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 N
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 E
n
N V E W n
Let W { n } = # of distinct microstates associated with a given set of { n }.
Let w(n ) = # of distinct microstates associated with level when it contains n particles.
W n w n
Bosons ( Bose-Einstein statistics) : W n w n
1 !! 1 !BE
n gw n
n g
See § 3.8
1 !! 1 !BE
n gW n
n g
Fermions ( Fermi-Dirac statistics ) :
n g w(n ) = distinct ways to divide g levels into 2 groups;
n of them with 1 particle, and g n with none.
!
! !FDgw n
n g n
!
! !FDgW n
n g n
Classical particles ( Maxwell-Boltzmann statistics ) :
!!
nMB
NW n gn
w(n ) = distinct ways to put n distinguishable particles into g levels.
nMBw n g
1!
ngn
Gibbs corrected
!
ngn
, , ln , ,S N V E k N V E
,ln N E
n
k W n
n N
n E
*lnk W n
Method of most probable value( also see Prob 3.4 )
n* extremize lnL W n n N n E
1 !! 1 !BE
n gW n
n g
ln ln ln lnBEW n n g n g n n g g
ln 1 ln 1g nn gn g
Lagrange multipliers
!
! !FDgW n
n g n
!
n
MBgW nn
ln ln 1 ln 1BEg nW n n gn g
ln ln ln lnFDW n g g n n g n g n
ln 1 ln 1g nn gn g
ln ln lnMBW n n g n n n
ln ln 1 ln 1g nW n n gn g
BE
FD
ln ln MBn gW n W n
ln ln 1 ln 1g nW n n gn g
lnL W n n N n E
ln 1 ln 1g nL n g n n N En g
1 1ln 1 0
1 1
g gg nn nn g
ln 1 0gn
1g en
* 11
ng e
BEFD Most probable occupation per level
e MB 1e
ln ln 1 ln 1g nW n n gn g
* 11
ng e
*
* **ln ln 1 ln 1S g nW n n g
k n g
BE
FD
* 1ln 1 1 ln 11
n e ge
* ln1
en ge
* ln 1n g e
ln 1N E g e
E TS PV N kT 1
kT ln 1PV kT g e
PV kT g e
MB: N kT*kT n
1e
6.2. An Ideal Gas in Other Quantum Mechanical Ensembles
Canonical ensemble : , , ,EN
E
Z N T V e Q T V
n N
n E
Ideal gas, = 1-p’cle energy :
, , expN
n
Z N T V g n n
g{ n } = statistical weight factor for { n }.
1BEg n 1 0, 10FD
all n org n
otherwise
1!MBg n
n
1g
, , expN
n
Z N T V g n n
1!MBg n
n
n E
n N
1, ,!
N n
n
Z N T V en
1 !! !
nN
n
N eN n
1!
N
eN
1 1, ,!
NZ T VN
Maxwell-Boltzmann :
11 ,
!NQ T V
N
,NQ T V
multinomial theorem
2 21
2 23
0
42
kmV dk k e
3/2
2 2
1 3 22 2 2V m
3/2
22mV
3
V
1/222mkT
2 2 20
22V m m d e
3/23/2
2 2
322 2V m
11, , ,Z T V Q T V e
2 21
2k
me
k
11, , ,
!NZ N T V Q T V
N
3
1!
NVN
partition function (MB)
0
, , , ,N
N
T V e Z N T V
Z 0
, ,N
N
z Z N T V
, ,z T VZ , ,z T VQ
30
1!
N
N
z VN
3exp z V
grand partition function (MB)
, , expN
n
Z N T V g n n
n E
n N
, , N n
n
Z N T V e
Bose-Einstein / Fermi-Dirac : 1BEg n 1 0, 10FD
all n org n
otherwise
Difficult to evaluate (constraint on N )
0
, , , ,N
N
T V e Z N T V
Z
0
expN N
nN
e g n n
expn
g n n
0
expN
nN
g n n
n
n
g n e
nn
g n e
; ,
z ; , n
n
g n e
z
1BEg n 1 0, 10FD
all n org n
otherwise
, , ; ,T V
Z z
B.E.
0
; , n
n
e
z 1
1 e
0 1e Z 0 0 0
F.D.
