Download 06._TheTheoryOfSimpleGases-1

6.
The Theory of Simple Gases
1. An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble
2. An Ideal Gas in Other Quantum Mechanical Ensembles
3. Statistics of the Occupation Numbers
4. Kinetic Considerations
5. Gaseous Systems Composed of Molecules with Internal Motion
6. Chemical Equilibrium
6.1.
An Ideal Gas in a Quantum Mechanical
Microcanonical Ensemble
N non-interacting, indistinguishable particles in V with E.
( N, V, E ) = # of distinct microstates
Let  be the average energy of a group of g >> 1 unresolved levels.
Let n be the # of particles in level .
n   E


n  N



Let W { n } = # of distinct microstates associated with a given set of { n }.

  N ,V , E  

 
N , E
W n 
n
Let w(n ) = # of distinct microstates associated
with level  when it contains n particles.

W n    w  n 

W n    w  n 

Bosons ( Bose-Einstein statistics) :
wBE  n  
 n  g  1  !
n !  g  1  !
WBE n   

See § 3.8
 n  g  1 !
n !  g  1  !
Fermions ( Fermi-Dirac statistics ) :
w(n ) = distinct ways to divide g levels into 2 groups;
n of them with 1 particle, and g  n with none.
wFD  n  
n !
WFD n   


g !
g  n !
n !

g !
g  n !
n  g
Classical particles ( Maxwell-Boltzmann statistics ) :
w(n ) = distinct ways to put n distinguishable particles into g levels.
wMB  n    g  
n
WMB  n  
N!
 n !
g


1
 n !
n
g
 
n



gn

 n !

Gibbs corrected
S  N ,V , E   k ln   N ,V , E 
 k ln W n*  
n* extremize
 N , E

 k ln 
W n 
 n 


n   E


Method of most probable value
( also see Prob 3.4 )




L  ln W n      n  N      n   E 
 

 

n  g  1  !

WBE n   
 n !  g  1  !
ln WBE n     n  g  ln  n  g

n  N


  n ln n  g ln g 



g 
n  
   n ln  1 
  g ln  1 

n
g
 
 
 


Lagrange
multipliers



g 
n 
ln WBE n     n ln  1     g ln  1    
n 
g  
 


WFD n   

n !

g !
g  n !
ln WFD n     g ln g  n ln n   g  n  ln  g  n  


 g


n 
   n ln    1   g ln  1    
g  
 
 n



 g


n 
ln W n     n ln    1   g ln  1    
g  
 
 n


gn
WMB n   
 n !
ln W n 
BE
FD
ln WMB n     n ln g  n ln n  n 


n
g 
ln WMB n 

 g


n 
ln W n     n ln    1   g ln  1    
g  
 
 n






L  ln W n      n  N      n   E 
 

 



 g


n 
L    n ln    1   g ln  1      n   n      N   E
g 
 
 n



 g

1 g
1
ln    1  

      0
g
n

 n

 1 n 1  
n
g

 g

ln    1        0
 n


n*
1
    
g e
1
 e     

BE
FD
MB
g
 e   
n
1
Most probable occupation per level
e
   
1

 g


n 
ln W n     n ln    1   g ln  1    
g  
 
 n



n*
1
    
g e
 *  g


S
n*  
*
 ln W n     n ln  *  1   g ln  1 

k
n
g
 
 
 



   n* ln  e   
 
1

1  1   g ln  1     
 e
1
BE
FD

1  
 *
 e     
    
*



n




g
ln
1
e
   n        g ln     









1 

 
e

 N   E
g ln 1


N
E PV
S
 
T
T
T
MB:
e
   
1
e     
  


PV 


kT
kT
1
kT
g 


PV 
kT
e       kT
g ln 1


n


*
e     
 N kT
6.2.
An Ideal Gas in Other Quantum Mechanical
Ensembles
Z  N , T ,V    e   E  QN T ,V 
Canonical ensemble :
E
Ideal gas,  = 1-p’cle energy :
 n   E


g n  exp   

Z  N , T ,V  
N 
 n 
 n  N

n

  
g n  = statistical weight factor for { n }.
g BE n   1
 1 all n  0, or 1
g FD n   
otherwise
0
g MB n   

1
n !
Actual g absorbed in  ( here is treated as non-degenerate: g = 1).
Z  N , T ,V  

N 
 n 

g n  exp   

g MB n   
Maxwell-Boltzmann :


Z  N , T ,V  

 
N 
n
 n   E

n

  
1
n !


1
  n  


e


  n !   




 

 QN T ,V 


1
N   N !  
   n 

e
   
N !  n    n !   
 


1 
  

e
 
N !  

