Solubility and precipitation equilibria

Kidney stones, the scale inside a kettle, the white crust on a showerhead and the stalactites of a limestone cave all form when an ionic solid precipitates from solution. Barium sulfate, swallowed as a suspension, lets radiologists image the digestive tract even though barium ions are toxic, because so little of it dissolves. Mining companies precipitate metals out of process water before releasing it. All of these depend on solubility equilibria, the equilibria between an ionic solid and its dissolved ions. CHEM-101 used simple solubility rules to predict whether a precipitate forms; this lesson makes those predictions quantitative.

The solubility product

When a sparingly soluble salt is placed in water, a small amount dissolves until the solution is saturated. At that point the dissolved ions are in dynamic equilibrium with the solid. For silver chloride,

The equilibrium constant is called the solubility product, K_sp. As for any heterogeneous equilibrium (lesson 7), the solid is left out. For a salt with more than one ion of a kind, each concentration is raised to the power of its coefficient: for calcium fluoride, CaF₂(s) ⇌ Ca²⁺ + 2 F⁻, K_sp = [Ca²⁺][F⁻]².

From solubility to K_sp and back

The molar solubility, s, is the number of moles of a salt that dissolve per litre of saturated solution. The relation between s and K_sp depends on the formula. For a 1 : 1 salt like AgCl, each mole that dissolves gives s of each ion, so K_sp = s². For a 1 : 2 salt like CaF₂, it gives s of Ca²⁺ and 2s of F⁻, so K_sp = s(2s)² = 4s³.

# Molar solubility from Ksp for salts of formula MxAy:  Ksp = (x s)^x (y s)^y
salts = {"AgCl": (1, 1, 1.8e-10), "CaF2": (1, 2, 3.9e-11), "Ag2CrO4": (2, 1, 1.1e-12),
         "BaSO4": (1, 1, 1.1e-10), "Ca3(PO4)2": (3, 2, 2.1e-33), "Fe(OH)3": (1, 3, 2.8e-39)}
for name, (x, y, Ksp) in salts.items():
    s = (Ksp / (x ** x * y ** y)) ** (1 / (x + y))
    print(f"{name:10s} Ksp = {Ksp:.1e}  molar solubility = {s:.2e} M")

AgCl       Ksp = 1.8e-10  molar solubility = 1.34e-05 M
CaF2       Ksp = 3.9e-11  molar solubility = 2.14e-04 M
Ag2CrO4    Ksp = 1.1e-12  molar solubility = 6.50e-05 M
BaSO4      Ksp = 1.1e-10  molar solubility = 1.05e-05 M
Ca3(PO4)2  Ksp = 2.1e-33  molar solubility = 1.14e-07 M
Fe(OH)3    Ksp = 2.8e-39  molar solubility = 1.01e-10 M

Will a precipitate form?

The ion product, Q, has the same form as K_sp but uses the actual concentrations in a solution. Comparing it with K_sp predicts precipitation, exactly as Q and K predicted direction in lesson 7. If Q < K_sp, the solution is unsaturated and no precipitate forms (more solid would dissolve). If Q = K_sp, the solution is saturated. If Q > K_sp, the solution is supersaturated and a precipitate forms until Q falls to K_sp. Supersaturated solutions can persist for a while if there is nothing for crystals to grow on, which is why seed crystals are added in industrial crystallization.

The common-ion effect

A salt is less soluble in a solution that already contains one of its ions. Dissolving silver chloride in a solution of sodium chloride, for instance, means the chloride concentration is already high; by Le Châtelier's principle, the equilibrium AgCl ⇌ Ag⁺ + Cl⁻ shifts left, and less silver chloride dissolves. This common-ion effect is used to precipitate ions as completely as possible and to control the solubility of medicines and minerals.

