Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistid Physica 201

Solution:
(a) The partition function of the system is

03

z = c(2J + 1) exp[-h2J(J + 1)/2IkT].

J=O

The internal energy is

dlnz

dT

U = NkT2-

(b) In the limit of high temperatures, kT >> h2/21, and the above sum

can be replaced by an integral. Letting 3: = J(J + l), we have

~=~~exp{-~i)dz=~, h2 2 IkT

U=NkT.

Thus the molar specific heat is C, = NAk = R.

2038
Consider a heteronuclear diatomic molecule with moment of inertia
I. In this problem, only the rotational motion of the molecule should be
considered.

(a) Using classical statistical mechanics, calculate the specific heat

C(T) of this system at temperature T.

(b) In quantum mechanics, this system has energy levels

h2..

E. - - 3 (3 + 1) , j = 0,1,2,....

• 21

Each j level is (2j + 1)-fold degenerate. Using quantum statistics, derive
expressions for the partition function z and the average energy (E) of this