction is known as thestability
y constant. A q
quantity n may
y now be defin
ned. It is the n umber of ligan
nd
mole
ecules bound to
t a metal ion. The average
e number of lig
gand groups bound
b
per mettal ion presentt is
there
efore designatted ñ(n bar) an
nd is written a
as
or
Altho
ough n has a definite
d
value for each speccies of complex (1 or 2 in this case), it maay have any va
alue
betw
ween 0 and the
e largest numb
ber of ligand m
molecules bou
und, 2 in this case. The num
merator of
equa
ation (10-11) gives
g
the total concentration
n of ligand spe
ecies bound. The
T second terrm in the
numerator is multiplied by 2 bec
cause two mollecules of ligand are contain
ned in each m
molecule of the
e
speccies, MA2. The
e denominatorr gives the tota
al concentratio
on of metal pre
esent in all forrms, both bound
and ffree. For the special
s
case in
n which ñ = 1, equation (10--11) becomes
Emp
ploying the ressults in equatio
ons (10-9) and
d (10-12), we obtain
o
the follo
owing relationn:
and ffinally
384
Dr. Murtadha Alshareifi e-Library
P.20
09
wherre p[A] is written for –log [A]]. Bjerrum also
o showed thatt, to a first app
proximation,
It sho
ould now be possible
p
to obttain the individ
dual complex formation
f
constants, K1 andd K2, and the
overrall stability constant, β, if on
ne knows two values: [n with
h bar above] and
a p[A].
Equa
ation (10-10) shows
s
that the
e concentratio
on of bound lig
gand must be determined
d
beefore ñ can be
e
evalu
uated. The ho
orizontal distan
nces represen
nted by the line
es in Figure 10
0-9 between thhe titration curve
for glycine alone (curve I) and fo
or glycine in th
he presence of
o Cu2+ (curve II) give the am
mount of alkalii
used
d up in the follo
owing reaction
ns:
This quantity of alkkali is exactly equal to the cconcentration of ligand boun
nd at any pH, aand, according to
equa
ation (10-10), when
w
divided by the total co
oncentration of
o metal ion, giives the valuee of [n with barr
abovve].
The concentration
n of free glycine [A] as the “b
base” NH2CH2COO- at any pH is obtainedd from the acid
disso
ociation expre
ession for glycine:
or
The concentration
n [NH3+ CH2CO
OO-], or [HA], of the acid species at any pH
p is taken as the difference
e
betw
ween the initiall concentration
n, [HA]init, of g lycine and the
e concentration, [NaOH], of alkali added. Then
or
oncentration of
o the ligand gllycine.
wherre [A] is the co
Exa
ample 10-1
Calc
culate Averrage Numbe
er of Ligand
ds
If 75
5-mL sample
es containing 3.34 × 10-2 mole/liter of glycine hydrochloride aloone and in
com
mbination with
h 9.45 × 10-3mole/liter of cupric ion arre titrated witth 0.259 N N
NaOH, the tw
wo
385
Dr. Murtadha Alshareifi e-Library
curvves I and II, respectively,
r
in Figure 10
0-9 are obtain
ned. Computte ñ and p[A]] at pH 3.50 and
a
pH 8
8.00. The pK
Ka of glycine is
i 9.69 at 30 °C.
a. From Fig
gure 10-9, th
he horizontal distance at pH 3.50 for the
t 75-mL saample is 1.60
0 mL
NaOH or
o 2.59 × 10-44 mole/mL × 1.60 = 4.15 × 10-4 mole. For a 1-liter sample, the
value is 5.54 × 10-3 mole.
m
The tottal concentra
ation of copp
per ion per lite
ter is 9.45 × 103
mole, and
a from equation (10-10
0), [n with barr above] is giiven by
From eq
quation (10-2
21),
b. At pH 8..00, the horiz
zontal distancce between the two curves I and II inn Figure 10-9
9 is
-4
equivale
ent to 5.50 mL
m of NaOH i n the 75-mL sample, or 2.59
2
× 10 × 5.50 × 1000
0/75
= 19.0 × 10-3 mole/litter. We have
e
The values of [n with
w bar above] and p[A] at vvarious pH values are then plotted as shoown in Figure 1010. T
The curve thatt is obtained is
s known as a fformation curv
ve. It is seen to reach a limitt at [n with bar
abovve] = 2, indicatting that the maximum
m
num
mber of glycine
e molecules tha
at can combinne with one ato
om
of co
opper is 2. Fro
om this curve at
a [n with bar a
above] = 0.5, at [n with bar above] = 3/2, and at [n with
h bar
abovve] = 1.0, the approximate
a
values
v
for log K 1, log K2, and
d log β, respec
ctively, are obbtained.
P.210
A typ
pical set of datta for the com
mplexation of g
glycine by copp
per is shown in Table 10-4. Values of log K1,
log K2, and log β for
f some meta
al complexes o
of pharmaceuttical interest are given in Taable 10-5.
386
Dr. Murtadha Alshareifi e-Library
Fig.. 10-10. Forrmation curv
ve for the c opper–glyccine complex
x.
Peca
ar et al.47 desscribed the ten
ndency of pyrrrolidone 5-hydroxamic acid to bind the ferrric ion to form
m
mono, bis, and triss chelates. These workers la
ater studied th
he thermodyna
amics of thesee chelates using a
potentiometric method to determ
mine stability cconstants. The
e method emp
ployed by Pecaar et al. is kno
own
as th
he Schwarzenbach methoda
and can be ussed instead of the potentiom
metric method described herre
when
n complexes are
a unusually stable. Sandm
mann and Luk
k48measured the
t stability coonstants for lithium
cateccholamine com
mplexes by po
otentiometric ttitration of the free lithium io
on. The resultss demonstrate
ed
that lithium forms complexes with the zwitterio
onic species of
o catecholamines at pH 9 too 10 and with
deprrotonated form
ms at pH value
es above 10. T
The interaction
n with lithium depends
d
on thhe dissociation
n of
the p
phenolic oxyge
en of catechollamines. At ph
hysiologic pH, the protonate
ed species shoow no significa
ant
comp
plexation. Som
me lithium saltts, such as lith
hium carbonatte, lithium chlo
oride, and lithiuum citrate, are
e
used
d in psychiatryy.
2
Table 10-44 Potentiom
metric Titr ation of Gllycine Hydrrochloride (3.34 × 10M
Mole/Liter,, pKa, 9.69)) and Cupriic Chloridee (9.45 × 10
0-3 Mole/Litter) in 75-m
mL
Sam
mples Usingg 0.259 N NaOH
N
at 30
0°C*
pH
H
Δ((mL NaOH
H)
(p
per 75-mL
Saample)
Molees OH,MAC
Complexed
d
(molle/liter)
[n
[ with barr
above]
a
p[A]
3.50
1.60
5.554 × 10-3
0.59
7.66
4.00
2.90
100.1 × 10-3
1.07
7.32
4.50
3.80
13 .1 × 10-3
1.39
6.85
5.00
4.50
15 .5 × 10-3
1.64
6.44
387
Dr. Murtadha Alshareifi e-Library
5.50
5.00
17.3 × 10-3
1.83
5.98
6.00
5.20
18.0 × 10-3
1.91
5.50
6.50
5.35
18.5 × 10-3
1.96
5.02
7.00
5.45
18.8 × 10-3
1.99
4.53
7.50
5.50
19.0 × 10-3
2.03
4.03
8.00
5.50
19.0 × 10-3
2.01
3.15
*From the data in the last two columns, the formation curve, Figure 10-10,
is plotted, and the following results are obtained from the curve: log K1 =
7.9, log K2 = 6.9, and log β = 14.8 (average log β from the literature at 25°C
is about 15.3).
Table 10-5 Selected Constants for Complexes Between Metal Ions and Organic
Ligand*
Organic Ligand
Metal Ion log K1
log K2
log, β =
log K1K2
Ascorbic acid
Ca2+
0.19
—
—
Nicotinamide
Ag+
—
—
3.2
Glycine (aminoacetic acid)
Cu2+
8.3
7.0
15.3
Salicylaldehyde
Fe2+
4.2
3.4
7.6
Salicylic acid
Cu2+
10.6
6.3
16.9
p-hydroxybenzoic acid
Fe3+
15.2
—
—
Methyl salicylate
Fe3+
9.7
—
—
388
Dr. Murtadha Alshareifi e-Library
Diethylbarrbituric acid
d
(barbital)
Ca2+
0.66
—
—
8-Hydroxyyquinoline
Cu2+
15
14
29
Pteroylgluttamic acid (folic
(
acid)
Cu2+
—
—
7.8
Oxytetracyycline
Ni2+
5.8
4.8
10.6
Chlortetraccycline
Fe3+
8.8
7.2
16.0
*From J. Bjerrum,
B
G. Schwarzennback, and L.
L G. Sillen,, Stability C
Constants,
Part I, Orgganic Ligan
nds, The Chhemical Sociiety, London, 1957.
Agra
awal et al.49 applied
a
a pH tittration method
d to estimate the
t average number of ligannd groups per
meta
al ion, [n with bar
b above], for several meta
al–sulfonamide chelates in dioxane–wate
d
er. The maximum
[n wiith bar above] values obtain
ned indicate 1::1 and 1:2 com
mplexes.
P.211
The linear relation
nship between the pKa of the
e drugs and th
he log of the sttability constaants of their
corre
esponding me
etal ion comple
exes shows th
hat the more basic ligands (d
drugs) give thee more stable
e
chela
ates with ceriu
um (IV), pallad
dium (II), and ccopper (II). A potentiometric
c method wass described in detail
by C
Connors et al.5
50for the inclus
sion-type com
mplexes formed between α-c
cyclodextrin annd substituted
d
benzzoic acids.
Dis
stribution
n Method
d
The method of disstributing a solute between ttwo immiscible
e solvents can
n be used to d etermine the
stabiility constant for
f certain com
mplexes. The ccomplexation of iodine by potassium iodidde may be use
ed as
an example to illusstrate the method. The equiilibrium reactio
on in its simple
est form is
Additional steps also occur in po
olyiodide form
mation; for example, 2I- + 2I2 [right harpooon over left
2harp
poon] I 6 may occur at highe
er concentratio
ons, but it nee
ed not be cons
sidered here.
Exa
ample 10-2
Free
e and Total Iodine
Whe
en iodine is distributed
d
be
etween wate
er (w) at 25°C
C and carbon
n disulfide ass the organic
phasse (o), as de
epicted inFigu
ure 10-11, th
he distribution
n constant K(o/w)
K
= Co/C
Cw is found to
o be
625. When it is distributed
d
be
etween a 0.1
1250 M solution of potass
sium iodide aand carbon
disu
ulfide, the con
ncentration of
o iodine in th
he organic so
olvent is foun
nd to be 0.18896 mole/liter.
Whe
en the aqueo
ous KI solutio
on is analyze
ed, the conce
entration of io
odine is founnd to be 0.02832
mole
e/liter.
In su
ummary, the
e results are as
a follows:


Total co
oncentration of
o I2 in the aq
queous
layer (fre
ee + complex
xed iodine): 0.02832, mo
ole/liter
389
Dr. Murtadha Alshareifi e-Library


Total co
oncentration of
o KI in the a
aqueous
o layer (free + complexed KI): 0.1250 mole/liter
m
Concenttration of I2 in
n the CS2 layyer (free): 0.1
1896 mole/litter
o Distribution coefficient, K
K(o/w) = [I2]o/[I
/ 2]w = 625
The species com
mmon to both
h phases is tthe free or un
ncomplexed iodine; the ddistribution la
aw
exprresses only the
t concentra
ation of free iodine, wherreas a chemical analysis yields
the ttotal concenttration of iodine in the aq
queous phase
e. The conce
entration of fr
free iodine in the
aque
eous phase is obtained as
a follows:
Fig.. 10-11. Thee distributio
on of iodinee between water
w
and carrbon disulfiide.
To o
obtain the co
oncentration of
o iodine in th
he complex and
a hence th
he concentraation of the
com
mplex, [I3-], on
ne subtracts the free iodin
ne from the total
t
iodine of
o the aqueouus phase:
Acco
ording to equ
uation (10-22
2), I2 and KI ccombine in equimolar
e
concentrationss to form the
com
mplex. Thereffore,
KI iss insoluble in
n carbon disu
ulfide and rem
mains entirely in the aque
eous phase. The
conccentration off free KI is thus
and finally
Higu
uchi and his asssociates inve
estigated the ccomplexing acttion of caffeine
e, polyvinylpyrrrolidone, and
d
polye
ethylene glyco
ols on a numb
ber of acidic drrugs, using the
e partition or distribution
d
meethod. According to
Higu
uchi and Zuck,,51 the reactio
on between ca
affeine and benzoic acid to form
f
the benzooic acid–caffe
eine
comp
plex is
390
Dr. Murtadha Alshareifi e-Library
and the stability co
onstant for the
e reactions at 0°C is
The results varied somewhat, th
he value 37.5 being an averrage stability constant.
c
Guttm
man and
Higu
uchi52 later sh
howed that cafffeine exists in
n aqueous solu
ution primarily
y as a monomeer, dimer, and
d
tetra
amer, which wo
ould account in
i part for the variation in K as observed by
b Higuchi an d Zuck.
Solubility Method
M
Acco
ording to the solubility
s
metho
od, excess qu
uantities of the
e drug are plac
ced in well-stooppered
conta
ainers, together with a soluttion of the com
mplexing agen
nt in various co
oncentrations,, and the bottles
are a
agitated in a constant-tempe
c
erature bath u
until equilibrium
m is attained. Aliquot
A
portionns of the
supe
ernatant liquid are removed and analyzed
d.
Higu
uchi and Lach5
53 used the so
olubility metho
od to investiga
ate the comple
exation of p-am
minobenzoic acid
a
(PAB
BA) by caffeine. The results
s are plotted in
n Figure 10-12
2. The point A at which the liine crosses th
he
verticcal axis is the solubility of
P.212
the d
drug in water. With the addittion of caffeine
e, the solubilitty of PABA rises linearly ow
wing to
plexation. At point
comp
p
B, the so
olution is saturrated with resp
pect to the com
mplex and to tthe drug itself.. The
comp
plex continuess to form and to precipitate from the saturrated system as
a more caffeeine is added. At
pointt C, all the exccess solid PAB
BA has passe
ed into solution
n and has been converted too the complex
x.
Altho
ough the solid drug is exhau
usted and the solution is no longer satura
ated, some of tthe PABA rem
mains
unco
omplexed in so
olution, and it combines furtther with caffe
eine to form hig
gher complexees such as (P
PABA2 cafffeine) as show
wn by the curv
ve at the rightt of the diagram
m.
Fig.. 10-12. Thee solubility of para-am
minobenzoicc acid (PABA) in the prresence of
cafffeine. (From
m T. Higuch
hi and J. L. L
Lack, J. Am
m. Pharm. Assoc.
A
Sci. E
Ed. 43, 525,
1954.)
Exa
ample 10-3
Stoichiometric
c Complex Ratio
R
The following ca
alculations arre made to o btain the sto
oichiometric ratio
r
of the coomplex. The
conccentration off caffeine, corresponding to the platea
au BC, equals the concenntration of
caffe
eine entering
g the complex over this ra
ange, and the quantity off p-aminobennzoic acid
ente
ering the com
mplex is obtained from the
e undissolve
ed solid rema
aining at poinnt B. It is
391
Dr. Murtadha Alshareifi e-Library
com
mputed by subtracting the
e acid in soluttion at the sa
aturation poin
nt B from thee total acid
initia
ally added to
o the mixture,, because th is is the amo
ount yet undissolved that can form the
e
com
mplex.
The concentratio
on of caffeine
e in the plate
eau region is found from Figure 10-122 to be 1.8 × 102
mo
ole/liter. The free, undisso
olved solid P
PABA is equa
al to the totall acid minus the acid in
solu
ution at point B, namely, 7.3
7 × 10-2 - 5
5.5 × 10-2, or 1.8 × 10-2 mole/liter, andd the
stoicchiometric ra
atio is
The complex forrmation is the
erefore writte
en as
and the stability constant for this 1:1 com
mplex is
K may be compu
uted as follow
ws. The conccentration off the complex
x [PABA-Caffffeine] is equal to
the ttotal acid con
ncentration at
a saturation less the solu
ubility [PABA
A] of the acid in water. Th
he
conccentration [C
Caffeine] in th
he solution att equilibrium is equal to the caffeine aadded to the
systtem less the concentratio
on that has be
een converte
ed to the com
mplex. The tootal acid
conccentration off saturation is
s 4.58 × 10-2 mole/liter wh
hen no caffeine is added (solubility off
PAB
BA) and is 5.3
312 × 10-2 mole/liter
m
whe
en 1.00 × 10-2 mole/liter of
o caffeine is added. We have
h
Therefore,
The stability consttants for a num
mber of caffein
ne complexes obtained principally by the distribution an
nd
the ssolubility meth
hods are given
n inTable 10-6 . Stability constants for a nu
umber of otherr drug comple
exes
were
e compiled by Higuchi and Connors.54
C
Ke
enley et al.55 studied waterr-soluble compplexes of vario
ous
ligan
nds with the an
ntiviral drug ac
cyclovir using the solubility method.
