Chemistry - Is cis-1,2-dimethylcyclohexane a meso compound?

Solution 1:

cis-1,2-Dimethylcyclohexane is achiral, not because there is a plane of symmetry, but because it consists of two enantiomeric conformations which interconvert rapidly via ring flipping at normal temperatures.

This is exactly the same case as amine inversion. "Chiral nitrogens" such as that in $\ce{NHMeEt}$ do not lead to chirality or optical activity because of rapid inversion of configuration at the nitrogen atom, leading to interconversion of the two enantiomeric forms 1a and 1b.

Conformers of N-methylethanamine

Likewise, the ring flip in cis-1,2-dimethylcyclohexane leads to two different conformers. I have deliberately chosen to depict the ring flip in the following fashion, to make the mirror image relationship more obvious. The green methyl group, equatorial in conformer 2a, is changed into an axial methyl group in conformer 2b. Likewise, the blue methyl group goes from axial to equatorial. The 1,2-cis relationship between the two methyl groups is retained in both conformers.

Conformers of cis-1,2-dimethylcyclohexane

Each individual conformer can be said to be chiral, but just like how the amine is considered achiral, cis-1,2-dimethylcyclohexane as a whole is considered achiral.

Is the compound meso? According to the IUPAC Gold Book, a meso-compound is defined as:

A term for the achiral member(s) of a set of diastereoisomers which also includes one or more chiral members.

1,2-Dimethylcyclohexane possesses two diastereomers, one cis and one trans form. The trans form is chiral, but the cis form is achiral, as explained above. Therefore, the cis form satisfies the above definition and is considered a meso-compound.

Solution 2:

I'd like to add a few points definition wise from the IUPAC goldbook.
(Because recently the original website seems to be broken I am using links via the internet archive.)

A prefix to indicate the presence of a symmetry element of the second kind (see chirality) in a chemical species which is usually one of a set of diastereoisomers that also includes a chiral member.

However, there is a more general definition, too:

A term for the achiral member(s) of a set of diastereoisomers which also includes one or more chiral members.

The first part clearly does not apply to the cis-isomer, (1R,2S)-1,2-dimethylcyclohexane, the second does because of the definition of chirality.

The geometric property of a rigid object (or spatial arrangement of points or atoms) of being non-superposable on its mirror image; such an object has no symmetry elements of the second kind (a mirror plane, σ = S1, a centre of inversion, i = S2, a rotation-reflection axis, S2n). If the object is superposable on its mirror image the object is described as being achiral.

I think the key point of this definition is in this case, that the object needs to be rigid, which (1R,2S)-1,2-dimethylcyclohexane is not at room temperature. If you cool down the solution, inhibiting the ring flip, you would probably be able to find both configurations; that is a racemic mixture of them.

Since (1R,2R)-1,2-dimethylcyclohexane and (1S,2S)-1,2-dimethylcyclohexane, the trans-conformers, are both chiral (and enantiomers), the remaining member of the set (1R,2S)-1,2-dimethylcyclohexane can be considered a meso-compound, since it is achiral.