Statistical Thermodynamics
Dr. Henry Curran
NUI Galway
1
Background
Thermodynamic parameters of stable
molecules can be found. However,
those for radicals and transition
state species cannot be readily
found. Need a way to calculate these
properties readily and accurately.
2
The Boltzmann Factor

f E E  
Physical Quantity  exp 

 const x fT T 
Boltzmann law for the population of quantised
energy states:
ni
nj
i  j
i
j
ni
  i   j / kT
 exp
nj
3
Average basis of the behaviour of matter
Thermodynamic properties are concerned
with average behaviour.
ni
nj
 exp


  i   j / kT
The instantaneous values of the occupation
numbers are never very different from
the averages.
4
Distinct, independent particles
• Consider an assembly of particles at
constant temperature. These particles are
• distinct and labelled (a, b, c, … etc)
• They are independent
– interact with each other minimally
– enough to interchange energy at collision
• Weakly coupled
– Sum of individual energies of labelled particles
E          ...   
a
b
c
d
i
i
5
Statistical weights
At any instant the distribution of particles
among energy states involve n0 with energy 0,
n1 with energy 1, n2 with energy 2 and so on.
We call the instantaneous distribution the
configuration of the system.
• At the next moment the distribution will be
different, giving a different configuration with
the same total energy.
• These configurations identify the way in which
the system can share out its energy among the
available energy states.
6
Statistical weights
A given configuration can be reached in a
number of different ways. We call the number
of ways W the statistical weight of that
configuration. It represents the probability
that this configuration can be reached, from
among all other configurations, by totally
random means.
For N particles arriving at a configuration in
which there are n0 particles with energy 0, n1
with 1 etc, the statistical weight is:
N!
W
n0 !n1!n2 !n3! ...
7
Principle of equal a priori probabilities
• Each configuration will be visited exactly
proportional to its statistical weight
N   ni
i
E   ni  i
i
• We must find the most probable configuration
– How likely is this to dominate the assembly?
– For an Avogadro number of particles with an
average change of configuration of only 1 part in
1010 reduces the probability by:
W max
434
 10
W
A massive collapse
in probability!
8
Maximisation subject to constraints
• The predominant configuration among N
particles has energy states that are populated
according to:
ni
   i
e
N
where  and  are constants under the
conditions of constant temperature
9
ni
 e    i
N
The lowest state has energy 0 = 0 and
occupation number n0
n0

e
N
which identifies the constant , and enables
us to write:
ni
n0    i
   i
e e
 e
N
N
ni
  i

e
n0
10
ni
  i
e
n0
this is a temperature dependent ratio since
the occupation number of the states vary
with temperature
The constant  can be stated as:
1

kT
where k is the Boltzmann constant
11
Molecular partition function
ni
  i   j / kT
e
nj
ni  n0 e
ni  n0 e
  i   0  / kT
 i 
Any state population (ni) is known if:
i, T, and n0 are known
12
Molecular partition function
If the total number of particles is N, then:
N  n0  n1  n2  n3  ... 
n
i
all states
N  n0  n0 e   1  n0 e   2  n0 e   3  ...  n0
  i
e

all states
N
 n0 
  i
e

all states
Ne   i
 ni 
  i
e

or writing
  i
e
 q
all states
all states
Ne   i
 ni 
q
13
Molecular partition function, q
Determines how particles distribute (or partition)
themselves over accessible quantum states.
q 1  e
  1
e
 2
e
  3
1 

 ...   0  0,  

kT 

An infinite series that converges more rapidly the
larger both the energy spacing between quantum
states and the value of  is. Convergence is
enhanced at lower temperatures since  = 1/kT.
when  >> 0, e- 0
14
Molecular partition function, q
• If 1-0 (D) is large (D >> kT)
– q
1 (lowest value of q)
• If 1-0 (D) ≈ kT (thermal energy)
– q
large number
magnitude of q shows how easily particles
spread over the available quantum states
and thus reflects the accessibility of the
quantum energy states of the particles
involved.
15
Energy states and energy levels
q
e
  i
all states
However, quantum states can be degenerate
with a number (g) of states all sharing the
same energy. States with the same energy
comprise an energy level and we use the
symbol gj to denote the degeneracy of the
jth level
q
g e
  j
j
all levels
16
The partition function explored
The total number of particles in our
assembly is N or, expressed intensively, NA
per mole
NA 
n   g n
i
all states
j
j
 n0 q
all levels
NA
 q
n0
The partition function is a measure of the
extent to which particles are able to escape
from the ground state
17
The partition function explored
The partition function q is a pure number
which can range from a minimum value of 1
at 0 K (when n0 = NA and only the ground
state is accessible) to an indefinitely large
number as the temperature increases
Fewer and fewer particles are left in the
ground state and an indefinitely large
number of states become available to the
system
18
The partition function explored
We can characterise the closeness of spacing
in the energy manifold by referring to the
density of states function, D(), which
represents the number of energy states in
unit energy level.
• If D() is high (translational motion in gas):
– particles find it easy to leave ground state
– q will rise rapidly as T increases
• If D() is low (vibrations of light diatomic molecules)
– small value of q (
1)
19
The partition function explored
• If q/NA (the number of accessible states per
particle) is small
– few particles venture out of the ground state
• If q/NA is large
– there are many accessible states and molecules are
well spread over the energy states of the system
• q/NA >> 1 for the valid application of the
Boltzmann law in gaseous systems
20
Canonical partition function
• Molecular  molar level
  ln q 

