1. The Statistical Basis of Thermodynamics

1. The Macroscopic & the Microscopic States

2. Contact between Statistics & Thermodynamics:

Physical Significance of the Number (N,V,E)

3. Further Contact between Statistics & Thermodynamics

4. The Classical Ideal Gas

5. The Entropy of Mixing & the Gibbs Paradox

6. The “Correct” Enumeration of the Microstates

1.1. The Macroscopic & the Microscopic States

System of N identical particles in volume V, with

, ,N

N V finiteV

(Thermodynamic limit )

E.g., Non-interacting particles:

1

N

ii

N n

1

N

i ii

E n

i = single particle energies ni = # of p’cles with energy i

A macrostate is specified by parameters ( N, V, E, ... ).

Postulate of equal a priori probabilities:All microstates satisfying the macrostate parameters are equally likely to occur.

, , ,N V E = # of all microstates that give rise to the macrostate (extensive) parameters N, V, E, ... .

Let

1.2. Contact between Statistics & Thermodynamics: Physical Significance of the Number (N,V,E)

Consider 2 systems A1 & A2 in thermal contact with each other,

i.e., partition is fixed, impermeable but heat conducting.

( Nj , Vj & E(0) = E1 + E2 are fixed )

A1

( N1 , V1 , E1 )

A2

( N2 , V2 , E2 )

01 2

0 01 1 2 2 E E E

E E E

Equilibrium is achieved if E1 ( with E2 = E(0) E1 ) maximizes (0) :

0

1

0E

1 1 2 2

2 2 1 11 2

E EE E

E E

2 2 2 2

1 2

E E

E E

(0) denotes properties of the composite system

1 1 2 22 2 1 1

1 2

0E E

E EE E

1 1 2 2

1 1 1 2 2 2

1 1E E

E E E E

1 1 2 2

1 2

ln lnE E

E E

ln E

E

Let 2 systems are in thermal equilibrium if they have the same .

Thermodynamics :,

1

N V

S

E T

Planck :

lnS k Boltzmann :

lnS k

1

k T

k = Boltzmann constant

3rd law

0th law ( thermal eqm.)

1.3. Further Contact between Statistics & Thermodynamics

For an impermeable but movable & heat conducting partition,

Nj , V(0) = V1 +V2 & E(0) = E1 + E2 are fixed.

Equilibrium is achieved, i.e., (0) is maximized, if

0

1

0E

0

1

0V

and

i.e., both system have the same values of &,

ln

N EV

1st law: dE T dS P dV dN

,N E

SP T

V

P

k T

chemical potential

~ mech. eqm.

For a permeable, movable & heat conducting partition,

N(0) = N1 + N2 , V(0) = V1 +V2 & E(0) = E1 + E2 are fixed.

Equilibrium is achieved, i.e., (0) is maximized, if

0

1

0E

0

1

0V

i.e., Both system have the same values of , , & ,

ln

V EN

1st law: dE T dS P dV dN

,V E

ST

N

k T

0

1

0N

~ chemical eqm.

Summary

Connection between statistical mechanics & thermodynamics is

lnS k

Once is written in terms of the independent thermodynamical variables,

all other thermodynamic quantities can be obtained via the Maxwell relations.

UInternal Energy

SV , X

P, YT

HEnthalpy

GGibbs free energy

FHelmholtz free energy

V

UT

S

Mnemonics for the Maxwell Relations

P

GS

T

U

S P

V T

G

T V

P S

dU TdS P dV Y dX 1

z x y

x y z

y z x

dH TdS Vd P Y dX

varvarvarF W

2

S V

PT

V S

U

S V

Good Physicists Have Studied Under Very Fine Teachers

= U ( P) V Y X

= F ( P) V Y X = H TS

= U TS

= U(V,S,X)

1.4. The Classical Ideal Gas

Non-interacting, classical ( distinguishable), point particles:

, , NN E V V

lnS k N V const

,N E

S P

V T

P V n R T

Nk

V

Cf n R

kN

A

R

N 231.38 10 /J K

23

8.31 /

6.02 10 /

J mol K

mol

58.62 10 /eV K

const here means indep. of V.

