Coprecipitation and Contamination

A gravimetric analyst dissolves 0.5000 g of a nickel alloy. He precipitates nickel as the bright red nickel dimethylglyoximate. He filters it and ignites nothing (it's weighed directly). He reports 24.15% Ni. The true value, from an independent certified method, is 24.38%. The precipitate looked perfect — bright red, well-crystallized, easy to filter. Where did 0.23% of the nickel, or rather 0.23% of something, go wrong? The answer almost never lies in the stoichiometry taught alongside gravimetric factors. It lies in coprecipitation: the silent, chemically illegitimate capture of foreign ions inside or on a precipitate. That precipitate is, by every visual and filtration test, behaving beautifully. Coprecipitation matters because it is the single largest source of systematic error in classical gravimetric analysis. It does not announce itself. A contaminated precipitate can look identical to a pure one under a microscope. It can filter at a normal rate and dry to a stable mass. Yet it may carry along mother-liquor ions never meant to be part of the stoichiometric compound being weighed. Unlike random error, which averages out with replicate trials, coprecipitation is often systematic. The same conditions of precipitation reproduce the same contamination. Repeating the experiment merely repeats the mistake with excellent precision and poor accuracy. The big picture connects directly to the physics of Nucleation and Particle Growth in Precipitates. Fast nucleation produces huge numbers of small, high-surface-area particles. That surface area is exactly the terrain on which coprecipitation operates. A precipitate is never a perfect single crystal isolated from solution. It is a solid grown from and bathed in an electrolyte-rich mother liquor. Every mechanism of contamination traces back to how ions from that liquor get stuck. They become trapped to, inside, or between the growing crystallites before the analyst can wash them away. This article works through the four classical mechanisms. These are surface adsorption, occlusion, inclusion (isomorphic and non-isomorphic), and mechanical entrapment. It covers the thermodynamics and kinetics that govern each. It presents quantitative treatments: the Otto–Weimarn ideas already implicit in particle-size control, the distribution law for adsorption, and Doerner–Hoskins for occlusion. It includes realistic worked numbers for a Ni, Ba, and Fe(OH)₃ system. Finally, it covers practical countermeasures — digestion, reprecipitation, washing with a common-ion electrolyte, precipitation from homogeneous solution — that every analytical laboratory uses to fight it.

Key takeaways

Coprecipitation is contamination of a precipitate by normally soluble species, occurring during precipitate formation, not after.
Four mechanisms: surface adsorption (fast, reversible, surface-area-dependent), occlusion (trapped in growing crystal, kinetic), inclusion (isomorphic substitution in the lattice, thermodynamic), and mechanical entrapment (physical pockets of mother liquor).
Adsorption follows the Paneth–Fajans–Hahn rule: the primarily adsorbed ion is the lattice ion in excess; the counter-ion layer follows.
The Doerner–Hoskins logarithmic distribution law describes how a coprecipitated trace ion partitions between crystal and solution as the precipitate grows.
Digestion (Ostwald ripening) reduces surface area and expels occluded liquor, dramatically cutting adsorption and occlusion errors — but does nothing for isomorphic inclusion.
Reprecipitation, appropriate washing electrolytes, and precipitation from homogeneous solution are the standard operational fixes.
Key idea: coprecipitation is a purity failure built into the crystal-growth process itself, not a filtration or washing failure — so its cure must intervene during nucleation and growth, not just after.

Where this lives today

Coprecipitation is not a historical curiosity confined to classical gravimetric methods; it is a design constraint across modern chemistry. In pharmaceutical manufacturing, controlling coprecipitation of process-related impurities and residual heavy metals during active pharmaceutical ingredient crystallization is a regulatory requirement (ICH Q3D elemental impurity guidelines), and crystallization engineers deliberately apply slow, controlled supersaturation — the same physical lever as digestion and precipitation from homogeneous solution — to keep drug substances pure. In environmental and nuclear chemistry, coprecipitation is exploited deliberately, exactly as Hahn used it: trace actinides and radionuclides in environmental water samples are concentrated onto carrier precipitates like Fe(OH)₃ or MnO₂ before analysis by mass spectrometry, using the same Doerner–Hoskins partition logic in reverse — maximizing rather than minimizing the coprecipitated fraction. In materials science, doped semiconductor and phosphor crystal growth relies on controlled isomorphic inclusion (deliberately engineered "coprecipitation" of a dopant ion into a host lattice) to achieve precise, homogeneous dopant distributions, while unwanted inclusion of trace metals during silicon crystal growth is a major yield-limiting defect mechanism that semiconductor fabs monitor obsessively. In biomineralization research, understanding how trace metals (Sr²⁺, Mg²⁺, Ba²⁺) coprecipitate into growing calcium carbonate biominerals (coral skeletons, foraminifera shells) underpins paleoclimate proxies like Mg/Ca and Sr/Ca thermometry, where the partition coefficient itself is the measured signal. And in the modern analytical laboratory, coprecipitation reasoning directly motivates the routine use of Digestion and Aging of Precipitates and, for the most demanding trace-level gravimetric and precipitation-titrimetric work, Precipitation from Homogeneous Solution, both of which exist specifically to suppress the mechanisms described here at the source, during nucleation and growth, rather than trying to wash the damage away afterward.

