Problem 18 - Basic Principles of Organic Chemistry

Question

Investigate the thermodynamic feasibility of the following propagation steps for opening the rings of cycloalkanes with \(n=2\) to \(n=6\) by a radical-chain mechanism: $$ \begin{array}{c} \left(\mathrm{CH}_{2}\right)_{n}+\mathrm{Br} \cdot \rightarrow \mathrm{BrCH}_{2}-\left(\mathrm{CH}_{2}\right)_{n-2}-\mathrm{CH}_{2} \\ \mathrm{BrCH}_{2}-\left(\mathrm{CH}_{2}\right)_{n-2}-\mathrm{CH}_{2} \cdot+\mathrm{Br}_{2} \rightarrow\left(\mathrm{CH}_{2}\right)_{n-2}\left(\mathrm{CH}_{2} \mathrm{Br}\right)_{2}+\mathrm{Br} \end{array} $$ Use 83 kcal \(\mathrm{mol}^{-1}\) for the bond-dissociation energy of a normal \(\mathrm{C}-\mathrm{C}\) bond and \(68 \mathrm{kcal} \mathrm{mol}^{-1}\) for the bond-dissociation energy of a \(\mathrm{C}-\mathrm{Br}\) bond. (An easy way to solve a problem of this type is first to calculate \(\Delta H\) of each step for cyclohexane, for which there is no strain, then to make suitable corrections for the strain that is present for smaller values of \(n .\) )

Step-by-Step Solution

Answer

Cyclohexane has positive \( \Delta H \), so less feasible; smaller rings could be feasible due to ring strain.

Step 1: Understanding the Problem

The task is to evaluate the thermodynamic feasibility of opening the rings of cycloalkanes using a radical-chain mechanism. We need to calculate the enthalpy change (\( \Delta H \)) for each step and assess the effect of ring strain on \(n\) for cycloalkanes ranging from 2 to 6.

Step 2: Calculate \( \Delta H \) for Cyclohexane

For cyclohexane (\(n=6\)), the first step involves breaking a \( \text{C-C} \) bond (83 kcal/mol) and forming a \( \text{C-Br} \) bond (68 kcal/mol). \( \Delta H = 83 - 68 = 15 \) kcal/mol for the first step. In the second step, two \( \text{C-Br} \) bonds are formed (\(2 \times 68\) kcal/mol) after breaking a \( \text{Br}_2 \) bond (46 kcal/mol). \( \Delta H = 2 \times 68 - 46 = 90 \) kcal/mol for the second step.

Step 3: Adjust for Cyclopentane and Smaller Rings

For rings smaller than cyclohexane, consider the strain. Calculate strain energy based on typical values: cyclopentane, cyclobutane, cyclopropane, etc., have known strain energies. For cyclopentane, add the strain energy (~6.3 kcal/mol), resulting in higher \( \Delta H \). Repeat similarly for each ring size.

Step 4: Evaluate Thermodynamic Feasibility

The reaction is feasible if the total \( \Delta H \) for both steps is negative or relatively small. For cyclohexane, because there is no ring strain, the process will be less feasible (as shown by positive \( \Delta H \)). Smaller rings with substantial ring strain can compensate, possibly making the reaction feasible for rings like cyclobutane.

Key Concepts

Radical-Chain Mechanism

In chemistry, radical-chain mechanisms are sequences of reactions that involve radicals, or atoms and molecules with unpaired electrons. These mechanisms are crucial in polymerization and chemical changes in hydrocarbons. The sequence typically comprises three main stages:

• Initiation: Occurs when radicals are created, often by applying heat or light to decompose a stable molecule. For example, a \( \mathrm{Br}_2 \) molecule can split into two \( \mathrm{Br} \cdot \) radicals.
• Propagation: The newly formed radicals react with other molecules to create new radicals. This step continues the chain reaction. Each radical formed reacts further, continuing the cycle.
• Termination: Radicals meet another radical leading to a stable molecule, thus ending the chain reaction.

Radical-chain mechanisms are efficient ways to perform chemical transformations, but they can be sensitive to reaction conditions, such as temperature or pressure. Such conditions affect the stability of radicals, impacting the reaction's overall rate and feasibility. Understanding these principles can help assess the thermodynamic feasibility of opening cycloalkane rings through a radical-chain mechanism.

Cycloalkanes

Cycloalkanes are hydrocarbons with a ring structure composed entirely of \( \text{C-C} \) single bonds and forming carbon rings. The general formula for a cycloalkane is \( \mathrm{C}_n\mathrm{H}_{2n} \), where \( n \) is the number of carbon atoms in the ring. Unlike linear alkanes, cycloalkanes have unique properties due to their cyclic shape.

Cycloalkanes can vary in size, with smaller rings such as cyclopropane and cyclobutane experiencing higher ring strain compared to larger rings like cyclohexane. The ring strain arises from bond angles deviating from the ideal 109.5° in tetrahedral structures, leading to increased energy and less stability. Cyclohexane, having practically no ring strain, is considered highly stable and is often used as a reference in chemical reactions.

Understanding cycloalkanes and their stability is essential for predicting the thermodynamic feasibility of reactions, such as those involving radical-chain mechanisms. Smaller cycloalkanes tend to release more energy upon opening due to ring strain, making some ring-opening reactions more thermodynamically feasible.

Bond-Dissociation Energy

Bond-dissociation energy (BDE) is a measure of the strength of a chemical bond within a molecule. It represents the amount of energy required to break a bond between two atoms in a molecule, measured in kilocalories per mole (kcal/mol). This property is crucial in understanding reaction mechanics and kinetics, particularly in radical-chain mechanisms.

Within a radical-chain mechanism, we look at the BDE to determine the energy changes that occur during bond breaking and formation. For example, breaking a \( \text{C-C} \) bond in cycloalkanes typically requires 83 kcal/mol, while a \( \text{C-Br} \) bond requires 68 kcal/mol. By calculating the differences in BDE when bonds are broken and formed during reactions, we can predict \( \Delta H \) or the change in enthalpy for each reaction step.

Understanding BDE helps in estimating the thermodynamic feasibility of a reaction. When the energy released in forming new bonds is higher than the energy required to break the initial bonds, the reaction is deemed exothermic and more likely to occur spontaneously.

Ring Strain

Ring strain is a concept often discussed when dealing with cyclic molecules like cycloalkanes. It refers to the extra energy contained within a molecule due to the geometric constraints of being in a ring form. The strain comes from angles and torsional strain, as well as steric strain if atoms are forced too closely together.

Small cycloalkanes, such as cyclopropane and cyclobutane, exhibit significant ring strain due to the impossibility of achieving ideal bond angles, which are less than the standard 109.5\(^\circ\). This causes an increase in the internal energy and instability of these molecules.

Larger rings, like cyclohexane, have minimal to no ring strain because they can adopt conformations like the "chair" form that alleviates these pressures. The strain plays a crucial role in the thermodynamics of reactions involving these cyclic compounds. High ring strain in smaller rings can make chemical reactions, like those involving ring opening via a radical-chain mechanism, more feasible, as the energy released can offset other enthalpy changes.