Problem 41 - Chemistry: The Central Science in SI Units, Global Edition

Question

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.

Step-by-Step Solution

Answer

The molar heat of combustion per CH2 group for cyclopropane is \(-696.3 \mathrm{~kJ/mol~CH_2}\) and for cyclopentane is \(-663.4 \mathrm{~kJ/mol~CH_2}\). The more negative value for cyclopropane indicates that more energy is released per CH2 group upon combustion compared to cyclopentane. This difference is attributed to the higher ring strain in cyclopropane, making it less stable than cyclopentane, causing more energy to be released upon combustion.

Step 1: Determine the number of CH2 groups in each molecule

For cyclopropane, a 3-membered ring with all single bonds between carbon atoms, we have 3 CH2 groups. Cyclopentane, on the other hand, is a 5-membered ring with all single bonds between carbon atoms, therefore having 5 CH2 groups.

Step 2: Calculate the molar heat of combustion per CH2 group for cyclopropane and cyclopentane

To find the heat of combustion per CH2 group, we need to divide the molar heat of combustion by the number of CH2 groups for each molecule. For cyclopropane: \(\frac{-2089 \mathrm{~kJ/mol}}{3 \mathrm{~CH_2 / mol}} = -696.3 \mathrm{~kJ/mol~CH_2} \) For cyclopentane: \(\frac{-3317 \mathrm{~kJ/mol}}{5 \mathrm{~CH_2 / mol}} = -663.4 \mathrm{~kJ/mol~CH_2} \)

Step 3: Compare and account for the difference

The heat of combustion per CH2 group is more negative for cyclopropane than for cyclopentane. This indicates that, for each CH2 group, more energy is released during combustion of cyclopropane relative to cyclopentane. This difference can be attributed to the relative stability of the molecules. Cyclopropane, having a 3-membered ring, experiences significant ring strain due to the angle between the carbon atoms being much smaller than the preferred tetrahedral angle, which makes it less stable. On the other hand, cyclopentane has a larger ring with reduced ring strain, making it more stable. Consequently, the difference in their heat of combustion per CH2 group is due to the difference in ring strain, resulting in the release of more energy upon combustion of cyclopropane relative to cyclopentane.