TBR O-Chem, Section 2, Passage I, Q3

[HooktOnFiziks]

I agree with D being the answer to this question....

What is the value of Keq(1,4)for the conversion of trans-1,4-dimethylcyclohexane from its conformation to its diequatorial conformation?

A . 0.0029
B. 2.16
C. 4.31
D. 345​

But I disagree with a part of the explanation:

The diequatorial orientation is better than the diaxial orientation, because with diaxial there are eclipsed interactions with the axial hydrogens. However, the anti orientation of the methyl groups is better than the gauche orientation. ​

My qualms pertain to that last sentence. I recognize that in the 1,2-diaxial orientation, the methyl substituents are anti to each other. However, in this orientation, there are also two gauche interactions between the methyl substituents and the ring bonds (the CH2's). In the 1,2-diequatorial orientation, there is only one gauche interaction between the methyl substituents, and their orientations relative to the ring bonds are anti (2 anti, 1 gauche).

So the book is saying the diequatorial orientation is better for the reason of there being no axial steric hindrance albeit the gauche interaction, but the diequatorial orientation has more favorable anti interactions and fewer gauche than the diaxial orientation does.

I've drawn a picture to explain what I mean (I don't account for the two ring CH2's as gauche since their orientation is inherent to both conformers.):

(http://imgur.com/sVHciNV)

​

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[Entadus]

You're neglecting the ECLIPSED interactions. Gauche and Anti are both still forms of STAGGERED conformation, so these are not nearly as unfavorable as 1,3-diaxial interactions (ECLIPSED!)

 

[Entadus]

Try drawing the relevant Newman projection. You should draw an extended Newman as to visualize the whole ring.
See:
http://www.chemstone.net/O_Chem/images/Stereo13.gif

 

[HooktOnFiziks]

I gotcha -- I see the eclipsed interactions and I know this steric hindrance among the axial groups is a primary reason for the equatorial preference. TBR however is mentioning that despite steric hindrance among "eclipsed" axial substituents, there is an "almost redeeming" quality to the axial configuration because the axial configuration places the methyl substituents anti to one another. I'm being nitpicky about the fact that they refer to this anti interaction as "almost redeeming" for the axial configuration, because they are ignoring the two gauche interactions between the methyl groups and the ring carbons -- these gauche interactions do not exist when the methyl groups are equatorial. So rather than gauche/anti interactions being a point in favor for the axial configuration, they're a point in favor for the equatorial configuration.

What TBR is saying:

What Vollhardt/Schore have to say about gauch/anti interactions in cyclohexanes:

 

[HooktOnFiziks]

So what I'm saying (I think correctly) is that TBR is wrong to say that the 1,2-diequatorial orientation results in gauche. 1,2-diequatorial results in ONE gauche, and TWO anti if you take into to account the "ring bonds" as shown in the textbook, whereas 1,2-diaxial results in ONE anti, TWO gauche.

It doesn't affect the answer, but it seems like TBR tried to go "above an beyond" by mentioning this as a side-note in their explanation.