Question

The difference in the number of gauche interactions present in diaxial and diequatorial chair conformations of trans-1,2-dimethylcyclohexane is (in integer).

Hint:

In a chair, 1,2-diaxial groups are anti to each other but each axial group has 2 extra gauche contacts with the ring; 1,2-diequatorial groups are gauche to each other but have no extra ring contacts.

Answer 1

Correct Answer: 3

A cleaner way to count gauche interactions is to build each conformer from a reference and count what changes, using the well known fact that a single axial methyl on cyclohexane carries 2 gauche (1,3-diaxial type) interactions relative to an equatorial one, worth about 1.8 kcal/mol combined (the methyl A-value), while two adjacent (1,2) equatorial substituents are gauche to each other, worth about 0.9 kcal/mol on their own.

1. Diaxial conformer: Both methyl groups point axial. On a 1,2 pair of carbons, axial-axial substituents are anti to one another (dihedral 180 degrees), so the methyls do not form a gauche pair between themselves. Each axial methyl still clashes with the axial hydrogens on the carbons two bonds away, giving 2 gauche interactions per methyl. With two axial methyls that is $2 \times 2 = 4$ gauche interactions.
2. Diequatorial conformer: Both methyl groups point equatorial. On adjacent (1,2) carbons, equatorial-equatorial substituents sit gauche to each other (dihedral 60 degrees), giving exactly 1 methyl-methyl gauche interaction. Equatorial groups have no 1,3-diaxial type clash with the ring, so there is nothing more to add. Total: 1 gauche interaction.

Comparing the two counts, $4 - 1 = 3$. This also matches the experimentally known conformational preference of trans-1,2-dimethylcyclohexane for the diequatorial form, worth roughly $3 \times 0.9 \approx 2.7$ kcal/mol, close to the measured value.

Let's summarize:

• Diaxial trans-1,2-dimethylcyclohexane: 0 methyl-methyl gauche interaction (anti pair) plus 4 axial-ring gauche interactions, total 4.
• Diequatorial trans-1,2-dimethylcyclohexane: 1 methyl-methyl gauche interaction (gauche pair) plus 0 axial-ring interactions, total 1.

The difference in gauche interactions is 3.