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Considering a general amphiprotic substance NaHA. I know that the maximum concentration of the specie $HA^-$ is found at $pH = \dfrac{pKa1+pKa2}{2}$. Why? How i can confirm this statment?

The equations are:

$$ \ce{HA^- + H_2O <=> A^{2-} + H_3O^{+}} \quad Ka2$$ and $$ \ce{HA^- + H_2O <=> H_2A + OH^{-}} \quad Kb=\dfrac{Kw}{Ka1}$$

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Let's take an aqueous solution of a salt $\ce{NaHA}$ with the initial concentration $C$ when added to water. It will completely dissociate according to the eaquation: $\ce{NaHA(s) \rightarrow Na^+ +HA^-}$.

$\ce{HA^-}$ will participate in three equilibria:

$\ce{2HA^- \leftrightarrows H2A +A^{2-}\quad \quad \quad }$ ${K_1^0=K_{A2}/K_{A1}}$

$\ce{HA^- +H2O\leftrightarrows H3O^+ +A^{2-}\quad }$ $K_2^0=K_{A2}$

$\ce{HA^- +H2O\leftrightarrows OH^- +H_2A^\quad }$ $K_3^0=K_{B1}={K_w}/{K_{A1}}$

In most cases, $K_1^0$ is far bigger than $K_2^0$ and $K_3^0$. So, the first equilibrium is the preponderant reaction, and this reaction will impose the pH of the solution.

Let's now calculate the product $K_{A2}\times K_{A1}$:

$K_{A2}\times K_{A1}= \frac{[\ce{}A^{2-}].[\ce{H3O+}]}{\ce{[HA^-]}}.\frac{[\ce{}HA^{-}].[\ce{H3O+}]}{\ce{[H_2A]}}$

According to the stoichiometry of the preponderant reaction, we have $\ce{[A^{2-}]=[H_2A^]}$. So the product $K_{A2}\times K_{A1}= \ce{[H3O+}]^2}$ . i.e. $\ce{pH}=0.5(\ce{p}K_{A2} +\ce{p} K_{A1})$

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  • $\begingroup$ Thanks. this is the only way to solve this problem. If you try to make the mass and charge balances and finding the roots of the derivative $\dfrac{d[HA^-]}{d[H_3O^+]}$, this don't work because your roots are not real. $\endgroup$ – Jorge Oct 25 '14 at 14:40
  • $\begingroup$ It's interesant to see (if i omit the first reaction) that: $\endgroup$ – Jorge Oct 25 '14 at 16:19
  • $\begingroup$ ${[HA^-]} = - \dfrac{K_{a1} ([H_3O^+]^2-K_w)}{[H_3O^+]^2- K_{a1}K_{a2}}$ is not defined at $[H_3O^+]=\sqrt{K_{a1}K_{a2}}$ $\endgroup$ – Jorge Oct 25 '14 at 16:22
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The usual answer for this kind of problem is to write down all species ($HA, A^-, H_2A^+, H_3O^+, OH^-$) and all equations for the various equilibria (don't forget autoionization!) and then find the maximum concentration of $HA$.

Try it! Try also various approximations. The exercise will increase your understanding of that kind of equilibrium problems, and of buffers, titration curves, acid-base reactions in general.

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