UNIVERSITY OF RWANDA (UR)
COLLEGE OF SCIENCE AND TECHNOLOGY (CST)
SCHOOL OF SCIENCE
CHEMISTRY DEPARTMENT
ACADEMIC YEAR 2018-2019
ADVANCED PHYSICAL CHEMISTRY
(15 Credits)
Level: Year 2 Env. Chem and Org. Chem
Instructor: Mr. Jean Bosco NKURANGA
1
COURSE OUTLINE
2
I.
II.
III.
IV.
V.
VI.
Introduction to statistical mechanics and statistical
thermodynamics, Boltzmann distribution
Partition function and molecular partition function
(translation, rotation, vibration and electronic) and Entropy in
statistical thermodynamics, calculation of thermodynamic
variables by using partition functions (ideal gas properties).
Heat capacities of solids
Classical statistical mechanics (phase space, Ensembles and
Ensembles average, equipartition, microcanonical, canonical
and grand canonical ensembles
Quantum Statistical Mechanics (B-E, F-D, phonons and
photons)
Colloidal Chemistry and its applications
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019
Recommended readings
3
1.
2.
3.
Alberty R.A and Silbey R.J. (2001) Physical
Chemistry, 3rd Edition, John Willey and Sons, Inc.
Puri, B.R., Sharama L.R. and Patthania M.S. (2008),
Principles of Physical Chemistry, 43rd Edition,
Washall publishing.co, Jalandhar-Delhi, India.
Atkins P. and De Paula J. (2010) Physical chemistry
, 9th Edition, Oxford University Press, Great Lakes
U.K. ISBN-13: 978-1-429-21812-2
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019
NKURANGA Jean Bosco
Address:
Office: Room P215, Muhabura block
Email:nkubo123@gmail.com
• Assessment:
1. CW: 50%
a. Test 1 (30 marks), Quiz (10 marks), Assignments (10 marks)
b.
Laboratory Practices:-
2.
UE: 50%
Prerequisites:
Basic Mathematics, Quantum Chemistry, Classical thermodynamics and Probability
and statistics
9/13/2019
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
4
Study Tips:
• You start passing/failing this course on
day one by attending/not attending all
class lectures.
• Practice makes perfect. If you are
interested in understanding this course
then confront as many readings as you
possibly can.
• Lecturers are here/there (office/lab)
to help you NOW, not after you have
done your exam and failed.
9/13/2019
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
5
I. Introduction to Statistical Thermodynamics
6





Statistical mechanics (SM) provides a link between quantum
mechanics (or wave mechanics) and classical thermodynamics .
Classical Thermodynamics (CT) deals with macroscopic properties of
matter and describes the behaviour of large number of molecules in
terms of state variables (P, V, n, T).
Quantum mechanics (QM) deals almost exclusively with matter at the
microscopic level, which is described by using a wave function, ψ .
However, QM does not indicate which wave function of a molecule will
represent the state of the system at a given time.
It is noteworthy that neither CT nor QM is able to calculate the
macroscopic properties of matter from microscopic structures of
individual molecules.
Since any observed equilibrium property of matter must be some kind
of an average of large number of molecules, it is plain that statistical
methods must be used to determine this property.
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019
I. Introduction to Statistical Thermodynamics ..
7


Statistical Thermodynamics/mechanics is the discipline, which
deals with the computation of the macroscopic properties of
matter from the data on the microscopic properties of individual
atoms (or molecules).
Statistical thermodynamics helps to understand the significance of
thermodynamic variables (T, S), which do not possess any microscopic
or mechanical interpretation.
Contributors to S T: J.C. Maxwell, L. Botzmann, Willard Gibss, Maxwell Planck, A. Einstein, S.N.
Bose, E. Fermi, P.M.A Dirac, Tolman, R.H Fowler, Guggenhein, Born, Debye, L. Onsager, L.D. Landau,
N. Bogoliubov, J.G. Kirkwood, I. Prigogine, N. Winner, J.E. Mayer, K.G. Wilson and R. Kubo and
Chandrasekhar (1910-1995), who applied quantum statistics to stellar dynamics (white dwarfs).
I. Introduction to Statistical Thermodynamics ..
Scope of Statistical Mechanics/Thermodynamics
8






