Effect of beta branching and carbocation rearrangement on SN1 reaction rates

My book (NCERT class 12 Chemistry) asks



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> Predict the order of reactivity of the four isomeric bromobutanes in SN1 and SN2 reactions.
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The answer given for SN1 is



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> 1-bromobutane < 1-bromo-2-methylpropane < 2-bromobutane < 2-bromo-2-methylpropane
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My confusion is between (I) 1-bromobutane and (II) 1-bromo-2-methylpropane. The book explains



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> the carbocation derivative derived from (II) is more stable than derived from (I) because of a **greater electron donating inductive effect** of (CH3)2CH- group.
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However, I have been taught that **hyperconjugation** is (usually) a stronger effect than induction. Since the carbocation from (I) would have a greater number of alpha hydrogens, that should be more stable.


However, the carbocation in (II) can rearrange to form a tertiary carbocation, whereas (I) can only rearrange to secondary. How significant will rearrangement be when considering the reaction rates? As an additional example, how will reaction rates of 2-bromopentane and 1-bromo-3-methylbutane (where tertiary carbocation is formed after two rearrangements) compare?


