MSEG 803Equilibria 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)

Example: a 2-particle system with 3 single particle states Single particle states are different from microscopic states of

a multi-particle system!

State 1 State 2 State 3

AB

AB

AB

A B

B A

A B

B A

A B

B A

Microscopic states of ideal gas systems

MB Statistics

State 1 State 2 State 3

AA

AA

AA

A A

A A

A A

BE Statistics

State 1 State 2 State 3

A A

A A

A A

FD Statistics

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:

Assumptions: no interactions between particles (ideal gas), all particles are distinguishable

1r

r

r

EE

r E

r

eP e

e Z

Two methods of deriving distribution functions

Model the systems as canonical ensembles

Model the systems as micro-canonical ensembles

Fix temperature

Calculate partition function

Z

Calculate average particle

# which minimizes Z

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 :

Canonical partition function:

Mean number of particles in a single particle state r :

1 1 2 2 3 3 ...R r rr

E n n n n

1 1 2 2 3 3exp exp ...RR R

Z E n n n

1 1 2 2 3 3

1 1 lnexp ...r r

R r

Zn n n n n

Z

Fermi-Dirac distribution

ns = 0 or 1, ; define:ss

n N

1 1 2 2

1 1 2 2

exp ...0 exp ( 1)

( ) exp ( 1)exp ...

rr rR

rr r r

R

n n nZ N

nZ N Z Nn n

( )1 1 1 1 1 1exp ... ...r

r r r r rR

Z N n n n

lnln 1 ln r

r r

ZZ N Z N

N

ln

1 exp rr r

ZZ N Z N

N

ln ln

~rZ Z

N N

In macroscopic systems:

exp ( 1)

( ) exp ( 11 p)

1

ex r

r rr

r r r

Z Nn

Z N Z N

The different number of ways to distribute ni particles over gi degenerate sub-levels of an energy level ei :

The total number of ways the set of occupation numbers { ni } can be realized:

Maximizing W subjected to the constraints:

Fermi-Dirac distribution: alternative derivation

!

! !i

ii i i

g

n g n

ii

n N i ii

n E

!

! !i

ii i i i i

g

n g n

1

exp 1i

ri i

nn

g

ln ln 0ii

d d

Partition function of Fermion ideal gas

Constraint:

Define sharply peaking at N’ = N

1 1 2 2 3 3( ) exp exp ...RR R

Z N E n n n

rr

n N

'

' exp 'N

Z N N

ln ~ ln exp ~ lnZ N N N Z N

ln ' exp ' ln0

'

Z N N Z

N N

1 2

1 1 2 20,1 0,1

exp exp ...

1 exp

n n

rr

n n

ln Z

N

ln ln ln 1 exp rr

Z N N

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 , FD distribution can be approximated by the classical MB distribution Electrons in the conduction

band of semiconductors or insulators

r kT

Bose-Einstein distribution

nr can be any positive integer,

Chemical potential m is determined by For a system comprised of conservative Bosons, m is

always lower than any single-particle energy level Partition function:

rr

n N

0 exp ( 1) 2 exp 2 ( 2) ...

( ) exp ( 1) exp 2 ( 2) ...

( ) 0 exp 2 exp 2 2 ...

( ) 0 exp exp 2 2 ...

exp

exp

r r r rr

r r r r r

r r r

r r r

rn

rn

Z N Z Nn

Z N Z N Z N

Z N

Z N

n n

n

1

exp 1r

rr

n N

ln ln 1 exp rr

Z N

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: m = 0

In equilibrium:

Note that the condition m = 0 only applies to photons in thermal equilibrium with a blackbody!

1 1

exp 1 exp 1rr h

n

0dG SdT VdP dN

0dN 0

Statistics of blackbody radiation

Blackbody: an idealized physical body that absorbs all incident electromagnetic radiation

Wave vector quantization condition:

Photon energy:

L

xn Z

2 2 2 2 2 22x y z x y zk k k k n n n

L

��������������������������������������� ���

2 2 22x y z

cE kc n n n

L

x xk nL

��������������

Statistics of blackbody radiation (cont’d)

The volume each state occupies in the phase space:

Photon Density of States (PDOS):

3 3 3

3

1

22 2stateVL L V

kx

ky

L

E

E + dE

E c

2

2 2

3 2 3 3

41

8

41

8 2

state

E c dkE dE

V

E c dE VEdE

V c c

Statistics of blackbody radiation (cont’d)

Total # of photons with energy between E to E + dE :

Total number of photons with frequency between n to n + dn per unit volume:

Total energy of photons having frequencies between n to n + dn per unit volume:

2

2 3 3

1

exp 1r

VEn E dE dE

E c

2 2

3 3 3

81 8 1

exp 1 1h kT

V hd h d

h h c c e

3

3

8 1,

1h kT

hu T d d

c e

Planck’s law and the Stefan-Boltzmann law

Blackbody spectral irradiance:

Total energy emitted from a blackbody per unit area per unit time:

3

3

8 1,

exp 1

hu T d d

c h

0

5 44 4

3 3

,

8

15

I T u T d

kT aT

c h

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.”