:::quiz[In the Doerner–Hoskins logarithmic distribution law, what happens to the coprecipitated fraction of a trace ion as the main precipitating ion approaches complete removal from solution ( $[B]_f/[B]_i \to 0$ ), given $\lambda > 0$ ?]

It approaches zero
It stays constant at 50%
It approaches unity (essentially complete coprecipitation) ✓
It becomes negative, indicating desorption

Ask the tutor: How would the digestion time needed to suppress adsorption change if I doubled the precipitant addition rate for a BaSO₄ gravimetric determination?

The physical picture

A precipitate destined for gravimetric weighing is never grown in isolation. It forms inside a solution that, at the instant of precipitation, is supersaturated in the analyte compound but also stuffed with every other ion originally present in the sample and every ion added as precipitant or held back as counter-ion. When BaSO₄ precipitates from a solution also containing Fe³⁺, NO₃⁻, and excess Cl⁻ or SO₄²⁻, the growing barium sulfate crystallites are bathed in that soup, and several physically distinct processes let foreign species hitch a ride into the final solid.

Surface adsorption is the fastest and most universal mechanism. A crystal surface is a discontinuity in the periodic lattice; ions at the surface have unsatisfied bonds (dangling coordination sites) and thus a residual electrostatic pull. The Paneth–Fajans–Hahn rule states that a precipitate preferentially adsorbs whichever of its own lattice ions is present in excess in solution (the "primary adsorbed layer"), and then attracts a secondary layer of oppositely charged ions from solution to balance charge — much like the electrical double layer of a colloid. If that secondary layer's counter-ion is a foreign species (say, adsorbed Fe³⁺ instead of Ba²⁺ balancing excess SO₄²⁻ on a BaSO₄ surface), the counter-ion becomes occluded as growth continues over it.

Occlusion happens when growth is too fast for the surface layers to re-equilibrate with bulk solution before being buried by the next layer of lattice ions. The adsorbed layer (primary + secondary) gets physically trapped inside the solid as the crystal face advances past it — a kinetic freezing-in of what would otherwise be a transient surface phenomenon. Occlusion scales with growth rate: the faster the precipitate forms (high supersaturation, high relative supersaturation $(Q-S)/S$ ), the more surface is created per unit time and the less time each surface patch has to desorb impurities before burial.

Inclusion is thermodynamically distinct: it happens when a foreign ion is similar enough in charge and radius to a native lattice ion that it can substitute directly into the crystal lattice, forming a mixed crystal or solid solution — genuinely isomorphic substitution, not a kinetic accident. Pb²⁺ (ionic radius ≈ 119 pm) substituting for Ba²⁺ (135 pm) in BaSO₄, or Ra²⁺ into BaSO₄, are classic examples; Sr²⁺ into KMnO₄ is another. Because inclusion is a lattice-energetics phenomenon governed by Goldschmidt's rules of substitution (similar ionic radius, within roughly 15%, and similar charge), it is essentially insensitive to digestion, washing, or precipitation rate — the impurity is not sitting at a grain boundary; it is a lattice atom.

Mechanical entrapment is the crudest mechanism: rapid crystal growth (especially of interlocking needles or dendrites) can physically pinch off a pocket of mother liquor between two crystal faces growing toward each other, sealing in solution — ions, solvent, everything — with no chemical selectivity at all. This is common in precipitates that grow as clusters of fine needles (e.g., some hydrous oxide gels) rather than as compact single crystals.

