Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 183

2023

A system of two energy levels Eo and El is populated by N particles

at temperature T. The particles populate the energy levels according to

the classical distribution law.

(a) Derive an expression for the average energy per particle.

(b) Compute the average energy per particle vs the temperature as

(c) Derive an expression for the specific heat of the system of N par-

(d) Compute the specific heat in the limits T + 0 and T -+ 00.

T + 0 and T + 00.

ticles.

( was co nsin)

Solution:
(a) The average energy of a particle is

Assuming El > Eo > 0 and letting AE = El - Eo, we have

EO + E1e-BAE

1 + e-BAE

U=

(b) When T + 0, i.e., p = l/kT -+ 00, one has

u M (EO + Ele-OAE)(l - e-oAE) = Eo + AEe-BAE ,

When T -+ co, or p + 0, one has

P

4

1

u CJ 2 -(Eo + El - PElAE) + El) - -(AE)2.

(c) The specific heat (per mole) is