Em   L
  V
– assume value of an extensive function for N
particles is just N times that for a single
particle
true for energy of non-interacting particles but not so
for other properties (e.g. entropy)
• Particles do interact! Thus we consider:
– every system has a set of system energy states
which molecules can populate
– these states are not restricted by the need for
additivity but can adjust to any inter-particle
interactions that may exist
21
Canonical partition function
• Molar sum over states
– each possible state of the whole system
involves a description of the conditions
experienced by all the particles that make
up a mole
Suppose N identical particles each with
the set of individual molecular states
available to them.
Particle labels
Molecular states
1, 2, 3, 4, …, N
a, b, c, d, …
22
Canonical partition function
Any given molar state can be described by a
suitable combination of individual molecular
states occupied by individual molecules. If
we call the ith state Ψi, we can begin to give
a description of this molar state by writing:
i  1a 2b 3h 4 f 5c 6 k 7 c 8t ...
with energy:
Ei                 ...
1
a
2
b
3
h
4
f
5
c
6
k
7
c
8
t
there is no restriction on the number of particles
that can be in the same molecular state
(e.g. particles 5 & 7 are both in molecular state c)
23
Canonical partition function (QN)
State Ψi, with energy Ei is just one of many
states of the whole system. The predominant
molecular configuration is called the canonical
distribution
– applies to states of an N-particle system
– at constant amount, volume, and Temperature
QN 
e
  i
system
states
with energy:
  ln QN
Em  Ei   
 


V
24
The Molar energy
  ln QN
Em  Ei   
 


V
The canonical partition function (QN) is much
more general than “the product of N molecular
partition functions q” since there is no need to
consider only independent molecules
  ln q 
N  q 
   N 

E   
q   V
  V
25
The Molar energy
If we are able to calculate thermodynamic
properties for assemblies of N
independent particles using q and for N
non-independent particles using QN, then,
in the limit of the particles of QN,
becoming less and less strongly
interdependent, the two methods should
eventually converge.
26
The Molar energy
Note that the expressions:
  ln q 

 N 
  V
and
  ln QN
 
 


V
are compatible if we assume that the two
different partition functions are related
simply by:
N
Qq
where Q is a function of an N-particle
assembly at constant T and V
27
Distinguishable and indistinguishable
particles
1
2
3
4
5
6
7
8
Ei   a   b   h   f   c   k   c   t ...
for the canonical partition function we can write:
 

Q   exp                  ...
1
a
2
b
3
h
4
f
5
c
6
k
7
c
8
t
i
In every one of the i system states, each particle (1, 2,
3, …) will be in one of its possible j molecular states
(a,b, c, d …) just once in each system state. If we
factorise out each particle in turn from the summation
over the system states and then gather together all
the terms that refer to a given particle, we get:
28
Distinguishable and indistinguishable
particles


 


  j  
  j  
  j 
Q  e
  e
  e
 ...
molecular
molecular
 molecular





states
states
states

1 
2 
3
If all molecules are of the same type and
indistinguishable by position they do not need labelling
N

  j 
N


Q e

q

 j

29
Distinguishable and indistinguishable
particles
If particles are indistinguishable the number of
accessible system states is lower than it is for
distinguishable ones. A system Ψi
i  1a 2b 3h ...
differs from a state Ψj
 j  1a 2 h 3b ...
if particles are distinguishable, because of the
interchange of particles 2 and 3 between states
b and h. However, Ψi is identical to Ψj if particles
are indistinguishable
30
Distinguishable and indistinguishable
particles
In systems which are not at too high a density and
are also well above 0 K, the correction factor for
this over-counting of configurations is 1/N!
Q q
N
( for distinguishable particles)
N
q
Q
N!
( for indistinguishable particles)
31
Two-level systems
The simplest type of system is one which
comprises particles with only two
accessible states in the form of two nondegenerate levels separated by a narrow
energy gap D:
nu
nl
D
 0  D
0
At temperatures that are comparable to D/k only
the ground state and first excited state are
appreciably populated
32
Effect of increasing Temperature
The average population ratio of the two
levels is an assembly of such two-level (or
two-state) particles is given by:
nu
 2 L / T
  D
e
e
nl
where the two-level temperature (2L) is defined as:
D
2L 
k
33
Temperature Dependence of the populations
Comparing energy gap
with background thermal
energy
D  2 L

kT T
Comparing characteristic T
(2L) with Temperature (T)
if T  5 2 L (high T )
nu
then  e  D  e  2 L / T  e 1/ 5  0.82 (1)
nl
if T  0.5 2 L (low T )
nu
then  e 1/ 0.5  0.14 ( 0)
nl
34
T dependence of populations
nl  e
 D
but total number of particles is constant
1 e n
 D
u
 1
nu  
 D
 1 e
 e  D
nl  
 D
 1 e
nu
nl  nu  N
N
 e   D 