Spectrosco
opy and Change
C
T
Transfer Complex
xation
Abso
orption spectro
oscopy in the visible and ulttraviolet region
ns of the spec
ctrum is comm
monly used to
invesstigate electro
on donor–acce
eptor or charge
e transfer com
mplexation.56,57 When iodinne is analyzed
d in a
nonccomplexing so
olvent such as CCl4, a curve
e is obtained with
w a single peak at about 5520 nm. The
soluttion is violet. A solution of io
odine in benze
ene exhibits a maximum shift to 475 nm, aand a new peak of
conssiderably highe
er intensity forr the charge-sshifted band ap
ppears at 300 nm. A solutioon of iodine in
dieth
hyl ether show
ws a still greate
er shift to lowe
er wavelength and the appe
earance of a neew maximum..
Thesse solutions arre red to brow
wn. Their curve
es are shown in Figure 10-13. In benzenee and ether, io
odine
is the
e electron accceptor and the organic solve
ent is the dono
or; in
P.213
CCl4, no complex is formed. The
e shift toward the ultraviolett region becom
mes greater ass the electron
dono
or solvent becomes a strong
ger electron-re
eleasing agent. These spectra arise from the transfer of
o an
electtron from the donor
d
to the acceptor
a
in closse contact in the
t excited sta
ate of the com
mplex. The more
easilly a donor succh as benzene
e or diethyl eth
her releases its electron, as measured byy its ionization
potential, the stron
nger it is as a donor. Ionizattion potentials of a series off donors produuce a straight line
when
n plotted again
nst the freque
ency maximum
m or charge tra
ansfer energie
es (1 nm = 18.663 cal/mole) for
f
,
soluttions of iodine
e in the donor solvents.56
s
57
7
T
Table 10-6 Approxima
A
ate Stabilitty Constantts of Some Caffeine C
Complexes in
i
W
Water at 30°°C
392
Dr. Murtadha Alshareifi e-Library
Compound Complexed with Caffeine
Approximate Stability Constant
Suberic acid
3
Sulfadiazine
7
Picric acid
8
Sulfathiazole
11
o-Phthalic acid
14
Acetylsalicylic acid
15
Benzoic acid (monomer)
18
Salicylic acid
40
p-aminobenzoic acid
48
Butylparaben
50
Benzocaine
59
p-hydroxybenzoic acid
>100
*Compiled from T. Higuchi et al., J. Am. Pharm. Assoc. Sci. Ed. 42, 138,
1953; 43,349, 524, 527, 1954; 45, 290, 1956; 46, 32, 1957. Over 500 such
complexes with other drugs are recorded by T. Higuchi and K. A. Connors.
Phase solubility Techniques, in C. N. Reilley (Ed.), Advances in Analytical
Chemistry and Instrumentation, Wiley, Vol. 4, New York, 1965, pp. 117–
212.
393
Dr. Murtadha Alshareifi e-Library
Fiig. 10-13. Absorption
A
curve
c
of ioddine in the noncomplex
n
xing solventt (1)
caarbontetrachhloride and the compleexing solven
nts (2) benzeene and (3) diethyl
etther. (From H. A. Beneesi and J. A.. Hildebrand
d, J. Am. Ch
hem. Soc. 770, 2832,
19948.)
The complexation constant, K, can
c be obtaine
ed by use of visible
v
and ultrraviolet spectro
roscopy. The
asso
ociation between the donor D and accepto
or A is represe
ented as
wherre K = k1/k-1 iss the equilibriu
um constant fo
or complexatio
on (stability constant) and k11 and k-1 are th
he
interraction rate constants. When
n two molecul es associate according
a
to th
his scheme annd the
abso
orbance A of th
he charge tran
nsfer band is m
measured at a definite wave
elength, K is rreadily obtaine
ed
from the Benesi–H
Hildebrand equ
uation58:
A0 an
nd D0 are initia
al concentratio
ons of the accceptor and don
nor species, re
espectively, inn mole/liter, ε is the
mola
ar absorptivity of the charge transfer comp
plex at its partticular wavelength, and K, thhe stability
consstant, is given in liter/mole or M-1. A plot o
of A0/A versus 1/D0 results in
n a straight linne with a slope
e of
1/(Kεε) and an interrcept of 1/ε, as
s observed in Figure 10-14.
394
Dr. Murtadha Alshareifi e-Library
Fig.. 10-14. A Benesi–Hild
B
debrand plo t for obtaining the stab
bility constaant, K, from
equaation(10-288) for chargee transfer coomplexation
n. (From M. A. Slifkinn, Biochim.
Biopphys. Acta 109, 617, 1965.)
Bora
azan et al.59 in
nvestigated th
he interaction o
of nucleic acid
d bases (electrron acceptorss) with catecho
ol,
epine
ephrine, and isoproterenol (electron
(
dono
ors). Catechols have low ion
nization potenntials and henc
ce a
tendency to donatte electrons. Charge
C
transfe
er complexatio
on was evident as demonstrrated by ultrav
violet
abso
orption measurements. With
h the assumpt ion of 1:1 com
mplexes, the equilibrium connstant, K, for
charrge transfer intteraction was obtained from
m Benesi–Hilde
ebrand plots at
a three or fourr temperatures
s,
and ∆
∆H° was obta
ained at these same temperratures from th
he slope of the
e line as plotteed in Figure 10
0-15.
The values of K an
nd the thermodynamic para
ameters ∆G°, ∆H°,
∆
and ∆S° are given in T
Table 10-7.
395
Dr. Murtadha Alshareifi e-Library
Fig. 10-15. Adenine–catechol stability constant for charge transfer complexation
measured at various temperatures at a wavelength of 340 nm. (From F. A. Al-Obeidi
and H. N. Borazan, J. Pharm. Sci. 65, 892, 1976. With permission.)
P.214
Table 10-7 Stability Constant, K, and Thermodynamic Parameters for Charge
Transfer Interaction of Nucleic Acid Bases with Catechol in Aqueous Solution*
Temperature (°C) K (M-)
ΔS° (cal/deg
ΔG° (cal/mole) ΔH° (cal/mole) mole)
Adenine–catechol
9
1.69
-294
18
1.59
-264
37
1.44
-226
-1015
-2.6
-3564
-14
Uracil–catechol
6
0.49
396
18
0.38
560
25
0.32
675
37
0.26
830
*From F. A. Al-Obeidi and H. N. Borazan, J. Pharm. Sci. 65, 892, 1976.
With permission.
Example 10-4
Calculate Molar Absorptivity
When A0/A is plotted against 1/D0 for catechol (electron-donor) solutions containing uracil
(electron acceptor) in 0.1 N HCI at 6°C, 18°C, 25°C, and 37°C, the four lines were observed
to intersect the vertical axis at 0.01041. Total concentration, A0, for uracil was 2 × 10-2 M,
and D0 for catechol ranged from 0.3 to 0.8 M. The slopes of the lines determined by the leastsquares method were as follows:
396
Dr. Murtadha Alshareifi e-Library
6°C
18°C
25°C
37°C
0.02125
0.02738
0.03252
0.0400
02
Calcculate the mo
olar absorptiv
vity and the sstability cons
stant,K. Know
wing K at theese four
temp
peratures, ho
ow does one
e proceed to obtain ∆H°, ∆G°, and ∆S
S°?
The intercept, fro
om the Bene
esi–Hildebran
nd equation, is the reciprocal of the m
molar
abso
orptivity, or 1/0.01041
1
= 96.1.
9
The mo
olar absorptiv
vity, ε, is a constant for a compound or a
com
mplex, indepe
endent of tem
mperature or concentratio
on. K is obtained from thee slope of the
e
four curves:
These K values are then plottted as their logarithms on
o the vertica
al axis of a grraph againstt the
recip
procal of the four temperatures, convverted to kelv
vin. This is a plot of equattion (10-49) and
yield
ds ∆H° from the slope of the line. ∆G ° is calculate
ed from log K at each of tthe four
temp
peratures ussing equation
n (10-48), in w
which the tem
mperature, T,
T is expresseed in kelvin. ∆S°
is fin
nally obtained using relattion: ∆G° = ∆
∆H° –T∆S°. The
T answers to this sampple problem are
a
give
en inTable 10
0-7.
Web
bb and Thompson60 studied
d the possible role of electro
on donor–acce
eptor complexxes in drug–
receptor binding using
u
quinoline
e and naphtha
alene derivativ
ves as model electron
e
donorrs and a
trinitrrofluorene derrivative as the electron acce
eptor. The most favorable arrangement foor the donor 8amin
noquinoline (heavy lines) an
nd the accepto
or 9-dicyanomethylene trinitrofluorene (ligght lines), as
calcu
ulated by a qu
uantum chemic
cal method, iss shown below
w:
397
Dr. Murtadha Alshareifi e-Library
Filled
d circles are nitrogen
n
atoms
s and open cirrcles oxygen atoms.
a
The donor lies abovee the acceptorr
mole
ecule at an inte
ermolecular distance of abo
out 3.35 Å and
d is attached by
b a binding ennergy of -5.7
kcal//mole. The negative sign sig
gnifies a positiive binding forrce.
Oth
her Methods
A nu
umber of otherr methods are available for sstudying the complexation
c
of
o metal and oorganic molecu
ular
comp
plexes. They include
i
NMR and
a infrared sspectroscopy, polarography, circular dichrroism, kinetics
s, xray d
diffraction, and
d electron diffrraction. Severa
ral of these willl be discussed
d briefly in thiss section.
Com
mplexation of caffeine
c
with L-tryptophan in
n aqueous solu
ution was inve
estigated by N
Nishijo et
1
al.61
1 using H-NM
MR spectroscopy. Caffeine in
nteracts with L-tryptophan
L
at
a a molar ratioo of 1:1 by parallel
stackking. Complexxation is a result of polarizattion and π - π interactions of
o the aromaticc rings. A possible
mode of parallel sttacking is shown in Figure 1
10-16. This stu
udy demonstrates that trypttophan, which is
presumed to be th
he binding site
e in serum albu
umin for certain drugs, can interact with ccaffeine even as
a
free amino acid. However,
H
P.215
caffe
eine does not interact with other
o
aromaticc amino acids such
s
as L-valine or L-leucinne.
398
Dr. Murtadha Alshareifi e-Library
Fig.. 10-16. Staacking of L-tryptophan (solid line) overlying caffeine
c
(daashed line). The
benzzene ring off tryptophan
n is located above the pyrimidine
p
ring
r of caffe
feine, and th
he
pyrrrole ring of L-tryptophan is above the imidazole ring of caffeine.
c
(Fr
From J. Nish
hijo,
I. Y
Yonetami, E.. Iwamoto, et al., J. Phaarm. Sci. 79
9, 18, 1990. With perm
mission.)
Bora
azan and Koum
mriqian62 stud
died the coil–h
helix transition
n of polyadeny
ylic acid induceed by the bind
ding
of the catecholamines norepinephrine and iso
oproterenol, using circular dichroism.
d
Mosst mRNA
mole
ecules contain
n regions of po
olyadenylic aciid, which are thought
t
to incrrease the stabbility of mRNA
A and
to favor genetic co
ode translation
n. The change
e of the circula
ar dichroism sp
pectrum of pollyadenylic acid
was interpreted ass being due to
o intercalative binding of catecholamines between
b
the sstacked adenin
ne
base
es. These rese
earchers sugg
gested that cattecholamines may exert a control
c
mechannism through
inducction of the co
oil-to-helix tran
nsition of polya
adenylic acid, which influences genetic coode translation
n.
De T
Taeye and Zee
egers-Huyskens63 used inffrared spectros
scopy to inves
stigate the hyddrogen-bonded
comp
plexes involvin
ng polyfunctio
onal bases succh as proton donors.
d
This is
s a very precisse technique fo
or
determining the th
hermodynamic
c parameters i nvolved in hyd
drogen-bond formation
f
and for characteriizing
the in
nteraction site
es when the molecule
m
has sseveral groups
s available to form
f
hydrogenn-bonded. Cafffeine
forms hydrogen-bo
onded comple
exes with vario
ous proton don
nors: phenol, phenol derivat
atives, aliphatic
c
alcoh
hols, and wate
er. From the in
nfrared techniq
que, the prefe
erred hydrogen
n-bonding sitees are the carb
bonyl
functtions of caffein
ne. Seventy percent of the ccomplexes are
e formed at the C==O groupp at position 6 and
30% of the comple
exes at the C=
==O group at p
position 2 of caffeine.
c
El-Sa
aid et al.64 useed conductom
metric
and infrared metho
ods to charactterize 1:1 com
mplexes betwe
een uranyl ace
etate and tetraacycline. The
struccture suggeste
ed for the uran
nyl–tetracyclin
ne complex is shown below.
Key Co
oncept
Drug–Protein Binding
B
The binding of drugs to prote
eins containe
ed in the bod
dy can influen
nce their actiion in a number
of w
ways. Proteins may (a) fac
cilitate the diistribution of drugs throug
ghout the boody, (b) inactiivate
the d
drug by not enabling
e
a su
ufficient conccentration of free drug to develop at tthe receptor site,
or (cc) retard the excretion of a drug. The interaction of
o a drug with
h proteins maay cause (a) the
disp
placement of body hormo
ones or a coa
administered agent, (b) a configuratioonal change in
i
the p
protein, the structurally
s
altered
a
form o
of which is ca
apable of bin
nding a coad ministered
agen
nt, or (c) the formation off a drug–prottein complex
x that itself is biologically active. Thes
se
,
topiccs are discusssed in a num
mber of revie
ews.65 66 Am
mong the pla
asma proteinns, albumin is
s the
mosst important owing
o
to its high
h
concent ration relativ
ve to the othe
er proteins annd also to its
s
399
Dr. Murtadha Alshareifi e-Library
ability to bind bo
oth acidic and
d basic drugss. Another pllasma protein, α1-acid glyycoprotein, has
h
been
n shown to bind
b
numerou
us drugs; thiss protein app
pears to have
e greater affiinity for basic
c
than
n for acidic drug molecule
es.
Pro
otein Binding
A co
omplete analyssis of protein binding,
b
includ
ding the multip
ple equilibria th
hat are involveed, would go
beyo
ond our immed
diate needs. Therefore,
T
onlyy an abbreviatted treatment is given here.
Bin
nding Eq
quilibria
We w
write the intera
action between a group or ffree receptor P in a protein and
a a drug moolecule D as
The equilibrium co
onstant, disreg
garding the diffference betwe
een activities and
a concentraations, is
or
wherre K is the asssociation cons
stant, [P] is the
e concentration of the proteiin in terms of ffree binding sites,
[Df] is the concentration, usually
y given in mole
es, of free drug, sometimes called the ligaand, and [PD] is
the cconcentration of the protein–
–drug complexx. K varies witth temperature
e and would bbe better
repre
esented as K((T); [PD], the symbol
s
for bou
und drug, is so
ometimes written as [Db], annd [D], the free
e
drug, as [Df].
If the
e total protein concentration
n is designated
d as [Pt], we can
c write
or
Subsstituting the exxpression for [P]
[ from equattion (10-32) in
nto (10-31)give
es
P.216
400
Dr. Murtadha Alshareifi e-Library
Let r be the numbe
er of moles off drug bound, [[PD], per mole
e of total prote
ein, [Pt]; then r = [PD]/[Pt], or
The ratio r can alsso be expresse
ed in other un its, such as milligrams
m
of drrug bound, x, pper gram of
prote
ein, m. Equatio
on (10-36) is one
o form of th e Langmuir ad
dsorption isoth
herm. Althouggh it is quite us
seful
for e
expressing pro
otein-binding data,
d
it must no
ot be conclude
ed that obedie
ence to this forrmula necessa
arily
requires that prote
ein binding be an adsorption
n phenomenon
n. Expression (10-36) can bbe converted to a
ar form, convenient for plotting, by invertin
ng it:
linea
If v in
ndependent binding sites arre available, th
he expression
n for r, equation (10-36), is ssimply v times that
for a single site, orr
and equation (10-3
37) becomes
Equa
ation (10-39) produces
p
what is called a K
Klotz reciprocal plot.67
An a
alternative man
nner of writing
g equation (10
0-38) is to rearrrange it first to
o
Data
a presented acccording to equation (10-41)) are known as
a aScatchard plot.67,68 Thee binding of
bishyydroxycoumarrin to human serum
s
albumin
n is shown as a Scatchard plot
p in Figure 110-17.
Grap
phical treatment of data usin
ng equation (1
10-39) heavily weights those
e experimentaal points obtain
ned
at low
w concentratio
ons of free dru
ug, D, and ma
ay therefore lead to misinterpretations reggarding the pro
oteinbindiing behavior at
a high concen
ntrations of fre
ee drug. Equattion (10-41) do
oes not have tthis disadvantage
and is the method of choice for plotting data. Curvature in these
t
plots usually indicatess the existence of
more
e than one typ
pe of binding site.
s
Equa
ations (10-39) and (10-41) cannot
c
be use
ed for the analysis of data if the nature an d the amount of
prote
ein in the expe
erimental syste
em are unkno
own. For these
e situations, Sa
andberg et al..69recommend
ded
the u
use of a slightlly modified forrm of equation
n (10-41):
wherre [Db] is the concentration
c
of bound drug
g. Equation (10
0-42) is plotted as the ratio [Db]/[Df] versu
us
[Db], and in this wa
ay K is determ
mined from the
e slope and vK
K[Pt] is determined from the intercept.