Figure

Four coprecipitation mechanisms on a growing crystal face

The practical consequence is that coprecipitation errors are almost always positive (the weighed precipitate is heavier than the pure stoichiometric compound would be) when the coprecipitated species has similar or greater molar mass and is not volatilized on ignition, but they can be negative if ignition converts a coprecipitated species to something volatile that is lost, or if the coprecipitated ion depresses the effective solubility product driving less analyte to precipitate. The direction and magnitude depend entirely on chemistry — there is no universal sign, which is precisely why every gravimetric method's classical literature specifies exact digestion times, wash-liquor compositions, and reprecipitation steps: these details are coprecipitation countermeasures baked into the method.

Figure

Paneth–Fajans–Hahn double layer on a BaSO₄ particle in excess SO₄²⁻

The core ideas

The quantitative heart of coprecipitation theory is the Doerner–Hoskins logarithmic distribution law, which describes how a trace contaminant ion partitions between a growing crystal and the surrounding mother liquor as precipitation proceeds — the regime relevant to occlusion and to some adsorption behavior in a well-mixed, slowly growing system.

\log\!\left(\frac{[A]_i}{[A]_f}\right) = \lambda \, \log\!\left(\frac{[B]_i}{[B]_f}\right)

where:

$[A]_i, [A]_f$ = initial and final concentration of the trace contaminant ion (e.g. Pb²⁺) remaining in solution
$[B]_i, [B]_f$ = initial and final concentration of the major precipitating ion (e.g. Ba²⁺) remaining in solution
$\lambda$ = the logarithmic distribution (partition) coefficient, characteristic of the ion pair and conditions

Reading this: as the main precipitate forms and $[B]$ falls (Ba²⁺ is consumed), the trace ion $A$ is swept into the solid in a fixed logarithmic ratio, not a fixed fractional ratio. This is the crucial physical content of the Doerner–Hoskins law versus its older cousin (the Berthelot–Nernst homogeneous distribution law, $D = [A]_{solid}/[A]_{solution} \times [B]_{solution}/[B]_{solid}$ , valid only for genuine equilibrium solid solutions with fast internal diffusion). The logarithmic law describes the more common real-world case: a growing crystal in which each newly deposited layer equilibrates with the current bulk solution but earlier layers are frozen (no back-diffusion) — the correct picture for occlusion. A second essential relationship is the qualitative adsorption-surface-area scaling, following from the fact that adsorption is a surface phenomenon while the analyte mass is a bulk (volume) phenomenon:

\frac{m_{\text{adsorbed impurity}}}{m_{\text{precipitate}}} \;\propto\; \frac{A_{\text{surface}}}{V_{\text{particle}}} \;\propto\; \frac{1}{r}

where:

$A_{\text{surface}}/V_{\text{particle}}$ = surface-to-volume ratio of the precipitate particles
$r$ = characteristic particle radius

Reading this: halving the particle radius doubles the specific surface area and (all else equal) roughly doubles the mass fraction of adsorbed contaminant — this is the direct quantitative link back to Nucleation and Particle Growth in Precipitates, since everything that controls particle size (relative supersaturation, digestion, precipitant addition rate) controls coprecipitation through this scaling.

Derivation AThe von Weimarn / surface-area argument for adsorption load

Goal: show quantitatively why small particles from fast precipitation carry far more adsorbed impurity, in relative terms, than large particles from slow, controlled precipitation, given the same total analyte mass.

Step 1 — geometry of a single particle. Model each precipitate particle as a sphere of radius $r$ . Its volume is $V = \tfrac{4}{3}\pi r^3$ and its surface area is $S = 4\pi r^2$ . The specific surface area (area per unit volume) is

\frac{S}{V} = \frac{4\pi r^2}{\tfrac{4}{3}\pi r^3} = \frac{3}{r}

Step 2 — total analyte mass is fixed. For a fixed total mass $m$ of precipitate of density $\rho$ , the total volume is $V_{\text{tot}} = m/\rho$ , independent of how finely it is divided. If that mass is divided into $N$ spheres of radius $r$ , then $N \cdot \tfrac{4}{3}\pi r^3 = m/\rho$ , so $N = \dfrac{3m}{4\pi \rho r^3}$ .

Step 3 — total surface area scales as $1/r$ . The total surface area of all $N$ particles is

S_{\text{tot}} = N \cdot 4\pi r^2 = \frac{3m}{4\pi\rho r^3}\cdot 4\pi r^2 = \frac{3m}{\rho r}

So for fixed total precipitate mass $m$ , halving the particle radius doubles the total surface area available for adsorption.