 N
 N     D
1 

e



1
 N     D
 N
1 
e

35
T dependence of populations
(Fig. 6.2)
number of particles / N
1.0
nl
0.5
nu
0.0
T
0 T
0.1
1
10
Reduced temperature T/2L
36
Two-level Molecular partition function
• The effect of
increasing T
– only two energy
states to consider
High T
T = 52L
q2L = 1 + 0.82
q2L
2
Both states equally
accessible
q2 L   e
  i
e
  0
e
  (  0  D )
both
states
 e   0 (1  e  D )
if  0  0  q2 L  (1  e
  D
)
Low T
T = 0.52L
q2L = 1 + 0.14
q2L
1
only lower state
accessible
37
The energy of a two-level system
E2 L   ni ei  nl x 0  nu x D  nu D
i
 1 
 N x D
 
 D 
 1 e 
 ND 
D
  ND 1  e
 
D 
 1 e 


1
At high T, half the particles occupy the
upper state and the total energy takes the
value ½ N D
38
Two-level heat capacity, CV
The spacing of energy levels in discussing
two-level systems is not affected by
changes in volume, so the relevant heat
capacity is CV not CP
 U  DU
CV  
 
 T V DT
Variation of CV with T is a measure of how
accessible the upper states becomes as T
increases.
39
Two-level heat capacity, CV
• Low T
– kT is small
– small DT has little tendency to excite particles
– overall energy remains constant => CV low
• Intermediate T
– kT is comparable to D
– small DT has larger effect in exciting particles
– CV is somewhat larger
• High T
– Almost half particles in excited state
– small DT causes very few particles excited to upper
– Overall energy remains constant => CV low
40
Two-level heat capacity, CV
E2 L  ND 1  e

D 1
dE
D 
D  2 D 

  ND 1  e  e   2 
dT
 kT 
D


e
2
CV  Nk ( D ) 

2
D
 e  1 
41
0.5
1.0
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0.0
0.0
10
0.1
0.42
1
E/Emax
CV/Nk
0.44Nk
Reduced temperature (T/)
Variation of CV and Energy of a two-level
system as a function of reduced Temperature
42
The effect of degeneracy
Consider a two-level system with degenerate
energy levels. If the degeneracy of the lower
level is g0 and that of the upper level is g1
then the two-level molecular partition
function is:
q2 L  ( g 0  g1e
  D
)
43
The effect of degeneracy
Energy of the degenerate two-level system:
  g1 

   ND 
g0 



E
  g1

 D 
   e  
  g 0
 
and the heat capacity:
 g 

  1 e  D 
CV

2   g0 
 D  
2
Nk
  g1  e  D  
 
  g 0
 
44
Toolkit equations

A  A(0) 
1. Massieu bridge :

 k ln Q
T
  ln Q 
2. Internal energy : U  U (0)  kT 

 T V
  ln Q 
3. Equation of state : p  kT 

 V T
2
4. Heat capacity :
5. Entropy :
2


ln
Q

ln Q 


2

CV  2kT 
  kT 
2
 T V
 T V
  ln Q 
S  k ln Q  kT 

 T V
45
Toolkit equations
  ln Q 
  ln Q 
6. Enthalpy :
H  H (0)  kT 
  kTV 

 T V
 V T
  ln Q 
7. Gibbs free energy : G  G (0)   kT ln Q  kTV 

 V T
2
In order to relate the partition function to
classical thermodynamic quantities, the equations
for internal energy and entropy are needed. Once
these have been expressed in terms of the
canonical partition function, Q, the Massieu bridge
can be derived. This in turn provides the most
compact link to classical thermodynamics.
46
Ideal monatomic gas
Consider an assembly of particles constrained to
move in a fixed volume. This system consists of
many, non-interacting, monatomic gas particles in
ceaseless translational motion. The only energy
that these particles can possess is translational
kinetic energy.
47
Translational partition function, qtrs
In classical mechanics, all kinetic energies are
allowed in a system of monatomic gas particles at a
fixed volume V and temperature T. Quantum
restrictions place limits on the actual kinetic
energies that are found.
To determine the partition function for such a
system to need to establish values for the allowed
kinetic energies.
We consider a particle constrained to move in a
cubic box with dimensional box with dimensions lx,
ly, and lz.
48
Particle in a one-dimensional box
The permitted energy levels, x, for a particle of
mass m that is constrained by infinite boundary
potentials at x = 0, and x = lx to exist in a onedimensional box of length lx are given by:
2
x
2
nh
x 
2
8mlx
and similarly for the y- and z-directions. The
translational quantum number, nx, is a positive
integer and the quantum numbers in the y- and zdirections are ny and nz, respectively.
49
One-dimensional partition function
The one-dimensional partition function, qtrs,x, is
obtained by summing over all the accessible energy
states. Thus:
qtrs, x   e
 n x2 h 2 / 8 mlx2
all n
an expression that is exact but cannot be
evaluated except by direct and tedious numerical
summation.
50
One-dimensional partition function
For all values of lx in any normal vessel, these
energy levels are very densely packed and lie
extremely close to each other. They form a virtual
continuum, so the summation can be replaced by an
integration with the running variable nx

qtrs, x   e
 n x2 h 2 / 8 mlx2
dnx
0
resulting in the expression:
1
2
 2m  lx

qtrs, x  
   h
51
Extension to three dimensions
We can factorise the translational partition
function:
3
qtrs  qtrs, x x qtrs, y
 2m  2 lx ly lz

x qtrs, z  
3

h


product of three dimensions gives the volume, V:
3
2
3
2
 2m 
 2mkT 
qtrs   2  V  
 V
2
 h

h  
52
Extension to three dimensions
For the canonical partition function
for N indistinguishable particles
N




q
1  2m  
1  2mkT  
 2  V 
Qtrs 


 V
2
N! N!  h   
N!  h
 




N
trs
3
2
3
2
N
53
Testing the continuum approximation
At its boiling temperature of 4.22 K, one mole of He
occupies 3.46 x 10-4 m3. How many translational
energy states are accessible at this very low
temperature, and determine whether the virtual
continuum approximation is valid.
3
2
3
 2k 
qtrs   2  ( MT ) 2 V
 h NA 
5
2

