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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3$\begingroup$ That's a good question! $\endgroup$– MithoronCommented Jan 29, 2015 at 17:34
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1$\begingroup$ Is what that this page says incorrect or incomplete? $\endgroup$– M.A.R.Commented Jan 29, 2015 at 18:40
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1$\begingroup$ Oh, wait. It's incomplete. $\endgroup$– M.A.R.Commented Jan 29, 2015 at 18:46
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1$\begingroup$ He's asking about protonation of hydronium and deprotonation of hydroxide - you won't find these in basic lists. $\endgroup$– MithoronCommented Jan 29, 2015 at 18:46
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1$\begingroup$ Sadly. :'(. Let's start a probe. $\endgroup$– M.A.R.Commented Jan 29, 2015 at 18:50
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2 Answers
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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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2$\begingroup$ 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 $\endgroup$– DavePhDCommented Jan 30, 2015 at 12:25
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1$\begingroup$ Ahhh, thanks Dave! I'll come back and fix that when I have time. $\endgroup$ Commented Jan 30, 2015 at 13:00
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1$\begingroup$ 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. $\endgroup$– MaxWCommented Nov 6, 2015 at 21:21
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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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1$\begingroup$ Just stepped in to say "thanks", though this comment is "too chatty"! $\endgroup$– M.A.R.Commented Jan 29, 2015 at 20:24
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1$\begingroup$ I must admit that I expected some more extreme values. $\endgroup$– EJCCommented Jan 30, 2015 at 19:45
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1$\begingroup$ some sources give even less extreme value for hydroxide pKa m.docente.unife.it/alberto.cavazzini/analitica-1-e-lab-2014/… $\endgroup$– DavePhDCommented Jan 30, 2015 at 20:13