organic chemistry - Determining chirality after considering conformational interconversion in a cyclohexane

Basically I want to know what does the question mean when it says "taking into consideration conformational interconversions."

I assume it means, when I draw the mirror image of the given compound, the mirror image can be same as the compound itself (the case with option (a) and option (c) ) or it (the mirror image) can be same as confomer of the compound (the case for option (a) and option (b) ), for the compound to be achiral.

Note: option (a) shows both the features because option (a) is same as it's confomer and it is same as it's mirror image as well.

Is my assumption correct?

Answer

Your analysis is correct. The stipulation "taking into consideration conformational interconversions" means that conformational interconversion is fast. That's important because compounds that we consider achiral can have chiral conformations. I used the example of butane in a previous question.

In this problem, three compounds are given that have multiple stereocenters, compounds b, c, and d. A common way to decide whether a compound with multiple stereocenters is chiral is to look for planes or points of symmetry within the compound. Compound c has a plane of symmetry, so it is clearly achiral. Neither of compounds b or d has a plane of symmetry in the drawn conformations, which means that those conformations are chiral. However, as you suggest, chair flip of b gives the mirror image of the original drawing, which means the original and mirror image are equivalent, assuming that the two conformations can interconvert.

There are many cases where conformations cannot interconvert, and that leads to very interesting compounds that don't even require a stereocenter to be chiral (again see earlier question).