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My book defines the basicity of the acid as the number of $\ce{H^+}$ ions furnished by one mole of the acid in solution. Well, I know that $\ce{B(OH)_3}$ is a weak Lewis acid and it accepts an $\ce{OH^-}$ to form an anion - thereby increasing the net $\ce{H^+}$ concentration in the solution.

So, what should exactly be its basicity? Should I just assume its basicity to be one as well - although it doesn't satisfy the definition?

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Boric acid does not dissociate in aqueous solution, rather it is acidic due to its interaction with water molecules (i.e accepts $\ce{OH-}$) to form the tetrahydroxyborate ion:

$\ce{B(OH)3 + H2O -> [B(OH)4]- + H+}$

$\ce{B(OH)3}$ does not act as a proton donor but behaves as a Lewis acid i.e. it accepts a pair of electrons from $\ce{OH-}$ ion and in this process $\ce{H+}$ ions are furnished.

I understand monobasic means there should be 1 replaceable hydrogen atom in the molecule, but $\ce{B(OH)3}$ in itself actually doesn't contain any replaceable hydrogen atom; rather, on interaction with water (i.e after accepting a $\ce{OH-}$ ion) a hydrogen ion is liberated. But if you see the after effect, say adding a monobasic acid (which holds the definition correctly) like $\ce{HCl}$ to water, per molecule it liberates one $\ce{H+}$ ion, similarly each $\ce{B(OH)3}$ also liberates one $\ce{H+}$ after it accepts $\ce{OH-}$ from water. So only looking at the after effects/ final result, both $\ce{HCl}$ and $\ce{B(OH)3}$ liberate 1 hydrogen per molecule though the process through which it was liberated is different.

So considering the after effect we consider $\ce{B(OH)3}$ as an acid and as it finally liberates 1 electron per molecule (though after accepting $\ce{OH-}$), we still can consider it to be a monobasic Lewis acid. At least, Wikipedia calls it a monobasic Lewis acid.

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