Quantum (Obeying Schrodinger Eq) Free Particles

Let these particles be confined within a cube of edge L.

Dirichlet boundary conditions: 0 at walls ( where x,y,z = 0,L ).

sin sin sinyx znn n

A x y zL L L

1,2,3, ; , ,in i x y z

Neumann boundary conditions: n 0 at walls.

cos cos cosyx znn n

A x y zL L L

,1,2,3,0in

22 2 2

2 2 2, ,2 2x y z x y z

kn n n n n n

m m L

1-particle energy :

2

2 2 22/38 x y z

hn n n

m V 3V L

2

2 2 22/38 x y z

hn n n

m V

* 2 2 2x y zn n n

2*

2/38

h

m V

i.e.

Let

( * is a positive integer )

1, ,N E V # of { nx, ny, nz } satisfying2/3

* 2 2 22

8x y z

m Vn n n

h

, ,N E V # of { nix, niy, niz } satisfying

For N non-interacting particles

2

2 2 22/3

18

N

i x i y i zi

hE n n n

m V

2

*2/38

hE

m V

* 2 2 2

1

N

i x i y i zi

E n n n

3

2

1

N

rr

n

2/3

* 2 2 22

1

8N

i x i y i zi

m V EE n n n

h

2/3, , ,N E V N V E

2/3, , ,N E V N V E 2/3, , ,S N E V S N V E

For reversible adiabatic processes, S & N are kept constant.

2/3V E const

2/31/3

,

20

3 N S

EdV V dE

V

,

2

3N S

E E

V V

,N S

EP

V

2

3

EP

V

,f N S

Valid for both classical & quantum statistics

(adiabatic processes)

5/3PV const

Better behaved quantity is ( N,E,V),

defined as the # of lattice points with non-negative coordinates & lying within

the volume bounded by the surface of a sphere, centered at the origin, and

with radius

Counting States: Distinguishable Particles

State labels { nix, niy, niz } form a lattice in the 3N-D n-space.

( N,E,V) = # of lattice points with non-negative coordinates & lying on the

surface of a sphere, centered at the origin, and with radius

*R

fluctuates wildly even for small E changes unless N >>1.

2/3*

2

8m V ER

h

* * * 3/21

1 41,

8 3N

As R , the lattice points become a continuum.

* 3/2

6

Better approximations:

Number of points on the x-y, y-z, z-x planes is

Since these points are shared by 2 neighboring sectors, the

volume integral counts each as half a point.

* * 3/2 *1

3

6 8

Dirichlet B.C.(exclude all nj = 0 points )

* * 3/2 *1

3

6 8

Neumann B.C.(include all nj = 0 points )

*13

4

( Density of states in n-space is 1. )

Volume of an n-D sphere of radius R is/2

!2

nn

sphV Rn

( see App.C )

Volume of points with non-negative coordinates

1

2

n

sphV V

( Take non-negative-half of every dimension )

3 3 /2

* * 3 /21,

32 !2

N NNN E E

N

2/3

*2

8, , ,

mV EN V E N E

h

3 /2

3

2

3!

2

NN m EVNh

! 1n n

3 /2

3

2, ,

3!

2

NN m EVN V E

Nh

Stirling’s formula: ln ! lnn n n n for n >>1

3/2

3

3ln , , ln 2 ln !

2

V NN V E N m E

h

3/2

3

4 3ln , , ln

3 2

V m E NN V E N

h N

3

23 3 3ln ! ln

2 2 2

N N NN

Let (N,V,E) = # of states lying between E ½ & E+ ½ .

