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From these two molecules in question, (C) should be more unstable as it is in a cis-configuration. The t-butyl and methyl groups being on the same side of the ring would exert a larger steric force on each other as compared to the trans configuration. However, (D) is mentioned as the correct answer.

How is a trans-1,3 disubstituted cyclohexane more unstable than the corresponding cis-isomer? I drew the chair projections of both compounds and the cis one has both groups on the same side, which would cause 1,3-diaxial interaction as mentioned here enter image description here

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    $\begingroup$ Look at that bottom picture and determine whether the methyl group and the protons that are drawn out are cis or trans (better names might by syn or anti)? $\endgroup$ – Jan Dec 17 '19 at 9:13
  • $\begingroup$ If I am not wrong, the groups in the axial projection are cis $\endgroup$ – Aniruddha Deb Dec 17 '19 at 11:54
  • $\begingroup$ If they are so in the axial projection, they are so too in the equatorial one. $\endgroup$ – Jan Dec 17 '19 at 12:03
  • $\begingroup$ So are the groups in the axial position cis? Was my comment right? $\endgroup$ – Aniruddha Deb Dec 17 '19 at 12:29
  • $\begingroup$ Yes, methyl and the two marked protons are cis in both the axial and equatorial pictures. $\endgroup$ – Jan Dec 17 '19 at 12:42
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First of all, as Jan pointed out on the comment, better nomenclature for thes compounds is syn- and anti- instead of cis- and trans-, respectively. However, OP is more familiar with cis- and trans- nomenclature I'd continue it with my answer.

Each substituted cyclohexane ring (e.g., di-substituted ones here) has two chair conformers as depicted in OP's bottom diagram of methylcyclohexane (a mono-substituted one). In one conformer, the methyl group is in equatorial position (left hand structure) and the methyl group in the other one is in axial position (right hand structure). Because of two 1,3-interactions cost additional approximately $2 \times \pu{3.8 kJ/mol}$ steric strain energy, the predominant conformer is equatorial one.

Now, if you put tert-butyl group in $\ce{C3}$-equotorial position on equatorial-methylcyclohexane (left hand structure, vide supra), you get cis-1,3-substituted compound (C). On the other hand, if you put tert-butyl group in $\ce{C3}$-equatorial position of axial-methylcyclohexane (right hand structure), you get trans-1,3-substituted compound (D). Because it is too bulky, the tert-butyl group is always in equotorial position in cyclohexane ring (1,3-interaction with one hydrogen would cost additional approximately $\pu{11.4 kJ/mol}$ steric strain energy), as depicted in following diagram (The larger the substituent, the more the equatorial substituted conformer will be favored):

t-Bu-Me-Cyclohexanes

The most predominant cis-isomer (C) has both substituents in equatorial-positions (the left hand side structure of (C)). Other conformer (get by the ring flip; the right hand side structure of (C)) has both substituents in axial-positions. Thus, as a rule of thumb, the conformer with both substituents in equatorial positions is more stable.

On the other hand, the both trans-isomers of (D) have one substituent in an equatorial-position and the other in an axial-position. Yet, the conformer with the tert-butyl group in the equatorial-position (left hand side structure of (D)) is more stable between two, because tert-butyl group is comparatively much larger than methyl group. However, because most predominant cis-isomer (C) has both groups in the equatorial-positions, it is relatively more stable than (D), which has methyl on axial-position, making additional steric strain energy.

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