1
0
; ,n
n
e
z 1 e
11 ze
1 ze
Grand potential : , , ln , ,F T V kT T V Z ln 1kT e
BEFD
q potential : , , ln , ,q z T V z T V Z ln 1 ze
; , n
n
g n e
z
1
1 e
Z
1 e
Z
, , ln 1F T V kT e
, , ln 1q z T V ze
BEFD
1e e MB : MBF kT e
MBq z e
1, ,kT e Z T V
1, ,z Z T V 1 ,z Q T Vc.f. §4.4
1TV
F eN kTe
11e
, ,
E NN N E
N N E N E
z Z N z e e Z
,
1 E N
N E
N N e Zln
T
kT
Z
Alternatively
,
1 N E
N E
E E z e Zln
z
Z1
z eze
z
q
1 1z e
1
11z e
TV
F
Mean Occupation Number
For free particles :
1 n
n
n n g n e
zZ
, ,
lnkT
Z
lnq Z
, ,
1 q
1
z ekTze
1
11
nz e
1
1e
BEFD
*n see §6.1
, ,
kT
ZZ
1 n
n
n g n e
z ,
kT
zz
, , ; ,T V
Z z 1; ,1
n
n
g n eze
z
6.3. Statistics of the Occupation Numbers
1
1n
e
BEFD
Mean occupation number :
n e
MB :
FD : 1n 1 1n forkT
BE : ~n B.E. condensation
1e
1
Classical : high T
must be negative & large3z nFrom §4.4 :
3 1NV same as §5.5
Statistical Fluctuations of n
1
1e
2 21 n
n
n n g n e
z
BEFD
2 2
2, ,
kT
ZZ
2 22n n n 22
22
,
1 1kT
z zz z
22
2,
lnkT
z
, ,
nkT
21
e
e
2e n
2 22
2
n nnn n
e
1 ez
1 n
n
n n g n e
z 1; ,1 ze
z
2 2
2,
kT
zz
2 22
2
n nn en n
1
1n
e
BEFD
1 1n
above normalbelow normal
Einstein on black-body radiation :
+1 ~ wave character
n 1 ~ particle character
see Kittel, “Thermal Phys.”
Statistical correlations in photon beams : see refs on pp.151-2
Probability Distributions of n
Let p (n) = probability of having n particles in state of energy .
np n C e
BEFD
11
ne
1 1en
1n
en
111
en
n
n
g n e
z
BE :
np n C e
0
111n
p n Ce
1 np n e e
1
1 1
nn
n n
BEFD
1
1n
e
1 1en
1n
en
111
en
11
n
n
n
n
FD : 1
0
1 1n
p n C e
1
1np n e
e
11
nn
nn
1 01
n nn n
MB :
0
1 expn
p n C e
!nCp n e
n
Gibbs’ correction
1
! expnp n e
n e
0n
n n p n
1
11 !
n
n
C en
0
1!
n
n
C e en
e
1!
n
np n nn e
!
nnn
en
Poisson distribution
0
1 exp!
n
n
e en
1n
e Alternatively
22
2 2
1n
2e e 2n n 2n n
1 11
p nn
p n n n
“normal” behavior of un-correlated events
1 11
p nn
p n n n
“normal” behavior of un-correlated events
11
n
n
np n
n
BE :
FD : 1 0
1n n
p nn n
1 1
np np n n
Geometric ( indep of n )> MB for large n :Positive correlation
11
10 1
nnp n
np n
n
< MB for large n :Negative correlation
n - Representation
Let n = number of particles in 1-particle state .
0,1, 2,0, 1
bosonsn
fermions
H
n
n e n
Z
Non-interacting particles :
0 1 2, , ,n n n n State of system in the n- representation :
n
n
n e n
Z
a n
n
e
Z
11
1
bosonse
e fermions
6.4. Kinetic Considerations
ln 1kTP g zeV
From § 6.1BEFD
ln 1kTP g zeV
kk
k
22 3
0
ln 12
pkT d p p ze
2
30
4 ln 12
kkT VP dk k zeV
Free particles :p k
3 3
2 30 0
1 1ln 12 3 3 1
pp
p
kT z e dp ze d p pdpze
k k
3
2 30
16 1
p
p
ze dd p pdpze
2
2 3 10
1 16 1p
dd p p pdpz e
2
2 30
1 16 1p
dP d p p pd pe
BEFD
3 2
3 2 30
= =22
V Vd p d p p
p
Let p( ) be the probability of a particle in state . Then
221
2 2m
m
p u
13
N dP pV d p
np
n
f p f
=nN
1
1n
e
22 3
0
16 p
dP d p p n pd p
1 n fN
=
13
n pu NnV
sp dp sd p
13
P ns 13
EnsN
13
EsV
s = 1 : phononss = 2 : free p’cles
All statistics
13
P n pu pressure is due to particle motion (kinetics)
Let n f(u) d3u = density of particles with velocity between u & u+du.
