Intro to Solutions
Sol-1
We are now going to use our knowledge of thermodynamics to
examine solutions…
Consider a solution of two components: 1 and 2
The Gibbs energy is a function of T, P, and the two mole numbers…
 G 
 G 
 G 
 G 



dG  
dT  
dP  
dn1  
dn2



 T  P ,n1 ,n2
 P T ,n1 ,n2
 n1 T , P ,n2
 n2 T , P ,n1
At constant T and P…
We can show by Euler’s theorem:
Differentiate:
(1)
G  1n1  2 n2
dG  1dn1  2dn2  n1d1  n2d2
(2)
(1) - (2)
Divide by n1 + n2:
Gibbs-Duhem Equation
(constant T and P)
Short Mathematics Review
Equations will we use today:
G  G  nRT ln


= standard (1 bar)
 
P
P
Sol-2
Eq 22.59
Eq 24.13
G  nA A  nB B
Eq 24.6
nAd A  nB dB  0
-OR-
xAd A  xB dB  0
-Divide everything by n (total number of moles)
Eq 24.10 or 24.11
Gibbs-Duhem Equation
Sol-3
Chemical Potential of Liquids
We need to know how the Gibbs energy of a liquid varies
with composition in order to discuss properties of liquid
mixtures (like solutions).
Vapor Pressure =
PA*
For vapor phase:
    RT ln
*
A

A
Pure Liquid
Solution

 A   A  RT ln
P
At equilibrium…
For liquid phase:
    RT ln
*
A

A

PA
P
At equilibrium…
=A
=A
=B
 A ( g )   A (l )
For solution:
PA*
P

For vapor phase:
PA*

 *A ( g )   *A (l )
Partial Pressure = PA
Combine these expressions…
 A    RT ln

A

PA
P
Sol-4
Ideal Solutions
• Two types of molecules are randomly distributed
• Typically, molecules are similar in size and shape
• Intermolecular forces in pure liquids & mixture are similar
• Examples: benzene & toluene, hexane and heptane
(more precise thermodynamic definition coming)
In ideal solutions, the partial vapor pressure of component
A is simply given by Raoult’s Law:
PA  x P
*
A A
mole fraction of A in solution
vapor pressure of pure A
Total VP of an ideal solution
PA
 A    RT ln *
PA
*
A
PA  x P
*
A A
 A   *A  RT ln x A
This serves to define an ideal
solution if true for all values of xA
The total vapor pressure of an ideal solution:
Ptotal  PA  PB
Ptotal  x A PA*  xB PB*
Ptotal  PA*  xB ( PB*  PA* )
Sol-5
40 °C
Deviations from Raoult’s Law
CS2 and dimethoxymethane: Positive
deviation from ideal (Raoult’s Law) behavior.
Sol-6
trichloromethane/acetone: Negative
deviation from ideal (Raoult’s Law) behavior.
Methanol, ethanol, propanol
mixed with water. Which one
is which? (All show positive
deviations from ideal behavior)
Sol-7
Raoult and Henry
PA  x A PA* as x A  1
Raoult’s law
(Raoult’s Law)
Henry’s law
PA  x A k H , A as x A  0
Henry’s behavior:
Henry’s law constant:
k H , A  PA*
 The Henry’s law constant reflects the intermolecular interactions between
the two components.
 Solutions following both Raoult’s and Henry’s Laws are called ideal-dilute
solutions.
DGmix, DSmix, and DHmix for ideal solution

DGmix  G

sol

G G
*
1
*
2

G sol  nA  A  RT ln x A   nB  B  RT ln xB 
&
D mixG id  n A RT ln x A   nB RT ln xB 
D mixG  0
id
id


D
G
D mix S id   mix
 T



 P ,n1 ,n2
D mix H id  D mixG id  TD mix S id 
Make sense?
Sol-8
G  ni 
*
i
*
i
“The Bends”
Sol-9
• If a deep sea or scuba diver
rises to the ocean surface
too quickly, he or she can
have great pain (mostly at
the joints) and may double
over in pain… they have “the
bends”.
• In terms of what we’ve
discussed today, brainstorm
some causes of “the bends”.
Temperature-Composition Diagrams
Sol-10
1-propanol and 2-propanol at ambient pressure (i.e., 760 torr)
Point a: On solution line …
760  x P  x P
*
1 1
*
2 2
P  760
x1 
*
P  P1
*
2
*
2
Point b: On vapor line …
How does this
relate to fractional
distillation?
*
1 1
P1
xP
y1 

760 760
Dalton’s Law
Distillation and Azeotropes
Sol-11
One example of non-ideal solutions: Benzene and Ethanol at 1 atm
Azeotrope: A mixture for which there is no change in composition upon boiling.
Can you separate these compounds by distillation?
Really non-ideal: immiscible mixtures
T3 > TC > T2 > T1
What is this line?
Sol-12
Deviations greater with increasing T
Temp-composition diagrams for immiscibles
Sol-13
Other thoughts on non-ideal solutions
Sol-14
Vapor Pressures can often be represented empirically… for example:
* x22  x23
1 1
P1  x P e
* x12 x13
2 2
P2  x P e
0  x1  1
0  x2  1
Activity
*
 sol


j
j (l )  RT ln x j
For ideal solutions:
For non-ideal solutions:
Activity defined as:
Sol-15
*
 sol


j
j (l )  RT ln a j
aj 
Pj
*
j
P
a1 
as
Activity
x1  1
With definitions for vapor pressure of non-ideal solutions on
Sol-15, what is a?
Activity coefficient (a measure of deviation from ideality):
j 
aj
xj
Typical non-ideal solution
Chlorobenzene + 1-nitropropane at 75 °C,
x1
1
P1*  119 torr
0.119 0.289 0.460 0.691 1.00
P1/torr 19.0
a1
Sol-16
41.9
62.4
86.4
119
Activities must be calculated wrt standard
states
Sol-17
Activity using Raoult’s law as standard state…

