Dynamical Approach to Multiequilibria Problems for Mixtures of Acids and their Conjugated Bases


Mathematical methods are described for the determination of steady-state concentrations of all species in multiequilibria systems consisting of several acids and their conjugated bases in aqueous solutions. The main example consists of a mixture of a diprotic acid H2A, a monoprotic acid HB, and their conjugate bases. The reaction equations lead to a system of autonomous ordinary differential equations for the species concentrations. The traditional equilibrium approach is briefly reviewed. Single variable polynomials are determined that are satisfied by the equilibrium proton concentrations. The remaining species are then given explicitly by rational functions of these proton concentrations, the equilibrium constants, and the initial concentrations of the other species. A quintic polynomial was derived to determine the equilibrium proton concentration for the example system. It is shown to reduce to a quartic polynomial in the absence of the second acid HB. An alternative dynamical approach to the equilibrium problem is described that involves the formulation of the kinetic rate equations for each species, which together constitute a nonlinear system of ordinary differential equations. The equilibrium concentrations are determined by evolving the initial concentrations via this dynamical system to their steady state. This dynamical approach is particularly attractive because it can easily be extended to determine equilibrium concentrations for arbitrarily large multiequilibria systems. With the equations provided here and some knowledge of computing software, the fast and accurate computation of equilibrium concentrations becomes feasible for the education of upper-division undergraduate and graduate students as well as for the study of research problems. This dynamical method also serves to introduce students to nonlinear dynamical systems, which are essential for the study of dynamic problems in chemistry, for example, oscillatory reactions.



Keywords and Phrases

Acids/Bases; Computer-Based Learning; Equilibrium; Graduate Education/Research; Interdisciplinary/Multidisciplinary; Mathematics/Symbolic Mathematics; Upper-Division Undergraduate

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Article - Journal

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