Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 171

Indicate clearly the reasoning you used to get this result.

(b) For m << N, this expression becomes

g(N, m) w g(N, 0) exp(-2m2/N).

Find the entropy of the system as a function of L for N >> 1, L < Nd.
(c) Find the force required to maintain the length L for L << Nd.
(d) Find the relationship between the force and the length, without
using the condition in (c), i.e., for any possible value of L, but N >> 1.

(UC, Berkeley)

N molecules

[ N = constant )

d = length of one link

___a

Fig. 2.3.

Solution:

angle then

N+ - N- = 2m,

Therefore N N

N+=-+m, N-=--m.

2

(a) Assume that there are N+ links of 0' angle and N- links of 180'

N+ + N- = N.

This corresponds to N!/(N+!N-$ arrangements. Note that for every

arrangement if the angles are reversed, we still get the overall length of

2md. Thus

2N!

g= (;+m)!(;-m)!

(b) When m << N, g(N, m) w g(N, 0) exp(-2m2/N), the entropy of

the system becomes

kL2

S = klng(N,m) = kIng(N,O) - 2~d2.

(c) From the thermodynamic relations dU = TdS + fdL and F =