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):

47 7 1 50 ~ 525heaven earth heavenT T T C

, ~ 445hell S bT T C

Appl. Opt. 11(8), A14 (1972).

Classical limit of quantum distribution

In a dilute ideal gas,

Partition function

Maxwell-Boltzmann partition function

1rn expr rn

exp exp expr r rr r r

n N

ln

exp rr

NkT

exp

expr

rr

r

n N

ln ln ln exp rr

Z N N N N N N

ln ln ln ( ln ) ln ln ! lnrMB

r

Z N e Z N N N Z N Z

Thermodynamic properties of ideal gas: MB classical treatment

Internal energy of monatomic gas:2

2kineticN

pE E

m

3 32 1

30

2 3 2 3 3 31 1 13

0

3 2

2 33 2

0 0

...1... exp

2

1exp ... exp ...

2 2

2exp

2

NMB N

N

N N NN

NNN

NN

d q d pZ p

m h

p d p p d p d q d qh m m

V mp d p V

h m h

20

3 3 2ln ln ln ln

2 2MB

mZ N V

h

: Single molecule partition function

Thermodynamic properties of ideal gas: MB classical treatment (cont’d)

Partition function:

Pressure:

Internal energy:

Entropy:

where

20

3 3 2ln ln ln ln

2 2MB

mZ N V

h

ln1 1MBZ N NkTP

V V V

ln 3 3

2 2MBZ N

E NkT

3

2Vc R

0

3ln ln ln

2MB MBS k Z E Nk V T

0 20

3 2 3ln

2 2

mk

h

Ideal gas equation

Thermodynamic properties of ideal gas: classical limit of quantum statistics

Partition function:

Internal energy:

Entropy:

where

2

ln ln ln exp

3 3 2ln ln ln 1

2 2

rr

Z N

V m

N h

N N N

N

ln 3 3

2 2

Z NE NkT

3ln ln ln

2

VS k Z E Nk T

N

2

3 2 5ln

2 2

mk

h

3ln ln

2MBS Nk V T S

!

N

ZN

Gibbs paradox

Mixing two parts of ideal gas of identical composition Maxwell-Boltzmann statistics

Before mixing:

After mixing:

Quantum statistics in the classical limit

Before mixing:

After mixing:

,

32 ln ln

2MB iS Nk V T

, ,

32 ln 2 ln

2MB f MB iS Nk V T S

0

32 ln ln

2i

VS Nk T

N

0

32 ln ln

2f i

VS Nk T S

N

Vapor-solid phase equilibrium

Equilibrium condition:

Partition function of vapor:

Chemical potential of vapor:

Chemical potential of solid:

!

N

vZN

3 2

2

,

ln 2lnv v

vv vT V

F Z kT mkTkT kT

N N hP

s v

,

lns ss

s sT V

F ZkT

N N

2ln lns ss

Z ZE kT

T

0

0 2ln ln

Ts

s s T

EZ Z T dT

kT

Different from MB !

Vapor-solid phase equilibrium (cont’d)

Chemical potential of solid:

0

0

T

s s s TE E T N c T dT

0 0T K 0 00

0 0

ln s ss

F T E TZ T

kT kT

0 0

02

lnT Ts

s s T T

E T dTZ N c T dT

kT kT

0 0

02

ln T Tsss T T

s s

E TZ dTkT T c T dT

N N kT

0 0

00 2

0

1 1ln ln

T Tss s s T T

E T dTZ Z T N c T dT

k T T kT

Vapor-solid phase equilibrium (cont’d)

Equilibrium condition:

Equilibrium vapor pressure:

0 0

3 20

2 2

2ln

T Ts

T Ts

E TkT mkT dTkT T c T dT

h N kTP

s v

0 0

3 20

2 2

2 1exp

T Ts

T Ts

E TmkT dTP kT c T dT

h kTN k kT

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:

Periodic boundary condition:

Quantization of wave vector k : Density of states:

expA ik r

xx x L

Lx

2x x

x

k nL

��������������

p k����������������������������

2 2

2

k

m

xn Z

33

2

Vk d k

3 2

1 22 3

2

4

mVd