Step 4 — link surface area to adsorbed impurity mass. If adsorption follows a roughly constant surface density of adsorbed foreign ions, $\sigma$ (mass of impurity per unit area, a reasonable first approximation for a fixed bulk impurity concentration and a saturated adsorbed monolayer), then the total adsorbed impurity mass is

m_{\text{impurity}} = \sigma \cdot S_{\text{tot}} = \sigma \cdot \frac{3m}{\rho r}

Step 5 — the fractional contamination. Dividing by the total precipitate mass $m$ gives the impurity mass fraction:

\frac{m_{\text{impurity}}}{m} = \frac{3\sigma}{\rho r}

This confirms the scaling $\propto 1/r$ stated above and shows it is a direct, unavoidable consequence of sphere geometry, not an empirical accident. A precipitate grown as 0.1 μm colloidal particles will carry roughly ten times the relative surface contamination of the same total mass grown as 1 μm crystallites — which is exactly the argument for digestion, since Ostwald ripening increases $r$ by dissolving small particles and redepositing that material on larger ones, driving $S_{\text{tot}}$ down toward the thermodynamic minimum for that total mass.

Limiting case: as $r \to \infty$ (an idealized single giant crystal holding the entire precipitate mass), $m_{\text{impurity}}/m \to 0$ — perfect digestion in the limit eliminates surface-adsorption contamination entirely, though occlusion and inclusion, which are not purely surface phenomena, are unaffected by this limit.

Go deeper · Derivation B · 2 minDoerner–Hoskins distribution applied to progressive precipitation

Derivation BDoerner–Hoskins distribution applied to progressive precipitation

Goal: derive the working form of the logarithmic distribution law and use it to predict how much of a trace contaminant, initially present at known concentration, ends up in a precipitate as the main precipitating ion is progressively consumed.

Step 1 — set up mass balance. Suppose a solution originally contains concentration $[B]_i$ of the major ion (e.g. Ba²⁺) and $[A]_i$ of a trace contaminant (e.g. Pb²⁺), both being swept down by a common precipitant (SO₄²⁻) added gradually. At any point during precipitation, some Ba²⁺ has been consumed, leaving $[B]_f$ in solution, and correspondingly some Pb²⁺ has coprecipitated, leaving $[A]_f$ in solution.

Step 2 — postulate local equilibrium at the growing surface, no back-diffusion in the bulk of the solid. Each infinitesimal shell of new precipitate that forms is assumed to equilibrate briefly with the solution at that instant, incorporating $A$ and $B$ in a ratio governed by a true (Nernst-type) equilibrium constant $\lambda$ for that instant, but once buried, the composition of that shell is frozen — it cannot re-equilibrate with later (more Ba²⁺-depleted, so more Pb²⁺-enriched relative to Ba²⁺) solution. Integrating this instantaneous partition over the whole precipitation event yields the logarithmic, not linear, relationship:

\log\!\left(\frac{[A]_i}{[A]_f}\right) = \lambda \log\!\left(\frac{[B]_i}{[B]_f}\right)

Step 3 — solve for the fraction of trace ion coprecipitated. Rearranging,

[A]_f = [A]_i \left(\frac{[B]_f}{[B]_i}\right)^{\lambda}

The fraction of the trace ion remaining in solution is $[A]_f/[A]_i = (\;[B]_f/[B]_i\;)^{\lambda}$ , so the fraction coprecipitated is

f_{\text{coppt}} = 1 - \left(\frac{[B]_f}{[B]_i}\right)^{\lambda}

Step 4 — interpret $\lambda$ . When $\lambda = 1$ , this reduces to the simple statement that the trace ion is removed from solution in exact proportion to the major ion — a "coprecipitates like a perfect isomorphic partner" limit. When $\lambda \to 0$ , no coprecipitation occurs regardless of how completely $B$ is precipitated (the impurity is essentially completely excluded — the desirable limit chemists seek by choice of precipitant and conditions). Real systems (Pb²⁺ into BaSO₄, Sr²⁺ into BaSO₄) show $\lambda$ typically in the range 0.1–1, determined empirically for each ion pair and each set of precipitation conditions (temperature, ionic strength, rate of addition).

Limiting case: as $[B]_f/[B]_i \to 1$ (precipitation barely begun, $B$ almost entirely still in solution), $f_{\text{coppt}} \to 1 - 1^{\lambda} = 0$ for any $\lambda$ — no coprecipitation before any precipitate has formed, as physically required. As $[B]_f/[B]_i \to 0$ (essentially complete precipitation of $B$ ), $f_{\text{coppt}} \to 1$ whenever $\lambda > 0$ : given enough time and complete consumption of the major ion, any nonzero partition coefficient eventually sweeps essentially all the trace ion into the solid, which is the practical warning behind never allowing supersaturation to run to full completion in the presence of a coprecipitating trace ion without countermeasures such as digestion or reprecipitation.

