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

Step 1: Identify the two chair forms of trans-1,2-dimethylcyclohexane.
For a trans-1,2-disubstituted cyclohexane, one chair form puts both methyl groups equatorial (diequatorial), and ring flipping puts both methyl groups axial (diaxial). These are the two conformations asked about.

Step 2: Recall the 1,2-dihedral relationships in a chair.
Looking along any C1-C2 bond of a chair, adjacent axial substituents are anti to each other (\(180^{\circ}\) dihedral), while adjacent equatorial substituents are gauche to each other (\(60^{\circ}\) dihedral). Also, every axial substituent carries two extra gauche interactions with the axial hydrogens on the carbons two positions away (its 1,3-diaxial partners), while an equatorial substituent has none of these extra interactions.

Step 3: Count gauche interactions in the diaxial form.
Both methyls are axial, so the direct methyl-methyl (1,2) interaction is anti, contributing 0 gauche interactions. Each axial methyl separately has 2 gauche interactions with the ring (its 1,3-diaxial type contacts), so the two axial methyls give \(2+2=4\) gauche interactions in total.

Step 4: Count gauche interactions in the diequatorial form.
Both methyls are equatorial, so the direct methyl-methyl (1,2) interaction is gauche, contributing 1 gauche interaction. Equatorial substituents have no 1,3-diaxial type gauche interactions with the ring. So the diequatorial form has just 1 gauche interaction in total.

Step 5: Find the difference.
\[ \text{Difference} = 4 - 1 = 3 \]

Final Answer:
The diaxial form has 3 more gauche interactions than the diequatorial form. \[ \boxed{3} \]