Problem 41 - 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 for cyclopropane is -696.33 kJ/mol per CH2 group, while for cyclopentane, it is -663.4 kJ/mol per CH2 group. The difference is due to the angle strain in cyclopropane's more compact structure, which results in instability and a higher heat of combustion per CH2 group. Cyclopentane has a lower angle strain, making it more stable and resulting in a lower heat of combustion per CH2 group.

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

Cyclopropane has a cyclic structure with three carbon atoms, meaning it has three CH2 groups. Cyclopentane, on the other hand, has a cyclic structure with five carbon atoms, meaning it has five CH2 groups.

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

To do this, we will divide the molar heat of combustion of cyclopropane by the number of CH2 groups in its structure: \( Heat\, of \,Combustion\, per\, CH_{2}\, for\, cyclopropane = \dfrac{-2089\mathrm{kJ/mol}}{3\, CH_{2}\, groups} \) \( Heat\, of\, Combustion\, per\, CH_{2}\, for\, cyclopropane = -696.33\mathrm{kJ/mol}\, per\, CH_{2}\, group \)

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

Similar to step 2, we will divide the molar heat of combustion of cyclopentane by the number of CH2 groups in its structure: \( Heat\, of\, Combustion\, per\, CH_{2}\, for\, cyclopentane = \dfrac{-3317\mathrm{kJ/mol}}{5\, CH_{2}\, groups} \) \( Heat\, of\, Combustion\, per\, CH_{2}\, for\, cyclopentane = -663.4\mathrm{kJ/mol}\, per\, CH_{2}\, group \)

Step 4: Comparing the heat of combustion per CH2 group

Now that we have calculated the heat of combustion per CH2 group for cyclopropane (-696.33 kJ/mol CH2) and cyclopentane (-663.4 kJ/mol CH2), we can observe the difference: Cyclopropane's heat of combustion per CH2 group is higher than that of cyclopentane.

Step 5: Account for the difference

The main reason behind this difference can be attributed to the energy content of the molecules. Cyclopropane has a more compact structure and more significant angle strain, as the ideal angle for an sp3 hybridized carbon is 109.5 degrees, while in cyclopropane, it's 60 degrees. This angle strain translates into instability, which in turn results in a higher heat of combustion per CH2 group. In cyclopentane, the angle strain is much lower, with the actual angle close to the ideal angle for sp3 hybridized carbons. As a result, cyclopentane has a lower heat of combustion per CH2 group.

Key Concepts

Cyclopropane

Cyclopropane is a type of hydrocarbon molecule characterized by a three-carbon ring structure. In terms of chemical composition, it consists of three carbon (C) atoms and six hydrogen (H) atoms, forming a triangular ring. This structure provides each carbon atom with two hydrogen atoms, denoted as CH2 groups.

The bond angles in cyclopropane are approximately 60 degrees, which is much sharper than the 109.5 degrees ideal for sp3 hybridized carbon atoms. This leads to what is known as angle strain, making cyclopropane a highly strained and reactive compound. Its unique structural properties have an impact on not only its chemical reactivity but also its heat of combustion, which is the energy released when the compound undergoes complete combustion in the presence of oxygen.

Cyclopentane

Cyclopentane, on the other hand, is a five-membered ring hydrocarbon with the formula C5H10. It consists of five carbon atoms linked together in a ring, with each carbon atom connected to two hydrogen atoms, forming a total of five CH2 groups.

Compared to cyclopropane, the bond angles in cyclopentane are closer to the ideal tetrahedral angle of 109.5 degrees. This results in less angle strain within the molecule, making it more stable than cyclopropane. The reduced strain in cyclopentane contributes to its lower heat of combustion per CH2 group. Understanding the structural differences between cyclopropane and cyclopentane provides insight into their distinct thermodynamic properties, including their respective heats of combustion.

Enthalpy Calculation

Enthalpy, a crucial concept in chemical thermodynamics, is the measure of total heat content in a thermodynamic system at constant pressure. The enthalpy change, denoted as \( \Delta H \), during a chemical reaction indicates the amount of heat absorbed or released. For combustion reactions, this value is referred to as the heat of combustion.

To calculate the heat of combustion per CH2 group in compounds like cyclopropane and cyclopentane, one divides the total heat of combustion of the molecule by the number of CH2 groups present. This calculation provides a normalized value to compare the energy released per CH2 unit, allowing for a fair comparison of energy content between different molecules.

Chemical Thermodynamics

Chemical thermodynamics is the branch of physical chemistry that studies the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It encompasses concepts like enthalpy, entropy, Gibbs free energy, and the first and second laws of thermodynamics.

The heat of combustion is a thermodynamic property reflecting the energy released when a substance undergoes complete oxidation. In the context of cyclopropane and cyclopentane, the heat of combustion values can be analyzed to understand the effect of molecular structure on stability and reactivity. This analysis falls within chemical thermodynamics, providing valuable insights into how the inherent energy of chemical bonds and the effects of molecular strain influence the thermodynamic behavior of substances during combustion.