Recall that in cyclic compounds, such as cyclohexane, the conformation of the ring predominantly determines the spatial arrangement of the constituent atoms. As such, the conformational stability of the ring influences the molecular symmetry as well as the stereoisomerism of cyclic compounds.
Consider the case of 1,2-dimethylcyclohexane, a disubstituted cyclohexane with two chiral centers and four possible configurations. Here, each configuration lacks a plane of symmetry, as the cyclohexane ring exists in the non-planar chair conformation.
Furthermore, the cyclohexane ring in these molecules can easily undergo ring-flipping at room temperature. As a result, each of the four possible configurations of 1,2-dimethylcyclohexane may exist as a mixture of two or more conformations.
This conformational flexibility of the cyclohexane ring affects the number of possible stereoisomers of 1,2-dimethylcyclohexane.
For instance, the cis configurations of 1,2-dimethylcyclohexane have conformations that are enantiomers of each other.
However, ring-flipping followed by a 180-degree rotation transforms one configuration into another. Therefore, these configurations simply represent conformations of the same molecule.
The trans isomers of 1,2-dimethylcyclohexane also have conformations that are enantiomers of each other.
These configurations cannot be superposed by rotation of the molecule or by undergoing a ring flip. Accordingly, each trans isomer of 1,2-dimethylcyclohexane exists as a unique compound.
Thus, 1,2-dimethylcyclohexane exhibits three stereoisomers: the cis-1,2-dimethylcyclohexane and the pair of enantiomers of trans-1,2-dimethylcyclohexane.
Similarly, 1,3-dimethylcyclohexane has two chiral centers but only exhibits three stereoisomers. Here, the two trans configurations are enantiomers of each other that cannot be interconverted. The other pair, the cis configurations, have a plane of symmetry and are achiral; that is, they are the same molecule.
