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

2 /

2/ 1

v

v

Tv

V vib T

eC NkT e

Very high T anharmonic effects Cvib T ( Prob 3.29-30)

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

Homopolar molecules

6.6. Chemical Equilibrium