Problem 37 - Chemistry: The Central Science

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 heat of combustion per CH2 group in cyclopropane is approximately \(-696.33 \mathrm{~kJ/mol}\) and in cyclopentane is approximately \(-663.40 \mathrm{~kJ/mol}\), resulting in a difference of approximately \(-32.93 \mathrm{~kJ/mol}\). This difference can be attributed to the higher bond strain and lower stability in cyclopropane due to its smaller ring size and angles compared to cyclopentane, leading to less energy being released during combustion for each CH2 group of cyclopropane than for cyclopentane.

Step 1: Determine the molar ratio of CH2 groups in cyclopropane and cyclopentane

To calculate the heat of combustion per CH2 group, we first need to determine the number of CH2 groups in each molecule. Cyclopropane has 3 CH2 groups, while cyclopentane has 5 CH2 groups.

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

To find the heat of combustion per CH2 group in cyclopropane, we'll divide the molar heat of combustion for cyclopropane by the number of CH2 groups it contains: \[ \text{Heat of combustion per CH2 group (cyclopropane)} = \frac{-2089 \mathrm{~kJ/mol}}{3 \mathrm{~CH}_{2}} \]

Step 3: Calculate the heat of combustion per CH2 group for cyclopentane

To find the heat of combustion per CH2 group in cyclopentane, we'll divide the molar heat of combustion for cyclopentane by the number of CH2 groups it contains: \[ \text{Heat of combustion per CH2 group (cyclopentane)} = \frac{-3317 \mathrm{~kJ/mol}}{5 \mathrm{~CH}_{2}} \]

Step 4: Calculate the heat of combustion per CH2 group and find the difference

Now we can calculate and compare the heat of combustion per CH2 group for cyclopropane and cyclopentane, and find the difference between them: \[ \text{Heat of combustion per CH2 group (cyclopropane)} \approx -696.33 \mathrm{~kJ/mol} \] \[ \text{Heat of combustion per CH2 group (cyclopentane)} \approx -663.40 \mathrm{~kJ/mol} \] \[ \text{Difference} \approx (-696.33 - (-663.40)) \mathrm{~kJ/mol} \approx -32.93 \mathrm{~kJ/mol} \]

Step 5: Account for the difference

The difference in heat of combustion per CH2 group between cyclopropane and cyclopentane can be attributed to the difference in stability of the two molecules. Cyclopropane has more bond strain due to its smaller ring size and angles, making it less stable than cyclopentane. Consequently, less energy is released during combustion for each CH2 group in cyclopropane compared to cyclopentane.