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What is the $K_\mathrm{a}$ of $\ce{OH^-}$ and $K_\mathrm{b}$ of $\ce{H_3O^+}$? Have these constants been determined?

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    $\begingroup$ That's a good question! $\endgroup$ – Mithoron Jan 29 '15 at 17:34
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    $\begingroup$ Is what that this page says incorrect or incomplete? $\endgroup$ – M.A.R. ಠ_ಠ Jan 29 '15 at 18:40
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    $\begingroup$ Oh, wait. It's incomplete. $\endgroup$ – M.A.R. ಠ_ಠ Jan 29 '15 at 18:46
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    $\begingroup$ He's asking about protonation of hydronium and deprotonation of hydroxide - you won't find these in basic lists. $\endgroup$ – Mithoron Jan 29 '15 at 18:46
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    $\begingroup$ Sadly. :'(. Let's start a probe. $\endgroup$ – M.A.R. ಠ_ಠ Jan 29 '15 at 18:50
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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...

suggests that at acidities of Hammet acidity function $H_o = -25$ to $-28$, protonated hydronium can be present.

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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 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).

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 including SCF calculations on $\ce{H4O^2+}$).

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    $\begingroup$ I think the ~picosecond lifetime of hydronium in pure water refers to lifetime in (OH-, H3O+) ion pairs, and the ion pairs don't count toward the 10^-7 M hydronium. On the occassion that the ion pair separates, the lifetime of hydronium is then 70 microseconds, and only these hydronium that are not part of the ion pairs are the 10^-7 M hydronium. See page 16 here: waterjournal.org/uploads/vol1/chaplin/WATER-Vol1-Chaplin.pdf $\endgroup$ – DavePhD Jan 30 '15 at 12:25
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    $\begingroup$ Ahhh, thanks Dave! I'll come back and fix that when I have time. $\endgroup$ – Fred Senese Jan 30 '15 at 13:00
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    $\begingroup$ Just be careful that all the water doesn't convert to polywater. ;-) All of you scalawags are probably too young to remember the media hooblah about that one. There were stories that if polywater were dropped into the oceans that it could catalyze the solidification of all the oceans. $\endgroup$ – MaxW Nov 6 '15 at 21:21

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