N
multinomial theorem
1
1
N
N
Z 1, T ,V  
Q1  T ,V 
N!
N!
 n  N
Z 1, T ,V   Q1 T ,V    e      e



V
 2 
V
 2
2
4  d k k e
2
3

1

2m
2
2
1

2m
2

k
V 1  3   2m 

   2 
2
2 2  2    
k2
 
d


e

0
V
m
 2 2 2 
2
 
1
N
1 V 
Z  N , T ,V  
Q1 T ,V  
 
N!
N !  3 
Z   , T ,V  

e
 N
Z  N , T ,V  
N 0

1  zV 

 3 
N
!
 
N 0
1/2
 2 2 
 

 mkT 
2 2
k
0
2m m

3/2
3/2
 m
V 
 2 

2 

3/2

V
3
 3
 3/2   
2
N
partition function (MB)

N
z
 Z  N , T ,V   Z  z, T ,V   Q  z, T ,V 
N 0
N
 zV 
 exp  3 
 
grand partition function (MB)
Z  N , T ,V  

N 
 n 

g n  exp   

Z  N , T ,V  

 
N 
n
Z   , T ,V  

e
 N


g n    e   n  
 

Z  N , T ,V 
N 0


N 
 
g n  exp   
N  0   n 




  g n  exp   

 n 


n
Difficult to evaluate
(constraint on N )


N    N
 
e
g n  exp   
N  0   n 




 n  


 n     


n





 



    g  n  e      
  n
 n  N
 1 all n  0, or 1
g FD n   
otherwise
0
g BE n   1
Bose-Einstein / Fermi-Dirac :

 n   E

n

  

 n 


     
g
n
e




 

   z  ;  ,  



n



z  ;  ,     g  n  e
n
      n
 1 n  0, or 1
g FD  n   
 0 otherwise
g BE  n   1
z  ;  ,     g  n  e
Z   , T ,V    z   ;  ,  

B.E.
z  ;  ,   
      n
e


n  0
F.D.
z  ;  ,   
 e
n  0
Grand potential :
1
1  e      

1
1  ze   


n
 1 e
     
 1  ze  
1  e      
Z   1  e     

F  , T ,V   kT ln Z  , T ,V    kT
q  z, T ,V   ln Z  z, T ,V  
1
 0  0
ln 1 e     

q potential :
      n
Z
 0   
Z  0  e        1
     
1
n !
n

1
g MB  n  
 ln 1
ze    
BE
FD
F  , T ,V    kT
e     e  
ln 1


FMB  kT
1  MB :
 ln 1
q  z , T ,V  
e     
 e    


ze    
BE
FD
 kT e  Z 1, T ,V 
c.f. §4.4
qMB  z  e     z Z 1, T ,V   z Q1 T ,V 

 F 
N  
 



TV
Alternatively
kT

 1
 e      
e
     
Z   zN Z  N  
N



e      1
 z N e  E 
e
N, E
N, E
1
  E  N 
 kT
N  N e 
Z N, E
  ln Z 
1
E   E z N e  E   



Z N, E

z
1


1
z 1e  
1
   E  N 
  ln Z 
 F 

  






T

TV
 q





z


 z e  
1 ze   



z 1e  
1
Mean Occupation Number
For free particles :
z  ; ,     g  n  e
Z   , T ,V    z   ;  ,  
n
 

1
BE
FD

1
      n 

  z       n g  n  e

Z    
n
 


1
 n g  n  e
z  
      n
n
kT

Z

 1 ze
n


      n
 Z 
 kT
  

, , 
n 
1
1  
z e
1


kT   z 
z     , 
  ln Z 
1


  


, , 
1
e     
n*

g
1
q  ln Z
 q
 z  e  
  kT
  

,

,
1 ze   

   
see §6.1
6.3.
Statistics of the Occupation Numbers
Mean occupation number :
e
BE :
 
FD :
n  1 
    
1


MB :
n  e    
1
Classical : high T
  must be negative & large
From §4.4 :

N 3

V
z  n 3
1
same as §5.5
n 
1
e
    
 

BE
FD
1
n ~ 
n  1 for
 
kT
B.E. condensation
1
BE
FD
Statistical Fluctuations of n
n 
n 
2

1
z  
1
z  
 n 
 n g  n  e
      n
1

 n g  n  e
2
      n
n
2
 n  n
2
 kT 

z  
  kT 
2
z  ;  ,    1 ze
e      1
n
2
2
kT    2 Z 
 2 z 


  2 
Z   2   ,  ,

 , 
   
2
 1  2 z  1  z 2 


 
2
z


z




   , 
e     
  n 

 kT 

e      1
  , , 


 n 


n
  
2

n2  n
n
2
 

2
2
1
 e       e  
z
  kT 
 e      n
2
2
  2 ln z 
  2 

 , 

1
n 
 n 


n
  
2

n2  n
n
2
2
e
    

1
1
n
1
e      1
BE
FD
above normal
below normal
Einstein on black-body radiation :
+1 ~ wave character
 n 1 ~ particle character
Statistical correlations in photon beams :
see Kittel,
“Thermal Phys.”
see refs on pp.151-2
Probability Distributions of n
Let p (n) = probability of having n particles in a state of energy  .
z     g  n  e
      n

n
n 
1
e      1

e
e
    
     
1 e

1
1
n
n

1  n
     

1
1  n
p  n   C e      n
BE
FD
p  n   C e
e
    
z  ;  ,    1 ze
  n    
1

1
n
e
     

1   p  n   C
BE :
n0


1  e      

1   p  n   C 1  e
n0

1
1 e
1
     
  n    



p  n   1  e
e
1  n
1
FD :
n

1  n
p  n  
1
1  e      
e
  n    
     


 

1
     
C  1 e
C


1  e      
n

BE
FD
1
z
n
1
 n 
 1  n  

1

n
 

e      1
1
1  n
     
 n 


1

n
 

1
n 
1 

n
n
n

n 1
1
z
 1  n

 n
n0
n 1
p  n  
MB :
C   n    
e
n!
Gibbs’ correction

1   p  n   C exp e      
n0
n 

 n p  n   C
n0
p  n  


1
  n    
     
e


C
e
n  1  n  1 !

