SCIENCE · INVESTIGATE AND EXPLAIN

Increase dissolved total while decreasing free metal

Complexed and free dissolved metal play different roles in a solubility product.

Learning outcome: Vary reservoir-controlled free ligand at two common-ion levels. Verify free-ion Ksp, total metal partition and the ligand-reservoir boundary rather than substituting total metal into 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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At free L=.002 M and beta10⁴ M⁻¹, total/free metal factor is21. With no common X and Ksp10⁻⁸, total solubility√(21×10⁻⁸)≈.000458258 M while free metal≈.0000218218 M.

Open the experiment

Predict

Can total metal rise while the free metal in Ksp falls?

Your prediction and reason

Experiment

Vary reservoir-controlled free ligand at two common-ion levels. Verify free-ion Ksp, total metal partition and the ligand-reservoir boundary rather than substituting total metal into Ksp.

Couple solubility with a controlled free-ligand reservoir

Worked teaching model · synthetic values · Fictional MX(s)⇌M++X− with Ksp10⁻⁸ and M++L⇌ML+ with concentration beta10⁴ M⁻¹. Free ligand is reservoir-controlled and neutral; excess solid. Total dissolved metal s=Mfree(1+beta L), total free X=X0+s; no other X species. Not a finite total-ligand addition.

Externally maintained free ligand: 2 mmol/L Initially supplied common anion: 0 mmol/L

Reset keeps your prediction, captures and practice history.

Total dissolved metal: solid
Free dissolved metal: 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 dissolved metal
externally maintained free ligand (mmol/L)	metal concentration (mol/L)
0	1.000000e-4
0.1666667	0.0001632993
0.3333333	0.0002081666
0.5	0.000244949
0.6666667	0.0002768875
0.8333333	0.000305505
1	0.0003316625
1.166667	0.0003559026
1.333333	0.0003785939
1.5	0.0004
1.666667	0.0004203173
1.833333	0.0004396969
2	0.0004582576
2.166667	0.0004760952
2.333333	0.0004932883
2.5	0.000509902
2.666667	0.0005259911
2.833333	0.0005416026
3	0.0005567764
3.166667	0.0005715476
3.333333	0.0005859465
3.5	0.0006
3.666667	0.0006137318
3.833333	0.0006271629
4	0.0006403124
4.166667	0.0006531973
4.333333	0.0006658328
4.5	0.000678233
4.666667	0.0006904105
4.833333	0.0007023769
5	0.0007141428
5.166667	0.000725718
5.333333	0.0007371115
5.5	0.0007483315
5.666667	0.0007593857
5.833333	0.0007702813
6	0.000781025
6.166667	0.0007916228
6.333333	0.0008020806
6.5	0.0008124038
6.666667	0.0008225975
6.833333	0.0008326664
7	0.000842615
7.166667	0.0008524475
7.333333	0.0008621678
7.5	0.0008717798
7.666667	0.0008812869
7.833333	0.0008906926
8	0.0009
8.166667	0.0009092121
8.333333	0.0009183318
8.5	0.0009273618
8.666667	0.0009363048
8.833333	0.0009451631
9	0.0009539392
9.166667	0.0009626353
9.333333	0.0009712535
9.5	0.0009797959
9.666667	0.0009882645
9.833333	0.0009966611
10	0.001004988

Free dissolved metal
externally maintained free ligand (mmol/L)	metal concentration (mol/L)
0	1.000000e-4
0.1666667	6.123724e-5
0.3333333	4.803845e-5
0.5	4.082483e-5
0.6666667	3.611576e-5
0.8333333	3.273268e-5
1	3.015113e-5
1.166667	2.809757e-5
1.333333	2.641353e-5
1.5	2.500000e-5
1.666667	2.379155e-5
1.833333	2.274294e-5
2	2.182179e-5
2.166667	2.100420e-5
2.333333	2.027212e-5
2.5	1.961161e-5
2.666667	1.901173e-5
2.833333	1.846372e-5
3	1.796053e-5
3.166667	1.749636e-5
3.333333	1.706640e-5
3.5	1.666667e-5
3.666667	1.629376e-5
3.833333	1.594482e-5
4	1.561738e-5
4.166667	1.530931e-5
4.333333	1.501879e-5
4.5	1.474420e-5
4.666667	1.448414e-5
4.833333	1.423737e-5
5	1.400280e-5
5.166667	1.377946e-5
5.333333	1.356647e-5
5.5	1.336306e-5
5.666667	1.316854e-5
5.833333	1.298227e-5
6	1.280369e-5
6.166667	1.263228e-5
6.333333	1.246757e-5
6.5	1.230915e-5
6.666667	1.215661e-5
6.833333	1.200961e-5
7	1.186782e-5
7.166667	1.173093e-5
7.333333	1.159867e-5
7.5	1.147079e-5
7.666667	1.134704e-5
7.833333	1.122722e-5
8	1.111111e-5
8.166667	1.099853e-5
8.333333	1.088931e-5
8.5	1.078328e-5
8.666667	1.068028e-5
8.833333	1.058018e-5
9	1.048285e-5
9.166667	1.038815e-5
9.333333	1.029597e-5
9.5	1.020621e-5
9.666667	1.011875e-5
9.833333	1.003350e-5
10	9.950372e-6

Calculated under the declared model
Quantity	Value	Unit
Total dissolved metal	0.0004582576	mol/L
Free metal concentration	2.182179e-5	mol/L
Complexed metal concentration	0.0004364358	mol/L
Free anion concentration	0.0004582576	mol/L
Recovered free-ion product	1.000000e-8	dimensionless
Total/free dissolved-metal ratio	21	ratio

Complexation can raise total solubility while lowering the free metal concentration entering Ksp.

Use total metal=free metal(1+beta L), but keep only free M and free X in the solubility product. Adding dissolution and complex formation multiplies the normalized constants after canceling free M; it does not mean total dissolved metal can replace free M in Ksp. The ligand reservoir must supply bound ligand; a finite dose requires the previous mass-balance model.

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

Complexation can raise total solubility while lowering the free metal concentration entering Ksp.

Observation to attach to a capture

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

Understand

Combine reactions without conflating species

The fictional solid MX produces M+ and X− with Ksp10⁻⁸. Free M binds neutral ligand L with concentration beta10⁴ M⁻¹. At controlled free L, ML=beta Mfree L, so total dissolved metal is Mfree(1+beta L). The ligand reservoir must supply any bound ligand while maintaining the free concentration.

If s is total dissolved formula-unit concentration and X0 is initial common anion, free X=X0+s. Therefore Ksp = (s/(1+beta L)) × (X0+s) under the normalized ideal convention. Only free M enters the original dissolution product.

Compare totals and free species

At L=.002 M and X0=0, the partition factor is21. Total s≈.000458258 M, free M≈.0000218218 M and bound ML≈.000436436 M. Their sum recovers total metal; free M times free X recovers10⁻⁸. Without ligand, both free concentrations would be .000100 M.

Adding dissolution and complex formation cancels free M and multiplies their normalized constants. That algebra predicts a coupled tendency but does not remove the need for material balances. A finite total-ligand supply would not remain unchanged as complex forms.

Transfer and check

At controlled L=.001 M and no common X, the factor is11. Predict total s=√(11×10⁻⁸)≈.000331662 M and free M=s/11≈.0000301511 M. Explain why a total-metal assay would suggest a falsely large Ksp if its reading were inserted as though it measured only free M.

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.

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Why would a finite ligand addition require a different mass-balance solution?

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Previous: Conserve a ligand before claiming quantitative complexation Next: Track a protonated anion family under an imposed pH

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