401
Dr. Murtadha Alshareifi e-Library
Fig.. 10-17. A Scatchard
S
pllot showingg the binding
g of bishydrroxycoumar
arin to humaan
seruum albumin at 20°C and 40°C plottted accordiing to equattion(10-41).. Extrapolattion
of thhe two liness to the horiizontal axis,, assuming a single class of sites w
with no
elecctrostatic intteraction, giives an apprroximate vaalue of 3 forr v. (From M
M. J. Cho, A.
A
G. M
Mitchell, annd M. Pernaarowski, J. P
Pharm. Sci.6
60, 196, 197
71; 60, 720,, 1971. With
h
perm
mission.) Thhe inset is a Langmuir adsorption isotherm off the bindingg data plotted
accoording to eqquation(10-3
36).
The Scatchard plo
ot yields a stra
aight line when
n only one clas
ss of binding sites
s
is presennt. Frequently in
drug-binding studies, nclasses of
o sites exist, e
each class i having vi sites with
w a unique association
consstant Ki. In succh a case, the plot of r/[Df] vversus r is not linear but exh
hibits a curvatuure that sugge
ests
the p
presence of more
m
than one class of bindin
ng sites. The data
d
in Figure 10-17 were aanalyzed in terrms
of on
ne class of site
es for simplific
cation. The plo
ots at 20°C an
nd 40°C clearly
y show that m
multiple sites arre
invollved. Blancharrd et al.70 rev
viewed the casse of multiple classes
c
of site
es. Equation (110-38) is then
writte
en as
or
previously note
ed, only v and K need to be evaluated wh
hen the sites are
a all of one cclass.
As p
Whe
en n classes off sites exist, equations(10-4
e
43 and 10-44) can be written as
The binding constant, Kn, in the term on the riight is small, indicating extremely weak aaffinity of the drug
d
for th
he sites, but th
his class may have a large n
number of site
es and so be considered
c
unssaturable.
P.217
Equilibrium
m Dialysis
s (ED) an
nd Ultrafilltration (UF)
402
Dr. Murtadha Alshareifi e-Library
A nu
umber of methods are used to determine tthe amount off drug bound to a protein. E quilibrium dialysis,
ultraffiltration, and electrophores
sis are the classsic technique
es used, and in
n recent yearss other method
ds,
such
h as gel filtratio
on and nuclea
ar magnetic ressonance, have been used with
w satisfactoory results. We
e
shalll discuss the equilibrium
e
dia
alysis, ultrafiltra
ration, and kinetic methods.
The equilibrium dialysis procedu
ure was refine
ed by Klotz et al.71 for study
ying the comp lexation betwe
een
meta
al ions or small molecules and macromole
ecules that ca
annot pass thro
ough a semipeermeable
mem
mbrane.
Acco
ording to the equilibrium
e
dia
alysis method, the serum alb
bumin (or othe
er protein undeer investigatio
on) is
place
ed in a Visking
g cellulose tub
bing (Visking C
Corporation, Chicago)
C
or sim
milar dialyzingg membrane. The
T
tubes are tied securely and suspended in vesssels containin
ng the drug in various conceentrations. Ion
nic
stren
ngth and some
etimes hydrog
gen ion concen
ntration are ad
djusted to definite values, annd controls an
nd
blankks are run to account
a
for the
e adsorption o
of the drug and
d the protein on
o the membra
rane.
If bin
nding occurs, the
t drug concentration in th
he sac containing the protein
n is greater at equilibrium th
han
the cconcentration of drug in the vessel outside
e the sac. Sam
mples are rem
moved and anaalyzed to obta
ain
the cconcentrationss of free and complexed
c
dru
ug.
Equilibrium dialysiis is the classic technique fo
or protein bind
ding and remains the most ppopular metho
od.
Som
me potential errrors associate
ed with this tecchnique are th
he possible bin
nding of drug tto the membra
ane,
transsfer of substan
ntial amounts of drug from tthe plasma to the buffer side
e of the membbrane, and osm
motic
volum
me shifts of flu
uid to the plasma side. Toze
er et al.72 dev
veloped mathe
ematical equattions to calculate
and correct for the
e magnitude of fluid shifts. B
Briggs et al.73 proposed a modified
m
equili brium dialysis
s
techn
nique to minim
mize experime
ental errors forr the determination of low le
evels of ligand or small
mole
ecules.
Ultra
afiltration meth
hods are perha
aps more convvenient for the
e routine deterrmination becaause they are less
time--consuming. The
T ultrafiltration method is similar to equilibrium dialys
sis in that maccromolecules such
s
as se
erum albumin are separated
d from small d
drug molecules
s. Hydraulic pressure or cenntrifugation is used
in ulttrafiltration to force
f
the solve
ent and the sm
mall molecules
s, unbound drug, through thhe membrane
while
e preventing th
he passage off the drug bou
und to the prottein. This ultrafiltrate is then analyzed by
specctrophotometryy or other suita
able technique
e.
The concentration
n of the drug th
hat is free and
d unbound, Df, is obtained by use of the B
Beer's law
equa
ation:
wherre A is the spe
ectrophotomettric absorbancce (dimensionless), ε is the molar absorpttivity, determin
ned
indep
pendently for each drug, c (D
( fin binding sstudies) is the concentration
n of the free drrug in the
ultraffiltrate in mole
es/liter, and b is the optical p
path length of the spectroph
hotometer celll, ordinarily 1 cm.
c
The following exam
mple outlines the steps invo
olved in calculating the Scattchard r value and the
percentage of drug
g bound.
Exa
ample 10-5
Bind
ding to Hum
man Serum
m Albumin
The binding of sulfamethoxypyridazine to
o human serum albumin was studied at 25°C, pH 7.4,
usin
ng the ultrafilttration technique. The co
oncentration of the drug under
u
study, [Dt], is 3.24 × 105
mo
ole/liter and the
t human se
erum albumiin concentrattion, [Pt], is 1.0 × 10-4 moole/liter. Afterr
equiilibration the ultrafiltrate has
h an absorrbance, A, off 0.559 at 540 nm in a ceell whose opttical
path
h length, b, iss 1 cm. The molar
m
absorp
ptivity, ε, of th
he drug is 5.6 × 104 liter/m
mole cm.
Calcculate the Sccatchard r value and the p
percentage of
o drug bound.
The concentratio
on of free (un
nbound) drug
g, [Df], is give
en by
The concentratio
on of bound drug,
d
[Db], iss given by
The r value is
403
Dr. Murtadha Alshareifi e-Library
The percentage of bound dru
ug is [Db]/[Dt] × 100 = 69%
%.
A po
otential error in
n ultrafiltration techniques m
may result from
m the drug bind
ding to the meembrane. The
choicce between ulltrafiltration an
nd equilibrium dialysis methods depends on
P.218
the ccharacteristicss of the drug. The
T two techn
niques have be
een compared
d in several prootein-binding
studiies.74,75,76
Fig.. 10-18. Thee dynamic dialysis
d
plott for determ
mining the co
oncentrationn of bound
drugg in a proteiin solution (From
(
M. C
C. Meyer an
nd D. E. Gutttman, J. Phharm. Sci. 57,
1627, 1968. Wiith permissiion).
Key Co
oncept
Protein Binding
Prottein binding (PB)
(
plays an
n important rrole in the ph
harmacokinetics and phaarmacodynam
mics
of a drug. The exxtent of PB in the plasma
a or tissue co
ontrols the vo
olume of disttribution and
affeccts both hepatic and rena
al clearance.. In many cas
ses, the free drug concenntration, rath
her
than
n the total concentration in plasma, iss correlated to
o the effect. Drug displaccement from
drug
g–protein com
mplex can oc
ccur by direcct competition
n of two drug
gs for the sam
me binding site
s
and is important with drugs that are highlly bound (>9
95%), for which a small diisplacement of
boun
nd drug can greatly incre
ease the free
e drug concentration in the plasma. Inn order to
mea
asure free fra
action or PB of a drug, ulttrafiltration (U
UF), ultracen
ntrifugation, eequilibrium
dialyysis (ED), ch
hromatograph
hy, spectroph
hotometry, electrophores
e
sis, etc. havee been used.
Esse
ential methodologic aspe
ects of PB stu
udy include the
t selection
n of assay proocedures,
deviices, and ma
aterials. The most commo
only used me
ethod for PB measuremeent is ED, wh
hich
404
Dr. Murtadha Alshareifi e-Library
is be
elieved to be
e less suscep
ptible to expe
erimental artiifacts. Howev
ver, it is timee consuming and
is no
ot suitable fo
or unstable co
ompounds b
because it req
quires substa
antial equilibbration time (3–24
hr) d
depending on
n drugs, mem
mbrane mate
erials, and de
evices. Many
y researchers
rs have used UF
centtrifugal devicces for PB me
easurement.. UF is a simple and rapid
d method in w
which
centtrifugation forces the bufffer containing
g free drugs through the size exclusioon membran
ne
and achieves a fast
f
separatio
on of free fro
om protein-bo
ound drug. However,
H
thee major
disa
advantage of this method is nonspeciffic binding off drugs on filtter membrannes and plastic
deviices. When the drug bind
ds extensivel y to the filtra
ation membra
ane, the ultraafiltrate
conccentration may deviate frrom the true free concenttration. (From
m K.-J. Lee, R. Mower, T.
T
Hollenbeck, J. Castelo,
C
N. Jo
ohnson, P. G
Gordon, P. J. Sinko, K. Ho
olme, and Y..-H. Lee, Pha
arm.
Res. 20, 1015, 2003.
2
With pe
ermission.)
Dy
ynamic Diialysis
Meye
er and Guttma
an77 develope
ed a kinetic m ethod for dete
ermining the concentrationss of bound drug in a
prote
ein solution. The
T method ha
as found favorr in recent years because it is relatively raapid, economic
cal in
terms of the amou
unt of protein required,
r
and rreadily applied
d to the study of competitivee inhibition of
prote
ein binding. It is discussed here
h
in some d
detail. The me
ethod, known as
a dynamic diialysis, is base
ed on
the rrate of disappe
earance of dru
ug from a dialyysis cell that is
s proportional to the concenntration of unb
bound
drug. The apparattus consists off a 400-mL jaccketed (tempe
erature-controlled) beaker innto which 200 mL
of bu
uffer solution is placed. A ce
ellophane dialyysis bag conta
aining 7 mL off drug or drug––protein solutiion is
susp
pended in the buffer solution
n. Both solutio
ons are stirred continuously. Samples of ssolution extern
nal to
the d
dialysis sac arre removed pe
eriodically and analyzed spe
ectrophotomettrically, and ann equivalent
amount of buffer solution
s
is retu
urned to the exxternal solution. The dialysis
s process folloows the rate la
aw
otal drug conc
centration, [Df] , is the concentration of free
e or unbound drug in the dia
alysis
wherre [Dt] is the to
sac, –d[Dt]/dt is the rate of loss of drug from tthe sac, and k is the first-ord
der rate constaant (seeChaptter
13) rrepresentative
e of the diffusio
on process. Th
he factor k can also be referred to as the apparent
perm
meability rate constant
c
for th
he escape of d
drug from the sac.
s
The conc
centration of unnbound drug, [Df],
in the
e sac (protein compartmentt) at a total dru
ug concentratiion [Dt] is calculated using eequation (10-4
45),
know
wing k and the
e rate –d[Dt]/dtt at a particula
ar drug concen
ntration, [Dt]. The
T rate consttant, k, is obta
ained
from the slope of a semilogarithmic plot of [Dt ] versus time when the experiment is connducted in the
e
abse
ence of the pro
otein.
Figure 10-18 illusttrates the type
e of kinetic plott that can be obtained
o
with this system. N
Note that in the
e
presence of proteiin, curve II, the
e rate of loss o
of drug from th
he dialysis sac
c is slowed coompared with the
t
rate in the absencce of protein, curve
c
I. To solvve equation (1
10-47)for free drug concentrration, [Df], it is
nece
essary to determine the slop
pe of curve II a
at various poin
nts in time. This is not donee graphically, but
b
inste
ead it is accura
ately accomplished by first ffitting the time
e-course data to a suitable eempirical equa
ation,
such
h as the follow
wing, using a co
omputer.
The computer fittin
ng provides es
stimates of C1 through C6. The
T values forr d[Dt]/dt can thhen be compu
uted
from equation (10--49), which represents the ffirst derivative of equation (1
10-48):
Finally, once we have
h
a series of
o [Df] values ccomputed from
m equations (1
10-49) and (10047) ccorresponding
g to experimen
ntally determin
ned values of [D
[ t] at each tim
me t, we can pproceed to
calcu
ulate the vario
ous terms for the Scatchard plot.
Exa
ample 10-6*
Calc
culate Scattchard Values
Assu
ume that the
e kinetic data illustrated in
n Figure 10-1
18 were obta
ained under tthe following
-3
cond
ditions: initial drug concentration, [Dt00], is 1 × 10 mole/liter an
nd protein cooncentration is 1
× 10
0-3 mole/liter. Also assum
me that the firrst-order rate
e constant, k, for the conttrol (curve I) was
405
Dr. Murtadha Alshareifi e-Library
dete
ermined to be
e 1.0 hr-1 and
d that fitting o
of curve II to equation (10
0-48) resulteed in the
-4
follo
owing empiriccal constants
s: C1 = 5 × 10
0 mole/liter, C2 = 0.6 hr-1
, C3 = 3 × 1104
mo
ole/liter, C4 = 0.4 hr-1, C5 = 2 × 10-4 m ole/liter, and
dC6 = 0.2 hr-1.
Calcculate the Sccatchard valu
ues (the Scattchard plot was
w discussed in the prevvious section
n)
for r and r[Df] if, during the diialysis in the presence off protein, the experimentaally determin
ned
valu
ue for [Dt] was 4.2 × 10-4 mole/liter
m
at 2 hr. Here, r = [Db]/Pt, wh
here [Db] is ddrug bound
and Pt is total prrotein concen
ntration. We have
Usin
ng equation (10-49),
(
whe
ere the (2) in the exponen
nt stands for 2 hr. Thus,
It folllows that at 2 hr,
Additional pointss for the Scattchard plot w
would be obta
ained in a sim
milar fashionn, using the data
d
obta
ained at vario
ous points throughout the
e dialysis. Ac
ccordingly, th
his series of ccalculations
perm
mits one to prepare
p
a Sca
atchard plot (seeFig. 10-17).
P.219
Judiss78 investigatted the binding
g of phenol an
nd phenol deriv
vatives by whole human seerum using the
e
dyna
amic dialysis te
echnique and presented the
e results in the
e form of Scatchard plots.
406
Dr. Murtadha Alshareifi e-Library
Fig.. 10-19. Schhematic view
w of hydropphobic interraction. (a) Two hydropphobic
mollecules are separately
s
enclosed
e
in ccages, surro
ounded in an
n orderly faashion by
hyddrogen-bondded moleculles of waterr (open circlles). The staate at (b) is somewhat
favoored by breaaking of thee water cagees of (a) to yield
y
a less ordered arrrangement and
a
an ooverall entroopy increasee of the systtem. Van deer Waals atttraction of tthe two
hyddrophobic sppecies also contributes
c
to the hydro
ophobic interaction.
Hy
ydrophob
bic Interac
ction
Hydrrophobic “bonding,” first pro
oposed by Kau
uzmann,79 is actually not bo
ond formation at all but rath
her
the tendency of hyydrophobic mo
olecules or hyd
drophobic parrts of molecule
es to avoid waater because they
are n
not readily acccommodated in the hydroge
en-bonding strructure of wate
er. Large hydrrophobic species
such
h as proteins avoid
a
the wate
er molecules in
n an aqueous solution insofar as possiblee by associatin
ng
into m
micellelike structures (Chap
pter 15) with th
he nonpolar portions in conttact in the inneer regions of the
t
“mice
elles,” the pola
ar ends facing
g the water mo
olecules. This attraction of hydrophobic
h
sppecies, resultiing
nown as hydro
from their unwelco
ome reception in water, is kn
ophobic bondiing, or, better,, hydrophobic
interraction. It invollves van der Waals
W
forces, hydrogen bon
nding of water molecules in a threedime
ensional structture, and othe
er interactions.. Hydrophobic
c interaction is favored therm
modynamically
y
beca
ause of an incrreased disorder or entropy o
of the water molecules
m
that accompaniess the association of
the n
nonpolar mole
ecules, which squeeze
s
out th
he water. Glob
bular proteins are thought too maintain the
eir
ball-llike structure in
i water becau
use of the hyd
drophobic effect. Hydrophob
bic interaction is depicted
in Fig
gure 10-19.
Nagw
wekar and Kostenbauder80
0 studied hydro
ophobic effectts in drug bind
ding, using as a model of the
prote
ein a copolymer of vinylpyrid
dine and vinyl pyrrolidone. Kristiansen
K
et al.81
a
studied tthe effects of
orga
anic solvents in
n decreasing complex
c
forma
ation between
n small organic
c molecules inn aqueous solu
ution.
