As far as I can tell, there doesn't seem to be a chiral carbon in this compound. C-4 of the cyclohexane ring has two groups with exactly the same connectivity, and the exocyclic double bond can't give rise to optical isomerism. What am I missing?
Answer
The strict criterion for a compound to display chirality is that it must not be superimposable upon its mirror image. Let's ignore the chair conformation of the ring for a while, and assume it adopts a planar conformation. You could draw a side-on view of the ring like this:
Its mirror image would look like this.
This is an example of axial chirality (Wikipedia; IUPAC). The molecule does not possess a chiral centre itself. The chirality in this case therefore does not arise from disposition of groups about a point (the more familiar case); it arises from the disposition of groups about an axis, in this case the $\ce{C=C}$ axis.
Of course, the cyclohexane ring does adopt a chair conformation. That does not affect the fact that the compound is chiral; it just means that the ring isn't entirely flat as depicted.
The two stereoisomers described can be named using Cahn-Ingold-Prelog rules as described by Loong in this answer.
Actually, this is very similar to the case of an allene, which you may or may not be familiar with. Notice how our planar conformation looks almost like this allene
with the planar ring taking the place of the second double bond. Further explanation of axial chirality in allenes can be found in this question.
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