Why/How does Carbon's 3rd p-orbital contribute to its bond -- sp3 -- if it's vacant?
Formally, you have to start from an excited state of carbon where both the s and each of the p orbitals is occupied with one electron. This is called "promotion". The energy necessary for promotion is made up by the formation of 4 bonds instead of only two.
I'm merely familiar with the concept of electron promotion but from what little I know your explanation does not seem to account for the promotion: does the promotion occur first or does the tetrahedral bonding occur first; if the former then where is the promotional energy coming from and if the latter then I refer to my original question.
There is no first and last. This is only a formal decomposition of the total binding energy.
Does this promotion+tetrahedron happen with Boron, Nitrogen, Oxygen, Fluorine, or Neon -- or with any element other than Carbon?
"Happen" might not be quite the right word to describe this (we are speaking about a way to decompose and understand the binding of molecules in intuitive terms, not about an actual physical process which is observable in any way). But if we ignore this: Yes, most elements would be considered to be in an atomic state closer to an excited state than the atomic ground state while in molecular bonds. So it does happen to many elements.
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