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Our teachers told us that greater the number of alpha H, greater is the stability of carbocation.

But consider this:

$\ce{CH3CH2+}$ and. $\ce{CH3CH2CH2+}$

The first one has 3 alpha H while the second one has 2. So first one is more stable, as our teachers say. But even C-C sigma bond can participate in hyperconjugation, isn't it?

Going with that, second one has equal no bond resonating structures and even higher inductive effect of ethyl group. So second one should be more stable (without overruling the dominance of hyperconjugation).

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Hyperconjugation

Hyperconjugation, as you point out, is able to stabilise carbocation intermediates, IUPAC define this as:

the interaction is ... between σ-bonds and an unfilled or partially filled π- or p-orbital

Ref: IUPAC Gold Book

In a simple sense, we can visualise this interaction by drawing the carbocation and considering the adjacent σ bonds able to interact.

enter image description here

This can also be visualised in a more 'quantitative' way by considering a simple mixing of orbitals, in which we can see that the σC-H -> p+ lowers the energy relative to the vacant p-orbital.

enter image description here

Ref: Molecular orbitals and organic chemical reactions, Fleming

Stability of carbocations

If we now consider the various types of cation, we should be able to rationalise the order of stability.

enter image description here

The methyl group is completely unstabilised through hyperconjugation, due to the stereoelectronic requirement that the σ bond be located on the adjacant carbon (of which the methyl cation has none).

Primary carbocations have a maximum of 3 σ-bonds capable of hyperconjugation, secondary a maximum of 6, tertiary a maximum of 9– hence why the tertiary is the most stable cation out of the simply alkyl groups (the allyl and benzyl systems in the diagram above are stabilised through delocalisation of the charge onto the pi system).

Going back to your question of ethyl vs propyl carbocations, you can hopefully now see that they have the same number of σ-bonds in the correct position to participate in hyperconjugation.

An additional complication

Primary carbocations are sufficiently unstable that if they form, they tend to rearrange into more stable carbocations, such that the primary propyl cation will quickly undergo a 1,2-hydride shift to give the secondary propyl cation. A more complex example is shown below:

enter image description here

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    $\begingroup$ And leaving that secondary carbocation at the very bottom lying around for too long should result in another Wagner-Meerwein this time with a methyl group migrating to give a tertiary carbocation $\endgroup$ – Jan Jun 18 '17 at 17:28

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