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Why isn't a mesomeric structure with carbon having only two single bonds and a lone pair used for depiction of vinyl carbocation?

Resonance forms

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The alternative answer to what @levineds said is that you're asking about a carbene. Carbenes are actually reasonably common in chemistry though we generally have them coordinated to metals for stability. There are a few reactions that proceed through putative naked carbenes.

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If I understand your drawing correctly, you are asking about the $\ce{C2}$ molecule. It is highly unstable but it does exist (and is a favorite molecule of quantum chemists because the bonding situation is quite controversial).

https://en.wikipedia.org/wiki/Diatomic_carbon

EDIT: To elaborate slightly on what I mean by controversial bonding. Shaik et al have argued that this a quadruple bond 1 on the basis of Valence-Bond calculations. Basic MO theory indicates that this is a bond somewhat intermediate to a double and a triple bond (but close to a double bond). The stretching frequency and bond dissociation energy are intermediate to a double and triple bond as well, and some have argued strongly that this implies that the bond is just a double bond as expected 2. Other sophisticated calculations indicate that the bond is strongly correlated and so the question of bond order is sorta complicated and probably best described as fractional (i.e. not just double or triple or quadruple).

1: Shaik, S., Danovich, D., Wu, W., Su, P., Rzepa, H., et al. Nat. Chem. 4, 195–200 (2012)

2: Hermann, M., Frenking, G. Chem. Eur. J. 22 (2016), 4100

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    $\begingroup$ I think it was meant to be the vinyl cation, not carbon dimer. $\endgroup$ – orthocresol May 29 '17 at 8:10
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In the vinyl cation, examine the geometry of the P-orbitals involved in the pi-bond and the empty P-orbital of the carbocation. You will see that the pi-system and the empty P-orbital are orthogonal/perpendicular. This means that the two pi-electrons cannot simply be dumped from the pi-system into the empty P-orbital to plug up the carbocation (orthogonal orbital systems do not normally exhibit resonance).

Now don't get me wrong, the resonance you showed can still occur. However, the empty P-orbital will remain empty, and that carbon's P-orbital previously used for the pi-bond will now be filled with a lone pair. Thus, the carbene will be a singlet carbene, which is indeed net neutral but more accurately thought of as both positive (empty P) and negative (lone pair in another P). This high-energy species might be even more unstable than the vinyl cation, in which case it may not be a significant resonance contributor.

For more, you might investigate the difference between singlet carbenes and triplet carbenes.

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In addition to @Zhe 's answer, let me add a bit more information about carbenes.

Carbenes are divalent carbon species, that are highly reactive, and rarely occur as stable compounds.

Recall that carbon has an electronic configuration of $\ce{[He]}2s^2 2p^2$. In most carbon compounds, an electron from the $2s$ orbital excites and lands in the $2p$ orbital. This makes the configuration as $\ce{[He]}2s^1 2p^3$ In this excitation, carbon produces 4 unpaired electrons that form covalent bonds with other atoms.

Carbenes don't excited in the same way. As a result, the have only 2 unpaired electrons, and in addition a vacant $p$ orbital and a lone $s$ pair.

The presence of both empty orbitals and a lone pair makes them liable to both nucleophilic and electrophilic attack. This makes them unstable.

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    $\begingroup$ This description of the atomic-like excitations is misleading. There is no excitation necessary (although I know that this is how some people teach hybridization), the adiabatic surface from symmetrically stretching the C-H bonds in methane connects directly to the ground atomic state. Moreover, there are both singlet and triplet carbenes in which the electrons not in a bond are either coupled as a pair or not. So this whole excitation argument is sort of irrelevant for discussing carbenes. They are just molecules in which carbon forms two bonds instead of four. $\endgroup$ – levineds May 29 '17 at 3:44
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    $\begingroup$ Ignoring the ideas about excitation of electrons, and ignoring triplet carbenes, the hybridisation description is still slightly incorrect - in (singlet) carbenes the lone pair is not pure 2s in character, there is still hybridisation, as is evidenced by the bond angles. $\endgroup$ – orthocresol May 29 '17 at 8:07

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