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Introduction to statistical
thermodynamics
The Boltzmann factor and
partition functions
How are the molecules
distributed over the various
possible energy states at a given
temperature?
• Boltzmann factor: probability that a system
exists in an energy Ej:
pj  e
 E j / k BT
• The normalization constant is:
Q  e
 E j / k BT
j
Q is the partition function
2
Consider a macroscopic system
• N particles, volume V, and given certain
forces among the particles
• Schrödinger equation:
Hˆ N  j  E j  j
• For an ideal gas:
J = 1, 2, 3,…
N
E j ( N ,V )    i
i 1
• Monoatomic gas:
n n n
x y z
Cubic container of side a
Ignore electronic states and focus on translational energies

h2
2
2
2

n

n

n
x
y
z
8ma2

3
We want to evaluate pj for the
monoatomic gas
• probability that a system exists in an
energy Ej: p  e E / k T
j
B
j
• We consider an ensemble :
Each system has same N, V, T
but is in a different quantum
state
System: Ej (N, V)
Infinite heat bath
(thermal reservoir)
Thermal insulation
There are aj systems with energy Ej; the total number of systems is A
4
Consider two specific energy
states: E1(N, V) and E2(N, V)
• Relative number of systems in states 1 and 2:
a2
 f ( E2 , E1 )  f ( E1  E2 )
a1
a3 a 2 a 3

a1 a1 a2
e x y  e xe y
and
f ( E1  E3 )  f ( E1  E2 ) f ( E2  E3 )
A good candidate for f is:
f (E)  e
 is an arbitrary constant
E
5
Relative number of systems in
states 1 and 2:
a2
 e  ( E1  E2 )
a1
And
In general,
a j  Ce
an
 ( Em  En )
e
am
 E j
Where C and  are constants that we need to find out
6
Determining C
a
j
 C e
j
C
j
a
j
j
e
 E j
A

 E j
e
j
Then
 E j
a j  Ce
j
 E j
 E j
Ae

 E j
e
j
aj
And
A

e
 E j
e
 E j
j
In the limit of A infinitely large, the fraction aj/A is a probability
7
Probability that a randomly chosen system is
in state j with energy Ej(N, V)
p j ( N ,V ,  ) 
aj
A

e
 E j
e
 E j
 E j
e

Q( N , V ,  )
j
We can express all the macroscopic thermodynamic properties in
terms of the partition function, Q
we will show later that
1

k BT
8
Average ensemble energy
E   p j ( N ,V ,  ) E j ( N ,V )  
j
j
Q  e
E j ( N , V )e
 E j ( N ,V )
Q( N , V ,  )
 E j
j
  ln Q 


   N ,V
so,
   e  E j
1 j
 
Q  



 E j

E
e
1
 E
    E j e j   j
Q j
Q

j
 N ,V
  ln Q 

E  
   N ,V
9
Example: monoatomic ideal gas
Ground state electronic state, only translational modes
N

q(V ,  )
Q( N ,V ,  ) 
N!
Get
3/ 2
with
 2m 
q(  ,V )   2  V
h  
  ln Q 

E  
   N ,V
10
For a diatomic ideal gas
N

q(V ,  )
Q( N ,V ,  ) 
N!
3/ 2
with
 2m 
8 2 I e  h / 2
q(  ,V )   2  V 2
 h
h

h

1

e


 h
3
h Nhe
U  E  NkBT  NkBT  N

 h
2
2
1 e
11
Heat Capacity at Constant Volume
 E
CV  
 T

U 
  


 N ,V  T  N ,V
  ln Q 
2   ln Q 
  k BT 
E  

 T  N ,V
   N ,V
With the equations that we found for U of a monoatomic ideal gas and U for a diatomic ideal
gas, we can get the heat capacities
12
Molar heat capacities of ideal gases
3
Cv  R
2
2
 h / k BT


5
h
e

C v  R  R
 h / k BT
2
k
T
 B  1 e


2
13
Einstein model of an atomic crystal
Qe
 U o
 h / 2
 e

 h
1

e




3N
N atoms in lattice sites; N independent
harmonic oscillators, each vibrating in 3 directions,
same frequency for all atoms
Uo is the sublimation energy (all atoms separated)
  ln Q 

U obs  E  
   N ,V
 h
3Nh 3Nhe
U  Uo 

 h
2
1 e
 E
CV  
 T

U 

 


 N ,V  T  N ,V
14
Specific heat of the Einstein
atomic crystal model
2

h
k BT
 h 
e


C v  3R

h

 k BT  
1  e k B T






2
15
Specific heat of the Einstein atomic
crystal model
high temperature limit
2

h
k BT
 h 
e


C v  3R

h
k
T

 B  
1  e k B T






2
at high T, exp (-h/kBT) is small, and exp (x) ~1+x for small x
2
 h 
1

C v  3R
 3R
2
 k BT   h 


 k BT 
law of Dulong-Petit
16
Evaluation of the pressure
 E j 

Pj ( N ,V )  
 V  N
P   p j ( N , V ,  ) Pj ( N , V )
j
 E j
 Q 

     
 V  N , 
j  V
1
Pobs  P 
Q
 E j

E

j  e

P    
V  N Q
j 

 E
 e j
 N ,
 Q 
  ln Q 

  k BT 

 V  N , 
 V  N ,
17
Ideal Gas Equation of State
a) Monoatomic ideal gas
N

q(V ,  )
Q( N ,V ,  ) 
N!
3/ 2
with
 2m 
q(  ,V )   2  V
h  
18
Example: identify the EOS
1  2m 
Q( N ,V ,  )   2 
N!  h  
3N / 2
(V  Nb) e
N
aN 2 / V
a and b are constants
19
A system of independent,
distinguishable molecules
• To evaluate the average Energy or the average pressure
we need the knowledge of the eigenvalues Ej (energy
states).
• In some cases, we can approximate the total energy as
the sum of individual energies.
• Lets consider the case of a perfect crystal, each atom
occupies one and only one lattice site and the lattice
sites are distinguishable, therefore the particles are
distinguishable.
• The atoms vibrate in their lattice sites and these
vibrations are independent (in the same way the modes
of a polyatomic molecule are independent)
20
A system of independent,
distinguishable molecules
 