Theyy attributed the
e interactions of the organicc species to a significant contribution by bboth hydropho
obic
bond
ding and the unique
u
effects of the water sstructure. They
y suggested th
hat some noncclassic “donorr–
acce
eptor” mechan
nism may be operating to len
nd stability to the complexes formed.
407
Dr. Murtadha Alshareifi e-Library
P.220
Feldman and Gibaldi82 studied the effects of urea, methylurea, and 1,3-dimethylurea on the solubility of
benzoic and salicylic acids in aqueous solution. They concluded that the enhancement of solubility by
urea and its derivatives was a result of hydrophobic bonding rather than complexation. Urea broke up
the hydrogen-bonded water clusters surrounding the nonpolar solute molecules, increasing the entropy
of the system and producing a driving force for solubilization of benzoic and salicylic acids. It may be
possible that the ureas formed channel complexes with these aromatic acids as shown inFigure 10-3 a,
b, and c.
The interaction of drugs with proteins in the body may involve hydrophobic bonding at least in part, and
this force in turn may affect the metabolism, excretion, and biologic activity of a drug.
Self-Association
Some drug molecules may self-associate to form dimers, trimers, or aggregates of larger sizes. A high
degree of association may lead to formation of micelles, depending on the nature of the molecule
(Chapter 16). Doxorubicin forms dimers, the process being influenced by buffer composition and ionic
strength. The formation of tetramers is favored by hydrophobic stacking aggregation.83 Self-association
may affect solubility, diffusion, transport through membranes, and therapeutic action. Insulin shows
concentration-dependent self-association, which leads to complications in the treatment of diabetes.
Aggregation is of particular importance in long-term insulin devices, where insulin crystals have been
observed. The initial step of insulin self-association is a hydrophobic interaction of the monomers to form
dimers, which further associate into larger aggregates. The process is favored at higher concentrations
of insulin.84 Addition of urea at nontoxic concentrations (1.0–3 mg/mL) has been shown to inhibit the
self-association of insulin. Urea breaks up the “icebergs” in liquid water and associates with structured
water by hydrogen bonding, taking an active part in the formation of a more open “lattice” structure.85
Sodium salicylate improves the rectal absorption of a number of drugs, all of them exhibiting selfassociation. Touitou and Fisher86chose methylene blue as a model for studying the effect of sodium
salicylate on molecules that self-associate by a process of stacking. Methylene blue is a planar aromatic
dye that forms dimers, trimers, and higher aggregates in aqueous solution. The workers found that
sodium salicylate prevents the self-association of methylene blue. The inhibition of aggregation of
porcine insulin by sodium salicylate results in a 7875-fold increase in solubility.87 Commercial heparin
samples tend to aggregate in storage depending on factors such as temperature and time in storage.88
Factors Affecting Complexation and Protein Binding
Kenley et al.55 investigated the role of hydrophobicity in the formation of water-soluble complexes. The
logarithm of the ligand partition coefficient between octanol and water was chosen as a measure of
hydrophobicity of the ligand. The authors found a significant correlation between the stability constant of
the complexes and the hydrophobicity of the ligands. Electrostatic forces were not considered as
important because all compounds studied were uncharged under the conditions investigated. Donor–
acceptor properties expressed in terms of orbital energies (from quantum chemical calculations) and
relative donor–acceptor strengths correlated poorly with the formation constants of the complex. It was
suggested that ligand hydrophobicity is the main contribution to the formation of water-soluble
complexes. Coulson and Smith89 found that the more hydrophobic chlorobiocin analogues showed the
highest percentage of drug bound to human serum albumin. These workers suggested that chlorobiocin
analogues bind to human albumin at the same site as warfarin. This site consists of two noncoplanar
hydrophobic areas and a cationic group. Warfarin, an anticoagulant, serves as a model drug in proteinbinding studies because it is extensively but weakly bound. Thus, many drugs are able to compete with
and displace warfarin from its binding sites. The displacement may result in a sudden increase of the
free (unbound) fraction in plasma, leading to toxicity, because only the free fraction of a drug is
pharmacologically active. Diana et al.90investigated the displacement of warfarin by nonsteroidal antiinflammatory drugs. Table 10-8 shows the variation of the stability constant, K, and the number of
binding sites, n, of the complex albumin–warfarin after addition of competing drugs. Azapropazone
408
Dr. Murtadha Alshareifi e-Library
markedly decreases the K value, suggesting that both drugs, warfarin and azapropazone, compete for
the same binding site on albumin. Phenylbutazone also competes strongly for the binding site on
albumin. Conversely, tolmetin may increase K, as suggested by the authors, by a conformational
change in the albumin molecule that favors warfarin binding. The other drugs (see Table 10-8) decrease
the K value of warfarin to a lesser extent, indicating that they do not share exclusively the same binding
site as that of warfarin.
Plaizier-Vercammen91 studied the effect of polar organic solvents on the binding of salicylic acid to
povidone. He
P.221
found that in water–ethanol and water–propylene glycol mixtures, the stability constant of the complex
decreased as the dielectric constant of the medium was lowered. Such a dependence was attributed to
hydrophobic interaction and can be explained as follows. Lowering the dielectric constant decreases
polarity of the aqueous medium. Because most drugs are less polar than water, their affinity to the
medium increases when the dielectric constant decreases. As a result, the binding to the
macromolecule is reduced.
Table 10-8 Binding Parameters (± Standard Deviation) for Warfarin in the
Presence of Displacing Drugs*
Competing Drug
Racemic Warfarin
K × 10-5 M-1
n
None
1.1 ± 0.0
6.1 ± 0.2
Azapropazone
1.4 ± 0.1
0.19 ± 0.02
Phenylbutazone
1.3 ± 0.2
0.33 ± 0.06
Naproxen
0.7 ± 0.0
2.4 ± 0.2
Ibuprofen
1.2 ± 0.2
3.1 ± 0.4
Mefenamic acid
0.9 ± 0.0
3.4 ± 0.2
Tolmetin
0.8 ± 0.0
12.6 ± 0.6
*From F. J. Diana, K. Veronich, and A. L. Kapoor, J. Pharm. Sci. 78, 195,
1989. With permission.
Protein binding has been related to the solubility parameter δ of drugs. Bustamante and Selles92 found
that the percentage of drug bound to albumin in a series of sulfonamides showed a maximum at ∆ =
12.33 cal1/2 cm-3/2. This value closely corresponds to the δ value of the postulated binding site on
albumin for sulfonamides and suggests that the closer the solubility parameter of a drug to the δ value of
its binding site, the greater is the binding.
409
Dr. Murtadha Alshareifi e-Library
Chapter Summary
Complexation is widely used in the pharmaceutical sciences to improve properties such as
solubility. The three classes of complexes or coordination compounds were discussed in the
context to identify pharmaceutically relevant examples. The physical properties of chelates
and what differentiates them from organic molecular complexes were also described. The
types of forces that hold together organic molecular complexes also play an important role in
determining the function and use of complexes in the pharmaceutical sciences. One widely
used complex system, the cyclodextrins, was described in detail with respect to
pharmaceutical applications. The stoichiometry and stability of complexes was described as
well as methods of analysis to determine their strengths and weaknesses. Protein binding is
important for many drug substances. The ways in which protein binding could influence drug
action were discussed. Also, methods such as the equilibrium dialysis and ultrafiltration were
described for determining protein binding.
Practice problems for this chapter can be found at thePoint.lww.com/Sinko6e.
References
1. L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, New York, 1940, p.
81.
2. A. Werner, Vjschr. nuturf. Ges. Zurich, 36, 129, 1891.
3. P. Mohanakrishnan, C. F. Chignell, and R. H. Cox, J. Pharm. Sci. 74, 61, 1985.
4. J. E. Whitaker and A. M. Hoyt, Jr., J. Pharm. Sci. 73, 1184, 1984.
5. A. E. Martell and M. Calvin, Chemistry of the Metal Chelate Compounds, Prentice Hall, New York,
1952.
6. L. B. Clapp, in J. R. Bailar, Jr. (Ed.), The Chemistry of the Coordination Compounds, Reinhold, New
York, 1956, Chapter 17.
7. G. Canti, A. Scozzafava, G. Ciciani, and G. Renzi, J. Pharm. Sci. 69, 1220, 1980.
8. F. J. Bullock, in M. Florkin and E. H. Stotz (Eds.),Comprehensive Biochemistry, Elsevier, New York,
1967, pp. 82–85.
9. J. Buxeraud, A. C. Absil, and C. Raby, J. Pharm. Sci. 73,1687, 1984.
10. T. Higuchi and J. L. Lach, J. Am. Pharm. Assoc. Sci. Ed. 43,349, 525, 527, 1954; T. Higuchi and D.
A. Zuck, J. Am. Pharm. Assoc. Sci. Ed. 42, 132, 1953.
11. T. Higuchi and L. Lachman, J. Am. Pharm. Assoc. Sci. Ed.44, 521, 1955; L. Lachman, L. J. Ravin,
and T. Higuchi, J. Am. Pharm. Assoc. Sci. Ed. 45, 290, 1956; L. Lachman and T. Higuchi, J. Am. Pharm.
Assoc. Sci. Ed. 46, 32, 1957.
12. T. Higuchi and I. H. Pitman, J. Pharm. Sci. 62, 55, 1973.
13. P. York and A. Saleh, J. Pharm. Sci. 65, 493, 1976.
14. A. Marcus, Drug Cosmet. Ind. 79, 456, 1956.
15. J. A. Plaizier-Vercammen and R. E. De Nève, J. Pharm. Sci.70, 1252, 1981; J. A. PlaizierVercammen, J. Pharm. Sci. 76,817, 1987.
16. K. H. Frömming, W. Ditter, and D. Horn, J. Pharm. Sci. 70,738, 1981.
17. D. S. Hayward, R. A. Kenley, and D. R. Jenke, Int. J. Pharm.59, 245, 1990.
18. T. Hosono, S. Tsuchiya, and H. Matsumaru, J. Pharm. Sci.69, 824, 1980.
19. B. J. Forman and L. T. Grady, J. Pharm. Sci. 58, 1262, 1969.
20. H. van Olphen, Introduction to Clay Calloid Chemistry, 2nd Ed., Wiley, New York, 1977, pp. 66–68.
21. Merck Index, 11th Ed., Merck, Rahway, N.J., 1989, p. 365.
22. J. A. A. Ketelaar, Chemical Constitution, Elsevier, New York, 1958, p. 365.
23. H. M. Powell and D. E. Palin, J. Chem. Soc. 208, 1947; J. Chem. Soc. 61, 571, 815, 1948; J. Chem.
Soc. 298, 300, 468, 1950; J. Chem. Soc. 2658, 1954.
24. (a) S. G. Frank, J. Pharm. Sci. 64, 1585, 1975; (b) M. E. Davis and M. E. Brewster, Nat. Rev. Drug
Discov. 3, 1023, 2004.
25. K. Cabrera and G. Schwinn, Am. Lab. 22, 22, 24, 26, 28, 1990.
410
Dr. Murtadha Alshareifi e-Library
26. D. Duchêne and D. Wouessidjewe, Pharm. Tech. 14(6), 26, 28, 32, 34, 1990; 14(8), 22, 24, 26, 28,
30, 1990.
27. O. Beckers, J. H. Beijnen, E. H. G. Bramel, M. Otagiri, A. Bult, and W. J. M. Underberg, Int. J.
Pharm. 52, 239, 1989.
28. T. Bakensfield, B. W. Müller, M. Wiese, and J. Seydel, Pharm. Res. 7, 484, 1990.
29. J. L. Lach and J. Cohen, J. Pharm. Sci. 52, 137, 1963; W. A. Pauli and J. Lach, J. Pharm.
Sci. 54, 1945, 1965; J. Lach and W. A. Pauli, J. Pharm. Sci. 55, 32, 1966.
30. D. Amdidouche, H. Darrouzet, D. Duchêne, and M.-C. Poelman, Int. J. Pharm. 54, 175, 1989.
31. M. A. Hassan, M. S. Suleiman, and N. M. Najib, Int. J. Pharm. 58, 19, 1990.
32. F. Kedzierewicz, M. Hoffman, and P. Maincent, Int. J. Pharm.58, 22, 1990.
33. M. E. Brewster, K. S. Estes, and N. Bodor, Int. J. Pharm. 59,231, 1990; M. E. Brewster, K. S. Estes,
T. Loftsson, R. Perchalski, H. Derendorf, G. Mullersman, and N. Bodor, J. Pharm. Sci. 77, 981, 1988.
34. A. R. Green and J. K. Guillory, J. Pharm. Sci. 78, 427, 1989.
35. B. W. Müller and U. Brauns, J. Pharm. Sci. 75, 571, 1986.
36. J. Pitha, E. J. Anaissie, and K. Uekama, J. Pharm. Sci. 76,788, 1987.
37. Y. Horiuchi, F. Hiriyama, and K. Uekama, J. Pharm. Sci. 79,128, 1990.
38. F. Hirayama, N. Hirashima, K. Abe, K. Uekama, T. Ijitsu, and M. Ueno, J. Pharm. Sci. 77, 233, 1988.
39. F. M. Andersen, H. Bungaard, and H. B. Mengel, Int. J. Pharm. 21, 51, 1984.
40. K. A. Connors, in A Textbook of Pharmaceutical Analysis, Wiley, New York, 1982, Chapter 4; K. A.
Connors, Binding Constants, Wiley, New York, 1987.
41. P. Job, Ann. Chim. 9, 113, 1928.
42. A. E. Martell and M. Calvin, Chemistry of the Metal Chelate Compounds, Prentice Hall, New York,
1952, p. 98.
43. M. E. Bent and C. L. French, J. Am. Chem. Soc. 63, 568, 1941; R. E. Moore and R. C. Anderson, J.
Am. Chem. Soc. 67,168, 1945.
44. W. C. Vosburgh, et al., J. Am. Chem. Soc. 63, 437, 1941; J. Am. Chem. Soc. 64, 1630, 1942.
45. A. Osman and R. Abu-Eittah, J. Pharm. Sci. 69, 1164, 1980.
46. J. Bjerrum, Metal Amine Formation in Aqueous Solution, Haase, Copenhagen, 1941.
47. M. Pecar, N. Kujundzic, and J. Pazman, J. Pharm. Sci. 66,330, 1977; M. Pecar, N. Kujundzic, B.
Mlinarevic, D. Cerina, B. Horvat, and M. Veric, J. Pharm. Sci. 64, 970, 1975.
P.222
48. B. J. Sandmann and H. T. Luk, J. Pharm. Sci. 75, 73, 1986.
49. Y. K. Agrawal, R. Giridhar, and S. K. Menon, J. Pharm. Sci.76, 903, 1987.
50. K. A. Connors, S.-F. Lin, and A. B. Wong, J. Pharm. Sci. 71,217, 1982.
51. T. Higuchi and A. A. Zuck, J. Am. Pharm. Assoc. Sci. Ed. 42,132, 1953.
52. D. Guttman and T. Higuchi, J. Am. Pharm. Assoc. Sci. Ed.46, 4, 1957.
53. T. Higuchi and J. L. Lach, J. Am. Pharm. Assoc. Sci. Ed. 43,525, 1954.
54. T. Higuchi and K. A. Connors, in C. N. Reilley (Ed.),Advances in Analytical Chemistry and
Instrumentation, Vol. 4, Wiley, New York, 1965, pp. 117–212.
55. R. A. Kenley, S. E. Jackson, J. S. Winterle, Y. Shunko, and G. C. Visor, J. Pharm. Sci. 75, 648,
1986.
56. R. Foster (Ed.), Molecular Complexes, Vol. 1, Elek, London, 1973.
57. M. A. Slifkin, Charge Transfer Interactions of Biomolecules, Academic Press, New York, 1971,
Chapters 1 and 2.
58. H. A. Benesi and J. H. Hildebrand, J. Am. Chem. Soc. 70,2832, 1948.
59. F. A. Al-Obeidi and H. N. Borazan, J. Pharm. Sci. 65, 892, 1976; J. Pharm. Sci. 65, 982, 1976; H. M.
Taka, F. A. Al-Obeidi, and H. N. Borazan, J. Pharm. Sci. 68, 631, 1979; N. I. Al-Ani and H. N. Borazan,
J. Pharm. Sci. 67, 1381, 1978; H. N. Borazan and Y. H. Ajeena, J. Pharm. Sci. 69, 990, 1980; J. Pharm.
Sci.77, 544, 1988.
411
Dr. Murtadha Alshareifi e-Library
60. N. E. Webb and C. C. Thompson, J. Pharm. Sci. 67, 165, 1978.
61. J. Nishijo, I. Yonetami, E. Iwamoto, S. Tokura, and K. Tagahara, J. Pharm. Sci. 79, 14, 1990.
62. H. N. Borazan and S. N. Koumriqian, J. Pharm. Sci. 72,1450, 1983.
63. J. De Taeye and Th. Zeegers-Huyskens, J. Pharm. Sci. 74,660, 1985.
64. A. El-Said, E. M. Khairy, and A. Kasem, J. Pharm. Sci. 63,1453, 1974.
65. A Goldstein, Pharmacol. Rev. 1, 102, 1949.
66. J. J. Vallner, J. Pharm. Sci. 66, 447, 1977.
67. C. K. Svensson, M. N. Woodruff, and D. Lalka, in W. E. Evans, J. J. Schentag, and W. J. Jusko
nd
(Eds.), Applied Pharmacokinetics, 2 Ed., Applied Therapeutics, Spokane, Wash., 1986, Chapter 7.
