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SCIENCE · INVESTIGATE AND EXPLAIN

Audit species selectivity before inferring a solubility constant

A filtered dissolved-total assay is not necessarily a free-ion measurement.

Learning outcome: Vary complexation factor and common gain separately. Compare total and free products, propagate independent2%/3% reading bounds and identify the external information needed to infer true Ksp.

Starting recap

Refresh normalized free-ion activities, charge-balanced stoichiometry, finite analytical inventories, coupled constants and independent versus shared assay errors.

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Try

With total/free metal factor3 and shared gain1.1, a total-metal product is3.63×10⁻⁸ while the indicated free-ion product is1.21×10⁻⁸; true supplied Ksp is10⁻⁸.

Open the experiment

Predict

Does the shared gain cancel in a concentration product as it did in the earlier isomer ratio?

Experiment

Vary complexation factor and common gain separately. Compare total and free products, propagate independent2%/3% reading bounds and identify the external information needed to infer true Ksp.

Test whether an assay reports the species in Ksp

Worked teaching model · synthetic values · Synthetic saturated packet has free M=free X=10⁻⁴ M and true Ksp10⁻⁸. Ligand binds metal so total M=factor*free M, without consuming X; independent reservoir supplies the excess metal/ligand inventory. Observed total/free metal and X share a chosen gain. No real assay, filtration or physical concentration procedure.

Reset keeps your prediction, captures and practice history.

apparent normalized producttotal/free dissolved-metal factor055.000000e-80
  • Total-metal apparent product versus complexation factor: solid
  • Free-metal apparent product: long dashes

Axes rescale. A solubility comparison, depletion constraint and conditional reservoir scan have different horizontal meanings. Read the axes; these are not measured time traces.

All calculated plot coordinates
Total-metal apparent product versus complexation factor
total/free dissolved-metal factorapparent normalized product
11.000000e-8
1.0666671.066667e-8
1.1333331.133333e-8
1.21.200000e-8
1.2666671.266667e-8
1.3333331.333333e-8
1.41.400000e-8
1.4666671.466667e-8
1.5333331.533333e-8
1.61.600000e-8
1.6666671.666667e-8
1.7333331.733333e-8
1.81.800000e-8
1.8666671.866667e-8
1.9333331.933333e-8
22.000000e-8
2.0666672.066667e-8
2.1333332.133333e-8
2.22.200000e-8
2.2666672.266667e-8
2.3333332.333333e-8
2.42.400000e-8
2.4666672.466667e-8
2.5333332.533333e-8
2.62.600000e-8
2.6666672.666667e-8
2.7333332.733333e-8
2.82.800000e-8
2.8666672.866667e-8
2.9333332.933333e-8
33.000000e-8
3.0666673.066667e-8
3.1333333.133333e-8
3.23.200000e-8
3.2666673.266667e-8
3.3333333.333333e-8
3.43.400000e-8
3.4666673.466667e-8
3.5333333.533333e-8
3.63.600000e-8
3.6666673.666667e-8
3.7333333.733333e-8
3.83.800000e-8
3.8666673.866667e-8
3.9333333.933333e-8
44.000000e-8
4.0666674.066667e-8
4.1333334.133333e-8
4.24.200000e-8
4.2666674.266667e-8
4.3333334.333333e-8
4.44.400000e-8
4.4666674.466667e-8
4.5333334.533333e-8
4.64.600000e-8
4.6666674.666667e-8
4.7333334.733333e-8
4.84.800000e-8
4.8666674.866667e-8
4.9333334.933333e-8
55.000000e-8
Free-metal apparent product
total/free dissolved-metal factorapparent normalized product
11.000000e-8
1.0666671.000000e-8
1.1333331.000000e-8
1.21.000000e-8
1.2666671.000000e-8
1.3333331.000000e-8
1.41.000000e-8
1.4666671.000000e-8
1.5333331.000000e-8
1.61.000000e-8
1.6666671.000000e-8
1.7333331.000000e-8
1.81.000000e-8
1.8666671.000000e-8
1.9333331.000000e-8
21.000000e-8
2.0666671.000000e-8
2.1333331.000000e-8
2.21.000000e-8
2.2666671.000000e-8
2.3333331.000000e-8
2.41.000000e-8
2.4666671.000000e-8
2.5333331.000000e-8
2.61.000000e-8
2.6666671.000000e-8
2.7333331.000000e-8
2.81.000000e-8
2.8666671.000000e-8
2.9333331.000000e-8
31.000000e-8
3.0666671.000000e-8
3.1333331.000000e-8
3.21.000000e-8
3.2666671.000000e-8
3.3333331.000000e-8
3.41.000000e-8
3.4666671.000000e-8
3.5333331.000000e-8
3.61.000000e-8
3.6666671.000000e-8
3.7333331.000000e-8
3.81.000000e-8
3.8666671.000000e-8
3.9333331.000000e-8
41.000000e-8
4.0666671.000000e-8
4.1333331.000000e-8
4.21.000000e-8
4.2666671.000000e-8
4.3333331.000000e-8
4.41.000000e-8
4.4666671.000000e-8
4.5333331.000000e-8
4.61.000000e-8
4.6666671.000000e-8
4.7333331.000000e-8
4.81.000000e-8
4.8666671.000000e-8
4.9333331.000000e-8
51.000000e-8
Calculated under the declared model
QuantityValueUnit
Observed total dissolved metal0.0003mol/L
Observed free metal0.0001mol/L
Observed free anion0.0001mol/L
Naive total-metal product3.000000e-8dimensionless
Free-species product with shared gain1.000000e-8dimensionless
Underlying supplied Ksp1.000000e-8dimensionless
Independent2%/3% lower product bound9.506000e-9dimensionless
Independent2%/3% upper product bound1.050600e-8dimensionless

