Resonance in carbocation derived from allenes

During a reaction if addition on one of the $\pi$ bonds of an allene (containing even number of $\pi$ bonds like $\ce {H2C=C=CH2}$) takes place then a carbocation is formed. To decide the position of the positive charge, we look at the stability of the carbocation. Example:
$$\ce {H2C=C=CH2 ->[\ce{H3O+}]} \quad?$$


In the above reaction the proton ($\small{\ce{H+}}$) attacks the electron rich $\pi$ bond (say the left one) and can form one of the following carbocation:
$$\ce{H2C+\bond{-}CH=CH2}\quad\quad \text{or }\quad\quad \ce{CH3\bond{-}C+=CH2}$$
$$ (\text{I})\quad\qquad\qquad\qquad (\text{II})$$




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Now the major product (acetone) is formed as a result of the carbocation on the right (II), which apparently is less stable than the one on the left (I) as the positive charge in I is in resonance/conjugation with the $\pi$ bond. But our teacher said that since in allenes the overlapping p-orbitals (say $\ce{p\_z}$) of one $\pi$ bond are perpendicular to the overlapping p-orbitals of the other (say $\ce{p\_y}$; which makes $\ce{p\_x}$ in the direction of internuclear axis $x$) and hence the positive charge cannot resonate with the $\pi$ bond (formed by overlap of $\ce{p\_z}$) as the positive charge is effectively an empty p-orbital (the $\ce{p\_y}$) and hence the carbocation (II) is **relatively** more stable.


But once addition has taken place the $\ce{C\_1-C\_2}$ $\pi$ bond no longer exists and the $\sigma$ bond is free to rotate so the empty p-orbital/positive charge can freely rotate and become coplanar with the p-orbitals of the $\pi$ bond. **But why does this not happen?**
The reason cannot be steric hindrance as the two groups on $\ce{C\_1}$ are the smallest, $\ce{H}$.




The answer to your question might need a few clarifying points. My answer will mostly focus on the protonation (and deprotonation) of allene based on gas-phase experiments and theoretical calculations. A few references that point to elements that will be used concerning the allyl and 2-propenyl cations are as follow : [G. van der Rest *et al. Eur. Mass Spectrom.* **3** 323 (1997)](http://journals.sagepub.com/doi/abs/10.1255/ejms.12) and [K. Raghavashari *et al. J. Am. Chem. Soc.* **103** 5649 (1981)](http://pubs.acs.org/doi/abs/10.1021/ja00409a004).


**Relative stability of the allyl and 2-propenyl cations**


Both articles make it clear that the allyl cation (I) is the most stable structure for the $\ce{C\_{3}H\_{5}^{+}}$ carbocations. It is around $\pu{21 kJ mol-1}$ more stable than its 2-propenyl (II) isomer. The other possible isomers are all higher in energy.


**Protonation of allene**


According to the first reference above, the protonation of allene on an extremal carbon to lead to the allyl cation requires to pass a high energy barrier, whereas no high barrier is present to lead to the 2-propenyl cation. Figure 4 gives the structure of the interaction complex between $\ce{H\_{3}O^{+}}$ and allene, showing clearly that the incoming proton is initially in interaction with the $\pi$ orbital on one side of the allene. Their is obviously no conjugation contribution between this protonation structure and the other $\pi$ orbital, since the proton arrival plane is perpendicular to the other $\pi$ orbital.


From this position, formation of the 2-propenyl cation (II) requires only a minimal energy barrier, requiring only a bending of the $\ce{C-C-C}$ bond and bending of the $\ce{CH\_{2}}$ group. Once this is done, the $\ce{CH\_{3}}$ group can rotate freely, but one should notice that it does not involve the $\ce{sp^{2}}$ vacant orbital located on the central carbon. This orbital will always remain orthogonal to the remaining $\pi$ orbital.


From this position, one could also consider direct protonation of the central carbon, followed by a rotation of the $\ce{CH\_{2}}$ group since no conjugation is present at this point. But here, one should consider that stabilization by conjugation can only proceed *after* formation of a (non stabilized) primary carbocation. These species are much less stable than secondary carbocations (on the order of $\pu{75 kJ mol-1}$), thus leading to the presence of a large barrier for the direct formation of an allyl cation.


A last pathway that one should consider is a 1,2 hydride transfer between (II) and (I). Although hydride transfers are generally easy between carbocations, this transfer proceeds through a three-center system which can be considered a proton attached to a $\pi$ orbital. Thus, this leads us back to the initial problem, which is that no conjugation can serve as a driving force leading to the allyl cation. And only once the unstable primary carbocation is formed can a rotation of the $\ce{CH\_{2}}$ group lead to formation of the allyl cation.


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