3
2
 (5.942 x 1030 / mol m 3 kg K

5
2
3
2
) ( MT ) V
3
2
 (5.942 x 1030 )( 4.003 x 10 3 x 4.22) (3.46 x 10  4 )
 4.5 x 10 24 (unitless )
54
Ideal monatomic gas: thermodynamic
functions
Since all classical thermodynamic functions are
related to the logarithm of the canonical
partition function, we start by taking the
logarithm of Qtrs:
3
3
ln Qtrs  N ln( 2m)  N ln T  N ln V  3 N ln h  ln N !
2
2
55
Ideal monatomic gas: thermodynamic
functions
  ln Qtrs  3 N

 
 T V 2T
  ln Qtrs 
3N

   2
2
2T
 T
V
  ln Qtrs  N

 
 V T V
2
56
The translational energy, Etrs
For a monatomic gas, the translational kinetic
energy, Etrs, is the only form of energy that the
monatomic particles possess, so we can equate
it directly to the internal energy, U, and
substitute the value of the derivative:
  ln Qtrs  3
Etrs  U  kT 
  NkT
 T V 2
2
(for one mole N = NA, NAk = R)
3
U m  RT
2
57
The equation of state
This can be derived using:
  ln Qtrs  NkT
p  kT 
 
V
 V T
(for one mole N = NA, NAk = R)
RT
p
or pVm  RT
Vm
58
The heat capacity, CV
This can be derived using:
  ln Qtrs 
2   ln Qtrs 

CV  2kT 
  kT 
2
 T V
 T
V
2
3
3
 3 Nk  Nk  Nk
2
2
(for one mole N = NA, NAk = R)
3
CV , m  R
2
59
Entropy of an ideal monatomic gas
All of the simplifying factors that result from
taking partial derivatives of ln Qtrs no longer hold
  ln Qtrs 
S  k ln Qtrs  kT 

 T V
3
3
ln Qtrs  N ln( 2m)  N ln T  N ln V  3N ln h  ln N!
2
2
lnQtrs is a direct term and so all variables appear
1 N
k ln Qtrs  k ln
q
N!
 1
N 
 k  ln
 ln q   k ( N ln q  ln N !)
 N!

60
Entropy of an ideal monatomic gas
Next, using Stirling’s approximation (lnN! ≈ NlnN – N)
q

k ln Qtrs  k ( N ln q  N ln N  N )  Nk 1  ln 
N

q 3
q

5
 S  Nk 1  ln   Nk  Nk   ln 
N 2
N

2
3



2
5
 2mkT  V  

 S  Nk   ln 


2


2
h
N




 

One mole: (N = NA, NAk = R, NAm = M and V = RT/p
3



2
 2MkT  RT  
5


 S m  R   ln  2
 h N A  N A p  
2







61
Entropy of an ideal monatomic gas
gathering together all experimental variables (M. T, p)
  3 5
2 2

M
T

S m  R ln 
p

 
3
 



5

  2  2


2

(ek ) 
  R ln 
2 

N
h

  A 

 

 
The Sackur-Tetrode Equation
The first term contains all of the experimental
variables, the second consists of constants
62
Using the Sackur-Tetrode equation
The second term has a value of 172.29 J K-1 mol-1
or 20.723 R. Thus we can write:
3
3

1 2
1 
2
S m / R  ln  M / kg mol
(T / K ) ( p / Pa)   20.723




Some calculated and measured entropies
Tb / K
Scalc / R
Scalor / R
Argon
87.4
15.542
15.60
Krypton
120.2
17.451
17.43
63
Using the Sackur-Tetrode equation
The Sachur-Tetrode equation can also be written
as using ln p-1 = ln V – ln R – ln T
3
3
S m  R ln V  R ln T  R ln M  18.605 R
2
2
where the variables are expressed in SI
units (V/m3, T/K, and M/kg mol-1)
64
Significance of Sackur-Tetrode equation
3
3
S m  R ln V  R ln T  R ln M  18.605 R
2
2
volume
temperature
dependence dependence
V2
DST  R ln
V1
3
T2
DSV  R ln
2
T1
T2
 CV ln
T1
the first two terms are
known from classical
thermodynamics
the last two terms (3/2 R lnM and 18.605 R) could not
have been foreseen from classical thermodynamics
65
Change in conditions
Effect of increase in T, V, or M on:
D()
T
Q
U
p
nil
CV
S
nil
V
nil
nil
nil
M
nil
nil
nil
66
Ideal diatomic gas: internal degrees
of freedom
• Polyatomic species can store energy in
a variety of ways:
– translational motion
– rotational motion
– vibrational motion
– electronic excitation
Each of these modes has its own manifold
of energy states, how do we cope?
67
Internal modes: separability of energies
• Assume molecular modes are separable
– treat each mode independent of all others
– i.e. translational independent of vibrational,
rotational, electronic, etc, etc
Entirely true for translational modes
Vibrational modes are independent of:
– rotational modes under the rigid rotor
assumption
– electronic modes under the BornOppenheimer approximation
68
Internal modes: separability of energies
Thus, a molecule that is moving at high speed is
not forced to vibrate rapidly or rotate very fast.
An isolated molecule which has an excess of any
one energy mode cannot divest itself of this
surplus except at collision with another molecule.
The number of collisions needed to equilibrate
modes varies from a few (ten or so) for rotation,
to many (hundreds) for vibration.
69
Internal modes: separability of energies
Thus, the total energy of a molecule j:
  
