Solubility Product K_sp, Molar Solubility and the Common Ion Effect

Substituting the ion concentrations gives a fixed relationship for each salt type. For AB salts K_sp = s², for AB2 or A2B salts K_sp = 4s³, and for AB3 salts K_sp = 27s⁴. Going backwards means taking the corresponding root.

The pure-water shortcuts assume no initial common ions, no significant hydrolysis or complex formation, and negligible activity-coefficient effects:

Coefficients create both concentration multipliers and exponents. For CaF₂, K_sp = [Ca²⁺][F⁻]² = s(2s)², not 2s². Solubility in grams per liter additionally requires multiplication by the salt's molar mass; a molar-solubility ranking is not automatically a mass-solubility ranking.

Do not rank salts of different stoichiometric types by K_sp alone. In ideal pure water at the same temperature, AgCl with K_sp = 1.8 × 10⁻¹⁰ has s ≈ 1.3 × 10⁻⁵ M, whereas Ag₂CrO₄ with K_sp = 1.1 × 10⁻¹² has s ≈ 6.5 × 10⁻⁵ M: the salt with the smaller K_sp is more soluble in this comparison.

For AgCl, K_sp ≈ s²; for Ag₂CrO₄, K_sp ≈ (2s)²s = 4s³. Derive each expression from its dissolution equation before comparing molar solubilities. A common ion or significant activity effects require a different calculation.

Dissolving AgCl in 0.10 M NaCl means starting with chloride already present. That extra product pushes the dissolution equilibrium back toward the solid, so far less AgCl dissolves. Put the supplied common-ion concentration in the initial row. Estimate s from the equilibrium expression, then test whether the added stoichiometric ion concentration is negligible relative to 0.10 M.

For AgCl in 0.100 M NaCl, the exact dilute-model relation is K_sp = s(0.100 + s). Estimate s = 1.80 × 10⁻¹⁰/0.100 = 1.80 × 10⁻⁹ M, then test s/0.100 = 1.80 × 10⁻⁸, far below 5%. The test, not a rule that common-ion approximations always work, justifies neglecting s. A much lower initial common-ion concentration may require the full equation.

To decide whether mixing two solutions produces a solid, calculate Q from the diluted ion concentrations and compare with K_sp. Q greater than K_sp means a precipitate forms until Q falls back to K_sp.

pH and ligands can change free-ion concentrations. Acid consumes CO₃²⁻ from a carbonate dissolution equilibrium, often increasing total dissolved salt; a complexing ligand can similarly lower free Ag⁺ while increasing total dissolved silver. K_sp still relates the free-ion activities at the fixed temperature. Do not insert the total concentration of free plus complexed metal into a free-ion K_sp expression. At high electrolyte concentration, activities can depart from concentrations and simple common-ion predictions need refinement.