Question

Investigate the thermodynamic feasibility of the following propagation steps for opening the rings of cycloalkanes with \(\mathrm{n}=2-6 \mathrm{by}\) a free-radical mechanism. (1) \(\left(\mathrm{CH}_{2}\right)_{\mathrm{n}}+\mathrm{Br}^{-} \rightarrow \mathrm{BrCH}_{2}\left(\mathrm{CH}_{2}+\mathrm{n}-2 \mathrm{CH}_{2}\right.\) (2) \(\mathrm{BrCH}_{2}\left(\mathrm{CH}_{2}+\mathrm{n}-2 \mathrm{CH}_{2}+\mathrm{Br}_{2} \rightarrow\left(\mathrm{CH}_{2}\right)_{\mathrm{n}-2}\left(\mathrm{CH}_{2} \mathrm{Br}\right)_{2}+\mathrm{Br}^{-}\right.\) Use 83 kcal for the bond-dissociation energy of a normal C-C bond and \(68 \mathrm{kcal}\) for the bond-dissociation energy of a C-Br bond. (An easy way to solve a problem of this type is to first calculate AH of each step for cyclohexane where there is no strain, and then make suitable corrections for the strain which is present for small values of n.) Refer to the following table for your answer. Table 1. Total Strain of Cycloalkanes $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Cycloalkane } \\ \left(\mathrm{CH}_{2}\right)_{\mathrm{n}} \end{array} & \mathrm{n} & \begin{array}{c} \text { Total strain } \\ (\mathrm{kcal} / \text { moles }) \end{array} \\ \hline \text { ethylene } & 2 & 22.4 \\ \hline \text { cyclopropane } & 3 & 27.6 \\ \hline \text { cyclobutane } & 4 & 26.4 \\ \hline \text { cyclopentane } & 5 & 6.5 \\ \hline \text { cyclohexane } & 6 & 0.0 \\ \hline \end{array} $$

Short answer

The propagation steps for opening the rings of cycloalkanes with $\mathrm{n}=2-5$ by a free-radical mechanism are not thermodynamically feasible due to the positive overall reaction enthalpy changes. However, cyclohexane ($\mathrm{n}=6$) appears to be thermodynamically feasible as its overall reaction enthalpy change is zero.

Step-by-step solution

Calculate enthalpy change of each step for cyclohexane

First, calculate the change in enthalpy, ΔH, for each propagation step for cyclohexane (n = 6). For step (1): The reaction involves breaking a C-C bond and forming a C-Br bond: ΔH(1) = bond energy for C-C bond - bond energy for C-Br bond = 83 kcal - 68 kcal = 15 kcal. For step (2): The reaction involves breaking a C-Br bond and forming a C-C bond: ΔH(2) = bond energy for C-Br bond - bond energy for C-C bond = 68 kcal - 83 kcal = -15 kcal.

Correct the enthalpy values for the strain present for small values of n

Now, we need to correct the calculated enthalpy values for each cycloalkane with n = 2-5 using the provided strain values in the table. Total strain for ethylene (n=2): ΔH(1) = 15 kcal + 22.4 kcal Total strain for cyclopropane (n=3): ΔH(1) = 15 kcal + 27.6 kcal Total strain for cyclobutane (n=4): ΔH(1) = 15 kcal + 26.4 kcal Total strain for cyclopentane (n=5): ΔH(1) = 15 kcal + 6.5 kcal