68. G. Scatchard, Ann. N. Y. Acad. Sci. 51, 660, 1949.
69. A. A. Sandberg, H. Rosenthal, S. L. Schneider, and W. R. Slaunwhite, in T. Nakao, G. Pincus, and
J. F. Tait (Eds.), Steroid Dynamics, Academic Press, New York, 1966, p. 33.
70. J. Blanchard, W. T. Fink, and J. P. Duffy, J. Pharm. Sci. 66,1470, 1977.
71. I. M. Klotz, F. M. Walker, and R. B. Puvan, J. Am. Chem. Soc.68, 1486, 1946.
72. T. N. Tozer, J. G. Gambertoglio, D. E. Furst, D. S. Avery, and N. H. G. Holford, J. Pharm.
Sci. 72, 1442, 1983.
73. C. J. Briggs, J. W. Hubbard, C. Savage, and D. Smith, J. Pharm. Sci. 72, 918, 1983.
74. A. Zini, J. Barre, G. Defer, J. P. Jeanniot, G. Houin, and J. P. Tillement, J. Pharm. Sci. 74, 530,
1984.
75. E. Okezaki, T. Teresaki, M. Nakamura, O. Nagata, H. Kato, and A. Tsuji, J. Pharm. Sci. 78, 504,
1989.
76. J. W. Melten, A. J. Wittebrood, H. J. J. Williams, G. H. Faber, J. Wemer, and D. B. Faber, J. Pharm.
Sci. 74, 692, 1985.
77. M. C. Meyer and D. E. Guttman, J. Pharm. Sci. 57, 1627, 1968.
78. J. Judis, J. Pharm. Sci. 71, 1145, 1982.
79. W. Kauzmann, Adv. Protein Chem. 14, 1, 1959.
80. J. B. Nagwekar and H. B. Kostenbauder, J. Pharm. Sci. 59,751, 1970.
81. H. Kristiansen, M. Nakano, N. I. Nakano, and T. Higuchi, J. Pharm. Sci. 59, 1103, 1970.
82. S. Feldman and M. Gibaldi, J. Pharm. Sci. 56, 370, 1967.
83. M. Menozzi, L. Valentini, E. Vannini, and F. Arcamone, J. Pharm. Sci. 73, 6, 1984.
84. S. Sato, C. D. Ebert, and S. W. Kim, J. Pharm. Sci. 72, 228, 1983.
85. A. M. Saleh, A. R. Ebian, and M. A. Etman, J. Pharm. Sci. 75,644, 1986.
86. E. Touitou and P. Fisher, J. Pharm. Sci. 75, 384, 1986.
87. E. Touitou, F. Alhaique, P. Fisher, A. Memoli, F. M. Riccieri, and E. Santucci, J. Pharm. Sci. 76, 791,
1987.
88. T. J. Racey, P. Rochon, D. V. C. Awang, and G. A. Neville, J. Pharm. Sci. 76, 314, 1987.
89. J. Coulson and V. J. Smith, J. Pharm. Sci. 69, 799, 1980.
90. F. J. Diana, K. Veronich, and A. Kapoor, J. Pharm. Sci. 78,195, 1989.
91. J. A. Plaizier-Vercammen, J. Pharm. Sci. 72, 1042, 1983.
92. P. Bustamante and E. Selles, J. Pharm. Sci. 75, 639, 1986.
Recommended Readings
H. Dodziuk, Cyclodextrins and Their Complexes: Chemistry, Analytical Methods, Applications, WileyVCH, Weinheim, Germany, 2006.
J. A. Goodrich and J. F. Kugel, Binding and Kinetics for Molecular Biologists, Cold Spring Harbor
Laboratory Press, New York, 2007.
Chapter Legacy
Fifth Edition: published as Chapter 11 (Complexation and Protein Binding). Updated by
Patrick Sinko.
Sixth Edition: published as Chapter 9 (Complexation and Protein Binding). Updated by
Patrick Sinko.
412
Dr. Murtadha Alshareifi e-Library
413
Dr. Murtadha Alshareifi e-Library
11 Diffusion
Chapter Objectives
At the conclusion of this chapter the student should be able to:
1. Define diffusion and describe relevant examples in the pharmaceutical sciences and
the practice of pharmacy.
2. Understand the processes of dialysis, osmosis, and ultrafiltration as they apply to the
pharmaceutical sciences and the practice of pharmacy.
3. Describe the mechanisms of transport in pharmaceutical systems and identify which
ones are diffusion based.
4. Define and understand Fick's laws of diffusion and their application.
5. Calculate diffusion coefficient, permeability, and lag time.
6. Relate permeability to a rate constant and to resistance.
7. Understand the concepts of steady state, sink conditions, membrane, and diffusion
control.
8. Describe the various driving forces for diffusion, drug absorption, and elimination.
9. Describe multilayer diffusion and calculate component permeabilities.
10. Calculate drug release from a homogeneous solid.
Introduction
The fundamentals of diffusion are discussed in this chapter. Free diffusion of substances through
liquids, solids, and membranes is a process of considerable importance in the pharmaceutical sciences.
Topics of mass transport phenomena applying to the pharmaceutical sciences include the release and
dissolution of drugs from tablets, powders, and granules; lyophilization, ultrafiltration, and other
mechanical processes; release from ointments and suppository bases; passage of water vapor, gases,
drugs, and dosage form additives through coatings, packaging, films, plastic container walls, seals, and
caps; and permeation and distribution of drug molecules in living tissues. This chapter treats the
fundamental basis for diffusion in pharmaceutical systems.
There are several ways that a solute or a solvent can traverse a physical or biologic membrane. The first
example (Fig. 11-1) depicts the flow of molecules through a physical barrier such as a polymeric
membrane. The passage of matter through a solid barrier can occur by simple molecular permeation or
by movement through pores and channels. Molecular diffusion or permeation through nonporous media
depends on the solubility of the permeating molecules in the bulk membrane (Fig. 11-1a), whereas a
second process can involve passage of a substance through solvent-filled pores of a membrane (Fig.
11-1b) and is influenced by the relative size of the penetrating molecules and the diameter and shape of
the pores. The transport of a drug through a polymeric membrane involves dissolution of the drug in the
matrix of the membrane and is an example of simple molecular diffusion. A second example relates to
drug and solvent transport across the skin. Passage through human skin of steroidal molecules
substituted with hydrophilic groups may predominantly involve transport through hair follicles, sebum
ducts, and sweat pores in the epidermis (Fig. 11-19). Perhaps a better representation of a membrane on
the molecular scale is a matted arrangement of polymer strands with branching and intersecting
channels as shown in Figure 11-1c. Depending on the size and shape of the diffusing molecules, they
may pass through the tortuous pores formed by the overlapping strands of polymer. If it is too large for
such channel transport, the diffusant may dissolve in the polymer matrix and pass through the film by
simple diffusion. Diffusion also plays an important role in drug and nutrient transport in biologic
membranes in the brain, intestines, kidneys, and liver. In addition to diffusion through the lipoidal
membrane, several other transport
P.224
mechanisms have been characterized in biologic membranes including energy-dependent and energyindependent carrier-mediated transport as well as diffusion through the paracellular spaces between
414
Dr. Murtadha Alshareifi e-Library
cells. The multitude of mechanisms of transport across various mucosal membranes will be introduced
later in this chapter. Several pharmaceutically important diffusion-based processes are covered in this
and subsequent chapters.
Key Concept
Diffusion
Diffusion is defined as a process of mass transfer of individual molecules of a substance
brought about by random molecular motion and associated with a driving force such as a
concentration gradient. The mass transfer of a solvent (e.g., water) or a solute (e.g., a drug)
forms the basis for many important phenomena in the pharmaceutical sciences. For example,
diffusion of a drug across a biologic membrane is required for a drug to be absorbed into and
eliminated from the body, and even for it to get to the site of action within a particular cell. On
the negative side, the shelf life of a drug product could be significantly reduced if a container
or closure does not prevent solvent or drug loss or if it does not prevent the absorption of
water vapor into the container. These and many more important phenomena have a basis in
diffusion. Drug release from a variety of drug delivery systems, drug absorption and
elimination, dialysis, osmosis, and ultrafiltration are some of the examples covered in this and
other chapters.
Fig. 11-1. (a) Homogeneous membrane without pores. (b) Membrane of dense
material with straight-through pores, as found in certain filler barriers such as
Nucleopore. (c) Cellulose membrane used in the filtration process, showing the
intertwining nature of the fibers and the tortuous channels.
Drug Absorption and Elimination
Diffusion through biologic membranes is an essential step for drugs entering (i.e., absorption) or leaving
(i.e., elimination) the body. It is also an important component along with convection for efficient drug
distribution throughout the body and into tissues and organs. Diffusion can occur through the lipoidal
bilayer of cells. This is termed transcellular diffusion. On the other hand, paracellular diffusion occurs
415
Dr. Murtadha Alshareifi e-Library
through the spaces between adjacent cells. In addition to diffusion, drugs and nutrients also traverse
biologic membranes using membrane transporters and, to a lesser extent, cell surface receptors.
Membrane transporters are specialized proteins that facilitate drug transport across biologic
membranes. The interactions between drugs and transporters can be classified as energy dependent
(i.e., active transport) or energy independent (i.e., facilitated diffusion). Membrane transporters are
located in every organ responsible for the absorption, distribution, metabolism, and excretion (ADME) of
drug substances. Specialized membrane transport mechanisms were covered in more detail in Chapter
12(Biopharmaceutics) and Chapter 13 (Drug Release and Dissolution).
Elementary Drug Release
Elementary drug release is an important process that literally affects nearly every person in everyday
life. Drug release is a multistep process that includes diffusion, disintegration, deaggregation, and
dissolution. These processes are described in this and other chapters. Common examples are the
release of steroids such as hydrocortisone from topical over-the-counter creams and ointments for the
treatment of skin rashes and the release of acetaminophen from a tablet that is taken by mouth. Drug
release must occur before the drug can be pharmacologically active. This includes pharmaceutical
products such as capsules, creams, liquid suspensions, ointments, tablets, and transdermal patches.
Osmosis
Osmosis was originally defined as the passage of both solute and solvent across a membrane but now
refers to an action in which only the solvent is transferred. The solvent passes through a semipermeable
membrane to dilute the solution containing solute and solvent. The passage of solute together with
solvent is now called diffusion or dialysis. Osmotic drug release systems use osmotic pressure as a
driving force for the controlled delivery of drugs. A simple osmotic pump
P.225
consists of an osmotic core (containing drug with or without an osmotic agent) and is coated with a
semipermeable membrane. The semipermeable membrane has an orifice for drug release from the
―pump.‖ The dosage form, after coming in contact with the aqueous fluids, imbibes water at a rate
determined by the fluid permeability of the membrane and osmotic pressure of core formulation. The
osmotic imbibition of water results in high hydrostatic pressure inside the pump, which causes the flow
of the drug solution through the delivery orifice.
Ultrafiltration and Dialysis
Ultrafiltration is used to separate colloidal particles and macromolecules by the use of a membrane.
Hydraulic pressure is used to force the solvent through the membrane, whereas the microporous
membrane prevents the passage of large solute molecules. Ultrafiltration is similar to a process
called reverse osmosis, but a much higher osmotic pressure is developed in reverse osmosis, which is
used in desalination of brackish water. Ultrafiltration is used in the pulp and paper industry and in
research to purify albumin and enzymes. Microfiltration, a process that employs membranes of slightly
larger pore size, 100 nm to several micrometers, removes bacteria from intravenous injections, foods,
and drinking water.1 Hwang and Kammermeyer2 defined dialysisas a separation process based on
unequal rates of passage of solutes and solvent through microporous membranes, carried out in batch
or continuous mode. Hemodialysis is used in treating kidney malfunction to rid the blood of metabolic
waste products (small molecules) while preserving the high-molecular-weight components of the blood.
In ordinary osmosis as well as in dialysis, separation is spontaneous and does not involve the highapplied pressures of ultrafiltration and reverse osmosis.
Diffusion is caused by random molecular motion and, in relative terms, is a slow process. In a classic
text on diffusion, E. L. Cussler stated, ―In gases, diffusion progresses at a rate of about 10 cm in a
minute; in liquids, its rate is about 0.05 cm/min; in solids, its rate may be only about 0.0001 cm/min.‖3 A
relevant question to ask at this point is, Can such a slow process be meaningful to the pharmaceutical
sciences? The answer is a resounding ―yes.‖ Although the rate of diffusion appears to be quite slow,
other factors such as the distance that a diffusing molecule must traverse are also very important. For
example, a typical cell membrane is approximately 5-nm thick. If it is assumed that a drug will diffuse
416
Dr. Murtadha Alshareifi e-Library
into a cell at a rate of 0.0005 cm/min, then it takes only a fraction of a second for that drug molecule to
enter the cell. On the other hand, the thickest biomembrane is skin, with an average thickness of 3 µm
(Fig. 11-19). For the same rate of diffusion, it would take 600 times longer for the same drug molecule to
diffuse through the skin. The time difference in the appearance of the drug on the other side of the skin
is known as the lag time. An even more extreme example is the levonorgestrel-releasing implant.4 This
long-acting contraceptive has been approved for 5 years of continuous use in human patients. To
achieve low constant diffusion rates, six matchstick-sized, flexible, closed capsules made of silicon
rubber tubing are inserted into the upper arm of patients. Annual pregnancy rates of Norplant users are
below 1 per 100 throughout 7 years of continuous use. Levonorgestrel implants provide low
progestogen doses: 40 to 50 µg/day at 1 year of use, decreasing to 25 to 30 µg/day in the fifth year.
Serum levels of levonorgestrel at 5 years are 60% to 65% of those levels measured at 1 month of
use.4 Although diffusion plays an important role in the successful delivery of levonorgestrel from the
Norplant system, drug release from long-acting delivery systems is a function of many other factors as
well.
Another pharmaceutically relevant example of diffusion relates to the mixing of a drug in solution with
intestinal contents immediately prior to drug absorption across the intestinal mucosa. At first glance,
mixing appears to be a simple process; however, several molecular- and macroscopic-level processes
must occur in parallel for efficient mixing to occur. It is important to remember that diffusion depends on
random molecular motions that take place over small molecular distances. Therefore, other processes
are responsible for the movement of molecules over much larger distances and are required for mixing
to occur. These processes are called macroscopic processes and include convection, dispersion, and
stirring. After the macroscopic movement of molecules occurs, diffusion mixes newly adjacent portions
of the intestinal fluid. Diffusion and the macroscopic processes all contribute to mixing, and, qualitatively,
the effects are similar. In 1860, Maxwell was one of the first to recognize this when he stated, ―Mass
transfer is due partly to the motion of translation and partly to that of agitation.‖5 Unlike many other
phenomena, diffusion in a solution always occurs in parallel with convection. Convection is the bulk
movement of fluid accompanied by the transfer of heat (energy) in the presence of agitation. An
example of convection relevant to intestinal absorption of drugs is fluid flow down the intestine.
Dispersion is also relevant to intestinal flow and is related to diffusion. ―The relation exists on two very
different levels. First, dispersion is a form of mixing, and so on a microscopic level it involves diffusion of
molecules. Second, dispersion and diffusion are described with very similar mathematics.‖3 Although it
is somewhat difficult to assess intestinal dispersion patterns in humans, they are most likely
characterized as ―turbulent.‖ In certain experimental models, such as the single-pass intestinal perfusion
procedure6 that is used to estimate the intestinal permeability of drugs in rats, flow conditions are
optimized to obtain laminar flow hydrodynamics. Laminar flow conditions are a special example of the
coupling of flow and diffusion. In contrast to turbulent flow, when operating under laminar flow conditions
a dispersion coefficient can be accurately predicted. In this system, mass transport occurs by radial
diffusion (i.e., movement toward the intestinal mucosa) and axial convection (i.e., flow down the length
of the intestine).
P.226
Steady-State Diffusion
Thermodynamic Basis
Mass transfer is the movement of molecules in response to an applied driving force. Convective and
diffusive mass transfer is important to many pharmaceutical science applications. Diffusive mass
transfer is the subject of this chapter, but convective mass transfer will not be covered in detail, and the
, ,
student is referred to other texts.7 8 9 Mass transfer is a kinetic process, occurring in systems that are
not in equilibrium.7 To better understand the thermodynamic basis of mass transfer, consider an
isolated system consisting of two sections separated by an imaginary membrane (Fig. 11-2).7 At
equilibrium, the temperatures, T, pressures, P, and chemical potentials, µ, of each of two species A and
417
Dr. Murtadha Alshareifi e-Library
B are equal in the two sections. If this isolated system is unperturbed, it will remain at this
thermodynamic equilibrium indefinitely. Suppose that the chemical potential of one of the species, A, is
now increased in section I so that µA,I > µA,II. Because the chemical potential of A is related to its
concentration, the ideality of the solution, and the temperature, this perturbation of the system can be
achieved by increasing the concentration of A in section I. The system will respond to this perturbation
by establishing a new thermodynamic equilibrium. Although it could reestablish the equilibrium by
altering any of the three variables in the system (T, P, or µ), let us assume that it will reequilibrate the
chemical potentials, leaving T and Punaffected. If the membrane separating the two sections will allow
for the passage of species A, then equilibrium will be reestablished by the movement of species A from
section I to section II until the chemical potentials of section I and II are once again equal. The
movement of mass in response to a spatial gradient in chemical potential as a result of random
molecular motion (i.e., Brownian motion) is called diffusion. Although the thermodynamic basis for
diffusion is best described using chemical potentials, it is mathematically simpler to describe it using
concentration, a more experimentally practical variable.
