Solved: Consider a gas of diatomic molecules (moment of

Chapter 42, Problem 42.39

(choose chapter or problem)

Consider a gas of diatomic molecules (moment of inertia I) at an absolute temperature T. If Eg is a groundstate energy and Eex is the energy of an excited state, then the MaxwellBoltzmann distribution (see Section 39.4) predicts that the ratio of the numbers of molecules in the two states is nex>ng = e-1Eex-Eg2>kT. (a) Explain why the ratio of the number of molecules in the lth rotational energy level to the number of molecules in the ground-state 1l = 02 rotational level is nl n0 = 12l + 12e-3l1l+12U2 4>2IkT (Hint: For each value of l, how many states are there with different values of ml?) (b) Determine the ratio nl >n0 for a gas of CO molecules at 300 K for (i) l = 1; (ii) l = 2; (iii) l = 10; (iv) l = 20; (v) l = 50. The moment of inertia of the CO molecule is given in Example 42.2 (Section 42.2). (c) Your results in part (b) show that as l is increased, the ratio nl >n0 first increases and then decreases. Explain why