The Solubility Product Constant: Ksp
The solubility product constant, Ksp, is the equilibrium constant for the dissolving of a sparingly soluble ionic solid, equal to the product of the ion concentrations raised to the powers of their coefficients: for AgCl, Ksp = [Ag+][Cl−] = 1.8 × 10−10 at 25 °C. In brief: a small Ksp means that very little of the salt dissolves, and for PbI2 a molar solubility of 1.35 × 10−3 M corresponds to Ksp = 4s3 = 9.8 × 10−9.
No salt is completely insoluble: even “insoluble” compounds such as silver chloride release a few ions into solution. The solubility product gives a quantitative measure of this, and it is the basis for predicting precipitation, for explaining the effect of a common ion and for the design of analytical separations. This article shows how to write Ksp expressions, gives a table of values and explains how Ksp is calculated from the solubility.
How do you write a Ksp expression?
When an excess of a slightly soluble salt is in contact with water, an equilibrium is established between the solid and its dissolved ions. For a salt AmBn:
AmBn(s) ⇌ mAn+(aq) + nBm−(aq)
Ksp = [An+]m [Bm−]n
The solid is left out, since its concentration is constant. Only ions that are in solution appear in the expression. For example:
- AgCl(s) ⇌ Ag+ + Cl−: Ksp = [Ag+][Cl−].
- PbI2(s) ⇌ Pb2+ + 2I−: Ksp = [Pb2+][I−]2.
- Ag2CrO4(s) ⇌ 2Ag+ + CrO42−: Ksp = [Ag+]2[CrO42−].
- Ca3(PO4)2(s) ⇌ 3Ca2+ + 2PO43−: Ksp = [Ca2+]3[PO43−]2.
Ksp values of common salts
| Salt | Dissolution | Ksp expression | Ksp |
|---|---|---|---|
| AgCl | Ag+ + Cl− | [Ag+][Cl−] | 1.8 × 10−10 |
| AgBr | Ag+ + Br− | [Ag+][Br−] | 5.4 × 10−13 |
| AgI | Ag+ + I− | [Ag+][I−] | 8.5 × 10−17 |
| BaSO4 | Ba2+ + SO42− | [Ba2+][SO42−] | 1.1 × 10−10 |
| CaCO3 | Ca2+ + CO32− | [Ca2+][CO32−] | 3.4 × 10−9 |
| CaF2 | Ca2+ + 2F− | [Ca2+][F−]2 | 3.9 × 10−11 |
| PbI2 | Pb2+ + 2I− | [Pb2+][I−]2 | 9.8 × 10−9 |
| Mg(OH)2 | Mg2+ + 2OH− | [Mg2+][OH−]2 | 5.6 × 10−12 |
| Ag2CrO4 | 2Ag+ + CrO42− | [Ag+]2[CrO42−] | 1.1 × 10−12 |
Values from different reference books vary by several percent, and sometimes more. For exact work, use the table prescribed by your course or laboratory.
How do you calculate Ksp from solubility?
Let s be the molar solubility (mol of salt that dissolves per liter). Each formula unit gives its ions in the proportions of the formula.
Example 1: lead(II) iodide
The solubility of PbI2 at 25 °C is 1.35 × 10−3 mol/L. Then [Pb2+] = s = 1.35 × 10−3 M and [I−] = 2s = 2.70 × 10−3 M. Ksp = s(2s)2 = 4s3 = 4 × (1.35 × 10−3)3 = 4 × 2.46 × 10−9 = 9.8 × 10−9.
Example 2: barium sulfate from a mass solubility
A saturated solution of BaSO4 contains 2.4 × 10−3 g/L. The molar mass is 233.4 g/mol, so s = 2.4 × 10−3 / 233.4 = 1.03 × 10−5 mol/L. Then [Ba2+] = [SO42−] = s and Ksp = s2 = 1.1 × 10−10.
Example 3: silver chromate
If s = 6.5 × 10−5 M for Ag2CrO4, then [Ag+] = 2s = 1.3 × 10−4 M and [CrO42−] = s = 6.5 × 10−5 M, so Ksp = (1.3 × 10−4)2 × 6.5 × 10−5 = 1.1 × 10−12.
Can you compare salts by Ksp?
Only if the salts have the same formula type (the same number of ions per formula unit). For two salts of the type AB, such as AgCl and AgBr, the one with the smaller Ksp is less soluble. For salts of different types, the molar solubility must be calculated. Silver chromate (Ksp = 1.1 × 10−12) has a smaller Ksp than silver chloride (1.8 × 10−10), yet it is more soluble: 6.5 × 10−5 M against 1.3 × 10−5 M.
What affects Ksp?
Ksp depends on the temperature, like any equilibrium constant. For most salts, dissolving is endothermic, and Ksp increases with temperature. The solubility is changed by the presence of other ions (the common-ion effect), by the pH for salts of weak acids and bases and by the formation of complex ions. At high ionic strengths, the activities of the ions differ from their concentrations, so the simple expression is less accurate. The concentration-based Ksp in this article is accurate for very dilute solutions of sparingly soluble salts.
Why does Ksp matter?
- Water treatment: softening removes Ca2+ and Mg2+ as CaCO3 and Mg(OH)2.
- Medicine: kidney stones are often calcium oxalate, which precipitates when its ion product exceeds Ksp. A barium sulfate “meal” is used for X-ray imaging of the digestive tract since BaSO4 is so insoluble that the toxic Ba2+ does not dissolve in appreciable amounts.
- Teeth: enamel, made of hydroxyapatite, Ca5(PO4)3OH, dissolves in acid, and fluoride converts it to the less soluble fluorapatite.
- Geology and environment: the formation of caves and stalactites from CaCO3, and the mobility of metal ions in soil.
- Analysis: precipitation is used for the gravimetric analysis of chloride and sulfate.
Who introduced the solubility product?
Walther Nernst developed the concept in 1889, as part of his theory of electrolytic solutions: he showed that the product of the ion concentrations in a saturated solution is a constant, which he called the solubility product. The idea was then applied by Wilhelm Ostwald and others to analytical chemistry, and it explained the earlier empirical observations on the precipitation of salts. Nernst received the Nobel Prize in Chemistry in 1920 for his work in thermochemistry.
Which mistakes are common?
- Including the solid in the expression: it is left out.
- Forgetting the exponents: the coefficients become powers.
- Comparing Ksp of different types: compare molar solubilities instead.
- Mixing up Ksp and solubility: Ksp is a constant, solubility changes with the conditions.
- Using Ksp for soluble salts: it is useful only for sparingly soluble compounds.