This question came up when answering Find the pH of an amphiprotic salt dissolved in water at room temperature. If we dissolve NaHA$\ce{NaHA}$ in water, the species $\ce{HA-}$ can act as acid or base:
$$\ce{H2A <=> AH- + H+}$$ $$\ce{HA- <=> A^2- + H+}$$$$ \begin{align} \ce{H2A &<=> AH- + H+}\\ \ce{HA- &<=> A^2- + H+} \end{align} $$
The pH$\mathrm{pH}$ of the solution can be estimated as
$$\mathrm{pH = \frac{pK_{a1} + pK_{a2}}{2}}$$$$\mathrm{pH} = \frac{\mathrm{p}K_\mathrm{a1} + \mathrm{p}K_\mathrm{a2}}{2}$$
My question is how accurate this estimate is, depending
- on concentration of the amphiprotic salt (as the concentration approaches zero, the pH$\mathrm{pH}$ should approach 7),
- on the average of the $\mathrm{pK_a}$$\mathrm{p}K_\mathrm{a}$ values (the closer these are to neutral, the smaller the difference between $\ce{[H2A]}$$[\ce{H2A}]$ and $\ce{[A^2-]}$$[\ce{A^2-}]$), and
- on the difference between the $\mathrm{pK_a}$$\mathrm{p}K_\mathrm{a}$ values (the bigger the difference, the lower the percentage of NaHA$\ce{NaHA}$ undergoing acid or base reactions).
So depending on these three variables, how accurate is the estimate? My guess is that high concentration, average of pKa$\mathrm{p}K_\mathrm{a}$ near neutral and difference between pKa$\mathrm{p}K_\mathrm{a}$ low would give good estimates. When any of these parameters a very different from those "ideal" conditions, the estimate probably gets worse.
