MSEG 803
Equilibria in Material Systems
9: Ideal Gas Quantum Statistics
Prof. Juejun (JJ) Hu
hujuejun@udel.edu
Ideal gas: systems consisting of particles with
negligible mutual interactions

Classical ideal gas (Maxwell-Boltzmann MB statistics)


All particles are distinguishable
Quantum statistics



Indistinguishable particles: interchanging two particles does not
lead to a new state
Particles with integral spin (Bosons in Bose-Einstein BE
statistics)
Particles with half-integral spin (Fermions in Fermi-Dirac FD
statistics): one single particle state can only be occupied by one
particle (Pauli exclusion principle)
Microscopic states of ideal gas systems


Example: a 2-particle system with 3 single particle states
Single particle states are different from microscopic states of
a multi-particle system!
MB Statistics
State 1
State 2
BE Statistics
State 3
AB
State 1
State 2
FD Statistics
State 3
AA
AB
AA
AB
B
A
B
A
A
A
B
B
A
A
B
B
A
State 2
A
A
A
AA
A
State 1
State 3
A
A
A
A
A
A
A
There is a greater tendency for particles
to bunch together in the BE case
Classical case: Maxwell-Boltzmann distribution

Take an individual system in a canonical ensemble as
one single classical ideal gas particle. The probability of
finding the particle in a specific single-particle level with
energy Er (in this case also the energy level of the
system) is given by the canonical probability distribution
function:
e  Er
1   Er
Pr 
 e
  Er
Z
e
r

Assumptions: no interactions between particles (ideal
gas), all particles are distinguishable
Two methods of deriving distribution functions

Model the systems as canonical ensembles
Fix
temperature

Calculate
partition
function Z
Calculate
average particle
# which
minimizes Z
Model the systems as micro-canonical ensembles
Fix internal
energy
Calculate # of
states W
Calculate
average particle
# which
maximizes W
Quantum statistics: problem formulation





Quantum states/levels of a single particle: r, s
Energy of single particle levels r or s: er or es
Number of particles in state r or s: nr or ns
Quantum state of the entire ideal gas system: R
Energy of the system in state R :
ER  n1e1  n2e 2  n3e 3  ...   nr e r
r

Canonical partition function:
Z   exp    ER    exp     n1e1  n2e 2  n3e 3  ... 
R

R
Mean number of particles in a single particle state r :
nr 
1
1  ln Z
  nr  exp     n1e1  n2e 2  n3e 3  ...    
Z R
 e r
Fermi-Dirac distribution

ns = 0 or 1,
n
 N ; define:
s
s
Z r  N    ( r ) exp     n1e1  ...  nr 1e r 1  nr 1e r 1  ... 
R
 n  exp   n e  n e  ...


 exp   n e  n e  ... Z
r
nr
1 1
0  exp   e r   Z r ( N  1)
2 2
R
1 1
2 2
r
( N )  exp   e r   Z r ( N  1)
R
 ln Z r
  ln Z r 
 Z r  N  1  Z r  N   exp  

N
N 

 ln Z r  ln Z
~
  
In macroscopic systems:
N
N
exp   e r   Z r ( N  1)
1
 nr 

Z r ( N )  exp   e r   Z r ( N  1) exp   e r     1
ln Z r  N  1  ln Z r  N  
Fermi-Dirac distribution: alternative derivation

The different number of ways to distribute ni particles
over gi degenerate sub-levels of an energy level ei :
i 

gi !
ni ! gi  ni !
The total number of ways the set of occupation numbers
{ ni } can be realized:
W  i  
i

i
gi !
ni ! gi  ni !
Maximizing W subjected to the constraints:
n
i
ne
N
i i
i
E
i
d ln W  d  ln i  0
i

ni
1

  nr 
gi exp   e i     1

Partition function of Fermion ideal gas
Z ( N )   exp    ER    exp     n1e1  n2e 2  n3e 3  ... 
R

R
n  N
  Z  N '  exp   N '
Constraint:
r
r

Define
sharply peaking at N’ = N
N'

 exp    e   n    exp    e   n
1
n1  0,1
1
n2  0,1
2
2
  ...
  1  exp    e r  
r
 ln  Z  N '  exp   N '
N '
ln
 ln Z
 ln Z



  

  0
N
N
~ ln  Z  N   exp   N    N  ~ ln Z   N
 ln Z   N  ln
  N   ln 1  exp    e r  
r
Properties of Fermi-Dirac distribution

At T = 0 K, FD distribution
is a step function and the
chemical potential defines
the Fermi surface


