Question

The heat of hydrogenation of cis-2,2,5,5-tetramethyl-3-hexene is \(-154 \mathrm{~kJ}(-36.7 \mathrm{kcal}) /\) \(\mathrm{mol}\), while that of the trans isomer is only \(-113 \mathrm{~kJ}(-26.9 \mathrm{kcal}) / \mathrm{mol}\). (a) Why is the heat of hydrogenation of the cis isomer so much larger (more negative) than that of the trans isomer? (b) If a catalyst could be found that allowed equilibration of the cis and trans isomers at room temperature (such catalysts do exist), what would be the ratio of trans to cis isomers?

Short answer

Short answer: The heat of hydrogenation of the cis isomer is more negative than the trans isomer because the cis isomer is less stable due to steric hindrance between the bulky groups. The ratio of trans to cis isomers in the presence of a catalyst that allows equilibration would be approximately 937:1. This difference in ratio is due to the difference in their stabilities, with the trans isomer being more stable and abundant in the mixture.

Step-by-step solution

Understand the difference in stability of cis and trans isomers

The stability of an isomer plays a key role in determining the heat of hydrogenation. Generally, cis isomers are less stable compared to their trans counterparts due to steric hindrance between the bulky groups on the same side of a molecule. As a result, the difference in heat of hydrogenation indicates the difference in stability between the cis and trans isomers.

Calculate the difference in heat of hydrogenation

We are given the heat of hydrogenation for both isomers. To find the difference, we will subtract the value of the trans isomer from that of the cis isomer. ΔH_difference = ΔH_cis - ΔH_trans = (-154 kJ/mol) - (-113 kJ/mol) = -41 kJ/mol

Calculate the equilibrium constant

The difference in heat of hydrogenation can be related to the equilibrium constant (K) by the equation: K = e^(-ΔH_difference / RT) where ΔH_difference is the difference in heat of hydrogenation, R is the gas constant (8.314 J/(mol·K)), and T is the temperature (assuming room temperature, T ≈ 298 K). K = e^(-(-41 × 10^3 J/mol) / (8.314 J/(mol·K) × 298 K))

Calculate the equilibrium constant value

Now we can plug in the values and calculate the equilibrium constant: K = e^(41 × 10^3 / (8.314 × 298)) K ≈ 937

Calculate the ratio of trans to cis isomers

Since we have the equilibrium constant, we can determine the ratio of trans to cis isomers using the formula: ratio_trans_to_cis = K So, the ratio of trans to cis isomers would be approximately 937:1. This means that under the influence of a suitable catalyst, the trans isomer would be significantly more abundant in the mixture as compared to the cis isomer due to the difference in their stabilities.

Key concepts

Cis and Trans Isomers

Cis and trans isomers represent types of stereoisomerism, where the molecules have the same formula but differ in the orientation of their substituent groups across a double bond. In cis isomers, the similar or identical groups are on the same side of the double bond, leading to increased steric hindrance and usually resulting in a higher energy state for the molecule. In contrast, trans isomers have the groups on opposite sides of the double bond, which generally causes less steric hindrance and, consequently, a more stable, lower energy configuration.

When comparing the heat of hydrogenation for these isomers, one can measure their relative stabilities. Hydrogenation refers to the addition of hydrogen to a molecule, typically double bonds in organic compounds, and the heat of hydrogenation is the change in enthalpy during this process. A higher heat of hydrogenation indicates a less stable molecule, as more energy is released when it converts to its saturated counterpart. Thus, with the cis isomer having a larger (more negative) heat of hydrogenation than the trans isomer, we can expect that the trans form is more stable due to reduced steric clashes.

Isomer Stability

The stability of isomers is profoundly influenced by the molecular structure and the spatial arrangement of their atoms. Factors affecting isomer stability, such as steric hindrance, as mentioned earlier, and torsional strain, are critical when evaluating isomeric compounds. In the given exercise, the bulkier groups in the cis isomer are closer together, leading to increased repulsive interactions and decreased stability.

Molecules naturally prefer to exist in the lowest possible energy state. Hence, more stable isomers are typically favored in a dynamic equilibrium because they have fewer internal conflicts and a lower overall energy. This preference is evident in the heat of hydrogenation values provided, where the cis isomer, being less stable, releases more energy upon hydrogenation compared to the trans isomer. Consequently, the stability of isomers directly correlates with their energy content and can be quantitatively ascertained through calorimetric measurements such as the heat of hydrogenation.

Equilibrium Constant

The equilibrium constant (K) is a number that reveals the ratio of the concentration of the products to the reactants at equilibrium for a chemical reaction. It is a direct reflection of the relative stabilities of the reactants and the products. In the context of isomerization between cis and trans forms, K can reveal which isomer is predominantly present at equilibrium. Since the equilibrium favors the formation of the more stable isomer, a higher value of K indicates a greater proportion of the more stable molecule in the equilibrium mixture.

In the exercise, the equilibrium constant calculation shows a large value, indicating a significant preference for the trans isomer over the cis isomer. The ratio of trans to cis isomers is remarkably high, at 937:1, which corresponds to the thermodynamic favorability of the trans isomer due to its lower energy state. The exercise improvement advice here is to remember that a large equilibrium constant value is associated with the formation of lower energy, more stable compounds at equilibrium.