Worked examples

Example 1 (foundational) — surface-adsorption mass balance from a fixed impurity monolayer. A chemist precipitates 0.7500 g of BaSO₄ (M = 233.39 g/mol, density 4.50 g/cm³) from a solution also containing trace Fe³⁺. Two precipitation protocols are compared: (a) rapid addition producing particles of average radius 0.20 μm, and (b) slow, digested precipitation producing average radius 2.0 μm. Assume a constant adsorbed-Fe³⁺ surface density $\sigma = 4.0 \times 10^{-8}\ \text{g/cm}^2$ (a typical order of magnitude for a submonolayer of a trivalent ion on a sulfate surface). Using the Step 5 result from the surface-area derivation,

\frac{m_{\text{impurity}}}{m} = \frac{3\sigma}{\rho r}

For protocol (a), $r = 0.20\ \mu\text{m} = 2.0\times10^{-5}\ \text{cm}$ :

\frac{m_{\text{impurity}}}{m} = \frac{3(4.0\times10^{-8}\ \text{g/cm}^2)}{(4.50\ \text{g/cm}^3)(2.0\times10^{-5}\ \text{cm})} = \frac{1.2\times10^{-7}}{9.0\times10^{-5}} = 1.33\times10^{-3}

So the adsorbed-Fe³⁺ mass is $1.33\times10^{-3} \times 0.7500\ \text{g} = 1.00\times10^{-3}\ \text{g} = 1.00\ \text{mg}$ of iron riding on the precipitate surface. For protocol (b), $r = 2.0\ \mu\text{m} = 2.0\times10^{-4}\ \text{cm}$ , exactly ten times larger, so the fractional contamination is exactly ten times smaller:

\frac{m_{\text{impurity}}}{m} = 1.33\times10^{-4}, \qquad m_{\text{impurity}} = 1.00\times10^{-4}\ \text{g} = 0.100\ \text{mg}

Digesting the precipitate to grow particles tenfold in linear dimension cuts the surface-adsorbed contamination by exactly a factor of ten — from 1.00 mg to 0.100 mg of iron on a 0.75 g BaSO₄ precipitate, a striking demonstration that particle size, not chemistry, is the primary lever against pure adsorption contamination.

Limiting-case check: if $r$ were allowed to grow without bound (a single monolithic crystal), $m_{\text{impurity}}/m \to 0$ , consistent with the derivation; if $r \to 0$ (colloidal, gelatinous precipitate), the fractional contamination diverges — exactly why gelatinous precipitates like Fe(OH)₃ or Al(OH)₃ are notoriously the worst offenders for coprecipitation and are essentially never digested to compact single crystals (they must instead be handled by dissolution/reprecipitation or avoided by choosing a different method entirely).

Example 2 (applied) — Doerner–Hoskins prediction of Pb²⁺ coprecipitated with BaSO₄. A water sample contains Ba²⁺ at 5.00 × 10⁻³ M and trace Pb²⁺ at 2.00 × 10⁻⁶ M (both about to be precipitated as sulfates by slow addition of dilute H₂SO₄, a controlled precipitation-from-homogeneous-solution style protocol). The empirical logarithmic distribution coefficient for Pb²⁺ into BaSO₄ under these conditions is $\lambda = 0.35$ . Precipitation is carried to 99.0% completion with respect to Ba²⁺, i.e. $[B]_f/[B]_i = 0.0100$ . Using

f_{\text{coppt}} = 1 - \left(\frac{[B]_f}{[B]_i}\right)^{\lambda}

first compute the ratio raised to the 0.35 power:

(0.0100)^{0.35} = 10^{-2 \times 0.35} = 10^{-0.700}

10^{-0.700} = 0.1995

f_{\text{coppt}} = 1 - 0.1995 = 0.8005 \approx 80.1\%

Despite Pb²⁺ being present at only 0.04% of the Ba²⁺ concentration (2.00×10⁻⁶ M vs 5.00×10⁻³ M), roughly 80% of that trace lead is swept into the BaSO₄ precipitate once barium precipitation reaches 99% completion. In absolute terms, if the precipitation is carried out in 500.0 mL, the total Pb²⁺ present is

n_{\text{Pb}} = (2.00\times10^{-6}\ \text{mol/L})(0.5000\ \text{L}) = 1.00\times10^{-6}\ \text{mol}

and the amount coprecipitated is $0.8005 \times 1.00\times10^{-6}\ \text{mol} = 8.01\times10^{-7}\ \text{mol}$ Pb, equivalent to $8.01\times10^{-7} \times 207.2\ \text{g/mol} = 1.66\times10^{-4}\ \text{g} = 0.166\ \text{mg}$ of lead contaminating the barium sulfate — a small absolute mass, but potentially a significant relative error if the BaSO₄ precipitate itself is only, say, 0.4 g (giving roughly 0.04% mass error from this single trace contaminant alone, before considering Fe³⁺, Sr²⁺, or other coprecipitating species also typically present).