j
tot
j
trs
j
rot

j
vib

j
el
70
Weak coupling: factorising the energy modes
• Admits there is some energy interchange
– in order to establish and maintain thermal equilibrium
• But allows us to assess each energy mode as if it
were the only form of energy present in the
molecule
• Molecular partition function can be formulated
separately for each energy mode (degree of
freedom)
• Decide later how individual partition functions
should be combined together to form the overall
molecular partition function
71
Weak coupling: factorising the energy modes
• Imagine an assembly of N particles that can store
energy in just two weakly coupled modes  and w
• Each mode has its own manifold of energy states
and associated quantum numbers
• A given particle can have:
- -mode energy associated with quantum number k
- w-mode energy associated with quantum number r
 tot   k   wr
72
Weak coupling: factorising the energy modes
The overall partition function, qtot:
qtot 
e
  (  i   w r )
all states
expanding we would get:
qtot  e   (  0   w 0 )  e   (  0   w 1 )  e   (  0   w 2 )  e   (  0   w 3 )
e
  (  1   w 0 )
e
  (  2   w 0 )
e
  (  1   w 1 )
e
  (  2   w 1 )
e
  (  1   w 2 )
e
  (  2   w 2 )
e
  (  1   w 3 )
e
  (  2   w 3 )
 e   (  3   w 0 )  e   (  3   w 1 )  e   (  3   w 2 )  e   (  3   w 3 )
 ...
73
Weak coupling: factorising the energy modes
but e(a+b) = ea.eb, therefore:
qtot  e
e
   0
   1
.e
.e
  w 0
  w 0
e
e
   0
   1
.e
.e
  w 1
  w 1
e
e
   0
   1
.e
.e
  w 2
  w 2
 ...
 ...
 e    2 .e   w 0  e    2 .e   w 1  e    2 .e   w 2  ...
 ...
each term in every row has a common factor of
   0
   1
in the first row, e
in the second, and
so on. Extracting these factors row by row:
e
74
Weak coupling: factorising the energy modes
qtot  e
   0
e
   1
e
   2
(e
(e
  w 0
  w 0
(e
  w 0
e
  w 1
e
  w 1
e
  w 1
e
  w 2
e
  w 2
e
  w 2
e
  w 3
 ...)
e
  w 3
 ...)
  w 3
 ...)
e
 e    3 (e   w 0  e   w 1  e   w 2  e   w 3  ...)
 ...
the terms in parentheses in each row are identical
and form the summation:
all
e

w
  wj
states
75
Weak coupling: factorising the energy modes

qtot  e
   0
e
   1

all
e
   2
e


  j
states
e
   3
x
all

  wj 

 ...   e

 allw states 
e

w

  wj
states
If energy modes are separable then we can
factorise the partition function and write:
qtot  q x qw
76
Factorising translational energy modes
Total translational energy of molecule j:

j
trs, tot

j
trs, x

j
trs, y

j
trs, z
which allows us to write:
qtrs 
e
  trs , tot
all states
qtrs 
e
  trs , x
all x states

e
  (  trs , x   trs , y   trs , z )
all states
x
e
  trs , y
all y states
e
x
  trs , z
all z states
qtrs  qtrs, x x qtrs, y x qtrs, z
77
Factorising internal energy modes
Total translational energy of molecule j:
qtot  qtrs . qrot .qvib .qel
using identical arguments the canonical partition
function can be expressed:
Qtot  Qtrs . Qrot . Qvib .Qel
but how do we obtain the canonical from the
molecular partition function Qtot from qtot? How
does indistinguishability exert its influence?
78
Factorising internal energy modes
When are particles distinguishable (having
distinct configurations, and when are they
indistinguishable?
• Localised particles (unique addresses) are always
distinguishable
• Particles that are not localised are indistinguishable
– Swapping translational energy states between such
particles does not create distinct new configurations
• However, localisation within a molecule can also
confer distinguishability
79
Factorising internal energy modes
When molecules i and j, each in distinct rotational and
vibrational states, swap these internal states with
each other a new configuration is created and both
configurations have to be counted into the final sum
of states for the whole system. By being identified
specifically with individual molecules, the internal
states are recognised as being intrinsically
distinguishable.
Translational states are intrinsically indistinguishable.
80
Canonical partition function, Q

