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My attempt: polar protic solvent, low temp, tertiary carbon, and a weak (?) nucleophile and a weak base.

It seems like this is a pretty standard SN1 reaction so "I and II" would be the correct answer?

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I think your answer is correct as long as you can justify it. (IV) is clearly absurd but why do you eliminate (III) ? However in my opinion at $\ce{20 ^{\circ} C}$ there would be no reaction from a practical point of view, i.e. one with reasonable yield and reaction time.

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    $\begingroup$ 20 C suggests a low temp which favors sn1 over e1 (which would give you III) since e1 has a higher deltaG $\endgroup$ – john clastings Nov 11 '15 at 2:23
  • $\begingroup$ The alcohol is indeed a weak nucleophile. A good general rule of thumb is that an alkoxide (a deprotonated alkyl alcohol such as ethanol or isobutanol) is a much better nucleophile than its alcohol. The $20^{o} C$ is also a good indication for a slow E1 product, as john has said (higher $\Delta G$). Since the question asks for specifically the major products, I would not include III in the answer. However, if the question had asked for all products, III is a possible answer to include. Its mechanism would involve first-order beta elimination due to the sterically-hindered alkyl halide. $\endgroup$ – timaeus222 Nov 11 '15 at 4:38

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