Problems and Solutions on Thermodynamics and Statistical Mechanics

200 Problems €4 Sdutiom on Thermodylom'cs d Statiaticd Mechanics

(b) At high temperatures BIT << 1 and exp[-Oj(j + 1)/T) changes

slowly as j changes, so that we can think of (2j + 1) exp[-Oj(j + 1)/T] as

a continuous function of j. Let x = j(j + l), then dx = 2j + 1, and we can

write zo as an integral:

(c) At high temperatures, the internal energy is

a

aB

U = --lnzo = IcT.

The heat capacity is

C,=k.

(d) At low temperatures, we have T << 6, and exp[-6j(j + 1)/T] is

very small. We need only take the first two terms of zo, SO

2037

The energy levels of a three-dimensional rigid rotor of moment of in-

ertial I are given by

EJ,M = h2J(J + 1)/2I ,

where J = 0,1,2,... ;M = -J, -J + 1,... , J. Consider a system of N

rotors:

(a) Using Boltzmann statistics, find an expression for the thermody-

(b) Under what conditions can the sum in part (a) be approximated

namical internal energy of the system.

by an integral? In this case calculate the specific heat C, of the system.

( wis co nsin)