The heat of hydrogenation of cis-2,2,5,5-tetramethyl-3-hexene is \(-154 \mathrm{~kJ}\) (-36.7 kcal)/ \(\mathrm{mol}\), while that of the trans isomer is only \(-119 \mathrm{~kJ}\) (-26.9 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 is isomers?

In order to understand the difference between the hydrogenation of cis and trans isomers, we should first discuss the concept of isomer stability. The more stable the alkene is, the less heat is released upon hydrogenation. Cis and trans isomers have different stability levels due to their molecular geometries, which lead to different steric hindrances and stronger or weaker non-bonding interactions.

In the case of cis-2,2,5,5-tetramethyl-3-hexene, the two larger alkyl groups (the 2 t-butyl groups) are on the same side of the double bond, which causes greater steric repulsion between these groups. Conversely, in the case of trans-2,2,5,5-tetramethyl-3-hexene, the methyl groups are positioned opposite each other, leading to a lower amount of steric hindrance. This means that the cis isomer is generally less stable than the trans isomer.

As we established earlier, the more stable an alkene is, the less heat is released upon hydrogenation. In this case, the cis isomer is less stable due to the greater steric hindrance between alkyl groups. Therefore, more heat is released upon hydrogenation of the cis isomer, leading to a larger (more negative) heat of hydrogenation for the cis isomer compared to the trans isomer.

We know that the ratio of isomers at equilibrium is given by the expression: \(K_\text{eq} = \frac{[\text{trans}]}{[\text{cis}]} = \exp{\left( -\frac{\Delta G^\circ}{RT}\right)}\) where \(K_\text{eq}\) is the equilibrium constant, \(\Delta G^\circ\) is the change in Gibbs free energy, \(R\) is the gas constant, and \(T\) is the temperature. To find the ratio of trans to cis isomers, we first need to find the change in Gibbs free energy: \(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ\) Since we are dealing with hydrogenation reactions, we can use the relationship between the heats of hydrogenation and \(\Delta H^\circ\): \(\Delta H^\circ_\text{cis} = -154~\text{kJ/mol}\) and \(\Delta H^\circ_\text{trans} = -119~\text{kJ/mol}\) We can find the difference between the heats of hydrogenation as: \(\Delta H^\circ_\text{cis-trans} = \Delta H^\circ_\text{trans} - \Delta H^\circ_\text{cis} = 35~\text{kJ/mol}\) Assuming that the difference in entropy, \(\Delta S^\circ\), is negligible (which is reasonable given the structural similarities between the isomers), we can calculate the \(\Delta G^\circ_\text{cis-trans}\): \(\Delta G^\circ_\text{cis-trans} = \Delta H^\circ_\text{cis-trans} = 35~\text{kJ/mol}\) Now we can insert the values into the expression for \(K_\text{eq}\) at room temperature, \(T = 298~\text{K}\), and \(R = 8.314~\text{J/(mol\cdot K)}\): \(K_\text{eq} = \exp{\left(-\frac{35\times10^3~\text{J/mol}}{8.314~\text{J/(mol\cdot K)}\times298~\text{K}}\right)} = 125\) So, the ratio of trans to cis isomers would be 125:1, meaning there would be 125 times more of the trans isomer than the cis isomer in equilibrium at room temperature with the given catalyst.