1   n    
 exp  e        
Alternatively    e
n  0 n!

2
n
 2
 2
   2
1
1
p  n  
n
n! e 
n
n

1   n      e       
e

n  0 n!
n  
1 
  
 e      
n
n

  e
n!
n
 e      e2     n  n
p  n 
1

n
p  n  1 n
1
  n    
e
n ! exp  e      
 prob of occupying state 
Poisson distribution
2

 n 
2
 n
“normal” behavior of
un-correlated events
p  n 
1

n
p  n  1 n
BE :
p  n  

FD :
n
1
n
“normal” behavior of
un-correlated events
n
1  n

n 1
n
p  n 

p  n  1 1  n
 1  n
p  n   
 n


Geometric ( indep of n )
> MB for large n :
Positive correlation
n0
n 1
 n
p  n 

  1  n
p  n  1 
 0
n 1
n 1
< MB for large n :
Negative correlation
n - Representation
Let n = number of particles in 1-particle state  .
  n   n0 , n1 , n 2 ,
State of system in the n- representation :
Z
  n 
e
   H  N 
 n 
Non-interacting particles :
 0, 1, 2,
n  
 0, 1
 n 
Z
  n 

e
      n 

 n 




Z    e        n       e       n 

 n    
  n

 n 
1

      

 
 1 e

  1  e       

 

bosons
fermions
bosons

fermions
Mean Occupation Number
F   f n
Let F be an operator of the form
H     n
e.g.,

1
F  Tr   F   
Z  n 
 n 
  e       n n f 
n

  
      n

e


 n




e
 H
F
 n 
  n f



   a    n 
Z    e

  n

 e  

n 
n
a 
 e  

n
 n
a 
n
 n
6.4.
From § 6.1
P
Free particles :
Kinetic Considerations
kT
V
P

P


g ln 1


kT
V
g
k
BE
FD
ze    

ln 1 ze
   k 
 k   k 

p k
k


kT V
   k 
2
4

d
k
k
ln
1
ze
0
V  2 3

kT  1 3
    p
p
ln
1
ze

2 2 3  3
 z  e   d
0 d p p 1 ze    p  dp

1
6 2
ze   d 
3
dp p
3 
  p
1 ze   dp
0

  p


0

1

3
1
6 2

kT
2 2
3

    p
2
d
p
p
ln
1
ze

0
  p

3

2
d
p
p
3 
0
1
z 1 e
   p



 d 
 p dp 
1


P
1
6 2

dp p
3 
1
2
e
0
   p    
 d 
p

d
p
1

n 
Let p( ) be the probability of a particle in state  .
n

N
n
p   
 n

1
e
    
1
BE
FD
Then
f =  p   f   =

1
N
n f  



P

1
6 2

dp p
3 
np
0
d
p
dp
1N
P
3V
  ps
2

p
s = 1 : phonons
s = 2 : free p’cles
 d 
p

 d p
V
 =  2   d
3
p
1
 n pu
3
d
 s
dp
N
n
V

P
1
ns 
3

3
p =

V
2 2
3
2
d
p
p

0
p2 1

 m u2
2m 2
1
E
1 E
ns
 s
3
N
3 V
All statistics
P
1
n pu
3

pressure is due to particle motion (kinetics)
Let n f(u) d3u = density of particles with velocity between u & u+du.
 d u n f  u  n
3

3
d
 u f  u  1
# of particles to strike wall area dA in time dt
= # of particles with u dA >0 within volume udA dt 

u dA  0
d 3u n f  u u  dA dt 
dA  nˆ dA
ˆ
Each particle imparts on dA a normal impluse = 2p  n
Total impulse imparted on dA =

P
I  2n 
unˆ  0
d 3u  f  u p  nˆ  u  nˆ  dA dt 
I
 n  d 3u f  u   p  nˆ   u  nˆ   n
dA d t
1
P

n pu

3
 p  nˆ   u  nˆ 
n
p u cos2 
Rate of Effusion
# of particles to strike wall area dA in time dt
n
unˆ  0
d 3u  f  u u  nˆ  dA dt 
 Rate of gas effusion per unit area through a hole in the wall is
Rn
unˆ  0
f  u  f  u 
nˆ  zˆ

R  2 n  du u

 d u f  u  1  4
3
d 3u  f  u u  nˆ 
2
0
1

0
0
3
d
cos

f
u
u
cos



n
du
u
f u 





2
du
u
f u 


R
0
1
n u
4
All statistics
R   u   Effused particles more energetic.
u>0
 Effused particles carry net momentum (vessel recoils)
Prob.
6.14