Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistid Physics 351

2163
A system consists of N very weakly interacting particles at a temper-
ature sufficiently high such that classical statistics are applicable. Each
particle has mass rn and oscillates in one direction about its equilibrium
position. Calculate the heat capacity at temperature T in each of the fol-
lowing cases:

(a) The restoring force is proportional to the displacement x from the

(b) The restoring force is proportional to x3.

equilibrium position.

The results may be obtained without explicitly evaluating integrals.

(UC, Berkeley)

Solution:

According to the virial theorem, if the potential energy of each parti-

cle is V cx xn, then the average kinetic energy T and the average poten-

tial energy v satisfy the relation 2T = nv. According to the theorem of

equipartition of energy. T = -kT for a one-dimensional motion. Hence we

can state the following:

1

2

1

2

(a) As f cx x, V cx x2, and n = 2. Then v = T = -kT, E = ST + T =

kT. Thus the heat capacity per particle is k and C, = Nk.

1- 1 3

2 4 4

(b) As f cx x3, V cx x4 and n = 4. Then v = -T = -kT, E = -kT.

Thus the heat capacity per particle is -k and C, = -Nk for the whole

system.

3 3

4 4

2164

By treating radiation in a cavity as a gas of photons whose energy

E and momentum k are related by the expression E = ck, where c is the

velocity of light, show that the pressure p exerted on the walls of the cavity

is one-third of the energy density.

With the above result prove that when radiation contained in a ves-

sel with perfectly reflecting walls is compressed adiabatically it obeys the

equation

PV7 = constant.

Determine the value of 7.

(UC, Berkeley)