Question

Following are the alternative chair conformations for trans-1,2-dimethylcyclohexane. (a) Estimate the difference in free energy between these two conformations. (b) Given your value in (a), calculate the percent of each chair present in an equilibrium mixture of the two at \(25^{\circ} \mathrm{C}\).

Short answer

Answer: At 25°C, the equilibrium mixture contains approximately 98.54% of the more stable conformation (where both methyl groups are equatorial) and 1.46% of the less stable conformation (where both methyl groups are axial).

Step-by-step solution

Identify the conformations

Draw the two chair conformations of trans-1,2-dimethylcyclohexane, and label the axial and equatorial positions for each substituent.

Estimate energy difference

Estimate the difference in free energy between the two conformations based on the axial and equatorial preferences of the substituents. In this case, both methyl groups are aligned axially in one conformation and equatorially in the other. Since axial methyl groups experience a 1,3-diaxial interaction, they have higher energy than the equatorial methyl groups. The energy difference due to one 1,3-diaxial interaction involving a methyl group is approximately 7.6 kJ/mol. Thus, the total energy difference between the two conformations is roughly 2 * 7.6 kJ/mol = 15.2 kJ/mol (\(\Delta G\)).

Calculate energy difference in kcal/mol

Convert the energy difference from kJ/mol to kcal/mol. 1 kcal/mol ≈ 4.184 kJ/mol. So, the energy difference is 15.2 kJ/mol * (1 kcal/mol / 4.184 kJ/mol) ≈ 3.63 kcal/mol.

Use the Boltzmann distribution

With the energy difference (\(\Delta G\)) and temperature given, we can use the Boltzmann distribution equation to calculate the ratio of the two conformations: \(K = e^{\frac{-\Delta G}{RT}}\) Here, K is the equilibrium constant, R is the gas constant (0.001986 kcal/mol•K), and T is the temperature in Kelvin (25 + 273.15 = 298.15 K). Plug in the values and solve for K: \(K = e^{\frac{-3.63 \text{ kcal/mol}}{(0.001986 \text{ kcal/mol•K})(298.15 \text{ K})}}\) \(K \approx 67.6\) This means that for every one molecule of the less stable conformation, there are approximately 67.6 molecules of the more stable conformation.

Calculate the percent of each chair conformation

To calculate the percent of each chair present in an equilibrium mixture, divide the number of each conformation by the total number of conformations and multiply by 100. Percent of more stable conformation: \((\frac{67.6}{67.6+1})\times100\% \approx 98.54\%\) Percent of less stable conformation: \((\frac{1}{67.6+1})\times100\% \approx 1.46\%\) Thus, the equilibrium mixture at 25°C contains approximately 98.54% of the more stable conformation and 1.46% of the less stable conformation.