Statistical mechanics can be applied to simple ideal systems such as monoatomic
and diatomic gases.
For application to interacting systems such as such as liquids, where strong
intermolecular forces exist), the details of the intermolecular potential energy, which
is not always known accurately, have also to be taken into account. That is why
statistical mechanics of liquids is a difficult but fascinating subject.
Gases under high pressures, too, are difficult to treat statistically since they deviate
strongly from ideality.
Recently, statistical methods have been applied to successfully to simple liquids and
dense gases.
Progress in this area has been made possible by application of both the advanced
mathematical methods and high speed computers, which otherwise highly intractable
differential and integro-differential equations involved in advanced theoretical
treatments.
The third law of Thermodynamics is well discussed in Statistical Thermodynamics
based on the partition function.
The goal of ST is to describe macroscopic, thermodynamic thermodynamic properties
Physical Chemistry
Year
2 Env. andproperties.
Org. Chem 9/13/2019
In termsAdvanced
of microscopic
atomic and
molecular
I. Introduction to Statistical Thermodynamics ..
9
1.2 Types of Statistics
Different physical situations encountered in nature are
described by three types of statistics:
 the Maxwell-Boltzmann (M-B) or Classical Statistics
 the Bose-Einstein (B-E) statistics,
 Fermi-Dirac (F-D) Statistics
The M-B statistics is also referred to classical statistics
while B-E and F-D statistics are called Quantum statistics.
These three types of statistics are described for
microcanonical ensemble.

Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019
I. Introduction to Statistical Thermodynamics ..
10
Maxwell-Boltzmann Probability
(ni  gi  1)!
wBE  
i 1 ni !( gi  1)!
ni
n
gi
wB  N !
i 1 ni !
n
n
gi !
wFD  
i 1 ni !( gi  ni )!
ni
n
gi
wMB  
i 1 ni !
wB and wMB yield the same distribution.
In M-B statistics, the particles are assumed to be distinguishable and any number of particles
May occupy the same energy level.
 Particles obeying M-B statistics are boltzons
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019
I. Introduction to Statistical Thermodynamics ..
11
ni
1
     fi
i
gi e
ni
1
   
 fi
i
gi e
1
ni
1
   
 fi
i
gi e
1
Boltzmann
Bose-Einstein
Fermi-Dirac
In B-E statistics, the particles are
indistinguishable
And any number of particles may occupy a
given Energy level, Ei.
This statistics is obeyed by particles having
integral spin such as H2 , N2, D2 and Helium-4
and photons.
• Particles obeying B-E statistics are bosons (
Symmetric wave function)
In F-D statistics, the particles are
indistinguishable, but one particle may
occupy a given Energy level, Ei.
This statistics is obeyed by particles having
half-integral spin (protones, electrons, He3, NO, etc.
• Particles obeying F-D statistics are
fermions (antisymetric wave function)
ni

gi
ni

gi
ni

gi
1
i 
e
1
kT
1
i 
e
Boltzmann
kT
i 
e
 fi
kT
 fi
Bose-Einstein
 fi
Fermi-Dirac
1
1
12
I. Introduction to Statistical Thermodynamics ..
Maxwell-Boltzmann Statistics



Consider a system of N distinguishable particles
occupying energy levels,  0 , 1,  2 ,...........,  n
The total number of arrangements for placing no
particles in the ground state energy level  0 , n1 particles
in the first excited energy level 1 , n2 particles in the
second energy level  2 , and so on, is known as the
thermodynamic probability, W, of a given macrostate.
In statistical thermodynamics, the problem is to determine
how many microstates correspond to a given macrostate.
W
N!
N!
 n
n1 !n2 !n3 !.....  n j
 ni
i 1
n
N   ni
i 1
13
I. Introduction to Statistical Thermodynamics ..
Maxwell-Boltzmann Statistics
…

In B-M distribution, it may happen that a given
energy level occur in more than one quantum state
with the same energy, so there is degeneracy or
multiplicity, of the energy,  i .
ni
n
gi
W  N !
 constant
i 1 ni !

The entropy, S and the probability, W of a given
state of a system are related by the Boltzmann
Equation, which is the most famous Equation in
Statistical mechanics.
S  k B ln W
How can determine the maximum probability, at
equilibrium?
S  k ln Wmax
14
I. Introduction to Statistical Thermodynamics ..
Maxwell-Boltzmann Statistics
…

In order to find a distribution that will make W a
maximum , it is more convenient, however, to
optimize the linearized W, so logarithm of W.
  ln W 
  ln W
d ln W  
dn

 1 

n
1 

 n2
n 
 ln W 
d ln W   
 dni

n
i 
i 1 

  ln W

dn

 2 

 n3
  ln W

dn

..............