qtrs 
N
N
N
qrot  qvib  qel 
Qtot 
N
and thus:
N!
1
N
Qtot  qtrs .qrot .qvib .qel 
N!
This conclusion assumes weak coupling. If
particles enjoy strong coupling (e.g. in liquids
and solutions) the argument becomes very
complicated!
81
Ideal diatomic gas: Rotational
partition function
Assume rigid rotor for which we can write
successive rotational energy levels, J, in terms
of the rotational quantum number, J.
EJ 
h2
J ( J  1) joules
8 I
EJ
h
1
J 
 2 J ( J  1) cm
hc 8 Ic
1
 BJ ( J  1) cm
2
where I is the moment of inertia of the
molecule, m is the reduced mass, and B the
rotational constant.
82
Ideal diatomic gas: Rotational
partition function
Another expression results from using the
characteristic rotational temperature, r,
h2
hcB
r  2 
 k r  hcB
8 Ik
k
E J  J ( J  1)k r ( joules )
• 1st energy increment = 2kr
• 2nd energy increment = 4kr
83
Ideal diatomic gas: Rotational
partition function
Rotational energy levels are degenerate and
each level has a degeneracy gJ = (2J+1). So:
qrot   g J e
 J / kT
  (2 J  1)e
 J ( J 1) r / T
If no atoms in the atom are too light (i.e. if the
moment of inertia is not too small) and if the
temperature is not too low (close to 0 K), allowing
appreciable numbers of rotational states to be
occupied, the rotational energy levels lie
sufficiently close to one another to write:
84
Ideal diatomic gas: Rotational partition
function

qrot   (2 J  1)e
 J ( J 1) r / T dJ
0
 qrot
8 2 IkT
 
r
h2
T
• This equation works well for heteronuclear
diatomic molecules.
• For homonuclear diatomics this equation
overcounts the rotational states by a factor of
two.
85
Ideal diatomic gas: Rotational partition
function
• When a symmetrical linear molecule rotates
through 180o it produces a configuration which is
indistinguishable from the one from which it
started.
– all homonuclear diatomics
– symmetrical linear molecules (e.g. CO2, C2H2)
• Include all molecules using a symmetry factor s
qrot 
T
sr
s = 2 for homonuclear diatomics, s = 1 for heteronuclear diatomics
s = 2 for H2O, s = 3 for NH3, s = 12 for CH4 and C6H6
86
Rotational properties of molecules at 300 K
r/K
H2
CH4
HCl
HI
N2
CO
CO2
I2
88
15
9.4
7.5
2.9
2.8
0.56
0.054
s
T/r
qrot
2
12
1
1
2
1
2
2
3.4
20
32
40
100
110
540
5600
1.7
1.7
32
40
50
110
270
2800
87
Rotational canonical partition function
Qrot  q
N
rot
relates the canonical partition function to the
molecular partition function. Consequently, for the
rotational canonical partition function we have:
N
Qrot
 T 
 8 IkT 
 T 


 

2

shcB 
 sh 
sr 
N
2
N
88
Rotational Energy
 8 2 Ik 

ln Qrot  N ln T  N ln 
2 
 sh 
this can differentiated wrt temperature, since the
second term is a constant with no T dependence
  ln Qrot 

2 
U rot  kT 
ln T 
  NkT 
 T

 T V
 U rot  NkT (for diatomic molecules)
2
89
Rotational heat capacity
U rot  NkT
(for diatomic molecules)
this equation applies equally to all linear molecules
which have only two degrees of freedom in rotation.
Recast for one mole of substance and taking the T
derivative yields the molar rotational heat capacity,
Crot, m. Thus, when N = NA, the molar rotational
energy is Urot,m
U rot, m  RT
Crot,m  R (linear molecules)
90
Rotational entropy
S rot
U rot
  ln Q 
 kT 
 k ln Q
  k ln Q 
T
 T V
 8 IkT 
NkT


 k ln 
2
T
 sh 
2
 S rot
N

 8 2 k 
IT
 Nk 1  ln
 ln  2 
s
 h 

Srot is dependent on (reduced) mass (I = mr2), and
there is also a constant in the final term, leading to:



Srot / R  ln I / kg m T / K s  106.53
2
1
91
Rotational entropy
Typically, qrot at room T is of the order of hundreds
for diatomics such as CO and Cl2. Compare this with
the almost immeasurably larger value that the
translational partition function reaches.
qrot  10
2
but qtrs  10
28
92
Extension to polyatomic molecules
• In the most general case, that of a non-linear
polyatomic molecule, there are three independent
moments of inertia.
• Qrot must take account of these three moments
– Achieved by recognising three independent characteristic
rotational temperatures r, x, r, y, r, z corresponding to the
three principal moments of inertia Ix, Iy, Iz
• With resulting partition function:
qrot


s
 T  T  T









 r , x  r , y   r , z




1
2
93
Conclusions
• Rotational energy levels, although more widely
spaced than translational energy levels, are still
close enough at most temperatures to allow us to
use the continuum approximation and to replace
the summation of qrot with an integration.
• Providing proper regard is then paid to rotational
indistinguishability, by considering symmetry,
rotational thermodynamic functions can be
calculated.
94
Ideal diatomic gas: Vibrational partition
function
Vibrational modes have energy level spacings that
are larger by at least an order of magnitude than
those in rotational modes, which in turn, are 25—
30 orders of magnitude larger than translational
modes.
– cannot be simplified using the continuum approximation
– do not undergo appreciable excitation at room Temp.
– at 300 K Qvib ≈ 1 for light molecules
95
The diatomic SHO model
We start by modelling a diatomic molecule on a
simple ball and spring basis with two atoms, mass
m1 and m2, joined by a spring which has a force
constant k.
The classical vibrational frequency, wosc, is given
by:
wosc
1

