We can eliminate S N 2 because water is a poor nucleophile and bromide is simply not a good enough leaving group to overcome this. If the reaction proceeded via the S N 1 mechanism, the primary carbocation would quickly rearrange to a carbocation that is tertiary with resonance. Capture of water by this carbocation would lead to a tertiary alcohol as shown below. Add the necessary reactants above the arrow so that the reaction does give the desired alcohol. The transition state is the highest potential energy point between the reactants and products along a single mechanism step.
Resonance is the most influential structural factor that influences nucleophilicity.
Acetate ion is therefore worst nucleophile in this group. All three carbocations are primary. Dielectric constant is a measure of compound's ability to separate ions and is related to the ratio of polar to nonpolar bonds in the molecule. The greater fraction of polar bonds the higher the dielectric constant. Thus methanol has the lowest dielectric constant of these three compounds.