3d u n f n u 3 1d u f u
# of particles to strike wall area dA in time dt
= # of particles with u dA >0 within volume udA
dt
3
0dd u n f d d t
u A
u u A
Total impulse imparted on dA =
Each particle imparts on dA a normal impluse = ˆ2 p n ˆd dAA n
3
ˆ 0ˆ ˆ2I n d u f dA d t
u n u p n u n
3 ˆ ˆn d u f u p n u nIP
dA dt ˆ ˆn p n u n 2cosn pu
13
P n pu
Rate of Effusion
3
ˆ 0ˆn d u f dA d t
u n u u n
Rate of gas effusion per unit area through a hole in the wall is
3
ˆ 0ˆR n d u f
u n u u n
# of particles to strike wall area dA in time dt
f f uu
1
2
0 0
2 cos cosR n du u d f u u
ˆ ˆn z 3
0
n du u f u
3 1d u f u 2
0
4 du u f u
14
R n u All statistics
R u Effused particles more energetic.
u > 0 Effused particles carry net momentum (vessel recoils)
Prob.6.14
6.5. Gaseous Systems Composed of Molecules with Internal Motion
Assumptions ( ideal Boltzmannian gas ) :
1. Molecules are free particles ( non-interacting).
2. Non-degeneracy (MB stat) :3 1n
22mkT
11 1, , , 1, , ,
! !N N
NZ N T V Q T V Z T V Q T VN N
internal1 1 11, , , translZ T V Q T V Q Q 3
V j T
j T g e
= quantum # for internal DoF
Internal DoF j T g e
int , ln lnA N T N kT j N kT j intint ln
T
A kT jN
intint
N
AST
int1 jU Nj
intint
,V
V N
UCT
2 ln jNkTT
lnln jN k j TT
2 ln jN k TT T
elec nucl vib rotj T j T j T j T j TMolecules :
Homopolar molecules (A-A) : elec nucl rot vibj T j T j T j T
6.5.A. Monatomic Molecules
Let 4 5~ 10 10ionT Kk
( All atoms are neutral & in electronic ground state )
Nuclear spin Hyperfine structure : T ~ 101 – 100 K.
Level-splitting treated as degeneracy : 2 1n ng S
Inert gases ( He, Ne, Ar, ... ) : Ground state L = S = 0 : 1eg
0/ /0
kT kTe e
j T g e
= 0 denotes ground state. 0 = 0.
0j T g
0
e ng g 2 1nS
L = 0; S 0 : 2 1 2 1nj T S S 2 1eg S
int , ln ln 2 1 ln 2 1nA N T N kT j N kT S S
intint ln
T
A kT jN
intint
N
AST
int1 jU Nj
intint
,V
V N
UCT
0 0
ln 2 1 ln 2 1nkT S S
ln 2 1 ln 2 1nN k S S
L = 0, S 0
2 1 2 1nj T S S
L 0, S 0
/2 1 J
L SkT
elecJ L S
j T J e
kT 2 1L S
elecJ L S
j T J
1 2 1 2 1 2 121 2 1 2 1 2 12
L S L S S L S
L S S L L S L
2 1 2 12 1 2 1L S L SS L S L
kT 0 /02 1 kT
elecj T J e Ground state 0 = 0.02 1J
CV, int = 0 in both limits CV has a maximum.
6.5.B. Diatomic Molecules
Let 4 5~ 10 10dissocT Kk
( All atoms are neutral & in electronic ground state )
Non-degenerate ground state ( most cases ) ge = 1 & jelec (T) = 1
j T g e
Degenerate ground state ( seldom ) :
1. Orbital angular momentum 0, but spin S = 0 :
In the absence of B, depends on |z|
doublet ( z = M ) is degenerate ( ge = 2 = j(T) ) CV = 0
2. = 0, S 0 : ge = 2S + 1 = j(T) CV = 0
3. 0 & S 0 :
Spin-orbit coupling Beff fine structure
0elec
fine struct
j T g g e
E.g., NO ( 1/2, 3/2 ) ( splitting of doublet ) :
178Kk /
0 1kT
elecj T g g e
0elecfine struct
j T g g e
1 elec
elecelec
jU Nj
/1
/0 1
kT
kT
g eNg g e
,
elecV elec
V N
UCT
// 1
1 22 / /0 1 0 1
1 kTkT
kT kT
g eN g ekT g g e g g e
2 /0 1
2/0 1
kT
kT
g g eN kkT g g e
//
2
kTkTe e
T kT
2
/ /1 0
0 1
1
1 1kT kT
N kkT g ge e
g g
1 12
1 0
0 1
2/1
0
1 1 0
0kT
g g kTkT g g
Nkg e kT
kT g
CV has max. for some kT ~
Vibrational States
310 Kk
for diatomic gases
Full contribution for T 104 K
No contribution for T 102 K
Harmonic oscillations (small amplitude) :12n n
From § 3.8 :
2 /
2/ 1
v
v
Tv
V vib T
eC N kT e
v
2
2
1 v
vV vib
v
TC NkT
T
vT Nk equipartition
value
vT 2
/v TvV vib
C N k eT
0 vib DoF frozen out
Nuclear Spin & Rotational States: Heteropolar Molecules
Heteropolar molecules ( AB ) : no exchange effects
interaction between nuclear spin & rotational states negligible.
2 1 2 1n A Bg S S From § 6.5.A :
Cnucl = 0
Molecule ~ rigid rotator with moment of inertia ( bond // z-axis )
, , 0diag I II 20I r A B
A B
m mm m
= reduced mass
r0 = equilibrium bond length
2
12rot l l
I
0, 1, 2,l