Inorganic chemistry
Precipitation equilibria, solubility product
Use solubility-product equilibria to predict precipitation, calculate solubility, and understand selective separation, complexation and crystal growth.
IntuitionA precipitate appears when dissolved ions become too concentrated
Two clear solutions can make a solid when mixed. The key comparison is not whether either solution was cloudy beforehand, but whether the product of the relevant ion concentrations after mixing exceeds the equilibrium solubility product. The golden-yellow PbI₂ crystals are a vivid example.
The scene depicts KI solution entering Pb(NO₃)₂: yellow PbI₂ forms and settles when the ionic product exceeds Ksp. The amount-added control changes the feed amount and simulated fall, not an analytically calibrated concentration or crystal yield.
SchoolDissolution equilibria, Ksp, and ion products
Definition: Solubility product
Ksp is the equilibrium constant for dissolution of a sparingly soluble ionic solid, written using ion activities; the pure solid has activity 1 and is omitted. In dilute solutions, activities are often approximated by molar concentrations. The reaction quotient Q has the same expression but uses current, not necessarily equilibrium, ion levels.
| Condition | State / tendency |
|---|---|
| Q<Ksp | Unsaturated; more solid may dissolve |
| Q=Ksp | Saturated equilibrium with solid present |
| Q>Ksp | Supersaturated; precipitation is thermodynamically favoured |
Precipitation begins at the threshold Q=Ksp, but a supersaturated solution can persist temporarily if nucleation is slow. When a solid is already present, equilibrium requires Q=Ksp. Mixing calculations must first account for dilution and any reaction that consumes the ions.
| Solid | Dissolution stoichiometry | Ksp |
|---|---|---|
| AgCl | Ag⁺ + Cl⁻ | 1.8×10⁻¹⁰ |
| AgBr | Ag⁺ + Br⁻ | 5.4×10⁻¹³ |
| AgI | Ag⁺ + I⁻ | 8.5×10⁻¹⁷ |
| BaSO₄ | Ba²⁺ + SO₄²⁻ | 1.1×10⁻¹⁰ |
| CaCO₃ | Ca²⁺ + CO₃²⁻ | 3.4×10⁻⁹ |
| PbI₂ | Pb²⁺ + 2 I⁻ | ≈9.8×10⁻⁹ |
| Mg(OH)₂ | Mg²⁺ + 2 OH⁻ | 5.6×10⁻¹² |
| Fe(OH)₃ | Fe³⁺ + 3 OH⁻ | 2.8×10⁻³⁹ |
| Ag₂CrO₄ | 2 Ag⁺ + CrO₄²⁻ | 1.1×10⁻¹² |
| CaF₂ | Ca²⁺ + 2 F⁻ | 3.5×10⁻¹¹ |
| Type | Ion concentrations | Ksp relation; s |
|---|---|---|
| AB | [A]=s, [B]=s | Ksp=s²; s=Ksp¹ᐟ² |
| AB₂ or A₂B | s and 2s | Ksp=4s³; s=(Ksp/4)¹ᐟ³ |
| AB₃ or A₃B | s and 3s | Ksp=27s⁴; s=(Ksp/27)¹ᐟ⁴ |
For MₐXᵦ in pure water, dissolution produces [M]=as and [X]=bs, so Ksp=(as)ᵃ(bs)ᵇ=aᵃbᵇsᵃ⁺ᵇ and . This assumes no common ions, side reactions, hydrolysis, or significant activity corrections.
Example: Molar solubility from Ksp
Find s for AgCl, Ag₂CrO₄ and PbI₂ in pure water, using Ksp=1.8×10⁻¹⁰, 1.1×10⁻¹² and 9.8×10⁻⁹.
Solution
AgCl ⇌ Ag⁺+Cl⁻: s=√Ksp=1.34×10⁻⁵≈1.3×10⁻⁵ M. For Ag₂CrO₄ and PbI₂, Ksp=4s³: s=(Ksp/4)¹ᐟ³ gives 6.50×10⁻⁵≈6.5×10⁻⁵ M and 1.348×10⁻³≈1.35×10⁻³ M, respectively.
The common-ion effect lowers solubility because some product ions are already present. Acid can increase the solubility of salts whose anions are protonated (for example CO₃²⁻ or F⁻), or hydroxides whose OH⁻ is neutralised; basic pH can instead drive hydroxide precipitation. Complex formation removes free metal ion and may pull more solid into solution.
Example: Common-ion solubility of AgCl
Estimate AgCl solubility in 0.10 M NaCl at 25 °C, Ksp=1.8×10⁻¹⁰.
Solution
Let s=[Ag⁺] and [Cl⁻]≈0.10 M. Then s(0.10)=1.8×10⁻¹⁰, so s≈1.8×10⁻⁹ M. Dissolution adds negligible chloride relative to the background.
Example: Will PbI₂ precipitate on mixing?
Mix 50 mL 0.010 M Pb(NO₃)₂ with 50 mL 0.020 M KI. Check against Ksp(PbI₂)=9.8×10⁻⁹.
