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When we are given this problem, there is a certain problem. We have a strong base, but hydrogen on the carbon of the phenyl. E2 is impossible. Also, SN2 is not possible since we have a hindered base.

The only product we could get is SN1. My teacher put up the answer in handwritten black in the above picture.

However can SN1 reactions happen when there is a strong base in the solution? Would that strong base be the nucleophile or the solvent tert-butyl alcohol be the nucleophile?

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The strong base is capable of being a nucleophile in this sort of situation; deprotonating tert-butyl alcohol will increase its nucleophilicity. In an SN1 reaction the overall rate is only dependent on the loss of a leaving group, so a strong base/poor nucleophile will react via SN1 as long as there are not plausible E1 pathways (e.g. sp3 bonded hydrogen alpha to carbocation).

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