Problems and Solutions on Thermodynamics and Statistical Mechanics

182 Problems €4 Solutions on Thermodynamics 8 statistical Mechanics

giving

2022
Suppose that the energy of a particle can be represented by the ex-
pression E(z) = az2 where z is a coordinate or momentum and can take
on all values from -00 to $00.
(a) Show that the average energy per particle for a system of such
particles subject to Boltzmann statistics will be E = kT/2.
(b) State the principle of equipartition of energy and discuss briefly its
relation to the above calculation.

Solution:

its distribution function is

(a) From Boltzmann statistics,

( wis co ns in)

whether z is position or momentum,

So the average energy of a single particle is

• E=

+m

L

1

= -kT.

2

Inserting E(z) = az2 in the above, we obtain

(b) Principle of equipartition of energy: For a classical system of par-

ticle in thermal equilibrium at temperature T, the average energy of each

degree of freedom of a particle is equal to -kT.
1
2
There is only one degree of freedom in this problem, so the average
1
2

energy is -kT.