Solubility Product

Precipitation reactions in qualitative inorganic analysis -- such as dilute $HCl$ precipitating $Pb^{2+}$ as sparingly soluble $PbCl_2$, or calcium oxalate precipitating in kidney stones -- are understood through the solubility equilibrium between an undissociated sparingly soluble salt and its constituent ions. For a general salt $X_mY_n(s) \rightleftharpoons mX^{n+}(aq)+nY^{m-}(aq)$, the equilibrium constant is $K=\dfrac{[X^{n+}]^m[Y^{m-}]^n}{[X_mY_n]}$; since the solid's concentration is itself constant in a heterogeneous equilibrium, it is absorbed to give the solubility product, $K_{sp}=[X^{n+}]^m[Y^{m-}]^n$ -- the product of the molar concentrations of the constituent ions, each raised to the power of its stoichiometric coefficient in the balanced equation. The solubility product decides whether a precipitate forms when solutions containing the constituent ions are mixed: computing the same expression using the actual (possibly non-equilibrium) concentrations present gives the ionic product, and comparing the two tells the outcome -- if the ionic product exceeds $K_{sp}$, the solution is supersaturated and precipitation occurs; if it is less than $K_{sp}$, the solution is unsaturated and no precipitation occurs; if the two are equal, the solution is exactly saturated and at equilibrium. …