What is the Ka of OH- and Kb of H3O+?

What is the $K\_\mathrm{a}$ of $\ce{OH^-}$ and $K\_\mathrm{b}$ of $\ce{H\_3O^+}$? Have these constants been determined?




In water both equilibrium constants will be extremely small; the concentrations of $\ce{O^2-}$ or $\ce{H4O^2+}$ ions in water are essentially zero. 


According to Cotton and Wilkinson's *Advanced Inorganic Chemistry*, for the hydrolysis of *solid* oxide ions you have


$$\ce{O^2- (s) + H2O (l) -> 2 OH- (aq)} \qquad \qquad K\_b > 10^{22}$$


so you can never have any significant concentration of oxide ions in aqueous solution. As DavePhD points out, $K\_\mathrm{a}$ for the hydroxide ion is less than $K\_\mathrm{w}/K\_\mathrm{b} = 10^{-36}$.


Now, $\ce{H3O+}$ may have some capacity to act as a base in media like $\ce{HF:SbF5}$ (see [this paper](http://pubs.acs.org/doi/abs/10.1021/ja00265a031) on the role of $\ce{H4O^2+}$ in isotopic exchange reactions between hydronium ions). $\ce{H4O^2+}$ can exist in sulfolane solution, too (see [this paper](http://www.sciencedirect.com/science/article/pii/0009261487804906)).


If $\ce{H4O^2+}$ exists in water *at all*, it's probably going to actually be two protons bridged and hydrated by a lot of waters, rather than an actual $\ce{H4O^2+}$ ion. It will have a *much* shorter lifetime than even a hydronium ion. The molecule can exist in theory, though, and its electronic structure has been studied for at least half a century (see [Rosenfeld's 1964 paper](http://actachemscand.dk/pdf/acta_vol_18_p1719-1738.pdf) including SCF calculations on $\ce{H4O^2+}$). 


 9

For $\ce{O^2-}$, the $\mathrm{p}K\_\mathrm{b}$ is approximately $-22$, so the $\mathrm{p}K\_\mathrm{a}$ of hydroxide is about $36$.


sources: <http://www.newworldencyclopedia.org/entry/Oxide> and <https://www.chem.tamu.edu/rgroup/hughbanks/courses/462/lecturenotes/class6-2.pdf>


For hydronium, the reference Fred Senese cited 


[Protonated hydronium dication...](http://pubs.acs.org/doi/pdf/10.1021/ja00265a031) 


suggests that at acidities of [Hammet acidity function](http://en.wikipedia.org/wiki/Hammett_acidity_function) $H\_o = -25$ to $-28$, protonated hydronium can be present.


 8