Problems and Solutions on Thermodynamics and Statistical Mechanics

338 Problems d Solutions on Thermodynamics d Statiaticd Mechunics

(c) The given potential is as shown in Fig. 2.32. Letting

27 a0 1

”[ a’’

dx 213

=Uo -12--2(-6)- =O,

we find that U(x) is minimum at x = a.

(d) According to classical mechanics, atoms are at rest at their equi-
librium positions when T = 0. The distance between neighboring atoms is
a. If T # 0, the interacting potential is

UT(5) = U(x + u) + U(a - x)

=UT(a)+UT,x2+U~3x3+...

where x is the displacement from equilibrium position, and UT, = 72Uo/a2,
UT, = -5040/a3 etc.
Using classical statistical mechanics, we obtain

Since kT << Uo, we obtain

= 7akT196Uo ,

7k

46U

and X = -.

2148

A classical system is described by its Hamiltonian H, which is a func-

tion of a set of generalized co-ordinates q; and momenta pi. The canonical

equations of motion are