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I am confused with this particular problem.
I began the problem with a mechanism as following. In the beginning $\ce{H+} $ protonates the alcohol and leaves as $\ce{H_2O}$. Then everything is about carbocation rearrangement.
But then my professor gave out a different answer saying that ring expansion doesn't happen.
Why doesn't ring expansion take place? Isn't a six-membered ring more stable than a five-membered ring? I am not sure where I am making the mistake.
Your proposed mechanism involves a ring expansion from cyclopentane to cyclohexane. This gives very little driving force in terms of relieving strain in the system. Cyclopentane, in a twisted or "envelope" conformation, has some angle strain ($102-106^o$ vs $109.5^o$ $^{ref~1}$) and it does also has some torsional strain down some of its bonds.
This is mostly alleviated by the move to cyclohexane, if it is in a full chair conformation, but the stabilization of the system is minimal ($25-29kJmol^{-1} $ $^{ref~2}$). Here is a link to a question that deals with ring strain and the calculation of it from $\Delta H$ of combustion.
Along with this, your mechanism proposes this ring opening involving an alkyl shift that moves the carbocation from tertiary to secondary. This involves a destabilizing of the cation due to the reduced hyperconjugation (and inductive effect). Studies have shown that a secondary carbocation is around $67-75kJmol^{-1} $less stable than a tertiary, for simple substituents (and in the gas phase$^{ref~3}$).
In total, the ring opening from a five to six-membered ring in your mechanism does not result in an overall increase in stability. Hence why the second mechanism is suggested to happen instead.
