Problems and Solutions on Thermodynamics and Statistical Mechanics

198 Problem3 U Solutiow on Thermodynamic8 U Statistical Mechanics

2035

The quantum energy levels of a rigid rotator are

E? = j(j + l)h2/8w2rna2 ,

where j = 0,1,2,... The degeneracy of each level is gj = 2j + 1.

(a) Find the general expression for the partition function, and show

that at high temperatures it can be approximated by an integral.

(b) Evaluate the high-temperature energy and heat capacity.

(c) Find the low-temperature approximations to z, U and C,.

(S VNY, BufuIo)

Solution:

(a) The partition function is

00 03

j=O j=O

(b) At high temperatures A, E (h2/8n2rna2kT)1/2 < 1,

where

Hence

~j = j+ - A,, AE~ = ~j+l - ~j = A,.

( 3

(a,) = 8w2ma2kT/h2.

The internal energy is

a

U = kT2-lnz = kT

8T