Consider a gas of diatomic molecules (moment of inertia I)

Chapter 42, Problem 42.43

(choose chapter or problem)

Consider a gas of diatomic molecules (moment of inertia I) at an absolute temperature T. If \(E_g\) is a ground-state energy and \(E_{\mathrm{ex}}\) is the energy of an excited state, then the Maxwell–Boltzmann distribution (see Section 39.4) predicts that the ratio of the numbers of molecules in the two states is

\(\frac{n_{\mathrm{ex}}}{n_{\mathrm{g}}}=e^{-\left(E_{\mathrm{cx}}-E_{\mathrm{g}}\right) / k T}\)

(a) Explain why the ratio of the number of molecules in the lth rotational energy level to the number of molecules in the ground \((l=0)\) rotational level is

\(\frac{n_{l}}{n_{0}}=(2 l+1) e^{-\left[l(l+1) \hbar^{2}\right] / 2 l k T}\)

(Hint: For each value of l, how many states are there with different values of )

(b) Determine the ratio for a gas of CO molecules at 300 K for the cases

(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 \(n_{l} / n_{0}\) first increases and then decreases. Explain why.