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What is the effect of hybridization and double-bond formation on the rest of the atoms/molecules attached to the central atom?

For example, in a benzene molecule, the Carbon-Carbon bonds are intermediate between single and double bonds. But what is the effect of the presence of the double bonds between the carbons on the bonds between the carbon and hydrogen? Do the bond lengths decrease because of the sp2 hybridized bonding orbitals?

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Yes, the bond length decreases. One way to think about it is that a sp3 center with 109.5 degree angles is more crowded than an sp2 center with 120 degree angles. More crowding = more repulsion = longer bond length.

Another way to think about it is that the e-pair in an s orbital is sitting on the nucleus. Adding a p-orbital moves the electron pair further away from the nucleus. More p-orbital character = longer bond lengths.

You can test it for yourself using http://molcalc.org/

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