a
j
individual particle energies
El ( N ,V )   (V )   (V )   (V )  ...
a
i
Q  e
 El
b
j
 e
l
c
k
N terms
  ( ia  bj  kc ...)
l
Q  e
i
  ia
e
j
  bj
e
  kc
...
molecular partition functions
k
Q  q(V , T )
N
independent, distinguishable molecules
21
example: Einstein model of atomic
crystals
Qe
 U o
 h / 2
 e

 h
1

e




N atoms in lattice sites; N independent
harmonic oscillators, each vibrating in 3 directions,
same frequency for all atoms
Uo is the sublimation energy (all atoms separated)
3N
if we write uo =Uo/N sublimation energy per atom at 0 K
 h / 2

 u o  e
Q  e 
 h

1 e



3



N
22
A system of independent,
indistinguishable particles
the total energy is
Eijk..   i   j   k  ...
Q( N , V , T ) 
e
N terms
  ( i  j  k ...)
i , j , k ..
two type of particles: fermions and bosons
Fermions have spin ½, 3/2, 5/2,… Pauli exclusion principle applies to them
Bosons have spin 0, 1, 2, 3… Pauli principle does not apply to them
23
spin of a particle
angular momentum in classical mechanics
quantization of an
standing wave
Fermions have spin ½, 3/2, 5/2,… Pauli exclusion principle applies to them
Bosons have spin 0, 1, 2, 3… Pauli principle does not apply to them
24
Evaluation of Q
Q( N , V , T ) 
e
  ( i  j  k ...)
i , j , k ..
Suppose 2 non interacting identical fermions (N =2),
each with energies 1, 2, 3, and 4
Enumerate the allowed total energies in Q
4
Q(2,V , T )   e
  (  i  j )
i , j 1
25
Evaluation of Q for bosons
• if there are only two states: 2(only one of them)
and all the others 10
• E = 2 + 10 + 10 + 10 +….
N terms
• or
• E = 10 + 2 + 10 + 10 +….
• etc
• However all these summations are identical
• If instead all states are different, to enumerate
all allowed states we have only one choice,
other permutations will represent identical
states, therefore we need to divide by N!
26
In conclusion, to evaluate Q
• we have problems when there are two or more
indices that are the same
• how realistic are these cases?
• if the # of available quantum states >> number of
particles, it would be unlikely for any two particles
to be in the same state
• From the systems that we studied, there are
infinite number of states. However, at a given
temperature not all these states are available,
because the energies of these states are >> kBT
(the average energy of a molecule).
27
Q for independent
indistinguishable systems
• So, if the # of available states with
energies < kBT is >> # of particles,
then all terms in Q will contain
energies with different indices, then a
good approximation is

q(V , T )
Q
N
N!
with
q(V , T )   e
  j / k BT
j
28
Is the assumption realistic?
• The # of translational states alone is sufficient to
guarantee that the # of available states is >> # of
particles.
• The criterion can be written as:

N h


V  8mkBT 
2
3/ 2
 1
• Therefore, low density, large mass, high T favors this
criterion
• Particles obey Boltzmann statistics
29
Table 3.1
30
Decomposition of a molecular
partition function

q(V , T )
Q
N
N!
with
q(V , T )   e
  j / k BT
j
  ln Q 
2   ln Q 
2   ln q 
  k BT 
E  
  NkBT 

 T  N ,V
 T  N ,V
   N ,V
  j / k BT
e
E  N  j
q(V , T )
j
31
Since the molecular partition
function was derived for a
system of independent particles
E N 
  j / k BT
e
E  N  j
q(V , T )
j
  j / k BT
e
   j
q (V , T )
j
We can obtain the probability that a molecule is in its jth state
32
probability that a molecule is in
its jth state
  j / k BT
e
j 
q (V , T )
 
trans
i

rot
i

vib
i

elec
i
Since all the terms are distinguishable
q(V , T )  q
trans rot
vib
q q q
elec
33
Example: diatomic molecule
N

q(V ,  )
Q( N ,V ,  ) 
N!
where
 vib
3/ 2
with
 2m 
8 2 I e  h / 2
q(  ,V )   2  V 2
 h
h

h

1

e


 h / 2
e
qvib (T ) 
 h
1 e
  ln qvib  h he  h
 
 

 h


2
1

e


3
h Nhe  h
U  E  NkBT  NkBT  N

2
2
1  e  h
Which agrees with
34
Set of states with the same
energy are called levels
q(V , T ) 
e
  j / k BT
states
Can be written as
q(V , T ) 
g e
  j / k BT
j
levels
35
Example: rigid rotator
2

 j  J ( J  1)
2I
g j  2J 1

qrot (T )   (2 J  1)e
  2 J ( J 1) / 2 Ik BT
J 0
36