 Solution

Use the supplied separable molecular model,with electronic factors unity. The integer masses are $2u$, $28u$ and $30u$. The rotational symmetry numbers are respectively $2,1,2$: rotating either hydrogen or methanal by $180^{\circ }$ about its appropriate symmetry axis exchanges identical hydrogens, whereas CO has distinct ends.

With the given conversion $k_{\mathrm{B}}/hc=0.6950\mathrm{c}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, a spectroscopic constant $\tilde{b}$ corresponds to a temperature $\theta =\tilde{b}/0.6950$. At $500$ K the partition-function factors areFor instance,The vibrational partition functions here include their zero-point Boltzmann factors. The given reference-energy change must therefore be used consistently with that convention; adding a second vibrational zero-point energy correction would double count it. Because the numerical energy change is per mole, its exponential uses $RT$, not $k_{\mathrm{B}}T$:Consequently,This tiny dimensionless value refers to the extremely dilute one-molecule-per-cubic-metre standard state. It is not a concentration-independent statement about the fraction of carbon monoxide converted in a particular cloud. The calculation follows the supplied classical rotational and harmonic vibrational formulas, without adding separate nuclear-spin factors.