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How does hydrogen bonding affect hybridisation of species? I mean in $\ce{B(OH)3}$ the hybridization of boron is $\mathrm{sp^2}$ but in aqueous phase, is the hybridization same or it changes?

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Good question! Hydrogen bonding in the solid appears to be at least partially responsible for the planar (sp2) structure. ("Crystalline boric acid consists of layers of B(OH)3 molecules held together by hydrogen bonds of length 272 pm." https://en.wikipedia.org/wiki/Boric_acid) If the empty orbital on boron (which is partially filled by donation from the oxygens) were active enough to connect to a fourth oxygen, the hybridization on boron would become sp3.

Boric acid in water might be expected to ionize to a proton and B(OH)4-, which is sp3 hybridized. However, boric acid is a very weak acid, so not much borate ion is created in aqueous solution (pKa:9.24, 12.4, 13.3 for the three OH groups - reference is the same Wikipedia article). Boric acid in water probably has the same kind of structure as it has in the solid; the hydrogen bonding has just shifted from other boric acid OH groups to HOH groups of water - except for a very few molecules that have ionized.

In solid H3BO3 and in aqeous solution, hydrogen bonding is essentially all there is; but if you raise the pH to 11 or 12, you can get hydroxide ions and sp3 hybridization because the extra negative charge on the OH- makes it a better donor into the empty boron p orbital.

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