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Lessons · Equilibrium & acid–base

Insoluble, but no precipitate? Dilute magnesium and hydroxide

Ion accounting machine-checked — mixing dilution + the quotient QKsp data-sourced (openstax-chemistry-2e)3 modeling assumptions (disclosed)

Milk of magnesia is a suspension of solid magnesium hydroxide, Mg(OH)₂ — the textbook “insoluble” base. So surely mixing a magnesium salt with a hydroxide makes the solid appear? Stir 50.0 mL of 0.00010 M Mg(NO₃)₂ (magnesium nitrate) into 50.0 mL of 0.00010 M NaOH — both perfectly clear. The dissolution equilibrium is Mg(OH)2(s)⇌Mg2+(aq)+2 OH−(aq)\mathrm{Mg(OH)_2(s)} \rightleftharpoons \mathrm{Mg^{2+}(aq)} + 2\,\mathrm{OH^-(aq)}, and whether a solid forms is decided by the reaction quotient: dilute each ion into the combined 100.0 mL, evaluate Q=[Mg2+][OH−]2Q = [\mathrm{Mg^{2+}}][\mathrm{OH^-}]^2, and compare it to KspK_{sp}. If Q<KspQ < K_{sp} the solution is unsaturated and stays clear — however “insoluble” the compound is called.

Mg(OH)2 (s)⇌Mg2+ (aq)+2 OH− (aq)\mathrm{Mg(OH)_{2}}\,\text{(s)} \rightleftharpoons \mathrm{Mg}^{2+}\,\text{(aq)} + 2\,\mathrm{OH}^{-}\,\text{(aq)}
Ksp=[Mg2+][OH−]2K_{sp} = [\mathrm{Mg}^{2+}][\mathrm{OH}^{-}]^{2}=8.9×10⁻¹²at 25 °C
Source solutionAddedProvidesAfter mixing (M)
Mg(NO₃)₂ magnesium nitrate50.0 mL · 0.00010 M[Mg²⁺] = 0.0001[Mg²⁺] = 0.00005
NaOH sodium hydroxide50.0 mL · 0.00010 M[OH⁻] = 0.0001[OH⁻] = 0.00005

machine-checkedCombined volume 100 mL. Each ion is diluted byVsource / Vtotal — the same dimensional-analysis move as any dilution. Nothing has reacted yet; these are the concentrations at the instant of mixing.

The reaction quotientQ has the same form as Ksp — evaluated now, not at equilibrium.

Q=[Mg2+][OH−]2Q = [\mathrm{Mg}^{2+}][\mathrm{OH}^{-}]^{2}=(0.00005)(0.00005)²=1.25×10⁻¹³

Q = 1.25×10⁻¹³<Ksp = 8.9×10⁻¹²
—No precipitate — the solution stays clear.

Q is about 71.2× below Ksp, so the solution is unsaturated —Mg(OH)₂ could still dissolve, and none precipitates.

VerificationProven at build time — not asserted.
  • ✓ Each mixed concentration = [ion]source × Vsource / Vtotal [mixing dilution]
  • ✓ Q = [Mg²⁺]1[OH⁻]2 at the mixed concentrations [quotient computed]
  • ✓ The verdict follows the comparison: a precipitate forms exactly when Q > Ksp [verdict consistent]
✗ Common misconception: “Magnesium hydroxide is a classic insoluble compound, so mixing any magnesium salt with hydroxide must produce a precipitate.”

“Insoluble” is not a yes/no label — it is quantitative. A precipitate appears only when the ion product of the mixture exceeds Ksp. Here Q = 1.25×10⁻¹³ sits about 71.2× below Ksp = 8.9×10⁻¹², so the solution is unsaturated and stays perfectly clear — the Mg²⁺ and OH⁻ stay dissolved. Dilute enough and even a famously insoluble salt like Mg(OH)₂ does not form. It is the Q-vs-Ksp comparison, not the compound's reputation, that decides.

Modeling assumptions — author-asserted, disclosed not discharged
  • model Both source salts are soluble strong electrolytes that dissociate completely, and volumes are additive — so mixing 50.0 mL and 50.0 mL dilutes each ion into the combined 100.0 mL. The spectators Na+\mathrm{Na^+} and NO3−\mathrm{NO_3^-} take no part, and water's own OH−\mathrm{OH^-} (about 10−710^{-7} M) is negligible beside the added hydroxide.
  • model Activities are approximated by molar concentrations (the ideal-dilute-solution model), which is why QQ and KspK_{sp} are written with concentrations.
  • model The thermodynamic prediction is taken to hold: Q<KspQ < K_{sp} means no solid forms (the solution is unsaturated), and nothing kinetic changes that.

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