Problems and Solutions on Thermodynamics and Statistical Mechanics

Statiaticol PhyYica 185

Summing up the terms for all such atoms, we have

where p(A) is the state density of distribution of A;.

2025
The three lowest energy levels of a certain molecule are El = 0, Ez =
E, E3 = 10~. Show that at sufficiently low temperatures (how low?) only
levels El, E2 are populated. Find the average energy E of the molecule at
temperature T. Find the contributions of these levels to the specific heat
per mole, C,, and sketch C, as a function of T.
( Wisconsin)
Solution:

levels for low temperatures.
according to the Boltzmann statistics, we have

We need not consider energy levels higher than the three lowest energy

Assuming the system has N particles and

NI + N2 + N3 = N,

hence

N

N3 = 1 + eQcfkT + elOa/kT *

When N3 < 1, there is no occupation at the energy level E3. That is, when

T < Tc, only the El and E2 levels are occupied, where T, satisfies

If N >> 1, we have
lo€
klnN ’

T, w -