Systematic Approach to Solving Equilibrium Problems
Calculating the solubility of Pb(IO3)2 in a solution of Pb(NO3)2 was more com- plicated than calculating its solubility in distilled water. The necessary calcula- tions, however, were still relatively easy to organize, and the assumption used to simplify the problem was fairly obvious. This problem was reasonably straight- forward because it involved only a single equilibrium reaction, the solubility of Pb(IO3)2. Calculating the equilibrium composition of a system with multiple equilibrium reactions can become quite complicated. In this section we will learn how to use a systematic approach to setting up and solving equilibrium problems.
As its name implies, a systematic approach involves a series of steps:
· Write all relevant equilibrium reactions and their equilibrium constant expressions.
· Count the number of species whose concentrations appear in the equilibrium constant expressions; these are your unknowns. If the number of unknowns equals the number of equilibrium constant expressions, then you have enough information to solve the problem. If not, additional equations based on the conservation of mass and charge must be written. Continue to add equations until you have the same number of equations as you have unknowns.
· Decide how accurate your final answer needs to be. This decision will influence your evaluation of any assumptions you use to simplify the problem.
· Combine your equations to solve for one unknown (usually the one you are most interested in knowing). Whenever possible, simplify the algebra by making appropriate assumptions.
· When you obtain your final answer, be sure to check your assumptions. If any of your assumptions prove invalid, then return to the previous step and continue solving. The problem is complete when you have an answer that does not violate any of your assumptions.
Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A–, must equal the weak acid’s initial concentration, CHA.*
The second type of equation is a charge balance equation. A charge balance equation is a statement of solution electroneutrality.
Total positive charge from cations = total negative charge from anions Mathematically, the charge balance expression is expressed as
where [Mz+]i and [Az–]j are, respectively, the concentrations of the ith cation and the jth anion, and (z+)i and (z–)j are the charges of the ith cation and the jth anion. Note that the concentration terms are multiplied by the absolute values of each ion’s charge, since electroneutrality is a conservation of charge, not con- centration. Every ion in solution, even those not involved in any equilibrium reactions, must be included in the charge balance equation. The charge balance equation for an aqueous solution of Ca(NO3)2 is
Note that the concentration of Ca2+ is multiplied by 2, and that the concentrations of H3O+ and OH– are also included. Charge balance equations must be written carefully since every ion in solution must be included. This presents a problem when the concentration of one ion in solution is held constant by a reagent of un- specified composition. For example, in many situations pH is held constant using a buffer. If the composition of the buffer is not specified, then a charge balance equa- tion cannot be written.
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