Electrons in metals
When e r    kT , FD
distribution can be
approximated by the
classical MB distribution

Electrons in the conduction
band of semiconductors or
insulators
Bose-Einstein distribution

nr can be any positive integer,
nr 

n
r
N
r
0  exp   e r   Z r ( N  1)  2  exp  2 e r   Z r ( N  2)  ...
Z r ( N )  exp   e r   Z r ( N  1)  exp  2 e r   Z r ( N  2)  ...
Z r ( N )  0  exp   e r     2  exp  2 e r  2    ...
Z r ( N )  0  exp   e r     exp  2 e r  2   ...
 n  exp  n    e 
1


 exp n    e  exp   e     1
r
n
r
r
n



Chemical potential  is determined by  nr  N
r
For a system comprised of conservative Bosons,  is
always lower than any single-particle energy level
Partition function: ln Z   N   ln 1  exp    e r  
r
Bose-Einstein condensate (BEC)

At low temperature, Bosons tends to cluster at the
lowest energy quantum mechanical state
Velocity-distribution data of a gas
of Rb atoms. Left: just before the
appearance of BEC. Center: just
after the appearance of BEC.
Right: nearly pure BEC.
Photon statistics (Planck statistics)


Photons are Bosons
The total number of photons is NOT conserved:  = 0
In equilibrium:
dG   SdT  VdP   dN  0 
  0
dN  0 
1
1
 nr 

exp  e r   1 exp   h   1

Note that the condition  = 0 only applies to photons in
thermal equilibrium with a blackbody!
Statistics of blackbody radiation


Blackbody: an idealized physical body that absorbs all
incident electromagnetic radiation
L
Wave vector quantization condition:
kx 
k 


L
 nx
2
nx  Z 
2
kx  k y  kz
2
2

 nx 2  n y 2  nz 2
L
Photon energy:
E

   kc 
2 c
 nx 2  n y 2  nz 2
L
Statistics of blackbody radiation (cont’d)

The volume each state occupies in
the phase space:
1 




 
2 L
2 L3 2V
3
Vstate 

3

L
ky
E + dE
3
E
Photon Density of States (PDOS):
1 4  E c  dk
W  E  dE  
8
Vstate
2
1 4  E c  dE
VE 2
 

 2 3 3  dE
3
8
 2V
c  c
2
kx
E c
Statistics of blackbody radiation (cont’d)

Total # of photons with energy between E to E + dE :
1
VE 2
nr  W  E  dE 
 2 3 3  dE
exp   E   1  c

Total number of photons with frequency between 
to  + d per unit volume:
8 V  h 
1
8 2
1

 d  h  
 h kT
 d
3 3
3
exp   h   1
hc
c
e
1
2

Total energy of photons having frequencies between
 to  + d per unit volume:
8 h 3
1
u  , T  d 

 d
3
h kT
c
e
1
Planck’s law and the Stefan-Boltzmann law

Blackbody spectral irradiance:
8 h 3
1
u  , T  d 

 d
3
c
exp   h   1

Total energy emitted from a blackbody per unit area per
unit time:

I T    u  , T  d
0
8 5 k 4
4
4


T

aT
15c3h3
a = 7.57 × 10-16 J·m-3·K-4
Heaven is hotter than Hell – a thermodynamic
proof

The Bible, Isaiah 30:26:

“Moreover, the light of the moon shall be as the light of the sun
and the light of the sun shall be sevenfold as the light of seven
days.”
Theaven

Tearth   7  7  1  50  Theaven ~ 525 C
4
The Bible, Revelations 21:8:


“But the fearful and unbelieving... shall have their part in the
lake which burneth with fire and brimstone.“
A lake of sulfur (brimstone):
Thell  TS ,b ~ 445 C
Appl. Opt. 11(8), A14 (1972).
Classical limit of quantum distribution

In a dilute ideal gas, nr  1  nr  exp     e r    
 n   exp   e
r
r
r
   kT ln
r
     exp      exp   e r   N
r
exp   e r 
N
 nr  N 
 exp  e r 
 exp  e r 
r
r

