Distinguish dissolved totals from the ions in a solubility product

The instrument may count an element, not one free ion

In a 1974 National Bureau of Standards brushite study, record E-6 has 4.82mM total phosphorus. That total includes phosphorus in different protonation states and ion pairs. The refined source model assigns 18.7% to NaHPO₄⁻. If you also call 100% of total phosphorus free HPO₄²⁻, the same phosphorus has been counted twice.

Start with that impossible proposal. Repair the fraction, then compare 80% and 40%. Both can avoid this one budget violation, but neither is established as the actual free fraction. Explain why the missing information matters.

Three quantities that must remain separate

For the source’s brushite equilibrium, the ion activity product involves free Ca²⁺ and free HPO₄²⁻. Its convention writes Ksp=[Ca²⁺][HPO₄²⁻]γCaγHPO₄, with the pure phase and water convention incorporated in the reported constant. Square brackets here denote concentrations; γ is an activity coefficient describing departure from an ideal solution.

The measured analytical totals Cₜ and Pₜ sum the relevant element over all its dissolved forms. A free fraction fCa or fP is the free-ion concentration divided by its element’s total concentration. Consequently Ksp=CₜPₜfCafPγCaγHPO₄. The activity correction and the species fraction are different steps; setting either to one requires justification.

For E-6, multiplying the totals gives (.00362)(.00482)=.0000174484mol²/L². The paper’s refined activity product is 2.44×10⁻⁷mol²/L². The large difference does not mean that arithmetic failed. It means the totals are not the free-ion activities in the equilibrium expression.

Make a conditional inference without inventing a species measurement

The model-derived sodium-bound concentration is 4.82×.187=.90134mM. Thus free HPO₄²⁻ cannot exceed 3.91866mM under that model, even before accounting for other phosphorus forms. The remainder is an upper bound, not a measured free-ion concentration.

The widget rearranges the reported activity product to show which remaining correction factor would be required for your chosen fP. This is an inverse consistency calculation using the source result as input, not an independent replication of its fit. Many proposals can require different remaining factors. Full acid equilibria, ion pairing and activity modeling would be needed to establish a physically consistent solution; this activity deliberately does not manufacture those missing species values.

Compare records and explanations

Compare E-1 with E-6: total calcium increases 1.60→3.62mM, while the refined activity product changes 2.52→2.44×10⁻⁷. Compare A-1 with A-6 at a different initial phosphorus level. The measured pH and calculated ionic strength also change. The comparisons therefore cannot identify one microscopic mechanism from a single concentration difference.

The paper reports that these brushite-saturated solutions were supersaturated with respect to hydroxyapatite. Saturation with respect to the studied phase does not establish global equilibrium among every possible solid phase.

The paper’s initial model is already more sophisticated than multiplying totals. Its refined model adds fitted sodium pairing and an activity-equation parameter. Because those parameters were estimated from the same observations, improved agreement is an in-sample diagnostic. It is not an unseen validation set or proof that every omitted interaction is negligible.

Capture an overallocated proposal and a proposal that remains unidentified. Write which observation or independent modeling information would distinguish the remaining alternatives. The practice case changes both the total and the paired fraction.

Source and continuation

Patel, Gregory and Brown, J.Res.NBS78A (1974),675–681, measured Table 1 and calculated Table 3. The original federal research is public domain; the learner task and inverse bookkeeping are original. Values are a historical case, not current recommended constants. No laboratory procedure or independent practical credit is supplied.

Return to the finite-solid investigation for ideal inventory constraints, or quotient construction to inspect which species belong in an expression.

Isolate a pH intervention in a qualitative supplied case

For a salt containing a weak-basic anion, acid can consume that anion and allow additional solid to dissolve. For example, H⁺ consumes OH⁻ released by a metal hydroxide, lowering the ion product at a held temperature; remaining solid can then dissolve toward saturation. This does not change Ksp merely by changing pH. Actual complexation, precipitation of another phase and activity changes can complicate the result.

Paired prediction: Consider a fictional salt containing a weak-acid cation BH⁺ and an inert anion X⁻. A maintained basic reservoir converts some BH⁺ to neutral B. Predict the qualitative effect on dissolution when solid remains and all competing reactions, gas loss and nonideal changes are explicitly excluded. Compare with the hydroxide salt under a maintained acidic reservoir. State which dissolved ion is removed in each case.

Compare the controlled interventions

Base consumes the acidic BH⁺ cation, reducing its free concentration and the salt's ion product; additional dissolution is favored. Acid consumes OH⁻ in the hydroxide case with the same qualitative consequence. The result follows from removal of a species in each dissolution expression, not a universal rule that acid or base always increases every salt's solubility. A finite exhausted solid supply cannot keep dissolving.

These are original theoretical comparisons, not instructions to adjust an actual solution. AP topic 8.11 requires the qualitative explanation; quantitative solubility-versus-pH calculations are optional enrichment, not a required assessment here. Reference: OpenStax precipitation and dissolution, checked 31 October 2026.