Question

The heat of hydrogenation of cis-2,2,5,5-tetramethyl-3-hexene is \(-154 \mathrm{~kJ}(-36.7 \mathrm{kcal}) /\) 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

Answer: The heat of hydrogenation of the cis isomer is larger (more negative) than that of the trans isomer because the cis isomer is generally less stable due to the presence of higher steric strain (repulsion between similar groups on the same side of the molecule). When hydrogenation occurs, the more unstable (cis) isomer will release more energy, resulting in a more negative heat of hydrogenation. Question: What is the ratio of trans to cis isomers if a catalyst is found that allows equilibration at room temperature? Answer: The ratio of trans to cis isomers is approximately 5:1 if a catalyst is found that allows equilibration at room temperature.

Step-by-step solution

(a) Heat of hydrogenation difference

The difference in heat of hydrogenation between the cis and trans isomers can be explained by considering their stability. Cis isomers generally have a higher steric strain due to the presence of similar groups on the same side of the molecule, causing repulsion. This makes them less stable compared to trans isomers, which have the similar groups on opposite sides of the molecule. When hydrogenation occurs, the more unstable (cis) isomer will release more energy, leading to a more negative heat of hydrogenation.

(b) Calculating the ratio of trans to cis isomers

To calculate the ratio of trans to cis isomers, we can use the relationship between the equilibrium constant (K) and the change in Gibbs free energy (ΔG) under standard conditions: ΔG = -RTln(K), where R is the gas constant (8.314 J/mol·K), and T is the temperature (assumed to be 25°C or 298K). First, let's find the difference in heat of hydrogenation (ΔH) between trans and cis isomers: ΔH = -113 kJ/mol (trans) - (-154 kJ/mol (cis)) = 41 kJ/mol Now, we assume that the enthalpy difference (ΔH) is approximately equal to the Gibbs free energy difference (ΔG) since changes in enthalpy and entropy typically have opposite signs and their magnitudes are usually small. Thus, ΔG ≈ ΔH. Now, we can plug the values into the equation to find K: ΔG = -RTln(K) 41,000 J/mol = -(8.314 J/mol·K)(298 K)ln(K) Solving for K: ln(K) ≈ 1.623 K ≈ 5.07 Therefore, the ratio of trans to cis isomers is approximately 5:1.