2
k
m
Hz
There is a quantum restriction on the available energies:
 vib
1

  v  hwosc
2

( v  0, 1, 2, ...)
96
The diatomic SHO model
1
The value hwosc is know as the zero point energy
2
• Vibrational energy levels in diatomic molecules
are always non-degenerate.
• Degeneracy has to be considered for polyatomic
species
– Linear: 3N-5 normal modes of vibration
– Non-linear: 3N-6 normal modes of vibration
97
Vibrational partition function, qvib
• Set 0 = 0, the ground vibrational state as the
reference zero for vibrational energy.
• Measure all other energies relative to reference
ignoring the zero-point energy.
– in calculating values of some vibrational thermodynamic
functions (e.g. the vibrational contribution to the
internal energy, U) the sum of the individual zero-point
energies of all normal modes present must be added
98
Vibrational partition function, qvib
The assumption (0 = 0) allows us to write:
1  hw ,  2  2hw ,  3  3hw ,  4  4hw , ...
Under this assumption, qvib may be written as:
qvib   e
  vib
 1 e
 hw
e
 2 hw
e
3 hw
 ...
a simple geometric series which yields qvib in closed
form:
qvib
1
1


 hw
1 e
1  e  vib / T
where vib = hw/k = characteristic vibrational
99
temperature
Vibrational partition function, qvib
• Unlike the situation for rotation, vib, can be
identified with an actual separation between
quantised energy levels.
• To a very good approximation, since the
anharmonicity correction can be neglected for low
quantum numbers, the characteristic temperature
is characteristic of the gap between the lowest and
first excited vibrational states, and with exactly
twice the zero-point energy, 1 hwosc .
2
100
Ideal diatomic gas: Vibrational
partition function
Vibrational energy level
spacings are much larger
Species
than those for rotation, so H
2
typical vibrational
HD
temperatures in diatomic
D2
molecules are of the order
N2
of hundreds to thousands
of kelvins rather than the CO
Cl2
tens of hundreds
characteristic of rotation. I2
vib/K
qvib
(@ 300 K)
5987
1.000
5226
1.000
4307
1.000
3352
1.000
3084
1.000
798
1.075
307
1.556
101
Vibrational partition function, qvib
• Light diatomic molecules have:
– high force constants
– low reduced masses
wosc
1

2
k
m
• Thus:
– vibrational frequencies (wosc) and characteristic
vibrational temperatures (vib) are high
– just one vibrational state (the ground state) accessible at
room T
• the vibrational partition function qvib ≈ 1
102
Vibrational partition function, qvib
• Heavy diatomic molecules have:
– rather loose vibrations
– Lower characteristic temperature
• Thus:
– appreciable vibrational excitation resulting in:
• population of the first (and to a slight extent higher)
excited vibrational energy state
• qvib > 1
103
Vibrational partition function, qvib
• Situation in polyatomic
species is similar
complicated only by the
existence of 3N-5 or 3N-6
normal modes of vibration.
Species
CO2
1.091
954(2)
NH3
4880(2)
1.001
4780
2330(2)
1360
CHCl3
tot
( n)
(1)
( 2)
( 3)
qvib
  qvib
 qvib
x qvib
x qvib
x ...
(1), (2), (3), … denoting individual normal
modes 1, 2, 3, …etc.
3360
1890
• Some of these normal modes
are degenerate
 
vib/K
∏(qvib)
(@ 300 K)
4330
2.650
1745(2)
1090(2)
938
523
374(2)
104
Vibrational partition function, qvib
As with diatomics, only the heavier species show
values of qvib appreciably different from unity.
Typically, vib is of the order of ~3000 K in many
molecules. Consequently, at 300 K we have:
qvib
1

1
10
1 e
in contrast with qrot (≈ 10) and qtrs (≈ 1030)
For most molecules only the ground state is
accessible for vibration
105
High T limiting behaviour of qvib
1

1  e  vib / T
At high temperature the equation qvib
gives a linear dependence of qvib with temperature.
If we expand 1  e
qvib 
 vib / T
, we get:
1
1  1  ( vib / T )  ...

T
 vib
High T limit
106
T dependence of vibrational partition
function
2.0
1.8
As T increases, the
linear dependence of
qvib upon T becomes
increasingly obvious
qvib
1.6
1.4
1.2
1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Reduced Temperature T/
1.4
107
The canonical partition function, Qvib
Qvib  q
N
vib
1



 vib / T 
1 e

N
Using U  kT 2   ln Q  we can find the first
 T V
differential of lnQ with respect to temperature
to give:
U vib
Nk vib
  ln Q 
 kT 
  vib / T
e
1
 T V
2


