Solving Equilibrium Problems for Complex Systems

Ladder diagrams are a useful tool for evaluating chemical reactivity, usually providing a reasonable approximation of a chemical system s composition at equilibrium. When we need a more exact quantitative description of the equilibrium condition, a ladder diagram may not be sufficient. In this case we can find an algebraic solution. Perhaps you recall solving equilibrium problems in your earlier coursework in chemistry. In this section we will learn how to set up and solve equilibrium problems. We will start with a simple problem and work toward more complex ones. 

So far, we have learned that if we know the chemical equilibria involved in a system, we can write a corresponding system of equations that allows us to solve for the concentrations of all species in the system. Although the systematic method gives us the means to solve equilibrium problems of great complexity, it is sometimes tedious and time consuming, particularly when a system must be solved for various sets of experimental conditions. For example, if we wish to find the solubility of silver chloride as a function of the concentration of added chloride, the system of five equations and five unknowns must be solved repetitively for each different concentration of chloride (see Example 11-9). 

Several programs have been written specifically for a very restricted class of equilibrium only problems. The Pit Method of Sillen and Warnquist has been widely used to solve for equilibrium constants in inorganic systems that have one or more simultaneous reversible reactions. DeLand uses goal-seeking routines to facilitate the matching of data, but free energy data for all reactants is required. Bos and Meershoek 24) have written a PL/1 program which uses the Newton-Raphson iteration to compute equilibrium constants in complex systems. 

The relative ease of solving the system of non-linear equations for rather complex equilibrium problems, as indicated by the shortness of the function NewtonRaphson. m and by the inconsequentiality of poor initial guesses, is misleading. As we will see shortly, this statement is particularly pertinent to cases of general systems of m equations with m parameters. Solving systems of equations is a common task and we give a short introduction. To start with, we investigate the simple case of one equation with one parameter. 

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