Filtering out solid does not automatically make a dissolved-metal assay specific to the free ion.

Complexes remain part of the dissolved total while only free ions enter this Ksp. A shared multiplicative concentration gain does not cancel in a product: it is squared. The2%/3% bounds describe separate relative reading limits around indicated free species, not calibration accuracy, confidence or true-K coverage when a shared bias persists. Independent speciation and calibration evidence are needed.

These calculated packets are not performed chemical measurements or separation procedures. Keep feasible inventories, activity conventions, fitting observations and independent checks distinct.

Filtering out solid does not automatically make a dissolved-metal assay specific to the free ion.

Complexes remain part of the dissolved total while only free ions enter this Ksp. A shared multiplicative concentration gain does not cancel in a product: it is squared. The2%/3% bounds describe separate relative reading limits around indicated free species, not calibration accuracy, confidence or true-K coverage when a shared bias persists. Independent speciation and calibration evidence are needed.

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Ask what the instrument reports

The synthetic saturated packet has true free metal and free anion each10⁻⁴ M. A dissolved complex raises total metal by a controlled factor while an external reservoir supplies the extra metal/ligand inventory. This is deliberately not the isolated dissolution-only amount balance from the preceding example.

A total dissolved-metal measurement counts free and complexed metal. Removing solid does not necessarily remove dissolved complexes or make the assay specific to the free ion. Inserting total metal into a free-ion solubility product therefore answers the wrong question even when the numerical assay is precise.

Separate species error from calibration error

At total/free factor3 and common positive gain1.1, the total-metal product is3×1.1²×10⁻⁸=3.63×10⁻⁸. Using the indicated free species removes the factor3 error but still gives1.21×10⁻⁸. A shared gain squares in a product; it does not cancel as it would in a same-power ratio.

Independent relative reading bounds of2% for metal and3% for anion give indicated-product multipliers .98×.97 and1.02×1.03. They do not include a hidden shared calibration bias or establish statistical confidence. More repeated readings with the same gain need not resolve either problem.

Transfer and check

Choose total/free factor4 and gain.90. The naive product is3.24×10⁻⁸, while the indicated free-ion product is.81×10⁻⁸. Explain why a free-species assay and an independent concentration reference address different uncertainties. Propose which check should be retained before treating a fitted product as an actual thermodynamic constant.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. All cases, prose and diagrams are original. Fictional species and explicitly ideal or supplied-activity constraints are not measured solubilities, real assay performance or safe physical separation protocols. No material handling, mixing, filtration or reagent procedure is requested. Retain free versus total species, inventory constraints, boundary conditions and independent evidence limits.

Apply

A shared concentration gain g in both free-ion readings multiplies their product by:

Public practice checks a specific claim. It does not establish independent mastery.

Why does filtering out solid leave the dissolved-complex versus free-ion ambiguity unresolved?

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Track a protonated anion family under an imposed pHNext: Count a disclosed entropy contribution without equating it to disorder
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