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

TL;DR

This paper presents a closed-form analytical expression for calculating the pH of aqueous solutions of diprotic acids, their buffers, and titrations, using geometric-algebraic methods to simplify analysis and understand buffer stability.

Contribution

It introduces a novel analytical formula for pH calculation in diprotic acid solutions and buffers, along with approximate equations for different acid strength ranges.

Findings

The formula enables direct pH computation from acid/base concentrations and pK values.

Titration paths are linear in the concentration plane, simplifying buffer stability analysis.

Two approximate equations are derived for different pK difference ranges.

Abstract

A closed-form analytical expression for $\ce{[H3O+]}$ has been obtained for aqueous solutions of diprotic acids and its soluble salts. This formula allows to calculate the pH of aqueous solutions of diprotic acids, their buffer solutions, and the titrations of these two by a strong base, from the values of p$K_1$, 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 a strong base titration of an acid, or its buffer solutions, is always a linear path in the $\bar{C}_\mathrm{a}$--$\bar{C}_\mathrm{b}$ plane, which allows a simple analysis of the pH stability of buffer solutions. The mathematical analysis of the equilibrium equations of the dissolution of a diprotic acid in water and the physical constraints allowed to obtain two approximate equations for 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 $\mathrm{p}K_2-\mathrm{p}K_1\le-\log_{10}4$.