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We were asked to find out the number of Gauche-Butane interactions for this compound:

enter image description here

I proceeded by drawing the Newman Projection for this chair conformation as follows and marked the Gauche-Butane interactions (as shown by red curves):

enter image description here

They told me that I was almost correct and asked me to try once again but I can't find out where I probably went wrong.

Can someone guide me out in this regard?

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    $\begingroup$ #3 looks dubious. Consider that the gauche interaction stretches over four carbons: C–C–C–C. $\endgroup$
    – orthocresol
    Jun 17 '20 at 16:09
  • $\begingroup$ @orthocresol, I thought that the methyl group involved in the step 3 suffers steric repulsion from that carbon like in step 1. Isn't it so? If not, what should it be? $\endgroup$
    – Sanu_012
    Jun 17 '20 at 16:14
  • $\begingroup$ Which 4 carbons should I choose? Should I start from the carbon in step 3 to all the way left ? $\endgroup$
    – Sanu_012
    Jun 17 '20 at 16:16
  • $\begingroup$ Ahem, well, I don't really want to give you the answer straight up (and indeed, I don't think you need it). The equatorial methyl group is only involved in one gauche interaction, which is your #2. The other carbons are anti to it, so that's not an issue. Your #1 and #2 are quite correct. Your #3 isn't. Have a think about it. By the way, are you supposed to ignore gauche interactions that are within the ring itself? Even without the methyl groups, unsubstituted cyclohexane has some gauche interactions, which you haven't included. $\endgroup$
    – orthocresol
    Jun 17 '20 at 16:36
  • $\begingroup$ Is this from an entrance exam preparation manual/worksheet? $\endgroup$
    – Micelle
    Jun 17 '20 at 16:59
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While cyclohexane does have gauche interactions, they are irrelevant because chair cyclohexane is taken as the reference point. The axial methyl group in cis isomer 1 has two gauche interactions with the ring as illustrated in red in 1a and 1b. A third gauche interaction is between the two methyl groups in 2. By comparison the trans isomer 3 only has gauche interactions between the methyl groups.

highlighted gauche interactions in cis-1,2-dimethylcyclohexane

The difference in the number of gauche interactions - namely, two - between these epimers is approximated by the difference in their heats of combustion[1] which is $\pu{1.54 kcal/mol}$. [1,2 cis: $\Delta H^\circ_\mathrm{comb} = \pu{-1248.31 kcal/mol}$; 1,2 trans: $\Delta H^\circ_\mathrm{comb} = \pu{-1246.77 kcal/mol}$.] A typical value for a gauche interaction is $\pu{0.9 kcal/mol/interaction}$.

References:

  1. Walter H. Johnson, Edward Prosen, and Frederick D. Rossini, Heats of Combustion and Isomerization of the Eight $\ce{C8H16}$ Alkylcyclohexanes. Research Paper RP1S12, Volume 39, July 1947. PDF via NIST
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  • $\begingroup$ Should I consider 1a and 1b as equal? $\endgroup$
    – Sanu_012
    Jun 18 '20 at 1:58
  • $\begingroup$ They are mirror images but additive. Axial methylcyclohexane is less stable than the equatorial conformation by 1.8 (2 x 0.9) kcal/mol. $\endgroup$
    – user55119
    Jun 18 '20 at 2:56
  • $\begingroup$ So you're saying that there are 4 gauche-butane interactions in total? $\endgroup$
    – Sanu_012
    Jun 18 '20 at 3:37
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    $\begingroup$ Not at all! Structures 1a, 1b and 2 are all the same cis isomer showing the three gauche interactions. Compound 3 is the trans isomer with one gauche. The difference between the two is a total of 2 gauche interactions. This difference is seen in the difference in the heat of combustion of the cis and trans isomers. See the 1,2-cis and trans isomers here: ursula.chem.yale.edu/~chem220/chem220js/… $\endgroup$
    – user55119
    Jun 18 '20 at 11:57
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    $\begingroup$ No! cis-1,2 has 3 gauche interactions as illustrated. Trans-1,2-has one. If you want to figure out the difference in their heats of combustion, the methyl/methyl interactions cancel each other leaving a net of 2 gauche interactions. The number 2 only applies to the net difference between the two isomers as far as the heats of combustion and heats of formation, for that matter., are concerned. You need to do some reading. Google the issue. $\endgroup$
    – user55119
    Jun 18 '20 at 17:46

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