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I saw a different mechanism for esterification of tertiary alcohol in a book.
They say that first the $\ce{OH}$ is protonated and it leaves to form the tertiary carbocation. Then the oxygen's lone pair from the carboxylic acid attacks to form the ester.
Is this mechanism actually correct?
Yes, the mechanism is correct and is different from the normal mechanism because here the more stable tertiary carbocation formation.
Upon esterification of a tertiary alcohol an alkene is formed as the major product, not an ester.
