Solubility Product and Common Ion Effect

Many ionic salts — such as $\text{AgCl}$, $\text{BaSO}_4$ and $\text{CaCO}_3$ — are described as

"insoluble," but this is only a matter of degree: a very small amount of every ionic solid does

dissolve, establishing a genuine equilibrium between the undissolved solid and its dissolved ions,

$\text{AgCl}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{Cl}^-(aq)$. Applying the equilibrium law,

and — exactly as for any heterogeneous equilibrium — omitting the constant concentration of the pure

solid, gives the solubility product,

More generally, for a salt $\text{A}_x\text{B}_y(s) \rightleftharpoons x\text{A}^{n+} + y\text{B}^{m-}$,

$K_{sp} = [\text{A}^{n+}]^x[\text{B}^{m-}]^y$. $K_{sp}$ is a true equilibrium constant, fixed at a given

temperature, and it can be measured directly from the salt's molar solubility, $s$ (the number of

moles of the salt that dissolve per litre of saturated solution, in pure water): for a 1:1 salt like

$\text{AgCl}$, dissolving $s$ mol of $\text{AgCl}$ produces $s$ mol of $\text{Ag}^+$ and $s$ mol of

$\text{Cl}^-$, so $K_{sp} = s \times s = s^2$; for a salt of a different stoichiometry the powers of $s$

must be adjusted to match its own dissociation equation.

The common ion effect on solubility. If a sparingly soluble salt is dissolved not in pure water

but in a solution that already contains one of its constituent ions (from some other, fully soluble

source), that pre-existing "common ion" pushes the dissolution equilibrium backward by Le Chatelier's

principle, and the salt's solubility is markedly reduced compared with its solubility in pure water.

For instance, dissolving $\text{AgCl}$ in a solution of $\text{NaCl}$ (which itself supplies a large,

independent concentration of $\text{Cl}^-$) suppresses how much additional $\text{Ag}^+$ (and hence how

much $\text{AgCl}$) can dissolve, since the product $[\text{Ag}^+][\text{Cl}^-]$ is still pinned at the

fixed value $K_{sp}$, and $[\text{Cl}^-]$ is already elevated by the added $\text{NaCl}$.

Predicting whether a precipitate will form. The same $K_{sp}$ expression, evaluated using

whatever ion concentrations actually exist in a solution at a given instant (not necessarily at

equilibrium), is called the ionic product, $Q_{sp}$. Comparing $Q_{sp}$ with the true $K_{sp}$

predicts what will happen: if $Q_{sp} > K_{sp}$, the solution is momentarily supersaturated and a …