1)Today I came across a problem which asks to find the order of dehydration of 2-hydroxycyclohexanone(1), 3-hydroxycyclohexanone(2),4-hydroxycyclohexanone(3). The answer is 2>3>1 But I thought answer should be 2>1>3 cause in 2 and 3 the product would be conjugated and hence more stable. 2)My second confusion is regarding the pathway taken by such hydroxyketones. Do they follow E1 pathway and form carbocation or do they take some other pathway? I have provided what I think but I am not sure whether the same mechanism will be followed or not.
I think your mechanism is reasonable.
Coming to rate of dehydration,
In first case Alpha hydroxy ketone will have the least rate of dehydration because a carbocation is formed which is destabilized by Ketone group and even notice there is hydrogen bonding in reactant making reactant more stable and thus rate of dehydration is decreased in first case.
In second case, though carbonation is not stabilized, here breaking of H will be easier because the H are more acidic being next to electron withdrawing group , so 2 has highest rate .
In third case, since carbocation is not so unstable and even the H is not much acidic. So 3 has moderate rate of dehydration.
Therefore, rate of dehydration:2>3>1
Product of all the three compounds would be same. You have missed space tautomerism in the third one.
Besides,charge beside a carbonyl is considered very unstable. Thus, first one is lowest in rate. Third one is placed second because there is an extra step of tautomerism.