Solution
Equal volumes halve both concentrations: [Pb²⁺]=0.0050 M and [I⁻]=0.010 M. Q=0.0050(0.010)²=5.0×10⁻⁷>Ksp, so PbI₂ precipitates. This onset test does not calculate the final equilibrium amount.
Example: pH onset for Mg(OH)₂ precipitation
At 25 °C, find the pH at which Mg(OH)₂ begins to precipitate from 0.010 M Mg²⁺, using Ksp=5.6×10⁻¹² and ideal concentrations.
Solution
At onset, [OH⁻]=√(Ksp/[Mg²⁺])=√(5.6×10⁻¹²/0.010)=2.37×10⁻⁵ M. Thus pOH=4.63 and pH=14.00−4.63=9.37.
UndergraduateCoupled equilibria, separation, and analytical use
Selective precipitation exploits different onset thresholds. In the Mohr method, chromate is an indicator: with 0.010 M Cl⁻ and 0.0010 M CrO₄²⁻, AgCl starts at [Ag⁺]=Ksp/[Cl⁻]=1.8×10⁻⁸ M, while Ag₂CrO₄ requires [Ag⁺]=√(Ksp/[CrO₄²⁻])=3.3×10⁻⁵ M. Thus chloride precipitates first; chromate appears only after most chloride is removed.
Example: Fractional precipitation: chloride then chromate
For the concentrations above, estimate residual [Cl⁻] when Ag₂CrO₄ just begins to form. Use Ksp(AgCl)=1.8×10⁻¹⁰ and Ksp(Ag₂CrO₄)=1.1×10⁻¹².
Solution
Chromate onset requires [Ag⁺]=√(1.1×10⁻¹²/0.0010)=3.32×10⁻⁵ M. While AgCl is present, [Cl⁻]=Ksp/[Ag⁺]=1.8×10⁻¹⁰/(3.32×10⁻⁵)=5.4×10⁻⁶ M. Relative to the initial 0.010 M, about 99.95% of chloride has precipitated.
Classical qualitative analysis separates cations by reagent and acidity. Group I cations form sparingly soluble chlorides; acidic H₂S precipitates very insoluble group II sulfides while suppressing S²⁻; a basic medium increases sulfide availability so less insoluble group III sulfides can form. The group labels depend on the scheme, and real separations require controlling pH, complexation and reagent concentrations.
Ammonia forms [Ag(NH₃)₂]⁺ (Kf≈1.1×10⁷), lowering free [Ag⁺] and dissolving AgCl through coupled equilibria. AgBr is less readily dissolved; AgI is effectively not dissolved under ordinary qualitative-analysis conditions. Mohr titration uses chromate to signal the first persistent Ag₂CrO₄; Volhard titration detects excess Ag⁺ by forming the red FeSCN²⁺ complex with thiocyanate.
Gravimetric analysis converts an analyte into a sparingly soluble solid of known composition, then filters, washes, dries or ignites and weighs it to infer analyte amount. Particle growth and purity matter: occlusion, adsorption and coprecipitation can bias the mass. Similar equilibria shape kidney-stone risk from calcium oxalate, hard-water CaCO₃ scale, and coagulation where electrolytes destabilise colloids.
AdvancedThermodynamic and nanoscale views of precipitation
Thermodynamic Ksp is defined with activities. At finite ionic strength, γᵢ<1 for many ions, so concentration products differ from activity products; adding an inert electrolyte can increase apparent molar solubility (the diverse-ion or salt effect) even though the thermodynamic Ksp is unchanged at fixed temperature. Specific interactions can require extended Debye–Hückel or Pitzer models.
The dissolution free energy reflects competition between lattice stabilisation and hydration/solvation of ions. Ion pairing (for example neutral CaSO₄⁰) means total dissolved analytical concentration can exceed the free-ion concentration that enters Ksp. Thus measured solubility need not equal the simple stoichiometric s calculated from Ksp alone.
Supersaturation drives nucleation and growth. Classical nucleation theory balances the positive surface cost of a new nucleus against its favourable bulk free-energy gain; this creates a critical radius below which a nucleus tends to dissolve and above which growth is favourable. Ostwald ripening then tends to dissolve smaller particles and grow larger ones.
Curvature raises the chemical potential of a small crystal and increases its equilibrium solubility. The Gibbs–Thomson (Ostwald–Freundlich) relation predicts a solubility increase approximately exponential in 1/radius for spherical particles, with a coefficient set by interfacial energy, molar volume, temperature and solvent. Bulk tabulated Ksp values therefore do not fully describe nanocrystals.
Walther Nernst introduced the solubility-product treatment in 1889, connecting ionic equilibria with quantitative precipitation reasoning. Ostwald’s work on solution equilibria and nucleation helped establish the physical-chemical framework used to interpret solubility, supersaturation and crystal growth.
References
- Inorganic Chemistry · Catherine E. Housecroft; Alan G. Sharpe, 2018
- Quantitative Chemical Analysis · Daniel C. Harris, 2010