Equilibria Involving Complex Ions

General Chemistry • Solubility and Complex Ion Equilibria

Written by STEM Calculators Team Published April 18, 2026 Updated April 18, 2026

Equilibria Involving Complex Ions — Precipitation or Prevention

A metal ion can stay dissolved by forming a complex with a ligand, or it can precipitate as a sparingly soluble salt. This calculator checks whether precipitation occurs in the presence of a complexing agent and estimates the minimum ligand concentration required to prevent precipitation.

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Recognized keys: mode, M, L, a, X, Ksp, Kf, Mtot, Ltot, Xconc, precision.

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Metal–ligand system

Metal symbol

Ligand formula

Complex stoichiometry a in ML_a

a = 1 a = 2 a = 3

Precipitating salt and constants

Anion symbol X

K_sp for MX

K_f for ML_a

Displayed significant figures

Mode 1 — Check if MX precipitates

Total metal concentration [M]_tot (mol·L⁻¹)

Total ligand concentration [L]_tot (mol·L⁻¹)

Anion concentration [X] (mol·L⁻¹)

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Complex ions vs precipitation

A metal ion can either form a complex with a ligand L or precipitate as a sparingly soluble salt MX(s) with an anion X⁻. Which process dominates depends on the formation constant \(K_f\) and the solubility product \(K_{sp}\).

Complex-ion formation

\[ \mathrm{M^{n+}} + a\,\mathrm{L} \rightleftharpoons \mathrm{ML_a} \]

\[ K_f = \frac{[\mathrm{ML_a}]}{[\mathrm{M^{n+}}][\mathrm{L}]^{a}} \]

With total concentrations \([\mathrm{M}]_{\text{tot}}\) and \([\mathrm{L}]_{\text{tot}}\):

\[ [\mathrm{M}]_{\text{tot}} = [\mathrm{M^{n+}}] + [\mathrm{ML_a}],\quad [\mathrm{L}]_{\text{tot}} = [\mathrm{L}] + a[\mathrm{ML_a}]. \]

Large \(K_f\) means most metal is complexed, so free \([\mathrm{M^{n+}}]\) is very small.

Solubility and ion product

\[ \mathrm{MX}(s) \rightleftharpoons \mathrm{M^{n+}} + \mathrm{X^-} \]

\[ K_{sp} = [\mathrm{M^{n+}}][\mathrm{X^-}],\qquad Q_{sp} = [\mathrm{M^{n+}}]_{\text{actual}}[\mathrm{X^-}]_{\text{actual}}. \]

\(Q_{sp} < K_{sp}\): unsaturated, no precipitate.
\(Q_{sp} = K_{sp}\): saturated, at equilibrium with MX(s).
\(Q_{sp} > K_{sp}\): MX(s) precipitates until \(Q_{sp} = K_{sp}\).

Mode 1 — Will MX(s) precipitate?

Given \(a\), \(K_f\), \(K_{sp}\), \([\mathrm{M}]_{\text{tot}}\), \([\mathrm{L}]_{\text{tot}}\), and \([\mathrm{X^-}]\), the calculator:

Solves the metal–ligand system (using \(K_f\) and mass balances) for free \([\mathrm{M^{n+}}]\).
Computes \(Q_{sp} = [\mathrm{M^{n+}}][\mathrm{X^-}]\).
Compares \(Q_{sp}\) with \(K_{sp}\) to decide if MX(s) precipitates and reports what fraction of metal is complexed.

Mode 2 — Minimum ligand to prevent precipitation

For fixed \([\mathrm{M}]_{\text{tot}}\) and \([\mathrm{X^-}]\), the maximum free metal allowed without precipitation is

\[ [\mathrm{M^{n+}}]_{\max} = \frac{K_{sp}}{[\mathrm{X^-}]}. \]

If \([\mathrm{M}]_{\text{tot}} > [\mathrm{M^{n+}}]_{\max}\) and \(K_f\) is large, the approximate minimum free-ligand concentration needed is

\[ [\mathrm{L}]_{\min} = \left( \frac{[\mathrm{M}]_{\text{tot}}} {K_f[\mathrm{M^{n+}}]_{\max}} \right)^{1/a}, \]

and the approximate minimum total ligand concentration is

\[ [\mathrm{L}]_{\text{tot,min}} \approx [\mathrm{L}]_{\min} + a[\mathrm{M}]_{\text{tot}}. \]

Assumes dilute solution, activities ≈ concentrations, and no additional side equilibria beyond the single complex ML_a and the salt MX(s).

Frequently Asked Questions

How does complex-ion formation affect precipitation of MX(s)?

Complex formation reduces the free metal-ion concentration by converting M^n+ into ML_a. Lower free [M^n+] lowers Qsp = [M^n+][X-], which can prevent MX(s) from precipitating.

What does the formation constant Kf represent?

Kf measures how strongly a metal ion binds a ligand to form a complex. For M^n+ + aL ⇌ ML_a, Kf = [ML_a] / ([M^n+][L]^a).

How does the calculator decide if MX(s) will precipitate?

It estimates the free metal-ion concentration from the metal–ligand equilibrium, then computes Qsp = [M^n+][X-]. If Qsp is greater than Ksp, precipitation is favored; if Qsp is less than Ksp, no precipitate is expected.

What is the minimum ligand concentration to prevent precipitation?

It is the minimum free ligand level required so that the free metal concentration stays at or below the maximum allowed by Ksp at the given [X-]. In Mode 2, the calculator estimates [L]min and an approximate [L]tot,min based on a and [M]tot.

What assumptions does this calculator use?

It assumes dilute solutions where activities are approximated by concentrations and considers only one complex ML_a and one precipitating salt MX(s). Mode 2 also assumes Kf is large so most metal is complexed near the precipitation threshold.

Equilibria Involving Complex Ions

Equilibria Involving Complex Ions — Precipitation or Prevention

Metal–ligand system

Precipitating salt and constants

Mode 1 — Check if MX precipitates

Mode 2 — Minimum ligand level to prevent precipitation

Calculation steps

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Frequently Asked Questions

Equilibria Involving Complex Ions — Precipitation or Prevention

Metal–ligand system

Precipitating salt and constants

Mode 1 — Check if MX precipitates

Mode 2 — Minimum ligand level to prevent precipitation

Calculation steps

Rate this calculator

Frequently Asked Questions

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