8/8/2019 Home Sol 5

1/2

Problem Set 5

Statistical MechanicsPhysics 341

Tim Pennycook

Problem 1.

To find the thermodynamic properties NA, NB , NAB, one can use the partition function, which iseasily constructed, since we are given the single particle partition functions for each type of particle:

Q =fNAA f

NBB f

NABAB

NA!NB!NAB !,

where the factorials come from the assumption that these particles are indistinguishable. Using theSterling approximation, the free energy,

F = kT ln Q

= kT(NA

ln fAN

Aln N

A+ N

A+ N

Bln f

BN

Bln N

B+ N

B+ N

ABln f

ABN

ABln N

AB+ N

AB)

= kT(NA ln fA/NA + NA + NB ln fB/NB + NB + NAB ln fAB/NAB + NAB)

Now to find the minimum of the free energy,

(ln Q) = (ln fA/NA 1)NA + (ln fB/NB 1)NB + (ln fAB/NAB 1)NAB .

Setting this equal to zero, and using the fact that NAB = NA NB ,

0 = (ln fA/NA 1)NA + (ln fB/NB 1)NB (ln fAB/NAB 1)(NA + NB)

= (ln fA/NA)NA + (ln fB/NB)NB (ln fAB/NAB)(NA + NB)

= lnfANAB

NAfAB

NA + lnfBNAB

NBfAB

NB .

Finally, using NA = NB

ln

fAfBNABNANBfAB

= 0 =

NABNANB

=fAB

fAfB,

or, multiplying by one, 1VnABnAnB

= fABfAfB , so

nABnAnB

= VfAB

fAfB.

Problem 2.

Q1 =1

h3

PS

epcd3pd3r

=4V

h3

0

epcp2dp

=8V

(ch)3,

1

8/8/2019 Home Sol 5

2/2

so for N indistinguishable particles,

QN(V, T) =1

N!

8V

kT

hc

3N,

so using the free energy F = kT ln Q,

P =

F

V

T,N

=kN T

V, U =

ln Q

=

3N

= 3NkT,

confirming P V =1

3U, and U/N = 3kT . The specific heats,

CV =

U

T

N,V

= 3kN, C P =

U + P V

T

N,P

= 4kN,

so we see that indeed = CP/CV = 4/3.

Problem 3.

The multiplicity ofj, gj , is obviously equivalent to the number of ways of putting j balls in s boxes,which we know to be

gj =(j + s 1)!

j!(s 1)!,

the desired expression. The one particle partition function,

Q1 = es/2

j=0

ej = e1

2sw1 ew

s,

so the system of N such oscillators will have the partition function

Q1 = e

1

2sNw

1 ew

sN

,

which, as expected, is the same as for a system of sN one-dimensional oscillators. The partitionfunction uniquely defines the thermodynamic properties, so those for the N s-dimensional oscillatorswill be identical with those of sN one-dimensional oscillators:

= sN12

+ kT ln(1 e) , S = sNk12

coth12

ln 2 sinh12

P = 0, U =

1

2sN coth

1

2

,

and

C = sNk

1

2

2cosech2

1

2

.

2