 4

 ni

 dni

For a closed system of independent particles:
n
n
(i ) N   ni  constant
i 1
n

  ln W
dn

 3 
 n4

(ii ) U   ni i  constant
(i ) dN   dni  0
Differentiate
i 1
Advanced Physical Chemistry
i 1
n
(ii ) U    i dni  0
i 1
Year 2 Env. and Org. Chem.
n
n
i 1
i 1
ln W  ln N !  ni ln gi   ln ni !  ct
15
I. Introduction to Statistical Thermodynamics ..
Maxwell-Boltzmann Statistics
…

The derivation of the B-M uses the striling’s
approximation (SA):
ln x !  x ln x  x
n
n
i 1
i 1
ln W  ln N !  ni ln gi   ln ni !  ct
Stirling Approx.
  gi 

 ln  n      i dni  0
i 1   i 

n
i 1
i 1
ln W  N ln N   ni ln gi   ni ln ni  ct
Remember the considered system is closed, so ni
and our N is constant, the values of dni are not
independent of one another and the energy of
the system is constant!
The introduction of the Lagrange’s
Undetermined multipliers α and β for
∑dni and ∑ɛidni, respectively.
n
Differentiating w.r.t ni,
N and gi are constants
!
?
n
i 1
n
i 1
d ln W   ln gi dni   ln ni dni
g 
d ln W   ln  i dni  0
 ni 
i 1
n
ni  gi e
n
   i
ni
1
   
i
gi
e
16
I. Introduction to Statistical Thermodynamics ..
M-B Statistics and Partition Function
ni  gi e
   i

gi
   i
e
This equation is one of the form of the Boltzmann distribution law, which is for the most
Probable distribution for a macrostatate, i.e., it gives the occupation numbers of the
molecular energy levels for the most probable distribution in terms of the energies, ɛi ,
The degeneracy, gi and the undetermined Lagrange’s multipliers, α and β.
The quantity Z is known
   i
  i
  i
n
n
ge
ge
ge
Pi  i  n i
 n i
 i
 Z  gi e   i as the partition function,
a function that enables
N
Z
   i
  i
i 1
gi e
gi e
the calculation of all
i 1
i 1
thermodynamic variables!


Advanced Physical Chemistry

Year 2 Env. and Org. Chem
9/13/2019
17
I. Introduction to Statistical Thermodynamics ..
Bose-Einstein Statistics
 0 , 1,  2 ,...........,  n




Consider a system of N indistinguishable particles such that ni
particles are in the ith energy level with degeneracy, gi .
The ni particles have to be distributed among gi states.
For the sake of simplicity, let us consider that the i th energy
level has (gi - 1) partitions, which are sufficient to separate
energy level into gi intervals.
In B-E statistical thermodynamics, the possible number of
distributions of ni particles among the gi states may be
determined by permuting the array of partitions and particles.
n
( g 1)
C( n i g 1)
i
i
ni  gi  1!


ni ! gi  1!
N   ni
i 1
n
U    i ni
i 1
18
I. Introduction to Statistical Thermodynamics ..
Bose-Einstein Statistics
In B-E statistics, the thermodynamic probability, W for the
system of N particles, i.e., the number of ways of distributing N
particles among the various energy levels is given by:

ni  gi  1!

W 
 constant
i 1 ni ! gi  1 !
n
n
ln W   ln  ni  gi  1! ln ni ! ln  gi  1!  constant
i 1
1) ni+ gi -1 ≈ ni+ gi since ni is large
2) gi -1 ≈ gi since gi is large
3) Stirling’s Approximation: lnx! =xlnx-x
Approxi
mations:
n
n
n
i 1
i 1
i 1
 n  gi
d ln W   ln  i
i 1  ni
n
Differentiate
w.r.t ni
ln W   (ni  gi ) ln  ni  gi    ni ln ni   gi ln gi  constant
B-E distribution of N particles among the various energy levels

dni  0

the Lagrange’s
Undetermined
multipliers
α ∑dni and β ∑ɛidni,
gi
ni    
i 1
e
19
I. Introduction to Statistical Thermodynamics ..
 0 , 1,  2 ,...........,  n
Fermi-Dirac (F-D) Statistics