Limiting-case check: if precipitation were stopped at only 10.0% completion ( $[B]_f/[B]_i = 0.900$ ), $f_{\text{coppt}} = 1 - (0.900)^{0.35} = 1 - 0.9635 = 0.0365$ , only 3.65% coprecipitated — confirming that early-stage precipitate (formed while $B$ is still abundant in solution) carries far less contaminant than late-stage precipitate, which is the chemical rationale behind reprecipitation: discard the first precipitate (which trapped a modest fraction of contaminant while accumulating most of the analyte), redissolve it, and reprecipitate from a solution now nearly free of the trace ion (since most of it was already removed with the first batch's mother liquor), sharply lowering the coprecipitated fraction the second time.

Example 3 (exam/competition-level) — combining occlusion and a gravimetric percent-error calculation with ignition chemistry. A student determines %Fe in an ore by precipitating Fe³⁺ as Fe(OH)₃ with excess NH₃ from a solution that also contains Mg²⁺ (as an interference, from dolomite gangue in the ore). The Fe(OH)₃ is filtered, ignited to Fe₂O₃ (M = 159.69 g/mol), and weighed. Some Mg²⁺ coprecipitates with the gelatinous Fe(OH)₃ (Mg(OH)₂ is much more soluble alone but is occluded/adsorbed onto the huge surface area of ferric hydroxide floc — a classic case). On ignition, occluded Mg(OH)₂ converts to MgO (M = 40.30 g/mol), which remains in the ignited residue (MgO is not volatile at ignition temperatures near 800–900 °C). Suppose true Fe present corresponds to 0.3000 g Fe₂O₃ exactly (i.e. the pure-precipitate mass expected), but the precipitate also occludes 0.0040 g of Mg(OH)₂ (M = 58.32 g/mol) before ignition. On ignition, this converts to MgO with mass

m_{\text{MgO}} = 0.0040\ \text{g} \times \frac{40.30\ \text{g/mol}}{58.32\ \text{g/mol}} = 0.0040 \times 0.6912 = 2.765\times10^{-3}\ \text{g}

The total ignited mass weighed is therefore

m_{\text{total}} = 0.3000\ \text{g} + 0.002765\ \text{g} = 0.3028\ \text{g (to 4 sig figs)}

If the analyst (unaware of the contamination) computes %Fe assuming the entire 0.3028 g is pure Fe₂O�3, using the gravimetric factor $2 \times 55.85/159.69 = 0.6994$ for Fe, and a sample mass of 1.000 g:

\%\text{Fe (reported, contaminated)} = \frac{0.3028\ \text{g} \times 0.6994}{1.000\ \text{g}} \times 100\% = 21.18\%

versus the true value based on the pure 0.3000 g Fe₂O₃:

\%\text{Fe (true)} = \frac{0.3000\ \text{g} \times 0.6994}{1.000\ \text{g}} \times 100\% = 20.98\%

The systematic positive error is

\Delta(\%\text{Fe}) = 21.18\% - 20.98\% = +0.20\%\ \text{Fe (absolute)}

or a relative error of $0.20/20.98 \times 100\% = +0.95\%$ — nearly 1% relative error from occluded magnesium hydroxide alone, entirely invisible to the analyst unless an independent method (or a blank/spike experiment) is used, because the ignited residue looks, weighs, and behaves exactly like pure Fe₂O₃.

Limiting-case check: if no Mg²⁺ had been present in the gangue (occluded mass → 0), $\Delta(\%\text{Fe}) \to 0$ and the reported and true values coincide exactly, confirming the error scales linearly with the mass of contaminant occluded, as it must for a simple additive mass-balance error of this kind.