At very high chloride concentrations the trend reverses slightly, because silver forms soluble complex ions such as AgCl₂⁻. Complex-ion formation is an equilibrium too, and it is how ammonia dissolves silver chloride and how cyanide extracts gold from ore, a process used, under strict regulation, at gold mines in Ontario, Quebec, Nevada and elsewhere.

pH and solubility

Salts whose anions are bases dissolve better in acid. Hydroxides are the clearest case: in Mg(OH)₂(s) ⇌ Mg²⁺ + 2 OH⁻, adding acid removes hydroxide ions as water, pulling the equilibrium to the right. That is how milk of magnesia neutralizes stomach acid. Carbonates, phosphates, sulfides and fluorides behave similarly, because their anions combine with H⁺. Tooth enamel, mostly hydroxyapatite, Ca₅(PO₄)₃OH, dissolves when bacteria in plaque produce acid; fluoride converts part of it to the less soluble and more acid-resistant fluorapatite, which is why fluoride toothpaste and community water fluoridation reduce tooth decay. Fluoridation is decided locally in both countries. Health Canada's recommended optimal level is 0.7 mg/L, the value the US Public Health Service also set in 2015; in both countries some communities have stopped fluoridating, and in the US some states have restricted it and federal guidance has been under review since 2025, so check current policy where you live.

# Solubility of Mg(OH)2 (Ksp = 5.6e-12) at different pH values
Ksp = 5.6e-12
for pH in (12.0, 11.0, 10.0, 9.5, 9.0):
    OH = 10 ** -(14 - pH)
    s = Ksp / OH ** 2          # [Mg2+] that can exist at this hydroxide concentration
    note = " (above 1 M: dissolves completely in practice)" if s > 1 else ""
    print(f"pH {pH:4.1f}: [OH-] = {OH:.1e} M, solubility of Mg(OH)2 = {s:.1e} M{note}")

pH 12.0: [OH-] = 1.0e-02 M, solubility of Mg(OH)2 = 5.6e-08 M
pH 11.0: [OH-] = 1.0e-03 M, solubility of Mg(OH)2 = 5.6e-06 M
pH 10.0: [OH-] = 1.0e-04 M, solubility of Mg(OH)2 = 5.6e-04 M
pH  9.5: [OH-] = 3.2e-05 M, solubility of Mg(OH)2 = 5.6e-03 M
pH  9.0: [OH-] = 1.0e-05 M, solubility of Mg(OH)2 = 5.6e-02 M

Each unit drop in pH multiplies the solubility of a hydroxide like Mg(OH)₂ by 100, because [OH⁻] appears squared in K_sp.

Selective precipitation

Because salts differ greatly in solubility, a reagent can precipitate one ion while leaving another in solution. Adding chloride slowly to a solution of Ag⁺ and Pb²⁺ precipitates AgCl long before PbCl₂, because AgCl's K_sp is so much smaller. The classical qualitative analysis scheme taught in many first-year labs separates groups of metal ions this way, and the same principle underlies industrial separations.

Solubility in the environment and industry

Hard water contains dissolved calcium and magnesium, picked up as rain and groundwater flow through limestone and dolomite, which are common across the Canadian Shield's margins, southern Ontario, the Prairies and much of the US Midwest. When hard water is heated, CO₂ escapes and calcium carbonate precipitates as scale in kettles, water heaters and pipes. Water softeners exchange the calcium and magnesium for sodium.

In caves, the process runs both ways: groundwater rich in CO₂ dissolves limestone, and where it drips into air and loses CO₂, calcium carbonate precipitates as stalactites and stalagmites. The overall equilibrium is

This same equilibrium governs ocean acidification and shell formation, the subject of lesson 12's capstone.

Mining and industry use precipitation to clean wastewater. Raising the pH with lime precipitates dissolved metals as hydroxides, whose K_sp values are tiny (Fe(OH)₃ is about 10⁻³⁹). Mines in Canada must meet the federal Metal and Diamond Mining Effluent Regulations, and US mines meet effluent limits under the Clean Water Act; lime treatment is one of the main tools for both. Because some metal hydroxides, such as aluminum and zinc hydroxides, redissolve at high pH by forming hydroxide complexes, plants control the pH carefully rather than simply adding as much lime as possible.