A More Complex Problem: The Common Ion Effect
Calculating the solubility of Pb(IO3)2 in distilled water is a straightforward prob- lem since the dissolution of the solid is the only source of Pb2+ or IO3–. How is the solubility of Pb(IO3)2 affected if we add Pb(IO3)2 to a solution of 0.10 M Pb(NO3)2? Before we set up and solve the problem algebraically, think about the chemistry occurring in this system, and decide whether the solubility of Pb(IO3)2 will increase, decrease, or remain the same. This is a good habit to develop. Knowing what answers are reasonable will help you spot errors in your calcula- tions and give you more confidence that your solution to a problem is correct.
We begin by setting up a table to help us keep track of the concentrations of P b2+ and IO3– in this system.
Substituting the equilibrium concentrations into the solubility product expression (equation 6.33)
(0.10 + x)(2x)2 = 2.5 x 10–13
and multiplying out the terms on the left leaves us with
4x3 + 0.40x2 = 2.5 x 10–13
This is a more difficult equation to solve than that for the solubility of Pb(IO3)2 in dis- tilled water, and its solution is not immediately obvious. A rigorous solution to equa- tion 6.34 can be found using available computer software packages and spreadsheets.
How might we solve equation 6.34 if we do not have access to a computer? One possibility is that we can apply our understanding of chemistry to simplify the algebra. From Le Châtelier’s principle, we expect that the large initial concentration of Pb2+ will significantly decrease the solubility of Pb(IO3)2. In this case we can reasonably expect the equilibrium concentration of Pb2+ to be very close to its initial concentration; thus, the following approximation for the equilibrium concentration of Pb2+ seems reasonable
[Pb2+] = 0.10 + x = 0.10 M
Substituting into equation 6.34
Before accepting this answer, we check to see if our approximation was reasonable. In this case the approximation 0.10 + x = 0.10 seems reasonable since the difference between the two values is negligible. The equilibrium concentrations of Pb2+ and IO3–, therefore, are
The solubility of Pb(IO3)2 is equal to the additional concentration of Pb2+ in solu- tion, or 7.9 x 10–7 mol/L. As expected, the solubility of Pb(IO3)2 decreases in the presence of a solution that already contains one of its ions. This is known as the common ion effect.
As outlined in the following example, the process of making and evaluating ap- proximations can be extended if the first approximation leads to an unacceptably large error.
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