, ,, ,

N V EN V E

E

3, ,

2

NN V E

E

3/2

3

4 3ln , , ln

3 2

V m E NN V E N

h N

ln , ,N V E ln , ,N V E

3/2

3

4 3ln , , ln

3 2

V m E NN V E N

h N

3/2

3

4 3, , ln ln

3 2

V m E N kS N V E k N k

h N

2

2/3

3 2, , exp 1

4 3

h N SE S V N

m V N k

3

2E N k T

3

2n R T ,N V

ET

S

2

3E

N k

,

V

N V

EC

T

3

2N k

3

2n R

,N S

EP

V

2

3E

V

2

3P V E N k T n R T

,

P

N P

HC

T

,N P

E PV

T

VC n R 5

2n R

5

3P

V

C

C

3/2

3

4 3, , ln

3 2

V m E N kS N V E N k

h N

3

2E N k T

Isothermal processes ( N, T = const ) : E const

,N E

SS V

V

N kV

V lnN k V

ln lnf i f iS S N k V V ln f

i

VN k

V

Adiabatic processes ( N, S = const ) :

3/2V E const 3/2V T const

P V n R T 5/2P V const P V

1V T

Alternatively, ,N SdE P dV

2

3

EdV

V also leads to 3/2V E const

5

3

1.5. The Entropy of Mixing & the Gibbs Paradox

3/2

3

4 3ln

3 2

V m E N kS N k

h N

This S is not extensive, i.e., , , , ,S N V E S N V E

Mixing of 2 ideal gases 1 & 2 (at fixed T ) :

3

3ln

2before itotal i i

i i i

VS S k N

2

3 4ln 1 ln

2 3

m EN k V N k

h N

3/2

2

2 3ln

2

m k TS N k V

h

3

2E N k T

3

3ln

2aftertotal mixed i

i i

VS S k N

ii

V V

3

3ln

2

VN k

22

mkT

Thermal wavelength

Entropy of mixing of gases :

after beforetotal totalmixing

S S S ln 0ii i

Vk N

V

Gibb’s paradox :

For the mixing of different parts of the same gas in equilibrium (Ni / Vi = N / V ,

i = ), the formula still applies & we also have S > 0, which is

unacceptable.

i

i

N N

V V

3

3ln

2before itotal i i

i i i

VS S k N

3

3ln

2aftertotal i

i i

VS k N

Irreversible process: S > 0 is expected.

For the mixing of different parts of the same gas in eqm., ,ii

i

N N

V V

lnimixi i

VS k N

V ln lni i

i

k N N k N N

ln ln 0mixed i i ii

S k N N S k N N

Thus, Gibbs’ paradox is resolved using Gibbs’ recipe :

3/2

3

4 3, , l 1n

3 2

V m ES N V E N k N

h NNk

Sackur-Tetrode eq.

S is now extensive, i.e., , , , ,S N V E S N V E

ln ln 0mixed i i i ii

S k N N N S k N N N or

ln ln !N N N N

ln !S S k N

3

3ln

2before itotal i

i

VS k N

3

3ln

2aftertotal

VS k N

lnii i

Nk N

N

Revised Formulae

3/2

3

4 3, , l 1n

3 2

V m ES N V E N k N

h NNk

2/323 2

, , e2

p4 3

x 13

h N SNE S V N

m V N k

extensive

In general, relations derived using the previous definition of S

are not modified if they do not involve explicit expression of S.

,V S

E

N

2 21

33

E S

N N k

intensive

Gibbs’ recipe is cancelled by removing all terms in red.

,N V

ET

S

2

3

E

N k

,N S

EP

V

2

3

E

V

2

3

ET

N k

2

3

EP

V

2 2

33

E

N

E E S

N N N k 1

PE STN

V

3/2

3

4 3l 1n

3 2

V m ES N k N k

h NN

3

3ln

21k

N

VN

3

2 2

3 3

3 21 ln 1

2 3 N

Vk T

3lnk TV

N

21

3

2

3

E S

N N k

3ln 1k TN

NV

A G P V N P V N N kT

21

3

2

3

E S

N N k

22

mkT

1.6. The “Correct” Enumeration of the Microstates

Elementary particles are all indistinguishable.

In the distribution of N particles such that ni particles occupy the i state,

!

!Di

i

N

n

for distinguishable particles

1 for indistinguishable particles

In the classical (high T ) limit, 0in i !D N

!D

N

Gibbs’ recipe corresponds to