Solution Mixing Calculations for Saturated AC–BC Solutions: A General Analytical Analysis

Abstract

A closed-form, analytical analysis of the physical and chemical processes that occur when a saturated solution of a slightly soluble ionic salt, AC, is titrated with a saturated solution of a slightly more soluble ionic salt, BC, is developed and applied to the two example systems considered by Schrier in the preceding paper (10.1021/acs.jchemed.0c01456J. Chem. Educ.2021). The analysis assumes ideal solutions and complete ionization of the dissolved AC and BC ionic compounds. A simple inequality involving the ratios of the respective solubility constants and volumes for AC and BC is derived. This inequality describes when and if a precipitate of the less soluble salt, AC, will occur during the titration. The chemical amount of AC precipitate, if any, and the ion-product Q are obtained as functions of the volume ratio, α. Both of these quantities exhibit maxima at some value of α which is determined in the analysis. The present analytic analysis is compared with the diagrammatic approach presented by Schrier. It is found that some aspects of the system are best addressed by one method. For different aspects, the second method is preferable. In general, the two approaches are found to be highly complementary. It is suggested that the study of this system can be a powerful pedagogical tool in the teaching of the principles involved in dilution, equilibrium, and coupled chemical reactions involving a common ion.

Solution Mixing Calculations as a Geometry, Linear Algebra, and Convex Analysis Problem

“…As there was none present, the amount must just remain zero. While all of these conclusions may be derived in the traditional way using algebra and calculus (see Raff's treatment in the next article of this issue 24 ), this illustrates how one can demonstrate the same ideas in an intuitive way using only a diagram and the geometric principles discussed in the article.…”

“…A closed-form analytical treatment of this general problem is presented by Raff in the next article of this issue. 24 The criterion for precipitation is determined by recalling that soluble A and C are in equilibrium with an insoluble product AC, described by the solubility equilibrium expression K 1 = [A][C]; rearranging this to the form [C] = K 1 /[A] results in an equilibrium curve, depicted as the blue line in Figure 6 for the specific case of K sp = 0.12. The A(aq) + C(aq) → AC(s) precipitation reaction corresponds to a vector pointed in the direction (−1, −1) in the ([A], [C]) concentrations space plane, as each unit of reaction consumes an equal amount of A and C. Consequently, any point above the equilibrium curve is supersaturated, and precipitation of solid AC will occur until equilibrium is reached.…”

“…A graphical approach allows us to answer the qualitative question of whether any mixture of the two solutions will result in precipitation of the less soluble salt, with a minimum of calculation and provides an opportunity to demonstrate the interplay of geometric and algebraic reasoning enabled by these types of diagrams. A closed-form analytical treatment of this general problem is presented by Raff in the next article of this issue …”