Complex-Ion Equilibria and Formation Constants, Kf

Core Concept

A complex ion forms when a metal cation (a Lewis acid) binds several ligands (Lewis bases) — molecules or anions with lone pairs. The equilibrium is described by a formation constant Kf, which is usually huge, meaning complex formation is strongly favored. This is why "insoluble" salts dissolve: a ligand such as ammonia or cyanide pulls free metal ions out of solution, driving the dissolution equilibrium forward. The overall reaction is the sum of the dissolution (Ksp) and complexation (Kf) steps, so its equilibrium constant is the product K = Ksp × Kf.

Key Ideas

• Complex ion: a central metal ion surrounded by ligands, e.g., [Ag(NH₃)₂]⁺, [Cu(NH₃)₄]²⁺, [Fe(CN)₆]³⁻.
• Lewis picture: the metal ion is a Lewis acid; each ligand is a Lewis base donating a lone pair.
• Kf (formation constant): equilibrium constant for assembling the complex from the free metal ion and ligands.
• Large Kf ⇒ the complex is very stable and its free metal-ion concentration is tiny.
• Solubility boost: ligands remove free metal ions, so more solid dissolves (Le Châtelier).
• Overall K: for dissolution followed by complexation, K = Ksp × Kf.

Equations and Variables

• Ag⁺ + 2 NH₃ ⇌ [Ag(NH₃)₂]⁺, Kf = [Ag(NH₃)₂⁺] / ([Ag⁺][NH₃]²)
• AgCl(s) ⇌ Ag⁺ + Cl⁻, Ksp
• Overall: AgCl(s) + 2 NH₃ ⇌ [Ag(NH₃)₂]⁺ + Cl⁻, K = Ksp × Kf
• Stepwise formations: Kf = K1 × K2 × K3 × … (each step adds one ligand)

How It Works

1. A ligand (NH₃, CN⁻, OH⁻, S₂O₃²⁻, EDTA) approaches the metal cation and donates a lone pair, forming a coordinate covalent bond.
2. Complexes form stepwise — one ligand at a time — and the overall Kf is the product of the individual step constants.
3. Because Kf is typically large (10⁵ to 10³⁰), nearly all free metal ions are captured once enough ligand is present.
4. In a solubility context, this removal of free metal ions shifts the dissolution equilibrium MₓAᵧ(s) ⇌ x Mⁿ⁺ + y Aᵐ⁻ to the right, so the salt dissolves much more than Ksp alone would predict.
5. Quantitatively, add the two equations and multiply their constants: K = Ksp × Kf.

Worked Example

Calculate the overall equilibrium constant for dissolving AgCl in aqueous ammonia.

AgCl(s) ⇌ Ag⁺ + Cl⁻, Ksp = 1.8 × 10⁻¹⁰

Ag⁺ + 2 NH₃ ⇌ [Ag(NH₃)₂]⁺, Kf = 1.7 × 10⁷

Sum the two steps (Ag⁺ cancels):

AgCl(s) + 2 NH₃ ⇌ [Ag(NH₃)₂]⁺ + Cl⁻

K = Ksp × Kf = (1.8 × 10⁻¹⁰)(1.7 × 10⁷) = 3.1 × 10⁻³

The overall K is far larger than Ksp alone — that is why AgCl, essentially "insoluble" in pure water, dissolves readily in concentrated NH₃. (This is the basis of the classic qualitative-analysis test for silver.)

How it works

1. A ligand (NH₃, CN⁻, OH⁻, S₂O₃²⁻, EDTA) approaches the metal cation and donates a lone pair, forming a coordinate covalent bond.
2. Complexes form stepwise — one ligand at a time — and the overall Kf is the product of the individual step constants.
3. Because Kf is typically large (10⁵ to 10³⁰), nearly all free metal ions are captured once enough ligand is present.
4. In a solubility context, this removal of free metal ions shifts the dissolution equilibrium MₓAᵧ(s) ⇌ x Mⁿ⁺ + y Aᵐ⁻ to the right, so the salt dissolves much more than Ksp alone would predict.
5. Quantitatively, add the two equations and multiply their constants: K = Ksp × Kf.

Common confusions

• "A complex ion and a precipitate are the same." — A complex ion is a soluble species; it actually prevents or reverses precipitation.
• "Kf and Ksp are unrelated." — They combine: the overall constant for dissolving a salt in a ligand solution is Ksp × Kf.
• "Ligands only bind transition metals." — Many cations (Mg²⁺, Ca²⁺, Al³⁺) form complexes too; transition metals are just especially good at it.
• "Large Kf means the complex dissociates easily." — Opposite: large Kf means it barely dissociates — the complex is stable.
• "Overall K = Ksp + Kf." — When you add the two reactions, you multiply their equilibrium constants, not add them.

Quick review

• Complex ion = metal cation + ligands (Lewis acid–base).
• Kf = [MLₙ]/([M][L]ⁿ); large Kf = stable.
• Ligands increase apparent solubility by removing free metal ions.
• Overall K (dissolve + complex) = Ksp × Kf.
• Stepwise formation; overall Kf = product of steps.