Limiting cases, scope & common traps

Regime / case	Behavior	Consequence
Very small particles ( $r \to 0$ ), gelatinous precipitate	$S/V \to \infty$ ; adsorption and occlusion dominate	Large, often unpredictable positive error; digestion alone may be insufficient
Very large single crystals ( $r \to \infty$ )	$S/V \to 0$ ; adsorption vanishes	Only inclusion (lattice substitution) can still occur
$\lambda \to 0$ in Doerner–Hoskins	Trace ion essentially excluded regardless of precipitation extent	Ideal — occurs when trace ion is chemically dissimilar to lattice ion
$\lambda \to 1$	Trace ion removed in direct proportion to main ion	Worst case for occlusion-type coprecipitation; common when ionic radii/charges match
Isomorphic ionic radius match (within ~15%, Goldschmidt)	True inclusion, thermodynamically stable mixed crystal	Digestion, washing, and rate control do not help; requires different precipitant/masking
Ignited contaminant is volatile (e.g., some ammonium salts, certain chlorides)	Contaminant lost on ignition	Can cause negative error, opposite sign from the more common positive-error case
Common-ion washing electrolyte used (e.g., dilute H₂SO₄ wash for BaSO₄)	Suppresses peptization, does not reverse chemisorption	Removes surface-adsorbed layer to first approximation but not occluded/included species

Common trapMisconception: "digestion (aging) fixes all coprecipitation."

Digestion is extremely effective against surface adsorption (by reducing total surface area through Ostwald ripening) and reasonably effective against occlusion (by allowing slow recrystallization to expel trapped mother liquor), which is why it is a standard step before filtration for crystalline precipitates like BaSO₄ or CaC₂O₄. But digestion does essentially nothing for true isomorphic inclusion, because the contaminant ion is already a stable, low-energy member of the crystal lattice — no amount of aging at constant composition will expel a species that the lattice thermodynamically prefers to hold. Ra²⁺ coprecipitated into BaSO₄ crystals, historically exploited to concentrate radium from pitchblende residues, is not removed by digestion; it requires a chemically different separation strategy entirely.

The chemist behind it

Coprecipitation theory as a quantitative discipline owes its central law to Otto Hahn, working initially with Friedrich Paneth in the 1910s on the behavior of radioactive trace elements — work that directly motivated the more general Paneth–Fajans–Hahn adsorption rule. Hahn, trained as a radiochemist, needed to understand precisely how a radioactive tracer present at almost immeasurably low concentration (femtomolar or lower) partitions between a macroscopic carrier precipitate and solution, because radiochemical separations of that era depended entirely on "carrying" trace radioisotopes down with a chemically similar bulk precipitate — a technique that is coprecipitation used deliberately rather than fought as an error source. Hahn's meticulous quantitative studies of how RaSO₄ traces followed BaSO₄ carriers led directly to the logarithmic distribution law refined by Doerner and Hoskins in 1925, still the standard quantitative treatment taught today. Hahn later became one of the discoverers of nuclear fission (with Fritz Strassmann, 1938), a discovery whose chemical proof rested on exactly the same careful radiochemical carrier-precipitation methodology he had developed studying coprecipitation decades earlier.

Origins · 1 minOtto Hahn and how this was discovered

Otto Hahn

1879–1968 · Radiochemistry, coprecipitation carrier theory, discovery of nuclear fission

Hahn's early career work on the adsorption and coprecipitation behavior of radioactive tracers with carrier precipitates established the quantitative and mechanistic foundation for understanding how trace ions partition into growing crystals — directly enabling the logarithmic distribution law and underpinning both classical gravimetric error analysis and modern radiochemical separations.

History

- 1913 — Kasimir Fajans and Otto Hahn formulate rules for the adsorption of ions on ionic crystal surfaces (the seed of the Paneth–Fajans–Hahn rule).

- 1918 — Otto Hahn and Lise Meitner discover protactinium, work relying heavily on carrier coprecipitation of trace radioactive species.

- 1925 — H. A. Doerner and W. M. Hoskins publish the logarithmic distribution law for trace-ion coprecipitation with BaSO₄, the quantitative treatment still used today.

- 1938 — Otto Hahn and Fritz Strassmann chemically identify barium among uranium fission products, using carrier-precipitation methodology directly descended from coprecipitation studies.

- 1950s–60s — precipitation from homogeneous solution (PFHS), developed largely by Lawrence Gordon and Louis Wilson, becomes a standard laboratory strategy for suppressing coprecipitation by keeping supersaturation permanently low.

What is the practical difference between coprecipitation and simple contamination from an unwashed precipitate?

