Selective Precipitation of Two Cations

Introduction

Many mixtures contain two metal ions that react with the same reagent. If both form sparingly soluble salts, adding that reagent slowly can remove one ion first while the other stays in solution. This is selective precipitation , a central idea in qualitative analysis, metal recovery and water treatment. The key question is simple: which salt reaches Q = Ksp first, and how far apart are the two onset points?

Core explanation

Compare thresholds, not raw Ksp values. For each cation, calculate the counter-ion concentration at which its salt begins to precipitate, using the method for onset of precipitation. The salt with the lower threshold precipitates first, because the counter-ion concentration rises from zero as reagent is added and meets that threshold before the other.

When Ksp comparison works. If both salts have the same formula type (for example both MX, or both MX₂) and the cation concentrations are equal, the smaller Ksp does correspond to the lower threshold. For Ba²⁺ and Sr²⁺, both 0.010 mol dm⁻³, with sulfate:

- BaSO₄ (Ksp ≈ 1.1 × 10⁻¹⁰): threshold [SO₄²⁻] = 1.1 × 10⁻⁸ mol dm⁻³ - SrSO₄ (Ksp ≈ 3.4 × 10⁻⁷): threshold [SO₄²⁻] = 3.4 × 10⁻⁵ mol dm⁻³

Barium sulfate starts to precipitate first, over three thousand times earlier in sulfate concentration.

When Ksp comparison fails. If the formula types differ, Ksp values have different units and cannot be compared directly. For Ag⁺ and Pb²⁺ (each 0.010 mol dm⁻³) with chloride: AgCl (Ksp 1.8 × 10⁻¹⁰) begins at [Cl⁻] = 1.8 × 10⁻⁸; PbCl₂ (Ksp ≈ 1.7 × 10⁻⁵) requires [Cl⁻]² = 1.7 × 10⁻³, so [Cl⁻] ≈ 0.041 mol dm⁻³. Here the order agrees with the Ksp order, but only a threshold calculation proves it — and with very different concentrations the order can reverse.

The separation window. Between the first threshold and the second lies a range of counter-ion concentration in which only the first salt forms. A useful separation needs this window to be wide enough that the first ion is almost completely removed before the second begins. A rule of thumb for equal concentrations of 1:1 salts is that the Ksp values should differ by a factor of at least 10³ to reach 99.9% removal, and preferably 10⁴ or more to allow a safety margin; the fraction-remaining calculation makes this precise.

Practical control. The counter-ion concentration must be held inside the window. Slow addition with stirring avoids local zones of high concentration where the second salt might briefly precipitate. When the counter-ion is H⁺-sensitive (OH⁻, S²⁻, CO₃²⁻), pH buffers provide much finer control, which is why many selective separations are pH-controlled.

Step-by-step reasoning

1. Write the Ksp expression for each possible precipitate. 2. Insert each cation's concentration and solve for its threshold counter-ion concentration. 3. Rank the thresholds: lowest precipitates first. 4. Note the second threshold as the upper edge of the separation window. 5. Test feasibility by calculating how much of the first ion remains at the second threshold.

Visual explanation

Draw a horizontal log[counter-ion] axis. Mark each salt's threshold as a vertical line. Moving from left to right as reagent is added, the region before the first line is clear solution, the region between the lines contains only the first solid, and beyond the second line both solids form. The width of the middle band, in log units, measures the selectivity.

Real-world analogy

Two hurdles of different heights stand on a track, and a rising water level represents the counter-ion. The lower hurdle is submerged first. If the heights are almost the same, both disappear at nearly the same time; if they are very different, there is a long period in which only the lower one is covered — the separation window.

Real-world example

In classical qualitative analysis, dilute hydrochloric acid is added to an unknown mixture to precipitate Ag⁺, Pb²⁺ and Hg₂²⁺ as chlorides, leaving most other cations in solution. Lead chloride is noticeably soluble, so some lead passes into later groups; this is predicted by its relatively large Ksp and high chloride threshold.

Why?

Why does the ion with the lower threshold come out first even though both ions "want" to precipitate? The counter-ion concentration builds up gradually from zero. Precipitation of any salt requires its own Q to reach its own Ksp, and the lower threshold is simply met sooner on this rising path.

Common misconception

"The salt with the smaller Ksp always precipitates first." This holds only for the same formula type and similar cation concentrations. Comparing Ksp for AgCl (mol² dm⁻⁶) with Ag₂CrO₄ (mol³ dm⁻⁹), for instance, compares quantities with different units and gives the wrong order in the Mohr titration.

Worked example

Question: A solution is 0.020 mol dm⁻³ in Ca²⁺ and 0.020 mol dm⁻³ in Ba²⁺. Sulfate is added slowly. Which precipitates first, and at what sulfate concentration does the second start? Ksp(BaSO₄) = 1.1 × 10⁻¹⁰; Ksp(CaSO₄) = 4.9 × 10⁻⁵.

Reasoning: BaSO₄ threshold = 1.1 × 10⁻¹⁰ ÷ 0.020 = 5.5 × 10⁻⁹ mol dm⁻³. CaSO₄ threshold = 4.9 × 10⁻⁵ ÷ 0.020 = 2.5 × 10⁻³ mol dm⁻³.

Answer: BaSO₄ precipitates first; CaSO₄ starts when [SO₄²⁻] reaches about 2.5 × 10⁻³ mol dm⁻³, giving a very wide separation window.

Quick check

1. Why can the Ksp values of AgCl and PbCl₂ not be compared directly to decide which precipitates first? Answer: The salts have different formula types, so their Ksp values have different units and powers; only threshold concentrations can be compared.

Exam focus

Always calculate both thresholds and state the order with a reason. Examiners frequently set pairs with different stoichiometries to test whether candidates avoid direct Ksp comparison. Finish with a statement of the window, for example "between 1.1 × 10⁻⁸ and 3.4 × 10⁻⁵ mol dm⁻³ sulfate, only BaSO₄ forms".

Advanced insight

Real separations are limited by co-precipitation: ions of the second metal can be trapped inside or adsorbed on the first solid even below their own threshold. Strontium, having the same charge and similar size to barium, substitutes into BaSO₄ crystals to some extent. Digesting the precipitate in warm mother liquor and reprecipitating it reduce this contamination.

Summary

To predict selective precipitation, compute each salt's onset threshold for the shared counter-ion. The lowest threshold precipitates first; the next threshold marks the upper limit of the separation window. Raw Ksp comparison is valid only for identical formula types and similar concentrations. A wide window, controlled addition and pH buffers make separations practical.

Practice questions

1. Solutions of 0.010 mol dm⁻³ Ag⁺ and 0.010 mol dm⁻³ Pb²⁺ are mixed with slowly added chloride. Give the chloride range in which only AgCl forms. Answer: From 1.8 × 10⁻⁸ mol dm⁻³ up to about 0.041 mol dm⁻³ chloride. 2. Two cations, M⁺ and N⁺, both 0.10 mol dm⁻³, form MX (Ksp 1 × 10⁻¹²) and NX (Ksp 1 × 10⁻¹¹). Is selective precipitation promising? Explain. Answer: No: the thresholds differ by only a factor of 10, so a large fraction of M⁺ (about 10%) would remain when NX starts. 3. Why is slow addition of reagent with good stirring important in selective precipitation? Answer: It prevents local regions of high counter-ion concentration where the second salt's threshold could be exceeded temporarily. 4. Suggest why pH control is often preferred to direct addition of hydroxide for separating metal hydroxides. Answer: Buffers fix [OH⁻] precisely at a chosen value inside the window, avoiding overshoot into the second threshold.