Question

Draw the alternative chair conformations for the product formed by the addition of bromine to 4 -tert-butylcyclohexene. The Gibbs free energy differences between equatorial and axial substituents on a cyclohexane ring are \(21 \mathrm{~kJ}\) (4.9 kcal)/mol for tert-butyl and \(2.0-2.6 \mathrm{~kJ}(0.48-0.62 \mathrm{kcal}) / \mathrm{mol}\) for bromine. Estimate the relative percentages of the alternative chair conformations you drew in the first part of this problem.

Short answer

Solution: Following the steps provided: 1. The reactants are 4-tert-butylcyclohexene and bromine. 2. The product formed is 4-tert-butyl-1,2-dibromocyclohexane. 3. The two chair conformations have the tert-butyl group in either axial or equatorial position, with bromine atoms in 1,2 positions. 4. The Gibbs free energy differences are 21 kJ/mol for the tert-butyl group in the axial position and 2.0-2.6 kJ/mol for bromine in the axial position. 5. The total Gibbs free energy difference is: ΔG = 21 kJ/mol + 2 × 2.3 kJ/mol = 25.6 kJ/mol 6. Using the Boltzmann distribution, the ratio is: \(ratio = e^{-\frac{25.6 \times 10^3}{8.314 \times 298}} = 0.012\) 7. The relative percentages of the two conformations are: - Lower-energy conformation: \(\frac{0.012}{1 + 0.012} \times 100 = 1.19 \% \) - Higher-energy conformation: \(100 - 1.19 = 98.81 \% \) Based on the given Gibbs free energy differences, the relative percentages of the two chair conformations of the product are approximately 1.19% for the lower energy conformation and 98.81% for the higher energy conformation.

Step-by-step solution

Draw the reactants

Draw the cyclic reactant, 4-tert-butylcyclohexene with the tert-butyl group at the 4th carbon, and the bromine molecule.

Addition of bromine to the double bond

Perform the halogen addition reaction, in which the double bond breaks and one bromine atom gets attached to each of the carbons that were initially part of the double bond. The product formed is 4-tert-butyl-1,2-dibromocyclohexane.

Draw the chair conformations

Draw the two chair conformations of 4-tert-butyl-1,2-dibromocyclohexane: one with the tert-butyl group in axial position and the other with it in equatorial position. Note that the bromine atoms will be in 1,2 positions in each chair conformation.

Calculate the Gibbs free energy differences

Use the given Gibbs free energy differences to calculate the total Gibbs free energy difference between the two chair conformations: 1. \(\Delta G_{1} = 21 \mathrm{~kJ/mol}\) for the tert-butyl group in axial position. 2. \(\Delta G_{2} = 2.0-2.6 \mathrm{~kJ/mol}\) for bromine in axial position.

Calculate the energy difference between the two conformations

The total Gibbs free energy difference of the two conformations is the sum of the individual energy differences for each substituted position: \(\Delta G = \Delta G_{1} + 2 \times \Delta G_{2}\). Calculate this value.

Convert the energy difference to a ratio

Use the Boltzmann distribution to convert the energy difference to percentages: \(ratio = e^{-\frac{\Delta G}{RT}}\), where R is the gas constant \((8.314 \mathrm{~J/(mol\cdot K)})\) and T is the room temperature (298 K).

Calculate the relative percentages

Use the ratio calculated in Step 6 to estimate the relative percentages of the two conformations. Add the ratio to 1 and divide the ratio by the sum, then multiply by 100 (for the lower energy conformation). Subtract this value from 100 to get the percentage of the higher energy conformation. After completing all these steps, you will have successfully drawn the alternative chair conformations of the product formed by the addition of bromine to 4-tert-butylcyclohexene and estimated the relative percentages of the two conformations using the given Gibbs free energy differences.