Transcript Introduction to stat..

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
8ma 2

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
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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 a3

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
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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
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Determining C
a
j
 C e
j
C
j
a
j
j
e
 E j

j
Then
 E j
a j  Ce
A
e
 E j
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
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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
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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
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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
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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  Nk BT  Nk BT  N

 h
2
2
1 e
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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
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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
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Einstein model of an atomic crystal
Qe
U obs
 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 

 E  
   N ,V
 h
3Nh 3Nhe
U  Uo 

 h
2
1 e
 E
CV  
 T

U 

 


 N ,V  T  N ,V
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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
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Specific heat of the Einstein atomic
crystal model
high temperature limit
2

h
k BT
 h 
e

C v  3R
h

 k BT  
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
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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
Pobs
1
 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 , 
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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  
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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
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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)
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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
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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


e
 u o
Q  e 
 h

1 e



3



N
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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
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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
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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
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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!
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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).
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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
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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
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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  Nk BT  Nk BT  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