Consider N indistinguishable particles such that ni particles are
in the ith energy level with degeneracy, gi and ni < gi .
The first particle have to be placed in any one of the gi states
and for each one of these choices, the second particle may be
placed in any one of the remaining, (gi - 1) states, and so on,
up to (gi - ni) states .
gi !
#
Arrangements

Thus, the number of arrangement is given by
 gi  ni !
In F-D statistical thermodynamics, the particles are
indistinguishabe, so the above expression has to be divided by
the possible number of permutations of ni particles.
g  ni
Cg i
i

gi !
ni ! gi  ni  !
n
W 
i 1
n
gi !
 constant
ni ! gi  ni !
N   ni
i 1
n
U    i ni
i 1
20
I. Introduction to Statistical Thermodynamics ..
Fermi-Dirac Statistics

In F-d statistics, the thermodynamic probability, W for the
system of N indistinguishable particles, i.e., the number of ways
of distributing N particles among the various energy levels is
n
given by:
gi !
W 
i 1
ni ! gi  ni !
 constant
n
ln W   ln gi ! ln ni ! ln  gi  ni !  constant
i 1
1) ni and gi are large
2) gi > ni since gi is large
3) Stirling’s Approximation: lnx! =xlnx-x
Approxi
mations:
n
n
n
i 1
i 1
i 1
 g n
d ln W   ln  i i
i 1  ni
n
Differentiate
w.r.t ni
ln W   (ni  gi ) ln  gi  ni    ni ln ni   gi ln gi  constant
F-D distribution of N particles among the various energy levels

dni  0

the Lagrange’s
Undetermined
multipliers
α ∑dni and β ∑ɛidni,
gi
ni    
i 1
e
21
I. Introduction to Statistical Thermodynamics ..
Fermi-Dirac Statistics
Fermi-Dirac distribution of plot for an electron in a metal
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019
I. Introduction to Statistical Thermodynamics ..
22
Evaluation of the Lagrange’s Undetermined Multiplayers
n
n
N   ni   gi e     N
 From B-M statistics,
i
ni  gi e
   i
i 1
e

i 1
n
  i

g
e

N

e

 i
i 1
Substitution
  i
gi e
ni  N
Z
n
 gi e 
i
N
Z
Z   gi e   i
i 1
• The partition function, Z is a quantity of immense importance
in statistical thermodynamics as it enables the calculation of the
Value of any thermodynamic function for the system.
• However, one need to determine the value of β prior to Z
evaluation.
?
Advanced Physical Chemistry

i 1
n
with
N
Year 2 Env. and Org. Chem
Determine the value of β
I. Introduction to Statistical Thermodynamics ..
Evaluation of the Lagrange’s Undetermined Multiplayers
23

From B-M distribution statistics, the thermodynamic probability in terms of ln
W is given by:
n
n
n
n
i 1
i 1
i 1
i 1
ln W  ln N !  ni ln gi   ln ni !  N ln N   ni ln gi   ni ln ni
gi e  i
ni  N
Z
ln ni  ln N  ln gi   i  ln Z
n
n
i 1
n
i 1
ln W  N ln N   ni ln gi   ni  ln N  ln Z  ln gi   i 
n
n
i 1
i 1
ln W  N ln N   ni ln gi  N ln N  N ln Z   ni ln gi    ni i
i 1
ln W  U  N ln Z
In ST (Boltzmann) : S  k B ln W
S  Nk B ln Z  k B U
I. Introduction to Statistical Thermodynamics ..
24
Evaluation of the Lagrange’s
Undetermined Multiplayers
n
Z   gi e   i
S  NkB ln Z  kB U
i 1
From the combined statement of the first and the second law of Classical Thermodynamics, the
Internal energy is an exact differential function of S and V, so U=f (S,V)
dU  TdS  pdV
 U 
 U 
dU  
 ds  
 dV
 S V
 V  S
S  NkB ln Z  kB U
 S 

  kB 
 U V
1
 S 

 
 U V T
Differentiate
w.r.t U
At constant volume, dV=0
Nk B  Z 
 S 
  


k


k
U
B
B 





Z  U V
 U V
 U V
Nk B  Z   
 S 


 


Z   V  U
 U V

dU  TdS
1
 S 

 
 U V T
1
k BT

  

k


k
U
B 
 B


 U V
 Z 
U


Z


N
  V
I. Introduction to Statistical Thermodynamics ..
25
Evaluation of the Lagrange’s Undetermined Multiplayers
Advanced Physical Chemistry
Year 2 Env. and Org. Chem
9/13/2019