Fig. 11-2. Isolated system consisting of two sections separated by an imaginary
permeable membrane. At equilibrium, the temperatures (T), pressures (P), and
chemical properties (µ) of each of the species in the system are equal in the two
sections. (Modified from G. L. Amidon, P. I. Lee, and E. M. Topp (Eds.), Transport
Processes in Pharmaceutical Systems, Marcel Dekker, New York, 2000, p. 13.)84
Fick's Laws of Diffusion
In 1855, Fick recognized that the mathematical equation of heat conduction developed by Fourier in
1822 could be applied to mass transfer. These fundamental relationships govern diffusion processes in
pharmaceutical systems. The amount, M, of material flowing through a unit cross section, S, of a barrier
in unit time, t, is known as the flux, J:
The flux, in turn, is proportional to the concentration gradient, dC/dx:
418
Dr. Murtadha Alshareifi e-Library
2
where D is the diffusion coefficient of a penetrant (also called thediffusant) in cm /sec, C is its
3
concentration in g/cm , and x is the distance in centimeter of movement perpendicular to the surface of
the barrier. In equation (11-1), the mass, M, is usually given in grams or moles, the barrier surface
2
2
area, S, in cm , and the time, t, in seconds. The units of J are g/cm sec. The SI units of kilogram and
meter are sometimes used, and the time may be given in minutes, hours, or days. The negative sign of
equation (11-2) signifies that diffusion occurs in a direction (the positive x direction) opposite to that of
increasing concentration. That is, diffusion occurs in the direction of decreasing concentration of
diffusant; thus, the flux is always a positive quantity. Diffusion will stop when the concentration gradient
no longer exists (i.e., when dC/dx = 0).
Although the diffusion coefficient, D, or diffusivity, as it is often called, appears to be a proportionality
constant, it does not ordinarily remain constant. D is affected by concentration, temperature, pressure,
solvent properties, and the chemical nature of the diffusant. Therefore, D is referred to more correctly as
a diffusion coefficientrather than as a constant. Equation (11-2) is known as Fick's first law.
Fick's Second Law
Fick's second law of diffusion forms the basis for most mathematical models of diffusion processes. One
often wants to examine the rate of change of diffusant concentration at a point in the system. An
equation for mass transport that emphasizes the change in concentration with time at a definite location
rather than the massdiffusing across a unit area of barrier in unit time is known as Fick's second law.
This diffusion equation is derived as follows. The concentration, C, in a particular volume element (Figs.
11-3 and 11-4) changes only as a result of net flow of diffusing molecules into or out of the region. A
difference in concentration results from a difference in input and output. The concentration of diffusant in
the volume element changes with time, that is, ΔC/Δt, as
P.227
the flux or amount diffusing changes with distance, ΔJ/Δx, in the xdirection, or*
Fig. 11-3. Diffusion cell. The donor compartment contains diffusant at
concentration C.
Differentiating the first-law expression, equation (11-2), with respect to x, one obtains
419
Dr. Murtadha Alshareifi e-Library
Substituting ∂C/∂t from equation (11-3) into equation (11-4) results in Fick's second law, namely,
Equation (11-5) represents diffusion only in the x direction. If one wishes to express concentration
changes of diffusant in three dimensions, Fick's second law is written in the general form
This expression is not usually needed in pharmaceutical problems of diffusion, however, because
movement in one direction is sufficient to describe most cases. Fick's second law states that the change
in concentration with time in a particular region is proportional to the change in the concentration
gradient at that point in the system.
Steady State
An important condition in diffusion is that of the steady state. Fick's first law, equation (11-2), gives the
flux (or rate of diffusion through unit area) in the steady state of flow. The second law refers in general to
a change in concentration of diffusant with time at any distance, x (i.e., a nonsteady state of flow).
Steady state can be described, however, in terms of the second law, equation (11-5). Consider the
diffusant originally dissolved in a solvent in the left-hand compartment of the chamber shown in Figure
11-3. Solvent alone is placed on the right-hand side of the barrier, and the solute or penetrant diffuses
through the central barrier from solution to solvent side (donor to receptor compartment). In diffusion
experiments, the solution in the receptor compartment is constantly removed and replaced with fresh
solvent to keep the concentration at a low level. This is referred to as ―sink conditions,‖ the left
compartment being the source and the right compartment the sink.
Fig. 11-4. Concentration gradient of diffusant across the diaphragm of a diffusion
cell. It is normal for the concentration curve to increase or decrease sharply at the
boundaries of the barrier because, in general, C1 is different from Cd, and C2 is
different from Cr. The concentration C1 would be equal to Cd, for example, only ifK -
420
Dr. Murtadha Alshareifi e-Library
C1/Cd had a value of unity.
Originally, the diffusant concentration will fall in the left compartment and rise in the right compartment
until the system comes to equilibrium, based on the rate of removal of diffusant from the sink and the
nature of the barrier. When the system has been in existence a sufficient time, the concentration of
diffusant in the solutions at the left and right of the barrier becomes constant with respect to time but
obviously not the same in the two compartments. Then, within each diffusional slice perpendicular to the
direction of flow, the rate of change of concentration, dC/dt, will be zero, and by Fick's second law,
3
C is the concentration of the permeant in the barrier expressed in mass/cm . Equation (112
2
7) demonstrates that because D is not equal to zero, d C/dx = 0. When a second derivative such as
this equals zero, one concludes that there is no change in dC/dx. In other words, the concentration
gradient across
P.228
the membrane, dC/dx, is constant, signifying a linear relationship between concentration, C, and
distance, x. This is shown in Figure 11-4 (in which the distance x is equal to h) for drug diffusing from left
to right in the cell of Figure 11-3. Concentration will not be rigidly constant, but rather is likely to vary
slightly with time, and then dC/dtis not exactly zero. The conditions are referred to as a ―quasistationary‖
state, and little error is introduced by assuming steady state under these conditions.
Diffusion Driving Forces
There are numerous diffusional driving forces in pharmaceutical systems. Up to this point the discussion
has focused on ―ordinary diffusion,‖ which is driven by a concentration gradient.8 However, other driving
forces include pressure, temperature, and electric potential. Examples of driving forces in
pharmaceutical systems are shown in Table 11-1.
Diffusion Through Membranes
Steady Diffusion Across a Thin Film and Diffusional Resistance
Yu and Amidon18 concisely developed an analysis for steady diffusion across a thin film as it relates to
diffusional resistance.Figure 11-4 depicts steady diffusion across a thin film of thickness h. In this case,
the diffusion coefficient is considered constant because the solutions on both sides of the film are dilute.
The concentrations on both sides of the film, Cd and Cr, are kept constant and both sides are well mixed.
Diffusion occurs in the direction from the higher concentration (Cd) to the lower concentration (Cr). After
sufficient time, steady state is achieved and the concentrations are constant at all points in the film as
shown in Figure 11-5. At steady state (dC/dt = 0), Fick's second law becomes
Key Concept
Membranes and Barriers
Flynn et al.19 differentiated between a membrane and a barrier. A membrane is a biologic or
physical ―film‖ separating the phases, and material passes by passive, active, or facilitated
transport across this film. The termbarrier applies in a more general sense to the region or
regions that offer resistance to passage of a diffusing material, the total barrier being the sum
of individual resistances of membranes.
421
Dr. Murtadha Alshareifi e-Library
Fig. 11-5. Diffusion across a thin film. The solute molecules diffuse from the wellmixed higher concentration, C1, to the well-mixed lower concentration, C2. The
concentrations on both sides of the film are kept constant. At steady state, the
concentrations remain constant at all points in the film. The concentration profile
inside the film is linear, and the flux is constant.
Integrating equation (11-8) twice using the conditions that at z = 0, C= Cd and at z = h, C = Cr, yields the
following equation:
The term h/D is often called the diffusional resistance, denoted by R. The flux equation can then be
written as
Although resistance to diffusion is a fundamental scientific principle, permeability is a term that is used
more often in the pharmaceutical sciences. Resistance and permeability are inversely related. In other
words, the higher the resistance to diffusion, the lower is the permeability of the diffusing substance. In
the next few sections the concepts of permeability and series resistance will be introduced.
Permeability
Fick adapted the two diffusion equations (11-2) and (11-5) to the transport of matter from the laws of
heat conduction. Equations of heat conduction are found in the book by Carslaw.20 General solutions to
these differential equations yield complex expressions; simple equations are used here for the most
part, and worked examples are provided so that the reader should have no difficulty in following the
discussion of dissolution and diffusion.
If a membrane separates the two compartments of a diffusion cell of cross-sectional area S and
thickness h, and if the concentrations in the membrane on the left (donor) and on the right (receptor)
sides are C1 and C2, respectively (Figure 11-4), the first law of Fick can be written as
P.229
Table 11-1 Driving Forces in Pharmaceutical Systems
422
Dr. Murtadha Alshareifi e-Library
Driving Force
Concentration
Pressure
Example
Description
References
Passive
diffusion
Passive diffusion is a
process of mass transfer
of individual molecules
of a substrate brought
about by random
molecular motion and
associated with a
concentration gradient
3
Drug
dissolution
Drug ―dissolution‖
occurs when a tablet is
introduced into a
solution and is usually
accompanied by
disintegration and
deaggregation of the
solid matrix followed by
drug diffusion from the
remaining small particles
10
Osmotic drug
release
Osmotic drug release
systems utilize osmotic
pressure as the driving
force for controlled
delivery of drugs; a
simple osmotic pump
consists of an osmotic
core (containing drug
with or without an
osmotic agent) coated
with a semipermeable
membrane; the
semipermeable
membrane has an orifice
for drug release from the
pump; the dosage form,
after contacting with the
aqueous fluids, imbibes
water at a rate
determined by the fluid
permeability of the
membrane and osmotic
pressure of core
formulation; this osmotic
11
423
Dr. Murtadha Alshareifi e-Library
imbibition of water
results in high
hydrostatic pressure
inside the pump, which
causes the flow of the
drug solution through
the delivery orifice
Temperature
Pressure-driven
jets for drug
delivery
Pressure-driven jets are
used for drug delivery; a
jet injector produces a
high-velocity jet (>100
m/sec) that penetrates
the skin and delivers
drugs subcutaneously,
intradermally, or
intramuscularly without
the use of a needle; the
mechanism for the
generation of highvelocity jets includes
either a compression
spring or compressed air
12
Lyophilization
Lyophilization (freezedrying) of a frozen
aqueous solution
containing a drug and a
inner-matrix building
substance involves the
simultaneous change in
receding boundary with
time, phase transition at
the ice–vapor interface
governed by the
Clausius–Clapeyron
pressure–temperature
relationship, and water
vapor diffusion across
the pore path length of
the dry matrix under low
temperature and vacuum
conditions
13
424
Dr. Murtadha Alshareifi e-Library
Electrical
potential
Microwaveassisted
extraction
Microwave-assisted
extraction (MAE) is a
process of using
microwave energy to
heat solvents in contact
with a sample in order to
partition analytes from
the sample matrix into
the solvent; the ability to
rapidly heat the sample
solvent mixture is
inherent to MAE and is
the main advantage of
this technique; by using
closed vessels, the
extraction can be
performed at elevated
temperatures,
accelerating the mass
transfer of target
compounds from the
sample matrix
14
Iontophoretic
dermal drug
delivery
Iontophoresis is used to
enhance transdermal
delivery of drugs by
applying a small current
through a reservoir that
contains ionized drugs;
one electrode (positive
electrode to deliver
positively charged ions
and negative electrode to
deliver negatively
charged ions) is placed
between the drug
reservoir and the skin;
the other electrode with
opposite charge is placed
a short distance away to
complete the circuit, and
the electrodes are
connected to a power
supply; when the current
flows, charged ions are
transported across the
15, 16
425
Dr. Murtadha Alshareifi e-Library
skin through a pore
Electrophoresis
Electrophoresis involves
the movement of
charged particles
through a liquid under
the influence of an
applied potential
difference; an
electrophoresis cell fitted
with two electrodes
contains dispersion;
when a potential is
applied across the
electrodes, the particles
migrate to the oppositely
charged electrode;
capillary electrophoresis
is widely used as an
analytical tool in the
pharmaceutical sciences
17
P.230
where (C1 - C2)/h approximates dC/dx. The gradient (C1 - C2)/hwithin the diaphragm must be assumed
to be constant for a quasistationary state to exist. Equation (11-11) presumes that the aqueous
boundary layers (so-called static or unstirred aqueous layers) on both sides of the membrane do not
significantly affect the total transport process. The potential influence of multiple resistances on diffusion
such as those introduced by aqueous boundary layers (i.e., multilayer diffusion) is covered later in this
chapter.
The concentrations C1 and C2 within the membrane ordinarily are not known but can be replaced by the
partition coefficient multiplied by the concentration Cd on the donor side or Cr on the receiver side, as
follows. The distribution or partition coefficient, K, is given by
Hence,
and, if sink conditions hold in the receptor compartment, Cr[congruent] 0,
where
It is noteworthy that the permeability coefficient, also called the permeability, P, has units of linear
velocity.*
426
Dr. Murtadha Alshareifi e-Library
In some cases, it is not possible to determine D, K, or hindependently and thereby to calculate P. It is a
relatively simple matter, however, to measure the rate of barrier permeation and to obtain the surface
area, S, and concentration, Cd, in the donor phase and the amount of permeant, M, in the receiving sink.
One can then obtain P from the slope of a linear plot of M versus t:
provided that Cd remains relatively constant throughout time. If Cdchanges appreciably with time, one
recognizes that Cd = Md/Vd, the amount of drug in the donor phase divided by the donor phase volume,
and then one obtains P from the slope of log Cd versus t:
Example 11-1
Simple Drug Diffusion Through a Membrane
A newly synthesized steroid is allowed to pass through a siloxane membrane having a cross2
sectional area, S, of 10.36 cm and a thickness, h, of 0.085 cm in a diffusion cell at 25°C.
From the horizontal intercept of a plot of Q = M/Sversus t, the lag time, tL, is found to be 47.5
3
min. The original concentration C0 is 0.003 mmole/cm . The amount of steroid passing
-3
through the membrane in 4.0 hr is 3.65 × 10 mmole.
a. Calculate the parameter DK and the permeability, P. We have
2
b. Using the lag time tL = h /6D, calculate the diffusion coefficient. We have
or
c.
Combining the permeability, Equation (11-15), with the value of D from (b), calculate
the partition coefficient, K. We have
Partition coefficients have already been discussed in the chapter on solubility.
Examples of Diffusion and Permeability Coefficients
Diffusivity is a fundamental material property of the system and is dependent on the solute, the
temperature, and the medium through which diffusion occurs.20 Gas molecules diffuse rapidly through
air and other gases. Diffusivities in liquids are smaller, and in solids still smaller. Gas molecules pass
slowly and with great difficulty through metal sheets and crystalline barriers. Diffusivities are a function
427
Dr. Murtadha Alshareifi e-Library
of the molecular structure of the diffusant as well as the barrier material. Diffusion coefficients for gases
and liquids passing through water, chloroform, and polymeric materials are given inTable 11-2.
Approximate diffusion coefficients and permeabilities for drugs passing from a solvent in which they are
dissolved (water, unless otherwise specified) through natural and synthetic membranes are given
in Table 11-3. In the chapter on colloids, we will see that the molecular weight and the radius of a
spherical protein can be obtained from knowledge of its diffusivity.
Multilayer Diffusion
There are many examples of multilayer diffusion in the pharmaceutical sciences. Diffusion across
biologic barriers may involve a number of layers consisting of separate membranes, cell contents, and
fluids of distribution. The passage
P.231
of gaseous or liquid solutes through the walls of containers and plastic packaging materials is also
frequently treated as a case of multilayer diffusion. Finally, membrane permeation studies using Caco-2
or MDCK cell monolayers on permeable supports such as polycarbonate filters are other common
examples of multilayer diffusion.
