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I just thought that the reaction should follow $\mathrm{S_{N}2}$ mechanism this is because the alkyl halide is primary but the book says that the reaction will follow $\mathrm{S_{N}1}$ mechanism resulting in the formation of tertiary carbonation and a ring expansion product. Can anyone please explain the reason behind this ?

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It's a tricky example, but the $S_N1$ process is favored because the solvent is protic and the resulting 6-member ring is more stable than a 5-member ring.

If ring expansion weren't possible, I believe your intuition is correct that the $S_N2$ mechanism would dominate. See this discussion. You're not alone!

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