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What would be the product(s) if 2,2-dimethylhex-5-en-3-ol reacts with Lucas reagent?

Would a Methyl shift happen? I've heard that in Groove's process, only primary and secondary alkyl chlorides are formed.

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Actually, in this case, a hydride shift from C4 seems more likely. Allylic carbocations tend to be more stable than even tertiary non-conjugated carbocations. Under SN1 circumstances you'd get a mixture of 5,5-dimethyl-3-chloro-1-hexene and 5,5-dimethyl-1-chloro-2-hexene. Under forcing conditions further addition might take place, mainly to the 1,3-dichloroalkane.

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A shift of the methyl group is likely, because this would yield a more stable tertiary carbocation. In concentrated hydrochloric acid, the double bond can also be protonated, producing a secondary carbocation which is captured by $\ce{Cl-}$. If both happens, the product would be 2,4-dichloro-4,5-dimethylhexane.

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