sol
j
  (l )  RT ln a j
*
j
aj 
Pj
*
j
P
a j  1 as x j  1
Activity using Henry’s law as standard state…

sol
j
  (l )  RT ln
*
j
Using Rauolt’s Law
kH , j
*
j
P
 RT ln a j
aj 
Pj
kH , j
a j  x j as x j  0
Using Henry’s Law
Gibbs Energy and Activity Coefficients
Sol-18
Ideal Solutions…
D mixG / RT  x A ln x A   xB ln xB 
id
(Slide Sol-8)
Non-ideal Solutions…
DGmix / RT  x1 ln x1  x2 ln x2  x1 ln  1  x2 ln  2
(Derivations on pg 994)
Activity etc with other concentration scales
Sol-19
Table 25.1
You need to know
how to convert
between mole
fraction, molality
and molarity!
Recall colligative properties?!
DP  P1*  P1  x2 P1*
DTb  K b m2
DT f  K f m2
  cRT
Sol-20
Boiling point elevation
Label gas, liquid and solid lines
Label melting and boiling pt
Sol-21
At equilibrium…
1 ( g )  1 (l )  1* (l )  RT ln a1
or
D1  1 ( g )   (l )  RT ln a1
*
1

T
Use Gibbs-Helmholtz equation (see A&G-18) and chemical potential def:
d ( DG / T )
DH
 2
dT
T
D1  DvapG
Boiling pt elevation con’t
Sol-22
Why these integrands?
 d ln a  
1
a1
1

D vap H  1
1

ln a1 
 * 
R  Tvap Tvap 
DT 
Let…
 
* 2
vap
RT
D vap H
Kb 
 M
RT
D vap H
Assumptions on this page
D vap H  DT 


ln a1  
*
R  TvapTvap 
x2  M1m2
x2
* 2
vap
Tva p D vap H
1
da1    *
dT
2
Tva p RT
a1
1
DT 
 M
* 2
vap
RT
D vap H
DT 
1
m2
Osmotic Pressure
Sol-23
1* (T , P)  1* (T , P  )  RT ln a1
1* (T , P  )  1* (T , P)  
P 
P
P 
 1* 

 dP   V1*dP  V1*
P
 P T
1* (T , P  )  1* (T , P)  RT ln a1  0
Assume the solution is dilute… ln a1 ~ x2 and x2 ~ n2/n1
RTx 2
 *
V1
Osmotic Pressure and Molecular Weight
Sol-24
It is found that 2.20 g of polymer dissolved in enough
water to make 300 mL of solution has an osmotic
pressure of 7.45 torr at 20 °C. Determine the
molecular mass of the polymer.
Why do we use osmotic pressure to find molecular
weight and not one of the other colligative properties?
Osmotic Pressure and Cells
Sol-25
In the figure, red blood cells are placed into saline solutions.
1. In which case (hypertonic, isotonic, or hypotonic) does the
concentration of the saline solution match that of the blood cells?
2. In which case is the saline solution more concentrated than the
blood cells?
Crenation
Hemolysis
Electrolyte Solutions
Sol-26
Electrolyte solutions deviate from ideal behavior more strongly and at
lower concentrations than nonelectrolyte solutions. (Why?)
Activities/activity coefficients are essential when working with electrolytes!
Examples of electrolytes… NaCl, MgSO4, MgCl2, Na2SO4
z
z
Cv  Av  ( s)  v C (aq)  v A (aq)
H 2O ( l )
a
From this reaction…
2  v   v  or
2  v (  RT ln a )  v (  RT ln a )

Also know… 2  2  RT ln a2
Therefore…
a2  av av or
Ionic Activity, Molality, & Activity Coefficients
Sol-27
We can define single-ion activity coefficients…
a  m 
a  m 
Mean ionic activity becomes…
m
v

Mean ionic molality

v

Mean ionic activity coefficient
Write out the mean ionic activity for CaCl2…
Table 25.3: Activity and electrolytes
Sol-28
Colligative Properties of Electrolytes
For a strong electrolyte…
x2  vmM1
Sol-29
v = total # of dissociated ions
m = molality
M1 = molar mass (in kg/mol)
If you use this definition in derivation of colligative properties …
Debye-Hückel Theory
Sol-30
Debye-Hückel Theory: Assumes ions are point ions (no radii) with
purely Coulombic interactions and activity coefficients depend
only on the ion charges and the solvent properties.
ln     z z AI
A  2N A 
1/ 2


e


 40 r k BT 
For Aqueous Solutions…
3/ 2
1/ 2
c
1 s 2
Ic   z j c j
2 j 1
Ionic Strength
Validity of Debye-Hückel Theory
Extended Debye-Hückel:
ln   
Sol-31
 Az z I
1/ 2
  c
1/ 2
c
1  BI
Why are activities so important anyway?!
Sol-32
The activity can be thought of as “the real
concentration”… anywhere concentrations are
used, activities should be used instead.
a1  [1] 1  {1}
Some examples:
Key Concepts
•
•
•
•
•
•
•
•
•
•
•
•
Gibbs-Duhem Equation
Partial Pressure
Ideal Solutions
Raoult’s Law
Henry’s Law
Azeotropes
Immiscible Solutions
Activity and Activity Coefficients
Collagative Properties
Electrolytes (and properties)
Debye-Huckel Theory
Importance of Activity
Sol-33