Systematic Treatment of Acid-Base Equilibria: An Unexpected Application of the Negative Root in Solving the pH of a Diluted Strong Acid or Base Solution

04 July 2022, Version 1
This content is a preprint and has not undergone peer review at the time of posting.


In undergraduate chemistry curricula, how to determine the concentration of hydrogen ions of a highly diluted strong acid or base solution is an important example to demonstrate the concept of the systematic treatment of chemical equilibria. For strong acid or base solutions below 10−6 M, to avoid excessive error in the resulting pH value, systematic treatment of two related chemical equilibria will be required. Such approach yields a quadratic equation with the concentration of hydrogen ions as the unknown. Solving such quadratic equation always leads to two roots with opposite signs: the positive root giving the value of the hydrogen ion concentration and the negative root was assumed to hold no significant value. This paper shows that the absolute value of this negative root is the hydrogen ion concentration from the autoprotolysis of water in such solution. Classroom application of this new approach demonstrated improved student learning outcome. Our application example can be an interesting case study in undergraduate curricula to improve students’ learning on the principle and application of the systematic treatment of equilibria.


analytical chemistry
physical chemistry
second-year undergraduate
upper-division undergraduate
inquiry-based learning
discovery learning
Acids and Bases
aqueous solution chemistry
systematic treatment of equilibria
quantitative analysis


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