An Unexpected Application of the Negative Root Arising from the Use of the Systematic Treatment of Equilibrium in Calculating pH Contribution from the Autoprotolysis of Water in Highly Diluted Strong Acid or Base Solutions

Abstract In the undergraduate analytical chemistry curriculum, determining the concentration of hydronium ions (H+) of a highly diluted strong acid or base solution (below 10–6 M) is an important example to demonstrate the concept of the systematic treatment of chemical equilibria using applicable charge balance, mass balance, and chemical equilibrium constants. In this case, to accurately describe the pH of the solution, the autoprotolysis of water must be considered in addition to the dissociation/protonation of the strong acid/base. Such systematic treatment yields a quadratic equation with [H+], the concentration of H+, as its unknown. Solving the quadratic equation always yields a positive root equaling the total [H+] of the solution and a negative one with no significant value. This article shows, based on the mass balance equations in such a chemical system, that the absolute value of this negative root is the [H+] from the autoprotolysis of water in the system. Our findings could be an interesting critical thinking activity in undergraduate curricula to assist students in their understanding of the principle and application of the systematic treatment of equilibria and practice critical thinking.