Chemistry » Equilibria of Other Reaction Classes » Precipitation and Dissolution

# The Solubility Product

## The Solubility Product

Silver chloride is what’s known as a sparingly soluble ionic solid (see the figure below). Recall from the solubility rules in an earlier tutorial that halides of Ag+ are not normally soluble. However, when we add an excess of solid AgCl to water, it dissolves to a small extent and produces a mixture consisting of a very dilute solution of Ag+ and Cl ions in equilibrium with undissolved silver chloride:

$$\text{AgCl}(s)\underset{\text{precipitation}}{\overset{\text{dissolution}}{⇌}}{\text{Ag}}^{\text{+}}(aq)+{\text{Cl}}^{\text{−}}(aq)$$

This equilibrium, like other equilibria, is dynamic; some of the solid AgCl continues to dissolve, but at the same time, Ag+ and Cl ions in the solution combine to produce an equal amount of the solid. At equilibrium, the opposing processes have equal rates.

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 the equilibrium between a slightly soluble ionic solid and a solution of its ions is called the solubility product (Ksp) of the solid. Recall from the tutorial on solutions and colloids that we use an ion’s concentration as an approximation of its activity in a dilute solution. For silver chloride, at equilibrium:

$$\text{AgCl}(s)⇌{\text{Ag}}^{\text{+}}(aq)+{\text{Cl}}^{\text{−}}(aq)\phantom{\rule{4em}{0ex}}{K}_{\text{sp}}=[{\text{Ag}}^{\text{+}}(aq)][{\text{Cl}}^{\text{−}}(aq)]$$

Note that the Ksp expression does not contain a term in the denominator for the concentration of the reactant, AgCl. According to the guidelines for deriving mass-action expressions described in an earlier tutorial on equilibrium, only gases and solutes are represented. Solids and liquids are assigned concentration values of one and thus do not appear in equilibrium constant expressions; therefore, [AgCl] does not appear in the expression for Ksp.

Some common solubility products are listed in the table below according to their Ksp values, whereas a more extensive compilation of solubility products appears in this appendix. Each of these equilibrium constants is much smaller than 1 because the compounds listed are only slightly soluble. A small Ksp represents a system in which the equilibrium lies to the left, so that relatively few hydrated ions would be present in a saturated solution.

Common Solubility Products by Decreasing Equilibrium Constants
SubstanceKsp at 25 °C
CuCl1.2 $$×$$ 10–6
CuBr6.27 $$×$$ 10–9
AgI1.5 $$×$$ 10–16
PbS7 $$×$$ 10–29
Al(OH)32 $$×$$ 10–32
Fe(OH)34 $$×$$ 10–38

## Example

### Writing Equations and Solubility Products

Write the ionic equation for the dissolution 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

(Hint: When determining how to break (d) and (e) up into ions, refer to the list of polyatomic ions in the section on chemical nomenclature.)

### Solution

(a)

$$\text{AgI}(s)⇌{\text{Ag}}^{\text{+}}(aq)+{\text{I}}^{\text{−}}(aq)\phantom{\rule{4em}{0ex}}{K}_{\text{sp}}={[\text{Ag}}^{\text{+}}][{\text{I}}^{\text{−}}]$$

(b)

$${\text{CaCO}}_{3}(s)⇌{\text{Ca}}^{\text{2+}}(aq)+{\text{CO}}_{3}{}^{\text{2−}}(aq)\phantom{\rule{4em}{0ex}}{K}_{\text{sp}}=[{\text{Ca}}^{\text{2+}}{][\text{CO}}_{3}{}^{\text{2−}}]$$

(c)

$${\text{Mg}(\text{OH})}_{2}(s)⇌{\text{Mg}}^{\text{2+}}(aq)+{\text{2OH}}^{\text{−}}(aq)\phantom{\rule{4em}{0ex}}{K}_{\text{sp}}=[{\text{Mg}}^{\text{2+}}]{[{\text{OH}}^{\text{−}}]}^{2}$$

(d)

$$\text{Mg}({\text{NH}}_{\text{4}}){\text{PO}}_{\text{4}}(s)⇌{\text{Mg}}^{\text{2+}}(aq)+{\text{NH}}_{4}{}^{\text{+}}(aq)+{\text{PO}}_{4}{}^{\text{3−}}(aq)\phantom{\rule{4em}{0ex}}{K}_{\text{sp}}=[{\text{Mg}}^{\text{2+}}{][\text{NH}}_{4}{}^{\text{+}}]{[\text{PO}}_{4}{}^{\text{3−}}]$$

(e)

$${\text{Ca}}_{5}({\text{PO}}_{4})3\text{OH}(s)⇌{\text{5Ca}}^{\text{2+}}(aq)+{\text{3PO}}_{4}{}^{\text{3−}}(aq)+{\text{OH}}^{\text{−}}(aq)\phantom{\rule{4em}{0ex}}{K}_{\text{sp}}={{[\text{Ca}}^{\text{2+}}]}^{5}[\text{P}{\text{O}}_{4}{}^{\text{3−}}{]}^{3}[{\text{OH}}^{\text{−}}]$$

Now we will extend the discussion of Ksp and show how the solubility product is determined from the solubility of its ions, as well as how Ksp can be used to determine the molar solubility of a substance.

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