In our first look at the ideal gas we considered only the translational energy of the particles. But molecules can rotate, with kinetic energy. The rotational motion is quantized: and the energy levels of a diatomic molecule are of the form

ε(j) = j(j + 1)ε0

Where j is any positive integer including zero; j = 0, 1, 2 . . . The multiplicity of each rotational level is g(j) = 2j + 1.

(a) Find the partition function ZR(τ) for the rotational states of one molecule. Remember that Z is a slim over all stales, not over all levels–this makes a difference.

(b) Evaluate ZR(τ) approximately for τ >> ε0, by converting the sum to an integral

(c) Do the same for τ << ε0, by truncating the sum after the second term

(d) Give expressions for the energy U and the heat capacity C, as functions of τ, in the both limits. Observe that the rotational contribution to the heat capacity of a diatomic molecule approaches 1 (or, in conventional units, ku) when r >> ε0.

(e) Sketch the behavior of U(τ) and C(τ) showing the limiting behaviors for τ → ∞ and τ → 0.