Question

Draw a chair conformation of 1,4 -dimethylcyclohexane in which one methyl group is equatorial and the other is axial. Draw the alternative chair conformation and calculate the ratio of the two conformations at \(25^{\circ} \mathrm{C}\).

Short answer

Answer: The ratio of the two conformations is 1, meaning that both conformations are equally probable at 25°C. 1,4-dimethylcyclohexane exists in an equal mixture of the two chair conformations at room temperature.

Step-by-step solution

Draw the first chair conformation of 1,4-dimethylcyclohexane

To draw the first chair conformation, place one methyl group on an equatorial position and the other on an axial position on the second carbon atom. The remaining carbon atoms are also in their respective conformations. Make sure to place the methyl groups at positions 1 and 4.

Draw the alternative chair conformation of 1,4-dimethylcyclohexane

For the alternative conformation, perform a chair flip to switch the positions of the methyl groups. Now, the first methyl group placed axially is in an equatorial position and the second methyl group placed equatorially is in an axial position. Confirm that the methyl groups are still at positions 1 and 4.

Calculate the ratio of the two conformations

First, we need to find the energy difference between the two conformations. The energy difference between an axial and an equatorial methyl group in cyclohexane is about 7.6 kJ/mol. In the first conformation, we have an equatorial and an axial methyl group, so the energy is \(0 + 7.6 = 7.6 \,\text{kJ/mol}\) . In the alternative conformation, the situation is the same, and the energy is also 7.6 kJ/mol. Now, we can calculate the ratio of the two conformations at 25°C or 298K using the Boltzmann distribution formula: $$\frac{C_1}{C_2} = e^{\frac{-\Delta E}{kT}}$$ where \(\Delta E\) is the energy difference between the conformations, \(k\) is the Boltzmann constant (8.314 J/mol K), and \(T\) is the temperature in Kelvin. Since the energy difference is 0 (both conformations have the same energy), the exponent becomes 0, and we get: $$\frac{C_1}{C_2} = e^0 = 1$$

Determine the ratio of the conformations

Since the ratio of the two conformations is 1, the conformations are equally probable at 25°C. This means that 1,4-dimethylcyclohexane exists in an equal mixture of the two chair conformations at room temperature.