The author replies to Julian L. Roberts

My approach shows that [H₂A] = [A^2-] if the [H₂A] from hydrolysis of HA^- and the [A^2-] from dissociation of HA^- are neglected. Intuition will usually show when to suspect that these by-products are significant and that actual calculation is called for. The mass and charge balance approach that Roberts prefers shows after four lines of algebra that [H₂A] = [A^2-] if [H⁺] and [OH^-] are neglected. However, the chemical significance of [H⁺] and [OH^-] is not obvious except to those of us who recognize that they represent [A^2-]_dissociation and [H₂A]_hydrolysis, a recognition that is not to be expected of most students and that is avoided by my approach.

Should it seem desirable to calculate the quantitative effect of neglecting the dissociation and hydrolysis reactions HA^- H⁺ + A^2- and HA^- + H₂O H₂A + OH^-, it can be done directly as follows:

[A^2-]_{autoprotolysis} = [H₂A]_{autoprotolysis}

so

[A^2-]_total - [A^2-]_dissociation = [H₂A]_total - [H₂A]_hydrolysis.

Inserting the four equalities:

[A^2-]_total = K_a2[HA^-]/[H⁺]

[A^2-]_dissociation = [H⁺]

[H₂A]_total = [H⁺][HA^-]/K_a1

[H₂A]_hydrolysis = [OH^-] = K_w/[H⁺]

the result is

K_a2[HA^-]/[H⁺] - [H⁺] = [H⁺][HA^-]/K_a2 - K_w/[H⁺]

which may be rearranged to

[H⁺]² = {K_a1 (K_a2[HA^-] + K_w)}/{K_a1 + [HA^-]}

which is the traditional result of the mass and charge balance approach, but is derived more directly from the chemistry of the system. It can be used to show when stoichiometric approximations are inadequate but not when the neglect of activity coefficients is important.

Is it sound pedagogy to start with a simple relationship and consider its limitations later? That is what we do with the gas laws and with pH calculations that assume pH ≈ -log [H⁺] or even that pH is equivalent to -log [H⁺]. Roberts correctly points out the disadvantage that students apply simplistic equations with unwarranted confidence even after they become professors. I am chagrined to meet this criticism of my work when I have so often raised it against other people's. Pass the crow.

At the same time, I take warning that I must cease to use the term "real world". Clearly, Roberts's understanding of this phrase is different from mine, which is cultivated by two decades in industry before becoming an academic. I do not quarrel with his interpretation, but will avoid the difficulty in future.