Explanation:1) Diatomic Gas Consider a gas of noninteracting diatomic molecules. The rotational energy of the molecules can be MODELLED as a rigid quantum rotator with energy levels Erot(l) with degeneracy g(l) given by: Erot(l) = l(l + 1) ~ 2 2I g(l) = 2l + 1 l = 0, 1, 2, ... Where I is the moment of inertia. The Hamiltonian for the system is then: H = P 2 2m + Erot a) Find the canonical partition function of a gas of N non-interacting diatomic molecules. Write this as a product of the partition function for an IDEAL monatomic gas, ZI , and a contribution from the rotational energy, Z = ZIZR. Leave the partition function in the form of a sum. b) This sum can be evaluated at the limits of high and low temperature. For the high temperature limit you can approximate the sum as an INTEGRAL to find the partition function. Use the partition function to find the energy and the specific heat of this gas. This should agree with the equipartition theorem for energy. c) For the low temperature limit, calculate the sum in the partition function for the lowest temperatures where the rotational motion of the molecules MAKES a contribution. This will be where only the lowest rotational levels are significant, for l = 0, 1. Calculate the energy and specific heat of the gas. SOLUTION: a) Z = 1 N! V N Λ3N ( X∞ l=0 (2l + 1)e −βl(l+1)~ 2/2I ) N b) U = 5/2N kT c) Keep only l = 0, 1, 2 in the sum for the partition function. Cv = 3/2N k + 6Nθ2 e 2βθ+3 1 kT2