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I was wondering whether the following order (from here) is correct:
In the first inequality, why did 9 hyperconjugations dominated resonance effect of benzene ring?
In the last inequality doesn't the 2 hyperconjugations of vinylic carbocation overpower the methyl carbocation?
In middle there is:
allyl = secondary
About the first two, it's an exception to the rule that resonance dominates hyperconjugation.
About the last inequality, the vinyl cation is sp hybridized which means it is highly electronegative, and a positive charge on it will be very unstable.
About the middle inequality, the benzyl carbocation (with five resonance structures) is more stable than allyl carbocation (two resonance structures). You may verify this here.
As for the 1st two:
Consider the 3 R's as independent(I am just explaining) ,each pushing a part of the electron cloud in its bond towards the central carbon through inductive effect as well as Hyper-conjugation. So in a way each R is reinforcing each other and an overall combined stability is achieved.
However consider the resonating structures of Benzyl carbocation. If you research a bit deeper , you will find that of all the 4 resonating structures possible, the form that is given ( + on the external carbon) contributes to almost 90% of the original stable structure. This is because, if you take any other resonating structure( say + charge on a carbon inside the ring ), you will find that the structure is quite unstable and very highly tends to resort back to the + charge outside the ring ( after all, breaking a Benzene ring isn't very favorable) . So in a way , the resonating structures do not 'reinforce' each other so "Resonance" does not stabilize the way it is expected to do (as in case of similar resonating structures)
Talking of the last inequality:
The "Hyper-conjugation" that you mentioned wouldn't play the role here as you imagine. You might- have overlooked the fact that vinyl cation is least stable MOSTLY because it has a very high tendency to get converted to another structure, namely the left carbon(in your diagram) with a + charge and the right one with a lone pair. This is because ,then it will have many more hyper-conjugations to get stable. This is what makes vinyl carbocation least stable, it very very rarely ever exists or tries to exist in this particular form.
As for the middle equality,
Yes, in a way your doubt is right. (Since there is an equivalent resonance structure in the left one (as given) and that is a powerful one!!).However the thing you have missed while comparing the 2nd one is the fact that "Inductive" effect plays a 'minor' role in contributing to the stability of the secondary carbocation.The 'Major' role is played by the 7 "Hyper-conjugations" which is no less than a single resonance. Of course 1 hyperconjugation is less than 1 resonance but basic intuition will tell you that 7 hyperconjugations will have a fair chance to cope with 1 resonance.
Resonance" does not stabilize the way it is expected to do. The benzyl cation has extensive resonance stabilization which accounts for the fact that it is far more stable than otherwise expected. It's just that tertiary alkyl cations are a little bit more stable.
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