Partition function

Maxwell-Boltzmann partition function


ln Z   N  N   N ln N  N  N ln   exp   e r  
 r



ln Z MB  N ln   e e r   ln Z  ( N ln N  N )  ln Z  ln N !  ln Z
 r

Thermodynamic properties of ideal gas: MB
classical treatment

Internal energy of monatomic gas: E  Ekinetic
p2

N 2m
3
3
1

2  d q1 ...d p N
Z MB   ... exp    
 p 
3N
2
m
h
N


0
1
  2 3
 

 3 N  exp  
p1   d p1  ...   exp  
pN 2   d 3 pN   d 3q1  ...  d 3qN
h0
 2m

 2m

N
V
 3N
h0
  2 m 

  2 3 
   exp  
p   d p   V   2 
 2m 
  h0  


 ln Z MB
N

3
3  2 m  
 N ln V  ln   ln  2  
2
2  h0  

32
N

  N

 : Single molecule
partition function
Thermodynamic properties of ideal gas: MB
classical treatment (cont’d)

Partition function: ln Z MB

Pressure: P 

Internal energy: E  

3
3  2 m  
 N ln V  ln   ln  2  
2
2  h0  

1  ln Z MB 1 N NkT

  

V
 V
V
Ideal gas equation
 ln Z MB 3 N 3
   NkT

2  2
cV 
3


S

k
ln
Z


E

Nk
ln
V

ln
T


 Entropy: MB
 MB

0

2


3  2 mk  3
where  0  ln  2  
2  h0  2
3
R
2
Thermodynamic properties of ideal gas:
classical limit of quantum statistics

Partition function:


ln Z   N ln N  N  N ln   exp   e r  
 r

 V 3
3  2 m  
 N ln  ln   ln  2   1
2  h  
 N 2

Internal energy: E  
Z
N
N!
 ln Z 3 N 3
   NkT

2  2
 V 3

 Entropy: S  k  ln Z   E   Nk  ln
 ln T   
 N 2

3  2 mk  5
3




ln

S

Nk
ln
V

ln
T


where


MB

S
2  h2  2
2


Gibbs paradox


Mixing two parts of ideal gas of identical composition
Maxwell-Boltzmann statistics
Before mixing: S MB ,i
3


 2 Nk  ln V  ln T   
2


After mixing: S MB , f  2 Nk ln  2V   ln T     S MB ,i


3
2

Quantum statistics in the classical limit
 V 3
Before mixing: Si  2 Nk  ln  ln T   0 
 N 2

After mixing: S f  2 Nk  ln

V 3

 ln T   0   Si
N 2

Vapor-solid phase equilibrium

Equilibrium condition: s  v

Partition function of vapor: Z v 

Chemical potential of vapor:
N
N!
Different from MB !
 kT
 Fv 
  ln Z v 
v  
  kT  
  kT ln 
 P
 N v T ,V
 N v 

 2 mkT 


2
 h

32



Chemical potential of solid:
 Fs 
  ln Z s 
s  
   kT  


N

N
s T ,V
s



  ln Z s 
T E
2   ln Z s 
s
Es   

kT

 ln Z s  ln Z s T0   
 dT



2
T



T
0 kT




Vapor-solid phase equilibrium (cont’d)

Chemical potential of solid:
Es  Es T0   N s   c T  dT
T
T0
T  dT 
T
Es T0   1 1 



ln Z s  ln Z s T0  
      Ns   

c
T
dT



T0 kT 2 T0
k
T
T

0 

ln Z s T0   
ln Z s  
Fs T0 
kT0
Es T0 
kT

Es T0 
kT0
T0  0K 
 dT  T 
 


 Ns   

c
T
dT



T0 kT 2 T0

T
T  dT 
T
Es T0 
  ln Z s 



s  kT  



T


c
T
dT



T0  kT 2 T0


N
Ns
s


Vapor-solid phase equilibrium (cont’d)

Equilibrium condition: s  v
 kT
kT ln 
 P

 2 mkT 


2
 h

32

T  dT 
T
Es T0 



T   

c
T
dT




T0 kT 2 T0
Ns



Equilibrium vapor pressure:
 2 mkT 
P  kT  

2
 h

32
 Es T0  1 T  dT  T 



 exp 
  
 c T  dT  
2 T
T
0
0

k
 kT

 kTN s
Saturated vapor (equilibrium):
the rate of molecules impinging on
the solid/liquid surface = the rate of
vaporization from solid/liquid
Density of states for single-particle levels


Consider ideal gas occupying a volume V
Plane wave solution time-independent Schrödinger
equation:
2 2
k
Lx
  A  exp  ik  r  p  k e 
2m

Periodic boundary condition:
  x     x  Lx 
 Quantization of wave vector k : k x 

Density of states:
W k  
V
 2 
3
 d 3k
W e  
V
4 2
2
 nx
Lx
2m 


nx  Z 
32
3
 e 1 2 de