Problems and Solutions on Thermodynamics and Statistical Mechanics

316 Problems €4 Solcltiona on Thermodynamics d Statistical Mechanics

2134
Consider N fixed non-interacting magnetic moments each of magnitude
po. The system is in thermal equilibrium at temperature T and is in a
uniform external magnetic field B. Each magnetic moment can be oriented
only parallel or antiparallel to B. Calculate:
(a) the partition function,
(b) the specific heat,
(c) the thermal average magnetic moment (a).
Show that in the high temperature limit the Curie Law is satisfied (i.e.,

(UC, Berkeley)

x = dM/dB a l/T).

Solution:

we can consider a single magnetic moment. As its partition function is

(a) Since the magnetic moments are nearly independent of one another,

1

z=a+-,

U

where a = exp(p0 B/kT), the partition function for the entire system is

(b)z=-poBN

= -poBN tanh(poB/kT) ,

giving the specific heat as

C=-=~N(~) dE PO B^2 1

dT cosh2 (g) *

• dM Npg 1
  x--=--
  dB IcT cosh2 (g).

1

At the high temperature limit, x a -.

T