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A closed-form analytical expression for $\ce{[H3O+]}$ has been obtained foraqueous solutions of diprotic acids and its soluble salts. This formula allowsto calculate the pH of aqueous solutions of diprotic acids, their buffersolutions, and the titrations of these two by a strong base, from the values ofp$K_1$, p$K_2$, and the effective concentrations of the acid and the base,$\bar{C}_\mathrm{a}$ and $\bar{C}_\mathrm{b}$ respectively. It is shown that astrong base titration of an acid, or its buffer solutions, is always a linearpath in the $\bar{C}_\mathrm{a}$--$\bar{C}_\mathrm{b}$ plane, which allows asimple analysis of the pH stability of buffer solutions. The mathematicalanalysis of the equilibrium equations of the dissolution of a diprotic acid inwater and the physical constraints allowed to obtain two approximate equationsfor the diprotic acids. One of the approximations is useful for acids with$\mathrm{p}K_2-\mathrm{p}K_1\le\log_{10}4$, the other for acids with$\mathrm{p}K_2-\mathrm{p}K_1\le-\log_{10}4$.

Geometric-algebraic approach to aqueous solutions of diprotic acids and its buffer mixtures