Unwashed contamination is mother liquor or wash solution physically adhering to the outside of particles, in principle fully removable by sufficient washing with the correct electrolyte. Coprecipitation is contamination incorporated during crystal formation — adsorbed on internal surfaces of an aggregate, buried by occlusion, substituted into the lattice by inclusion, or sealed in mechanically entrapped pockets. Washing removes the first category almost completely; it removes only part of the adsorbed-layer contribution to the second category and essentially none of occlusion, inclusion, or mechanical entrapment. This is why analytical procedures specify digestion times and reprecipitation steps as separate, necessary operations beyond washing.

Why does the Paneth–Fajans–Hahn rule say the lattice ion "in excess" adsorbs first?

A crystal surface has unsatisfied bonds; an ion identical to a native lattice ion fits into the existing lattice geometry with minimal distortion and lowest adsorption free energy, so lattice ions are always preferred adsorbates over unrelated foreign ions. Whichever lattice ion (cation or anion) is present in excess in solution at that moment dominates the primary adsorbed layer simply because it is more abundant and chemically matched. The secondary layer of counter-ions, needed to satisfy the resulting surface charge, is drawn from whatever ions are available in solution and need not be the "correct" stoichiometric counter-ion — this is the point of entry for foreign-ion contamination.

Does coprecipitation always cause a positive (too-high) gravimetric error?

No, though positive error is the more common outcome in classical examples like Fe(OH)₃/Mg(OH)₂ or BaSO₄/PbSO₄, where the coprecipitated species adds mass that survives ignition or drying. Negative errors occur when the coprecipitated species is volatilized during ignition and its removal leaves a void or structural loss greater than its own mass, or when coprecipitation of a foreign ion depresses the effective concentration of analyte ions available to precipitate (competing complexation or a shift in an equilibrium that reduces analyte recovery). The correct approach is always to trace the specific chemistry of the contaminant through drying/ignition, not to assume a universal sign.

How does precipitation from homogeneous solution reduce coprecipitation?

Precipitation from homogeneous solution (PFHS) generates the precipitating anion or cation slowly and uniformly throughout the solution via a controlled chemical reaction (e.g., hydrolysis of urea to slowly release OH⁻, or hydrolysis of an ester to slowly release SO₄²⁻) rather than by direct, localized mixing of concentrated reagents. This keeps relative supersaturation low everywhere at all times, which by the nucleation-and-growth relationship favors slow growth of a few large, well-ordered crystals over the explosive nucleation of many small, high-surface-area particles. Lower surface area directly suppresses adsorption (by the $1/r$ scaling), and slower growth gives surface layers time to re-equilibrate with bulk solution before burial, directly suppressing occlusion as well.

Can coprecipitation be exploited deliberately rather than avoided?

Yes — this is the basis of "carrier precipitation," historically essential in radiochemistry and still used in trace environmental and geochemical analysis. When a species of interest is present at concentrations too low to precipitate on its own (below its own solubility product under any practical condition), chemists deliberately add a chemically similar "carrier" ion at macroscopic concentration and precipitate the carrier, allowing the trace species to coprecipitate (often via genuine isomorphic inclusion) and concentrate into an easily separated solid. Otto Hahn's radium/barium sulfate carrier experiments and modern trace-actinide preconcentration with iron or manganese hydroxide carriers both exploit exactly the physics described here, run in the opposite direction from a gravimetric analyst's goal.

Why is Doerner–Hoskins called "logarithmic" rather than a simple linear partition law?

A simple linear (Berthelot–Nernst homogeneous) distribution law assumes the whole solid behaves as a single equilibrium phase continuously exchanging with solution — appropriate only if diffusion within the solid is fast enough to maintain uniform composition throughout precipitation, which is rarely true for a growing crystal at room temperature. The Doerner–Hoskins model instead assumes each newly formed surface shell equilibrates with the current bulk solution and is then frozen in place with no further exchange as it is buried by subsequent growth; integrating this "instantaneous equilibrium, no back-diffusion" picture over the whole precipitation event produces a logarithmic relationship between initial and final concentrations of the trace and major ions, not a linear one.

What experimental evidence would tell an analyst that coprecipitation, rather than incomplete precipitation or a filtration loss, is the source of a gravimetric bias?

A radiotracer spike experiment is the classical diagnostic: add a small known amount of a radioactively labeled form of the suspected contaminant ion to the solution before precipitation, then measure how much activity ends up in the washed, dried precipitate versus the filtrate. If a reproducible, non-negligible fraction of the tracer activity appears in the precipitate despite thorough washing, that fraction is direct, quantitative evidence of coprecipitation, and its magnitude can even be used to back out an empirical Doerner–Hoskins $\lambda$ for that specific ion pair and set of conditions.