The molar heat of combustion of gaseous cyclopropane is \(-2089 \mathrm{~kJ} / \mathrm{mol} ;\) that for gaseous cyclopentane is \(-3317 \mathrm{~kJ} / \mathrm{mol}\). Calculate the heat of combustion per \(\mathrm{CH}_{2}\) group in the two cases, and account for the difference.

In order to determine the number of \(\mathrm{CH}_{2}\) groups present in each compound, we need to know their molecular formulas. Cyclopropane is a cyclic compound with three carbon atoms and six hydrogen atoms, having the formula \(\mathrm{C}_{3}\mathrm{H}_{6}\). Since each carbon atom is bonded to two hydrogen atoms in the molecule, there are three \(\mathrm{CH}_{2}\) groups in cyclopropane. Cyclopentane is a cyclic compound with five carbon atoms and ten hydrogen atoms, having the formula \(\mathrm{C}_{5}\mathrm{H}_{10}\). Similar to cyclopropane, each carbon atom is bonded to two hydrogen atoms in the molecule; hence, there are five \(\mathrm{CH}_{2}\) groups in cyclopentane.

Now that we know the number of \(\mathrm{CH}_{2}\) groups in each compound, we can calculate the heat of combustion per \(\mathrm{CH}_{2}\) group. For cyclopropane: \[ \text{Heat of combustion per CH}_{2}\text{ group} = \frac{-2089~\mathrm{kJ/mol}}{3~\mathrm{CH}_{2}\text{ groups}} = -696.33~\mathrm{kJ/mol} \] For cyclopentane: \[ \text{Heat of combustion per CH}_{2}\text{ group} = \frac{-3317~\mathrm{kJ/mol}}{5~\mathrm{CH}_{2}\text{ groups}} = -663.4~\mathrm{kJ/mol} \]

Comparing the heat of combustion per \(\mathrm{CH}_{2}\) group for the two compounds, we find that cyclopropane has a slightly higher value (-696.33 kJ/mol) than cyclopentane (-663.4 kJ/mol). The difference in heat of combustion per \(\mathrm{CH}_{2}\) group can be attributed to the ring strain present in the molecules. Cyclopropane has a smaller, less stable ring structure compared to cyclopentane. The bond angles in cyclopropane are considerably smaller than the ideal tetrahedral bond angle (109.5°). As a result, the molecule experiences increased ring strain, which makes it more reactive and provides a higher heat of combustion per \(\mathrm{CH}_{2}\) group. In contrast, the bond angles in cyclopentane are closer to the ideal tetrahedral bond angles, making the molecule less strained and having a lower heat of combustion per \(\mathrm{CH}_{2}\) group.