Table 11-2 Diffusion Coefficients of Compounds in Various Media*
Diffusant
Partial Molar
Volume
(cm3/mole)
D×
Medium or Barrier
106(cm2/sec) (Temperature, °C)
Ethanol
40.9
12.4
Water (25)
n-Pentanol
89.5
8.8
Water (25)
Formamide
26
17.2
Water (25)
Glycine
42.9
10.6
Water (25)
Sodium lauryl
sulfate
235
6.2
Water (25)
Glucose
116
6.8
Water (25)
Hexane
103
15.0
Chloroform (25)
Hexadecane
265
7.8
Chloroform (25)
Methanol
25
26.1
Chloroform (25)
Acetic acid
64
14.2
Chloroform (25)
428
Dr. Murtadha Alshareifi e-Library
dimer
Methane
22.4
1.45
Natural rubber (40)
n-Pentane
—
6.9
Silicone rubber (50)
Neopentane
—
0.002
Ethycellulose (50)
*From G. L. Flynn, S. H. Yalkowsky, and T. J. Roseman, J. Pharm.
Sci. 63, 507, 1974. With permission.
Higuchi32 considered the passage of a topically applied drug from its vehicle through the lipoidal and
lower hydrous layers of the skin. Two barriers in series, the lipoidal and the hydrous skin layers of
thickness h1 and h2, respectively, are shown in Figure 11-6. The resistance, R, to diffusion in each layer
is equal to the reciprocal of the permeability coefficient, Pi, of that particular layer. Permeability,P, was
defined earlier [equation (11-15)] as the diffusion coefficient,D, multiplied by the partition coefficient, K,
and divided by the membrane thickness, h. For a particular lamina i,
Fig. 11-6. Passage of a drug on the skin's surface through a lipid layer, h1, and a
hydrous layer, h2, and into the deeper layers of the dermis. The curve of concentration
against the distance changes sharply at the two boundaries because the two partition
coefficients have values other than unity.
429
Dr. Murtadha Alshareifi e-Library
and
where Ri is the resistance to diffusion. The total resistance, R, is the reciprocal of the total
permeability, P, and is additive for a series of layers. It is written in general as
where Ki is the distribution coefficient for layer i relative to the next corresponding layer, i + 1, of the
system.19 The total permeability for the two-ply model of the skin is obtained by taking the reciprocal of
equation (11–20c), expressed in terms of two layers, to yield
The lag time to steady state for a two-layer system is
When the partition coefficients, Ki, of the two layers are essentially the same and one of the h/D terms,
say 1, is much larger than the other, however, the time lag equation for the bilayer skin system reduces
to the simple time lag expression
P.232
Table 11-3 Drug Diffusion and Permeability Coefficients*
Drug
Membrane Membrane
Diffusion Permeability
Coefficient Coefficient
(cm2/sec) (cm/sec)
Pathway
References
Amiloride
—
1.63 ×
10-4
Absorption
from
human
jejunum
21
Antipyrine
—
4.5 × 10-
Absorption
from
human
jejunum
22
Absorption
from
human
jejunum
22
4
Atenolol
—
0.2 × 104
430
Dr. Murtadha Alshareifi e-Library
Benzoic acid
—
36.6 ×
10-4
Absorption
from rat
jejunum
23
Carbamazepine
—
4.3 × 10-
Absorption
from
human
jejunum
22
4
Chloramphenicol
—
1.87 ×
10-6
Through
mouse skin
24
Cyclosporin A
4.3 ×
10-6
—
Diffusion
across
cellulose
membrane
25
Desipramine·HCl
—
4.4 × 10-
Absorption
from
human
jejunum
22
Absorption
from
human
jejunum
22
4
Enalaprilat
—
0.2 × 104
Estrone
—
20.7 ×
10-4
Absorption
from rat
jejunum
23
Furosemide
—
0.05 ×
10-4
Absorption
from
human
jejunum
22
Glucosamine
9.0 ×
10-6
—
Diffusion
across
cellulose
membrane
25
431
Dr. Murtadha Alshareifi e-Library
Glucuronic acid
9.0 ×
10-6
—
Diffusion
across
cellulose
membrane
25
Hydrochlorothiazide
—
0.04 ×
10-4
Absorption
from
human
jejunum
22
Hydrocortisone
—
0.56 ×
10-4
Absorption
from rat
jejunum
23
—
5.8 × 10-
Absorption
from rabbit
vaginal
tract
26
5
Ketoprofen
2.1 ×
10-3
—
Diffusion
across
abdominal
skin from a
hairless
male rat
27
Ketoprofen
—
8.4 × 10-
Absorption
from
human
jejunum
22
4
Mannitol
8.8 ×
10-6
—
Diffusion
across
cellulose
membrane
25
Mannitol
—
0.9 × 10-
Diffusion
across
excised
bovine
nasal
28
4
432
Dr. Murtadha Alshareifi e-Library
mucosa
Metoprolol·1/2tartrate
—
Naproxen
—
1.3 × 104
8.3 × 104
Absorption
from
human
jejunum
22
Absorption
from
human
jejunum
22
Octanol
—
12 × 10-4
Absorption
from rat
jejunum
23
PEG 400
—
0.58 ×
10-4
Absorption
from
human
jejunum
29
Piroxicam
—
7.8 × 10-
Absorption
from
human
jejunum
22
4
Progesterone
—
7 × 10-4
Absorption
from rat
jejunum
23
Propranolol
—
3.8 × 10-
Absorption
from
human
jejunum
29
Absorption
from rabbit
vaginal
tract
30
4
Salycylates
1.69 ×
10-6
—
433
Dr. Murtadha Alshareifi e-Library
Salycylic acid
—
10.4 ×
10-4
Absorption
from rat
jejunum
23
Terbutaline·1/2sulfate
—
0.3 × 10-
Absorption
from
human
jejunum
22
Testosterone
7.6 ×
10-6
—
Diffusion
across
cellulose
membrane
25
Testosterone
—
20 × 10-4
Absorption
from rat
jejunum
23
Verapamil·HCl
—
6.7 × 10-
Absorption
from
human
jejunum
22
Diffusion
into human
skin layers
31
4
4
Water
2.8 ×
10-10
2.78 ×
10-7
*All at 37°C.
Example 11-2
Series Resistances in Cell Culture Studies
Cell culture models are increasingly used to study drug transport; however, in many instances
only the effective permeability, Peff, is calculated. For very hydrophobic drugs, interactions
with the filter substratum or the aqueous boundary layer (ABL) may provide more resistance
to drug transport than the cell monolayer itself. Because the goal of the study is to assess the
cell transport properties of drugs,Peff may be inherently biased due to drug interactions with
the substratum or ABL. Reporting Peff is of value only if the monolayer is the rate-limiting
transport barrier. Therefore, prior to reporting the Peff of a compound, the effect of each of
these barriers should be evaluated to ensure that the permeability relates to that across the
cell monolayer. In cell culture systems the resistance to drug transport, Peff, is composed of a
series of resistances including those of the ABL (Raq), the cell monolayer (Rmono), and the filter
resistance (Rf) (Fig. 11-7). Total resistance is additive for a series of layers:
434
Dr. Murtadha Alshareifi e-Library
This can be written in terms of the reciprocal of the total permeability:
where Peff is the measured effective permeability, Paq is the total permeability of the ABL
(adjacent to both the apical surface of the cell monolayer and the free surface of the filter),
and Pf is the permeability of the supporting microporous filter.
Fig. 11-7. Diffusion of drug across the aqueous boundary layer (ABL) and cell
monolayer (M) in a cell culture system.
Permeability across the filter, Pf, can be obtained experimentally by measuring the Peff across
blank filters:
Because Paq is dependent on the flow rate,
(where k is a hybrid constant that takes into account the diffusivity of the compound,
kinematic viscosity, and geometric factors of the chamber; V is the stirring rate in mL/min;
and n is an exponent that varies between 0 and 1 depending on the hydrodynamic conditions
in the diffusion chamber), Pf can be calculated using nonlinear regression by
obtaining Peff across blank filters at various flow rates.
435
Dr. Murtadha Alshareifi e-Library
Similarly, 1/Pf + 1/Pmono can be determined by measuring the Peff through the cell monolayer
at various flow rates and by using nonlinear regression and the equation.
The implicit assumption of this method is that each resistance in series is independent of the
other barriers. Therefore, Pmono is calculated by difference, using the independently
determined Pf. Because Paq is independent of the presence of the monolayer, Pmono can be
calculated as follows:
Because the contributions of Rf and Raq vary depending on the nature of the drug, it is
important to correct for these biases by reporting Pmono. The deviation
between Pmono andPeff becomes more significant if the flow rate is low (i.e.,Raq is high) or if the
filter has low effective porosity (i.e., Rfis high). In addition, the permeability of the drug also
plays a major role such that the deviation between Pmono and Peffbecomes more significant for
highly permeable compounds.33
P.233
Membrane Control and Diffusion Layer Control
A multilayer case of special importance is that of a membrane between two aqueous phases with
stationary or stagnant solvent layers in contact with the donor and receptor sides of the membrane (Fig.
11-8).
The permeability of the total barrier, consisting of the membrane and two static aqueous diffusion layers,
is
This expression is analogous to equation (11-21). In equation (11-30), however, only one partition
coefficient, K, appears that giving the ratio of concentrations of the drug in the membrane and in the
aqueous solvent, K = C3/C4 = C3/C2. The flux J through this three-ply barrier is simply equal to the
permeability, P, multiplied by the concentration gradient, (C1 - C5), that is, J = P(C1 - C5). The receptor
serves as a sink (i.e., C5 = 0), and the donor concentration C1 is assumed to be constant, providing a
steady-state flux.34 We thus have
In equations (11-30) and (11-31), Dm and Da are membrane and aqueous solvent diffusivities, hm is the
membrane thickness, and hais the thickness of the aqueous diffusion layer, as shown in Figure 118. M is the amount of permeant reaching the receptor, and S is the cross-sectional area of the barrier. It
is important to realize thatha is physically influenced by the hydrodynamics in the bulk aqueous phases.
The higher the degree of stirring, the thinner is the stagnant aqueous
P.234
diffusion layer; the slower the stirring, the thicker is this aqueous layer.
436
Dr. Murtadha Alshareifi e-Library
Fig. 11-8. Schematic of a multilayer (three-ply) barrier. The membrane is found
between two static aqueous diffusion layers. (From G. L. Flynn, O. S. Carpenter, and
S. H. Yalkowsky, J. Pharm. Sci. 61, 313, 1972. With permission.)
Equation (11-31) is the starting point for considering two important cases of multilayer diffusion, namely,
diffusion under membrane control and diffusion under aqueous diffusion layer control.
Membrane Control
When the membrane resistance to diffusion is much greater than the resistances of the aqueous
diffusion layers, that is, Rm is greater than Ra by a factor of at least 10, or correspondingly, Pm is much
less than Pa, the rate-determining step (slowest step) is diffusion across the membrane. This is reflected
in equation (11-31) whenhmDa is much greater than 2haDm. Thus, equation (11-31) reduces to
Equation (11-32) represents the simplest case of membrane control of flux.
Aqueous Diffusion Layer Control
When 2haKDm is much greater than hmDa, equation (11-31)becomes
and it is now said that the rate-determining barriers to diffusional transport are the stagnant aqueous
diffusion layers. This statement means that the concentration gradient that controls the flux now resides
in the aqueous diffusion layers rather than in the membrane. From the relationship 2haKDm ≫ hmDa, it is
observed that membrane control shifts to diffusion layer control when the partition coefficient K becomes
sufficiently large.
Example 11-3
Transfer from Membrane to Diffusion-Layer Control
Flynn and Yalkowsky34 demonstrated a transfer from membrane to diffusion-layer control in a
homologous series of n-alkyl p-aminobenzoates (PABA esters). The concentration gradient is
almost entirely within the silicone rubber membrane for the short-chain PABA esters. As the
437
Dr. Murtadha Alshareifi e-Library
alkyl chain of the ester is lengthened proceeding from butyl to pentyl to hexyl, the
concentration no longer drops across the membrane. Instead, the gradient is now found in the
aqueous diffusion layers, and diffusion-layer control takes over as the dominant factor in the
permeation process. The steady-state flux, J, for hexyl p-aminobenzoate was found to be
-7
2
-6
2
1.60 × 10 mmole/cm sec. Da is 6.0 × 10 cm /sec and the concentration of the PABA
ester, C, is 1.0 mmole/liter. The system is in diffusion-layer control, so equation (11-33)
applies. Calculate the thickness of the static diffusion layer, ha. We have
One observes from equations (11-32) and (11-33) that, under sink conditions, steady-state
flux is proportional to concentration, C, in the donor phase whether the flux-determining
mechanism is under membrane or diffusion-layer control. Equation (11-33) shows that the flux
is independent of membrane thickness, hm, and other properties of the membrane when
under static diffusion layer control.
Fig. 11-9. Steady-state flux of a series of p-aminobenzoic acid esters. Maximum flux
occurs between the esters having three and four carbons and is due to a change from
membrane to diffusion-layer control, as explained in the text. (From G. L. Flynn and
S. H. Yalkowsky, J. Pharm. Sci. 61, 838, 1972. With permission.)
The maximum flux obtained in a membrane preparation depends on the solubility, or limiting
concentration, of the PABA homologue. The maximum flux can therefore be obtained using
equation (11-31) in which C is replaced byCs, the solubility of the permeating compound:
The maximum steady-state flux, Jmax, for saturated solutions of the PABA esters is plotted
against the ester chain length in Figure 11-9.34 The plot exhibits peak flux between n = 3
and n = 4 carbons, that is, between propyl and butyl p-aminobenzoates. The peak in Figure
438
Dr. Murtadha Alshareifi e-Library
11-9suggests in part the solubility characteristics of the PABA esters but primarily reflects the
change from membrane to static diffusion-layer control of flux. For the methyl, ethyl, and
propyl esters, the concentration gradient in the membrane gradually decreases and shifts, in
the case of the longer-chain esters, to a concentration gradient in the diffusion layers.
By using a well-characterized membrane such as siloxane of known thickness and a
homologous series of PABA esters, Flynn and Yalkowsky34 were able to study the various
factors: solubility, partition coefficient, diffusivity, diffusion lag time, and the effects of
membrane and diffusion-layer control. From such carefully designed and conducted studies, it
is possible to predict the roles played by various physicochemical factors as they relate to
diffusion of drugs through plastic containers, influence release rates from sustained-delivery
forms, and influence absorption and excretion processes for drugs distributed in the body.
Lag Time Under Diffusion-Layer Control
Flynn et al.19 showed that the lag time for ultrathin membranes under conditions of diffusionlayer control can be represented as
Fig. 11-10. Change in lag time of p-aminobenzoic acid esters with alkyl chain length.
(From G. L. Flynn and S. H. Yalkowsky, J. Pharm. Sci. 61, 838, 1972. With
permission.)
where ∑ ha is the sum of the thicknesses of the aqueous diffusion layers on the donor and
receptor sides of the membrane. The correspondence between tL in equation (11-35) with that
for systems under membrane control, equation (11-32), is evident. The lag time
for thick membranes operating under diffusion layer control is
When the diffusion layers, ha1 and ha2, are of the same thickness, the lag time reduces to
439
Dr. Murtadha Alshareifi e-Library
The partition coefficient, which was shown earlier to be instrumental in converting the flux
from membrane to diffusion-layer control, now appears in the numerator of the lag-time
equation. A large K signifies lipophilicity of the penetrating drug species. As one ascends a
homologous series of PABA esters, for example, the larger lipophilicity increases the onset
time for steady-state behavior; in other words, lengthening of the ester molecule increases the
lag time once the system is in diffusion-layer control. The sharp increase in lag time for PABA
esters with alkyl chain length beyond C4 is shown in Figure 11-10.34
P.235
Procedures and Apparatus For Assessing Drug Diffusion
A number of experimental methods and diffusion chambers have been reported in the literature.
Examples of those used mainly in pharmaceutical and biologic transport studies are introduced here.
Fig. 11-11. Simple diffusion cell. (From M. G. Karth, W. I. Higuchi, and J. L. Fox, J.
Pharm. Sci. 74, 612, 1985. With permission.)
Diffusion chambers of simple construction, such as the one reported by Karth et al.35 (Fig. 11-11), are
probably best for diffusion work. They are made of glass, clear plastic, or polymeric materials, are easy
to assemble and clean, and allow visibility of the liquids and, if included, a rotating stirrer. They may be
thermostated and lend themselves to automatic sample collection and assay. Typically, the donor
chamber is filled with drug solution. Samples are collected from the receiver compartment and
subsequently assayed using a variety of analytical methods such as liquid scintillation counting or highperformance liquid chromatography with a variety of detectors (e.g., ultraviolet, fluorescence, or mass
spectrometry). Experiments may be run for hours under these controlled conditions.
Biber and Rhodes36 constructed a Plexiglas three-compartment diffusion cell for use with either
synthetic or isolated biologic membranes. The drug was allowed to diffuse from the two outer donor
compartments in a central receptor chamber. Results were reproducible and compared favorably with
those from other workers. The three-compartment design created greater membrane surface exposure
and improved analytic sensitivity.
The permeation through plastic film of water vapor and of aromatic organic compounds from aqueous
solution can be investigated in two-chamber glass cells similar in design to those used for studying drug
440
Dr. Murtadha Alshareifi e-Library
solutions in general. Nasim et al.37 reported on the permeation of 19 aromatic compounds from
aqueous solution through polyethylene films. Higuchi and Aguiar38 studied the permeability of water
vapor through enteric coating materials, using a glass diffusion cell and a McLeod gauge to measure
changes in pressure across the film.
