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Conformations of Cyclohexane doesn't answers my question

This is what Organic Chemistry by David R. Klein states:

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Notice that the chlorine atom is on a wedge, which means that it is coming out of the page: it is UP. The methyl group is on a dash, which means that it is below the ring, or DOWN. The two chair conformations for this compound are as follows:

enter image description here

Notice that the chlorine atom is above the ring (UP) in both chair conformations, and the methyl group is below the ring (DOWN) in both chair conformations. The confguration (i.e., UP or DOWN) does not change during a ring flip. It is true that the chlorine is axial in one conformation and equatorial in the other conformation, but a ring flip does not change confguration. The chlorine atom must be UP in both chair conformations. Similarly, the methyl group must be DOWN in both chair conformations.

I am having a confusion in the last paragraph. Why should the Chlorine be up even after the ring flip? Moreover, why are we even discussing if it is above the ring or not, as it has already come in the equatorial position?

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Essentially, it is stating that a ring flip retains the absolute stereochemical configuration. Ring flipping is a physical phenomenon of 6-membered rings, not a chemical reaction that interconverts stereoisomers. No bonds are broken, though many do get strained in the process.

I'm not sure what you mean by your question about the importance of being in the axial/equatorial positions. Both the methyl and chloro groups are vying to reduce 1,3-diaxial groups by placing themselves in the equitorial position, but doing so for one places the other in the axial position. I think a natural question that arises "is which one wins?"

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