Question

On a cyclohexane ring, an axial carboxyl group has a conformational energy of \(5.9 \mathrm{~kJ}\) (1.4 kcal)/mol relative to an equatorial carboxyl group. Consider the equilibrium for the alternative chair conformations of trans- 1,4 -cyclohexanedicarboxylic acid. Draw the less stable chair conformation on the left of the equilibrium arrows and the more stable chair on the right. Calculate \(\Delta \mathrm{G}^{0}\) for the equilibrium as written and calculate the ratio of the more stable chair to the less stable chair at \(25^{\circ} \mathrm{C}\).

Short answer

In summary, we drew the chair conformations of trans-1,4-cyclohexanedicarboxylic acid, with the less stable conformation having both carboxyl groups axial and the more stable conformation having both carboxyl groups equatorial. The energy difference between the conformations is 11.8 kJ/mol. By calculating the equilibrium constant, we found it to be approximately 95.6. Then, we calculated the Gibbs free energy change for the equilibrium, which is approximately -8.15 kJ/mol. Finally, we determined that the ratio of the more stable chair conformation to the less stable chair conformation is around 95.6, meaning there are about 95.6 times more molecules in the more stable chair conformation than the less stable chair conformation at 25°C.

Step-by-step solution

Drawing Chair Conformations

The first step is to draw the less stable and more stable chair conformations of trans-1,4-cyclohexanedicarboxylic acid. In the less stable chair conformation, both carboxyl groups will be axial, while in the more stable chair conformation, both carboxyl groups will be equatorial. Less stable chair conformation (both carboxyl groups axial): H | COOH-1,3,5-cyclohexane-COOH | H More stable chair conformation (both carboxyl groups equatorial): COOH | 1,3,5-cyclohexane | COOH

Calculating Energy Difference

The conformational energy of an axial carboxyl group relative to an equatorial carboxyl group is given as 5.9 kJ/mol. Since the less stable chair conformation has two axial carboxyl groups and the more stable chair conformation has zero axial carboxyl groups, the energy difference between the two conformations can be calculated as: \(\Delta E = 2 \times 5.9 \mathrm{~kJ/mol} = 11.8 \mathrm{~kJ/mol}\)

Calculating Gibbs Free Energy Change

The equilibrium constant (\(K\)) and the Gibbs free energy change (\(\Delta G^0\)) are related by the following equation: \(\Delta G^0 = -RT \ln K\) Where \(R\) is the gas constant (8.314 J/mol·K) and \(T\) is the temperature in Kelvin. Since we are asked to calculate the ratio of the more stable chair to the less stable chair, we can define the equilibrium constant as: \(K = \frac{[\text{More stable chair}]}{[\text{Less stable chair}]}\) To calculate the Gibbs free energy change for the equilibrium, we will first need to find the value of \(K\).

Calculating the Equilibrium Constant

Given the energy difference between the two conformations (\(\Delta E = 11.8 \mathrm{~kJ/mol}\)), we can calculate the equilibrium constant using the Boltzmann distribution: \(K = e^{-\frac{\Delta E}{RT}}\) Where \(\Delta E\) is in Joules per mole, so we need to convert it: \(\Delta E = 11.8 \mathrm{~kJ/mol} \times 10^3 \mathrm{J/kJ} = 11800 \mathrm{~J/mol}\) At \(25^{\circ} \mathrm{C}\), the temperature in Kelvin is: \(T = 273.15 + 25 = 298.15 \mathrm{K}\) Plugging the values into the equation: \(K = e^{-\frac{11800 \mathrm{~J/mol}}{(8.314 \mathrm{J/mol\cdot K})(298.15 \mathrm{K})}} = 95.6\) So, the equilibrium constant is approximately 95.6.

Calculating Gibbs Free Energy Change

Now that we have the equilibrium constant, we can calculate the Gibbs free energy change as: \(\Delta G^0 = -RT \ln K = -(8.314 \mathrm{J/mol\cdot K})(298.15 \mathrm{K})\ln(95.6) ≈ -8147.68 \mathrm{~J/mol}\) The \(\Delta G^0\) for the equilibrium is approximately -8147.68 J/mol or -8.15 kJ/mol.

Calculating the Ratio of the More Stable Chair to the Less Stable Chair

The ratio of the more stable chair to the less stable chair can be determined using the equilibrium constant: \(\text{Ratio} = \frac{[\text{More stable chair}]}{[\text{Less stable chair}]} = K = 95.6\) So, the ratio of the two conformations is approximately 95.6, meaning that there are about 95.6 times more molecules in the more stable chair conformation than the less stable chair conformation at \(25^{\circ} \mathrm{C}\).