SCIENCE · INVESTIGATE AND EXPLAIN

Separate common-ion suppression from a prescribed activity correction

A change in concentration needed for saturation need not mean the thermodynamic constant changed.

Learning outcome: Vary common-ion supply at gamma1, then vary gamma at a fixed supply. Recover activities and product rather than inferring Ksp from bare total amounts.

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

For ideal MX with Ksp10⁻⁸ and initial X=.001 M, solve s(.001+s)=10⁻⁸ to obtain s≈9.90195×10⁻⁶ M.

Open the experiment

Predict

Will lowering both prescribed ionic coefficients require higher concentrations to achieve the same activity product?

Your prediction and reason

Experiment

Vary common-ion supply at gamma1, then vary gamma at a fixed supply. Recover activities and product rather than inferring Ksp from bare total amounts.

Separate a common-ion inventory from an activity correction

Worked teaching model · synthetic values · Fictional MX⇌M++X−, Ksp10⁻⁸, c°1 M, excess solid and inert counterions. Initially X=a mM, M=0. Both coefficients equal the prescribed gamma; supporting electrolyte is an external stipulated condition, not inferred from these concentrations. No ion pairs or acid/base reactions.

Initially supplied common anion: 1 mmol/L Prescribed common ionic activity coefficient: 1 ratio

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Solubility versus supplied common anion: solid

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

Solubility versus supplied common anion
initial common anion (mmol/L)	formula-unit solubility (mol/L)
0	1.000000e-4
0.06666667	7.207592e-5
0.1333333	5.351838e-5
0.2	4.142136e-5
0.2666667	3.333333e-5
0.3333333	2.769840e-5
0.4	2.360680e-5
0.4666667	2.052577e-5
0.5333333	1.813346e-5
0.6	1.622777e-5
0.6666667	1.467688e-5
0.7333333	1.339181e-5
0.8	1.231056e-5
0.8666667	1.138880e-5
0.9333333	1.059404e-5
1	9.901951e-6
1.066667	9.294020e-6
1.133333	8.755883e-6
1.2	8.276253e-6
1.266667	7.846135e-6
1.333333	7.458281e-6
1.4	7.106781e-6
1.466667	6.786777e-6
1.533333	6.494234e-6
1.6	6.225775e-6
1.666667	5.978554e-6
1.733333	5.750155e-6
1.8	5.538514e-6
1.866667	5.341856e-6
1.933333	5.158649e-6
2	4.987562e-6
2.066667	4.827433e-6
2.133333	4.677245e-6
2.2	4.536102e-6
2.266667	4.403211e-6
2.333333	4.277871e-6
2.4	4.159458e-6
2.466667	4.047413e-6
2.533333	3.941237e-6
2.6	3.840481e-6
2.666667	3.744741e-6
2.733333	3.653653e-6
2.8	3.566885e-6
2.866667	3.484137e-6
2.933333	3.405138e-6
3	3.329638e-6
3.066667	3.257410e-6
3.133333	3.188245e-6
3.2	3.121954e-6
3.266667	3.058361e-6
3.333333	2.997305e-6
3.4	2.938637e-6
3.466667	2.882219e-6
3.533333	2.827925e-6
3.6	2.775638e-6
3.666667	2.725247e-6
3.733333	2.676652e-6
3.8	2.629759e-6
3.866667	2.584479e-6
3.933333	2.540732e-6
4	2.498439e-6

Calculated under the declared model
Quantity	Value	Unit
Formula-unit solubility	9.901951e-6	mol/L
Free metal concentration	9.901951e-6	mol/L
Total free anion concentration	0.001009902	mol/L
Metal activity	9.901951e-6	dimensionless
Anion activity	0.001009902	dimensionless
Recovered activity product	1.000000e-8	dimensionless

A common ion suppresses dissolution at fixed activity coefficients, while a prescribed coefficient change alters the concentration needed for the same product.

Solve gamma² s(X0+s)=Ksp rather than replacing total anion by its initial value without checking. Gamma is externally stipulated, not derived from this packet or held constant by a claim about real ionic strength. Neither extra solid mass nor a faster stirring rate changes the equilibrium constant in this model.

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

A common ion suppresses dissolution at fixed activity coefficients, while a prescribed coefficient change alters the concentration needed for the same product.

Observation to attach to a capture

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

Understand

Account for the added ion and the ion released

For MX⇌M++X−, initial common-anion concentration X0 is supplied with an inert counterion. Dissolving s adds s of each free ion, giving M=s and X=X0+s. With equal prescribed activity coefficients gamma, saturation requires gamma²s(X0+s)=Ksp under the1 M standard.

At gamma1, X0=.001 M and Ksp10⁻⁸, the stable positive root is about9.90195×10⁻⁶ M. Replacing X0+s by X0 gives10⁻⁵ M, a useful approximation here but not an identity. Without added common ion the solubility is10⁻⁴ M.

Keep activity and inventory effects distinct

A smaller supplied gamma requires a larger concentration product for the same Ksp. This does not imply Ksp changed. Nor does the control derive gamma from the displayed concentrations: an external supporting-electrolyte condition is stipulated. In an actual ionic solution, coefficient estimates need independent justification and may vary with composition.

Adding more pure solid while a pure solid phase already remains does not alter its unit activity or the fixed-temperature constant. Stirring can change the approach to equilibrium but is not a new thermodynamic constant.

Transfer and check

At gamma=.50 with no common ion, gamma²s²=10⁻⁸ gives s=.000200 M. At the same gamma with X0=.001 M, solve s(.001+s)=4×10⁻⁸, obtaining s≈3.85165×10⁻⁵ M. Explain how common-ion suppression and the coefficient correction can act in opposite directions without contradiction.

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

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

What supporting-electrolyte evidence would be needed to justify holding gamma fixed in a real comparison?

Your explanation and limitations

Continue

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Previous: Compare solubilities only after reconstructing ion inventories Next: Dilute both packets before deciding how much can precipitate

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