Determine the feasibility of the reactions for each cycloalkane

Determine the thermodynamic feasibility of the propagation steps based on the calculated enthalpy values for each cycloalkane: A reaction is thermodynamically feasible if the overall reaction enthalpy change is negative. For ethylene (n=2): ΔH(total) = Total strain for ethylene + ΔH(2) = (15 + 22.4) + (-15) = 22.4 kcal. The overall reaction enthalpy change is positive, and therefore not thermodynamically feasible. For cyclopropane (n=3): ΔH(total) = Total strain for cyclopropane + ΔH(2) = (15 + 27.6) + (-15) = 27.6 kcal. The overall reaction enthalpy change is positive, and therefore not thermodynamically feasible. For cyclobutane (n=4): ΔH(total) = Total strain for cyclobutane + ΔH(2) = (15 + 26.4) + (-15) = 26.4 kcal. The overall reaction enthalpy change is positive, and therefore not thermodynamically feasible. For cyclopentane (n=5): ΔH(total) = Total strain for cyclopentane + ΔH(2) = (15 + 6.5) + (-15) = 6.5 kcal. The overall reaction enthalpy change is positive, and therefore not thermodynamically feasible. For cyclohexane (n=6): ΔH(total) = ΔH(1) + ΔH(2) = 15 kcal + (-15 kcal) = 0 kcal. The overall reaction enthalpy change is zero, and therefore considered thermodynamically feasible. In conclusion, according to the calculations, the propagation steps of opening the rings of cycloalkanes with n=2-5 by a free-radical mechanism are not thermodynamically feasible, and only cyclohexane appears to be feasible.

Key concepts

Free-Radical Mechanism

The free-radical mechanism is a sequence of steps that involve uncharged, highly reactive atoms or molecules with unpaired electrons, known as free radicals. These radicals can initiate chain reactions, such as the breaking and forming of bonds, in processes like the opening of cycloalkane rings. In the exercise, the opening of cycloalkanes through a free-radical mechanism involves a free radical reacting with a cycloalkane to create a new radical, which in turn reacts with another molecule to produce a final product and another radical, perpetuating the chain reaction.

The mechanism typically includes three basic types of steps: initiation, propagation, and termination. Initiation creates the reactive radicals, propagation involves the radicals reacting with stable molecules to create new radicals, and termination occurs when radicals combine to form stable products. Understanding the intricacies of this mechanism is essential for predicting the behavior of chemicals under various conditions and plays a vital role in synthetic chemistry, such as in the creation of cycloalkanes.

Bond-Dissociation Energy

Bond-dissociation energy (BDE) is the energy required to break a specific chemical bond in one mole of gaseous molecules, forming separated atoms or radicals. In the given exercise, BDE is a crucial concept when examining the thermodynamic feasibility of reactions involving cycloalkanes. The BDE values provided for a C-C bond and a C-Br bond are used to calculate the enthalpy changes during the free-radical mechanism processes.

Using BDE, we can determine the energy cost or release when a bond is broken or formed, which is a key factor in the stability of the products formed during the reaction. A low BDE indicates that less energy is needed to break a bond, making it more susceptible to chemical reactions. Conversely, a high BDE signifies that more energy is required, suggesting a more stable bond, less prone to react.

Enthalpy Change

Enthalpy change, denoted as \( \Delta H \), represents the heat absorbed or released during a chemical reaction at constant pressure. It's used here to assess the thermodynamic feasibility of the opening rings of cycloalkanes via a free-radical reaction. Positive \( \Delta H \) values suggest that a reaction is endothermic, absorbing heat from its surroundings, while negative \( \Delta H \)'s indicate an exothermic reaction that releases heat.

The calculation of \( \Delta H \) for each step involves the difference in bond dissociation energies of the bonds being broken and formed. The feasibility of a reaction is generally indicated by a negative overall \( \Delta H \), which means the reaction would spontaneously proceed in the direction written. However, as shown in the exercise, positive enthalpy changes for the opening of small cycloalkane rings indicate that these reactions are not thermodynamically feasible under standard conditions.

Ring Strain in Cycloalkanes

Ring strain occurs due to the peculiar bond angles and torsional strain in a cyclic structure, which results in higher potential energy compared to linear alkanes. In cycloalkanes, this strain can significantly influence the stability and reactivity of the molecule. In the exercise, the impact of ring strain on thermodynamic feasibility is evident when comparing the enthalpy changes calculated for cyclohexane (with no ring strain) to smaller cycloalkanes.

Each cycloalkane, from cyclopropane to cyclopentane, has an associated strain energy that must be considered when determining the total enthalpy change of the reaction. The smaller the ring, the higher the strain energy, as the ideal tetrahedral bond angles and staggered conformations cannot be achieved. The additional corrective values for strain energy result in higher enthalpy changes and indicate that breaking these smaller rings would not be thermodynamically favorable. In contrast, cyclohexane, which can adopt a near strain-free chair conformation, does not have this added barrier, demonstrating that strain can have a profound effect on the chemical behavior of cycloalkanes.