Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physica

(b) kT << h2/I. We have

ZR = 1 + 3e- a( h' /4na I) +. ,. + 3e-h1~4n2k~~

313

• " e-h214n21kT
  I 3Nh2 e-h'14n21kT
  En = 3N 1 + 3e-ha/4n21kT - I

CR = - 3N'4 e-h2 /4naIkT

PkT2

Nh2 - h' /4n2 IkT

S, = kN ln(1 + 3e-ha/4n21kT

h2/4naIkT 3Nh2 e -h2/4n21kT

IT

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2

An assembly of N fixed particles with spin - and magnetic moment

/.LO is in a static uniform applied magnetic field. The spins interact with the
applied field but are otherwise essentially free.

(a) Express the energy of the system as a function of its total magnetic

(b) Find the total magnetic moment and the energy, assuming that

(c) Find the heat capacity and the entropy of the system under these

(UC, Berkeley)

moment and the applied field.

the system is in thermal equilibrium at temperature T.

same conditions.

Solution:
(a) E = -MH.
(b) Assume that ii is the average magnetic moment per particle under
the influence of the external field when equilibrium is reached, then M =
NjZ and
e~oH/kT - e-PoH/kT
P=po e~oH/kT + e-@oH/kT = /.Lo