My copy the Pearson"s Organic dearteassociazione.org (7e), Morrison and Boyd, under the section "Reaction intermediates", accords a succinct description on the structure of carbocations:

The central $C$-atom (of carbocations) is in one $\mathrmsp^2$ hybridized state, because that which the carbocations have actually planar geometry. The $\mathrmp_z$-AO (atomic orbital) stays empty.

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The ingredient in parentheses was included by me

Aided by this description, ns conjured increase the adhering to "general" structure of carbocations:

Though i pulled out the above image native Google Images, it was pretty much the very same structure I"ve to be visualizing this whole time...drawing my very own would it is in messy

And together you deserve to see, I"ve related the "planar structure" pointed out in the book to "trigonal planar structure" (with an axial vacant $p$ orbital). This photo of a carbocation"s framework in mind confirmed rather handy, and didn"t it seems ~ to it is in incorrect in ~ all.

Wikipedia, top top the other hand, doesn"t sound for this reason confident around the central $C$-atom"s $\mathrmsp^2$ hybridized state.

One can reasonably i think a carbocation to have $\mathrmsp^3$ hybridization with an empty $\mathrmsp^3$ orbital giving positive charge. However, the reactivity the a carbocation an ext closely resembles $\mathrmsp^2$ hybridization with a trigonal planar molecular geometry.

(Emphasis, mine)

As you can see, Wikipedia doesn"t appear to (completely) endorse the $\mathrmsp^2$ framework of the central $C$-atom.

I ongoing to save the "trigonal planar" structure of carbocations in psychic while studying them. This do no hindrance until, ns came across these carbocations (in a book not yes, really worth mentioning):

produced using PubChem Sketcher V2.4

I"ve confronted multiple difficulties while trying to ascertain the hybridization cum geometry/structure that the central, confident $C$-atoms in those carbocations. I shall perform them separately,

1) issue with the Aryl carbocation

I visualized this as a particular Kekule structure of benzene having lost one hydrogen anion, thereby leaving a positively charged carbon atom in the ring. Considering the bonds including the confident $C$-atom (in the specific Kekule framework I put up), ns see two $σ$ bonds and also one $π$ bond. Also, the $\mathrmC=C^+-C$ link angle appears to be $\mathrm120^o$ (just favor the regular benzene molecule. Ns honestly can"t number out the hybridization or structure/geometry of the hopeful $C$-atom here. I guess ns should element in the "delocalization the the optimistic charge" across the ring, yet that hasn"t borne fruit (for me).

2) worry with the vinyl carbocation

I visualized this together an ethene molecule, having actually lost one hydrogen anion, in order to leaving a positively charged carbon atom (seen top top the right end in the image). Here again, ns see two $σ$ bonds and also one $π$ bond. From my expertise of the VSEPR theory, I intend the $\mathrmC=C^+-H$ bond edge is $\mathrm180^o$ (i.e- linear). Yet I can"t for the people figure the end what the hybridization of the optimistic $C$-atom here, is. Heck, I"m not totally sure if i predicted the geometry (linear) properly in the very first place...well, this case is extraterrestrial to me.

3) issue with the ethynyl carbocation

I visualized this as an ethyne molecule, having lost one hydrogen anion, thereby leaving a positively charge carbon atom (seen top top the appropriate end). Considering the bonds including the positive $C$-atom, I view one $σ$ bond and two $π$ bonds. Hybridization? No clue. Geometry around the positive $C$-atom? Um...kinda looks favor a ball at the end of a stick...not sure if there"s any type of "angle" existing ._.

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Could someone please attend to these "issues" I"ve encountered for the above mentioned (aryl, vinyl, ethynyl) carbocations? I"m not certain if suspect "planar" structure" necessarily means "trigonal planar structure"... Or if there"s something about "hybridization" that I"ve grossly overlooked.

My question(s), an ext explicitly put:

1) What is the hybridization state that the carbon atom carrying positive charges in the three instances I"ve supplied above? how is the determined?

2) What is the geometry/structure of the claimed hybridized carbon atoms? If the isn"t clear: I expected along the currently of "If it"s $\mathrmsp^3$ it"s tetrahedral, if it"s $\mathrmsp^2$ it is trigonal planar, if it"s $sp$ it"s linear"

I"m still in High-school, so ns feel a little bit overwhelmed at the minute (trying to wrap mine head approximately this...hopelessly)