108
The vibrational energy, Uvib
U vib, m
R vib
  vib / T
(e
 1)
This is not nearly as simple as:
3
U trs, m  kT
2
U rot, m  RT linear molecules
109
The vibrational energy, Uvib
U vib, m
R vib
  vib / T
(e
 1)
This does reduce to the simple form at
equipartition (at very high temperatures) to:
U vib, m  RT
Normally, at
room T:
U vib, m
(equipartition)
3000 R 1
 10
 R
(e  1) 7
110
The zero-point energy
• So far we have chosen the zero-point energy
(1/2hw) as the zero reference of our energy scale
• Thus we must add 1/2hw to each term in the
energy ladder
• For each particle we must add this same amount
– Thus, for N particles we must add U(0)vib, m = 1/2Nhw
U vib, m
R vib
  vib / T
 U (0) vib, m
(e
 1)
R vib
1
  vib / T
 N A hw
(e
 1) 2
111
Vibrational heat capacity, Cvib
The vibrational heat capacity can be found using:
 vib / T
 U vib, m 
e
  vib 
  R
 
  vib / T
2
 1)
 T  (e
 T V
2
Cvib, m
The Einstein Equation
This equation can be written in a more compact
form as:
Cvib, m
  vib 
 RF E 

 T 
112
Vibrational heat capacity, Cvib
FE with the argument vib/T is the Einstein function
2 u
ue
FE  u
2
(e  1)
 vib 

u 

T 

The Einstein function
113
The Einstein heat capacity
1.0
FE
0.5
low T
High T
0.0
0.1
1
10
Reduced temperature T/
114
The Einstein function
• The Einstein function has applications beyond
normal modes of vibration in gas molecules.
• It has an important place in the understanding
of lattice vibrations on the thermal behaviour
of solids
• It is central to one of the earliest models for
the heat capacity of solids
115
The vibrational entropy, Svib
S vib
U vib  U vib (0) Avib  Avib (0)


T
T
U vib  U vib (0)

 k ln Qvib
T
N and N = N for one mole,
• We know Qvib  qvib
A
thus:
ln Qvib  N A k ln qvib  R ln qvib

S vib, m
R

 vib / T
 vib / T
(e
 1)

 ln 1  e
 vib / T

116
Variation of vibrational entropy with
reduced temperature
3.0
2.5
Svib/R
2.0
1.5
1.0
0.5
T
0 T
0.0
0.1
1
10
Reduced temperature T/
117
Electronic partition function
• Characteristic electronic temperatures, el,
are of the order of several tens of thousands
of kelvins.
• Excited electronic states remain unpopulated
unless the temperature reaches several
thousands of kelvins.
• Only the first (ground state) term of the
electronic partition function need ever be
considered at temperatures in the range
from ambient to moderately high.
118
Electronic partition function
It is tempting to decide that qel will not be a
significant factor. Once we assign 0 = 0, we
might conclude that:
qel   e
  el , i / kT
0
 e  0 (higher terms)  1
i
To do so would be unwise!
One must consider degeneracy of the ground
electronic state.
119
Electronic partition function
The correct expression to use in place of the
previous expression is of course:
qel   g i e
  el , i / kT
 g 0 e 0  0 (higher terms)  g 0
i
Most molecules and stable ions have nondegenerate ground states.
A notable exception is molecular oxygen, O2,
which has a ground state degeneracy of 3.
120
Electronic partition function
Atoms frequently have ground states that are
degenerate.
Degeneracy of electronic states determined
by the value of the total angular momentum
quantum number, J.
Taking the symbol G as the general term in the
Russell—Saunders spin-orbit coupling
approximation, we denote the spectroscopic
state of the ground state of an atom as:
spectroscopic atom ground state =
(2S+1)G
J
121
Electronic partition function
spectroscopic atom ground state = (2S+1)GJ
where S is the total spin angular momentum
quantum number which gives rise to the term
multiplicity (2S+1). The degeneracy, g0, of the
electronic ground states in atoms is related to
J through:
g0 = 2J+1 (atoms)
122
Electronic partition function
For diatomic molecules the term symbols are
made up in much the same way as for atoms.
• Total orbital angular momentum about the
inter-nuclear axis.
Determines the term symbol used for the
molecule (S, P, D, etc. corresponding to S, P, D,
etc. in atoms).
As with atoms, the term multiplicity (2S+1) is
added as a superscript to denote the
multiplicity of the molecular term.
123
Electronic partition function
In the case of molecules it is this term
multiplicity that represents the degeneracy of
the electronic state.
For diatomic molecules we have:
spectroscopic molecular ground state = (2S+1)G
for which the ground-state degeneracy is:
g0 = 2S + 1 (molecules)
124
Electronic partition function
Species
Li
C
N
Term
Symbol
2S
1/2
3P
0
4S
3/2
gn
g0 = 2
g0 = 1
g0 = 4
O
3P
2
g0 = 5
F
2P
3/2
g0 = 4
2P
1/2
g1 = 2
2P
1/2
g0 = 2
3/2
g1 = 2
NO
2P
O2
el/K
3Sg
g0 = 3
1D
g
g1 = 1
590
178
11650
125
Electronic partition function
Where the energy gap between the ground and
the first excited electronic state is large the
electronic partition function simply takes the
value g0.
When the ground-state to first excited state
gap is not negligible compared with kT (el/T is
not very much less than unity) it is necessary
to consider the first excited state.
The electronic partition function becomes:
qel  g 0  g1e
 el / T
126
Electronic partition function
For F atom at 1000 K we have:
qel  g 0  g1e
 el / T
 4  2e
590 / 1000
 5.109
For NO molecule at 1000 K we have:
qel  g 0  g1e
 el / T
 2  2e 178/1000  3.674
127