The Solubility Product
Recall from the chapter on solutions that the solubility of a substance can vary from essentially zero (insoluble or sparingly soluble) to infinity (miscible). A solute with finite solubility can yield a saturated solution when it is added to a solvent in an amount exceeding its solubility, resulting in a heterogeneous mixture of the saturated solution and the excess, undissolved solute. For example, a saturated solution of silver chloride is one in which the equilibrium shown below has been established. $$AgCl(s)\; \overset{dissolution}{\underset{precipitation}\rightleftharpoons} \; Ag^+ (aq) + Cl^- (aq) $$
In this solution, an excess of solid AgCl dissolves and dissociates to produce aqueous Ag+ and Cl– ions at the same rate that these aqueous ions combine and precipitate to form solid AgCl, following the reaction above. Because silver chloride is a sparingly soluble salt, the equilibrium concentration of its dissolved ions in the solution is relatively low.
Silver chloride is a sparingly soluble ionic solid. When it is added to water, it dissolves slightly and produces a mixture consisting of a very dilute solution of Ag+ and Cl– ions in equilibrium with undissolved silver chloride.
The equilibrium constant for solubility equilibria such as this one is called the solubility product constant, Ksp. In this case.
$$AgCl(s)⇌Ag^+(aq)+Cl^-(aq)\qquad K_{sp}=[Ag^+(aq)][Cl^-(aq)]$$
Recall that only gases and solutes are represented in equilibrium constant expressions, so the Ksp does not include a term for the undissolved AgCl. A listing of solubility product constants for several sparingly soluble compounds is provided in an Appendix.
A reaction described by the solubility product Ksp will always have the form: $$\textbf{solid} \rightleftharpoons \textbf{aqueous ions}$$
Writing Equations and Solubility Products
Write the dissolution equation and the solubility product expression for each of the following slightly soluble ionic compounds:
(a) AgI, silver iodide, a solid with antiseptic properties
(b) CaCO3, calcium carbonate, the active ingredient in many over-the-counter chewable antacids
(c) Mg(OH)2, magnesium hydroxide, the active ingredient in Milk of Magnesia
(d) Mg(NH4)PO4, magnesium ammonium phosphate, an essentially insoluble substance used in tests for magnesium
(e) Ca5(PO4)3OH, the mineral apatite, a source of phosphate for fertilizers
Solution
(a) $AgI(s)⇌Ag^+(aq)+I^-(aq)$ $K_{sp}=[Ag^+][I^-]$
(b) $CaCO_3(s)⇌Ca^{2+}(aq)+CO_3^{2-}(aq)$ $K_{sp}=[Ca^{2+}][CO_3^{2-}]$
(c) $Mg(OH)_2(s)⇌Mg^{2+}(aq)+2OH^-(aq)$ $K_{sp}=[Mg^{2+}][OH^-]^2$
(d) $Mg(NH_4)PO_4(s)⇌Mg^{2+}(aq)+NH_4^+(aq)+PO_4^{3-}(aq)$ $K_{sp}=[Mg^{2+}][NH_4^+][PO_4^{3-}]$
(e) $Ca_5(PO_4)_3OH(s)⇌5Ca^{2+}(aq)+3PO_4^{3-}(aq)+OH^-(aq)$ $K_{sp}=[Ca^{2+}]^5[PO_4^{3-}]^3[OH^-]$
Check Your LearningWrite the dissolution equation and the solubility product for each of the following slightly soluble compounds:
(a) BaSO4
(b) Ag2SO4
(c) Al(OH)3
(d) Pb(OH)Cl
Answer:
(a) $BaSO_4(s)⇌Ba^{2+}(aq)+SO_4^{2-}(aq)$ $K_{sp}=[Ba^{2+}][SO_4^{2-}]$;
(b) $Ag_2SO_4(s)⇌2Ag^+(aq)+SO_4^{2-}(aq)$ $K_{sp}=[Ag^+]^2[SO_4^{2-}]$;
(c) $Al(OH)_3(s)⇌Al^{3+}(aq)+3OH^-(aq)$ $K_{sp}=[Al^{3+}][OH^-]^3$;
(d) $Pb(OH)Cl(s)⇌Pb^{2+}(aq)+OH^-(aq)+Cl^-(aq)$ $K_{sp}=[Pb^{2+}][OH^-][Cl^-]$