Problems and Solutions on Thermodynamics and Statistical Mechanics

170 ProMema 6, Solutions on Thermodynam'ca 8 Statistical Mechanics

2010
A solid contains N magnetic atoms having spin 1/2. At sufficiently
high temperatures each spin is completely randomly oriented. At suffi-
ciently low temperatures all the spins become oriented along the same di-
rection (i.e., Ferromagnetic). Let us approximate the heat capacity as a
function of temperature T by

[ c1 (% - 1) ifTL/2 < T < TI

C(T) =

lo otherwise ,

where TI is a constant. Find the maximum value c1 of the specific heat

(use entropy considerations).

(UC, Berkeley)

Solution:

dS

dT

From C = T - , we have

S(o0) - S(0) = -dT = cl(1- ln2).

l-

On the other hand, we have from the definition of entropy S(0) = 0, S(o0) =

Nk In 2, hence

Nkln2

1-In2

c1=-.

2011

The elasticity of a rubber band can be described in terms of a one-

dimensional model of polymer involving N molecules linked together end-

to-end. The angle between successive links is equally likely to be 0' or

180'.

(a) Show that the number of arrangements that give an overall length

of L = 2md is given by

2 N!

g(N,m) = , where m is positive

(; + (; - m)!