The sorption of gases and vapors can be determined by use of a microbalance enclosed in a
temperature-controlled and evacuated vessel that is capable of weighing within a sensitivity of ± 2 × 10
6
g. The gas or vapor is introduced at controlled pressures into the glass chamber containing the
-
P.236
polymer or biologic film of known dimensions suspended on one arm of the balance. The mass of
diffusant sorbed at various pressures by the film is recorded directly.39 The rate of approach to
equilibrium sorption permits easy calculation of the diffusion coefficients for gases and vapors.
Fig. 11-12. Diffusion cell for permeation through stripped skin layers. The permeant
may be in the form of a gas, liquid, or gel. Key: A, glass stopper; B, glass chamber; C,
aluminum collar;D, membrane and sample holder. (From D. E. Wurster, J. A.
Ostrenga, and L. E. Matheson, Jr., J. Pharm. Sci. 68, 1406, 1410, 1979. With
permission.)
In studying percutaneous absorption, animal or human skin, ordinarily obtained by autopsy, is
employed. Scheuplein31described a cell for skin penetration experiments made of Pyrex and consisting
of two halves, a donor and a receptor chamber, separated by a sample of skin supported on a
perforated plate and securely clamped in place. The liquid in the receptor was stirred by a Teflon-coated
bar magnet. The apparatus was submerged in a constant-temperature bath, and samples were removed
periodically and assayed by appropriate means. For compounds such as steroids, penetration was slow,
and radioactive methods were found to be necessary to determine the low concentrations.
Wurster et al.40 developed a permeability cell to study the diffusion through stratum corneum (stripped
from the human forearm) of various permeants, including gases, liquids, and gels. The permeability cell
is shown in Figure 11-12. During diffusion experiments it was kept at constant temperature and gently
shaken in the plane of the membrane. Samples were withdrawn from the receptor chamber at definite
times and analyzed for the permeant.
The kinetics and equilibria of liquid and solute absorption into plastics, skin, and chemical and other
biologic materials can be determined simply by placing sections of the film in a constant-temperature
bath of the pure liquid or solution. The sections are retrieved at various times, excess liquid is removed
441
Dr. Murtadha Alshareifi e-Library
with absorbant tissue, and the film samples are accurately weighed in tared weighing bottles. A
radioactive-counting technique can also be used with this method to analyze for drug remaining in
solution and, by difference, the amount sorbed into the film.
Partition coefficients are determined simply by equilibrating the drug between two immiscible solvents in
a suitable vessel at a constant temperature and removing samples from both phases, if possible, for
analysis.41 Equilibrium solubilities of drug solutes are also required in diffusion studies, and these are
obtained as described earlier (Chapter 8).
Addicks et al.42 described a flowthrough cell and Addicks et al.43designed a cell that yields results
more comparable to the diffusion of drugs under clinical conditions. Grass and Sweetana44 proposed a
side-by-side acrylic diffusion cell for studying tissue permeation. In a later paper, Hidalgo et
al.45 developed and validated a similar diffusion chamber for studying permeation through cultured cell
monolayers. These chambers (Fig. 11-13 a and b), derived from the Ussing chamber, have the
advantage of employing laminar flow conditions across the tissue or cell surface allowing for an
assessment of the aqueous boundary layer and calculation of intrinsic membrane drug permeability.
Biologic Diffusion
Example 11-4
Intestinal Drug Absorption and Secretion
-
The apparent permeability, Papp, of Taxol across a monolayer of Caco-2 cells is 4.4 × 10
6
cm/sec in the apical to basolateral direction (i.e., absorptive direction) and is 31.8 × 10
6
cm/sec from basolateral to apical direction (i.e., secretory direction). Assuming that both
absorptive and secretory drug transport occurs under sink conditions (Cr ≪Cd), what is the
amount of Taxol absorbed through the intestinal wall by 2 hr after administering an oral dose?
Assume that the Taxol concentration in the intestinal fluid is 0.1 mg/mL, and following
intravenous administration, the initial Taxol concentration in the plasma is 10 µg/mL. How
much Taxol will be secreted into the feces 2 hr after dosing? Assume that the effective area
,
,
for intestinal absorption and secretion is 1 m.2 46 47 We have
For intestinal absorption,
For intestinal secretion,
Gastrointestinal Absorption of Drugs
Drugs pass through living membranes according to two main classes of transport, passive and carrier
mediated. Passive transfer involves a simple diffusion driven by differences in drug concentration on the
two sides of the membrane. In intestinal absorption, for example, the drug travels in most cases by
passive transport from a region of high concentration
P.237
in the gastrointestinal tract to a region of low concentration in the systemic circulation. Given the
instantaneous dilution of absorbed drug once it reaches the bloodstream, sink conditions are essentially
maintained at all times.
442
Dr. Murtadha Alshareifi e-Library
443
Dr. Murtadha Alshareifi e-Library
Fig. 11-13. (a) Sweetana/Grass diffusion cell. Tissue is mounted between acrylic halfcells. Buffer is circulated by gas lift (O2/CO2) at the inlet and flows in the direction of
arrows, parallel to the tissue surface. Temperature control is maintained by a heating
block.
Carrier-mediated transport can be classified as active transport (i.e., requires an energy source) or as
facilitated diffusion (i.e., does not depend on an energy source such as adenosine triphosphate). In
active transport the drug can proceed from regions of lowconcentration to regions of high concentration
through the ―pumping action‖ of these biologic transport systems. Facilitative-diffusive carrier proteins
cannot transport drugs or nutrients ―uphill‖ or against a concentration gradient. We will make limited use
of specialized carrier systems in this chapter and will concentrate attention mainly on passive diffusion.
Many drugs are weakly acidic or basic, and the ionic character of the drug and the biologic
compartments and membranes have an important influence on the transfer process. From the
Henderson–Hasselbalch relationship for a weak acid,
-
where [HA] is the concentration of the nonionized weak acid and [A ] is the concentration of its conjugate
base. For a weak base, the equation is
+
where [B] is the concentration of the base and [BH ] that of its conjugate acid. pKa is the dissociation
exponent for the weak acid in each case. For the weak base, pKa = pKw - pKb.
The percentage ionization of a weak acid is the ratio of concentration of drug in the ionic form, I, to total
concentration of drug in ionic, I, and undissociated, U, form, multiplied by 100:
Therefore, the Henderson–Hasselbalch equation for weak acids can be written as
or
Substituting U into the equation for percentage ionization yields
Similarly, for a weak molecular base,
P.238
Table 11-4 Percentage Sulfisoxazole, pKa[congruent] 5.0, Dissociated and
Undissociated at pH Values
444
Dr. Murtadha Alshareifi e-Library
pH
Percentage DissociatedPercentage Undissociated
2.0
0.100
99.900
4.0
9.091
90.909
5.0
50.000
50.000
6.0
90.909
9.091
8.0
99.900
0.100
10.0
99.999
0.001
In equation (11-41), pKa refers to the weak acid, whereas in (11-42), pKa signifies the acid that is
conjugate to the weak base.
The percentage ionization at various pH values of the weak acid sulfisoxazole, pKa [congruent] 5.0, is
given in Table 11-4. At a point at which the pH is equal to the drug's pKa, equal amounts are present in
the ionic and molecular forms.
The molecular diffusion of drugs across the intestinal mucosa was long thought to be the major pathway
for drug absorption into the body. Drug absorption by means of diffusion through intestinal cells (i.e.,
enterocytes) or in between those cells (i.e., paracellular diffusion) is governed by the state of ionization
of the drug, its solubility and concentration in the intestine, and its membrane permeability.
pH-Partition Hypothesis
Biologic membranes are predominantly lipophilic, and drugs penetrate these barriers mainly in their
molecular, undissociated form. Brodie and his associates48 were the first workers to apply the principle,
known as the pH-partition hypothesis, that drugs are absorbed from the gastrointestinal tract by passive
diffusion depending on the fraction of undissociated drug at the pH of the intestines. It is reasoned that
the partition coefficient between membranes and gastrointestinal fluids is large for the undissociated
drug species and favors transport of the molecular form from the intestine through the mucosal wall and
into the systemic circulation.
The pH-partition principle has been tested in a large number of in vitro and in vivo studies, and it has
,
been found to be only partly applicable in real biologic systems.48 49 In many cases, the ionized as well
as the un-ionized form partitions into, and is appreciably transported across, lipophilic membranes. It is
found for some drugs, such as sulfathiazole, that the in vitro permeability coefficient for the ionized form
may actually exceed that for the molecular form of the drug.
Transport of a drug by diffusion across a membrane such as the gastrointestinal mucosa is governed by
Fick's law:
where M is the amount of drug in the gut compartment at time t, Dmis diffusivity in the intestinal
membrane, S is the area of the membrane, K is the partition coefficient between membrane and
aqueous medium in the intestine, h is the membrane thickness, Cgis the concentration of drug in the
intestinal compartment, and Cp is the drug concentration in the plasma compartment at time t. The gut
compartment is kept at a high concentration and has a large enough volume relative to the plasma
445
Dr. Murtadha Alshareifi e-Library
compartment so as to make Cg a constant. Because Cp is relatively small, it can be omitted.
Equation(11-43) then becomes
The left-hand side of (11-44) is converted into concentration units, C(mass/unit volume) × V (volume).
On the right-hand side of (11-44), the diffusion constant, membrane area, partition coefficient, and
membrane thickness are combined to yield a permeability coefficient. These changes lead to the pair of
equations
where Cg and Pg of equation (11-45) are the concentration and permeability coefficient, respectively, for
drug passage from intestine to plasma. In equation (11-46), Cp and Pp are corresponding terms for the
reverse passage of drug from plasma to intestine. Because the gut volume, V, and the gut
concentration, Cg, are constant, dividing (11-45) by (11-46) yields
Equation (11-47) demonstrates that the ratio of absorption rates in the intestine-to-plasma and the
plasma-to-intestine directions equals the ratio of permeability coefficients.
The study by Turner et al.49 showed that undissociated drugs pass freely through the intestinal
membrane in either direction by simple diffusion, in agreement with the pH-partition principle. Drugs that
are partly ionized show an increased permeability ratio, indicating favored penetration from intestine to
plasma. Completely ionized drugs, either negatively or positively charged, show permeability
ratios Pg/Pp of about 1.3, that is, a greater passage from gut to plasma than from plasma to gut. This
suggests that penetration of ions is associated with sodium ion flux. Their forward passage, Pg, is
apparently due to a coupling of the ions with sodium transport, which mechanism then ferries the drug
ions across the membrane, in conflict with the simple pH-partition hypothesis.
Colaizzi and Klink50 investigated the pH-partition behavior of the tetracyclines, a class of drugs having
three separate pKa values, which complicates the principles of pH partition. The lipid solubility and
relative amounts of the ionic forms of a tetracycline at physiologic pH may have a bearing on the
P.239
biologic activity of the various tetracycline analogues used in clinical practice.
Modification of the pH-Partition Principle
Ho and coworkers51 also showed that the pH-partition principle is only approximate, assuming as it
does that drugs are absorbed through the intestinal mucosa in the nondissociated form alone.
Absorption of relatively small ionic and nonionic species through the aqueous pores and the aqueous
diffusion layer in front of the membrane must be considered.23 Other complicating factors, such as
metabolism of the drug in the gastrointestinal membrane, absorption and secretion by carrier-mediated
processes, absorption in micellar form, and enterohepatic circulatory effects, must also be accounted for
in any model that is proposed to reflect in vivo processes.
Ho, Higuchi, and their associates23 investigated the gastrointestinal absorption of drugs using
diffusional principles and a knowledge of the physiologic factors involved. They employed an in situ
preparation, as shown in Figure 11-14, known as the modified Doluisio method for in situ rat intestinal
absorption. (The original rat intestinal preparation52 employed two syringes without the mechanical
pumping modification.)
The model used for the absorption of a drug through the mucosal membrane of the small intestine is
shown in Figure 11-15. The aqueous boundary layer is in series with the biomembrane, which is
composed of lipid regions and aqueous pores in parallel. The final reservoir is a sink consisting of the
blood. The flux of a drug permeating the mucosal membrane is
446
Dr. Murtadha Alshareifi e-Library
Fig. 11-14. Modified Doluisio technique for in situ rat intestinal absorption. (From N.
F. H. Ho, J. Y. Park, G. E. Amidon, et al., in A. J. Aguiar (Ed.), Gastrointestinal
Absorption of Drugs, American Pharmaceutical Association, Academy of
Pharmaceutical Sciences, Washington, D. C., 1981. With permission.)
Fig. 11-15. Model for the absorption of a drug through the mucosa of the small
intestine. The intestinal lumen is on the left, followed by a static aqueous diffusion
layer (DL). The gut membrane consists of aqueous pores (a) and lipoidal regions (l).
The distance from the membrane wall to the systemic circulation (sink) is marked off
from 0 to -L2; the distance through the diffusion layer is 0 to L1. (From N. F. Ho, W. I.
Higuchi, and J. Turi, J. Pharm. Sci. 61, 192, 1972. With permission.)
or, because the blood reservoir is a sink, Cblood [congruent] 0, and
where Papp is the apparent permeability coefficient (cm/sec) and Cbis the total drug concentration in bulk
solution in the lumen of the intestine. The apparent permeability coefficient is given by
447
Dr. Murtadha Alshareifi e-Library
where Paq is the permeability coefficient of the drug in the aqueous boundary layer (cm/sec) and Pm is
the effective permeability coefficient for the drug in the lipoidal and polar aqueous regions of the
membrane (cm/sec).
The flux can be written in terms of drug concentration, Cb, in the intestinal lumen by combining with it a
term for the volume, or
where S is the surface area and V is the volume of the intestinal segment. The first-order disappearance
-1
rate, Ku (sec ), of the drug in the intestine appears in the expression
Substituting equation (11-52) into (11-51) gives
P.240
Key Concept
Transport Pathways
Parallel transport pathways are all potential pathways encountered during a particular
absorption step. Although many pathways are potentially available for drug transport across
biologic membranes, drugs will traverse the particular absorption step by the path of least
resistance.
For transport steps in series (i.e., one absorption step must be traversed before the next one),
the slower absorption step is always the rate-determining process.
448
Dr. Murtadha Alshareifi e-Library
and from equations (11-49) and (11-50), together with (11-53), we find
or
Consideration of two cases, (a) aqueous boundary layer control and (b) membrane control, results in
simplification of equation (11-55).
a.
When the permeability coefficient of the intestinal membrane (i.e., the velocity of drug passage
through the membrane in centimeter per second) is much greater than that of the aqueous
layer, the aqueous layer will cause a slower passage of the drug and become a rate-limiting
barrier. Therefore, Paq/Pm will be much less than unity, and equation(11-55) reduces to
Ku is now written as Ku,max because the maximum possible diffusional rate constant is
determined by passage across the aqueous boundary layer.
b.
If, on the other hand, the permeability of the aqueous boundary layer is much greater than that
of the membrane,Paq/Pm will become much larger than unity, and equation(11-55) reduces to
The rate-determining step for transport of drug across the membrane is now under membrane control.
When neither Paq norPm is much larger than the other, the process is controlled by the rate of drug
passage through both the stationary aqueous layer and the membrane. Figures 11-16 and 11-17 show
the absorption studies of n-alkanol and n-alkanoic acid homologues, respectively, that concisely
illustrate the biophysical interplay of pH, pKa, solute lipophilicity via carbon chain length, membrane
permeability of the lipid and aqueous pore
P.241
pathways, and permeability of the aqueous diffusion layer as influenced by the hydrodynamics of the
stirred solution.
449
Dr. Murtadha Alshareifi e-Library
Fig. 11-16. First-order absorption rate constant for a series of n-alkanols under
various hydrodynamic conditions (static or low stirring rates and oscillation or high
stirring fluid at 0.075 mL/sec) in the jejunum, using the modified Doluisio technique.
(From N. F. H. Ho, J. Y. Park, W. Morozowich, and W. I. Higuchi, in E. B. Roche
(Ed.), Design of Biopharmaceutical Properties Through Prodrugs and Analogs,
American Pharmaceutical Association, Academy of Pharmaceutical Sciences,
Washington, D. C., 1977, p. 148. With permission.)
Fig. 11-17. First-order absorption rate constants of alkanoic acids versus buffered pH
of the bulk solution of the rat gut lumen, using the modified Doluisio technique.
Hydrodynamic conditions are shown in the figure. (From N. F. H. Ho, J. Y. Park, W.
Morozowich, and W. I. Higuchi, in E. B. Roche, (Ed.), Design of Biopharmaceutical
Properties Through Prodrugs and Analogs, American Pharmaceutical Association,
Academy of Pharmaceutical Sciences, Washington, D. C., 1977, p. 150. With
permission.)
Example 11-5
Small Intestinal Transport of a Small Molecule
450
Dr. Murtadha Alshareifi e-Library
Calculate the first-order rate constant, Ku, for transport of an aliphatic alcohol across the
-1
-4
mucosal membrane of the rat small intestine if S/V = 11.2 cm , Paq = 1.5 × 10 cm/sec,
-4
and Pm = 1.1 × 10 cm/sec. We have
For a weak electrolytic drug, the absorption rate constant, Ku, is23
where Pm of the membrane is now separated into a term P0, the