SCIENCE · INVESTIGATE AND EXPLAIN

Conserve a ligand before claiming quantitative complexation

A large formation constant cannot supply ligand that is not present.

Learning outcome: Vary total ligand through the metal amount and vary beta. Recover both free inventories, compare the excess-ligand approximation and reject any complex amount above either supply.

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 Mtotal=Ltotal=.001 M and concentration beta10⁴ M⁻¹, solve x=beta(.001−x)² to obtain ML≈.000729844 M, not all .001 M.

Open the experiment

Predict

Can treating total ligand as free ligand overstate the bound-metal fraction near equal supplies?

Your prediction and reason

Experiment

Vary total ligand through the metal amount and vary beta. Recover both free inventories, compare the excess-ligand approximation and reject any complex amount above either supply.

Conserve both metal and a finite ligand supply

Worked teaching model · synthetic values · Fictional M++L⇌ML+, neutral monodentate ligand, total M=.001 M and total L selected. Concentration-form beta=[ML]/([M][L]) has units M⁻¹; thermodynamic normalized formation constant is beta*c°. Ideal solution, fixed temperature, no other complexes, protonation or solids.

Total ligand amount per volume: 1 mmol/L Log10 concentration formation constant: 4 log10(beta / 1 M⁻¹)

Reset keeps your prediction, captures and practice history.

Exact bound fraction versus total ligand: 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

Exact bound fraction versus total ligand
total ligand (mmol/L)	metal bound fraction
0.5	0.4258343
0.5416667	0.4573767
0.5833333	0.4880151
0.625	0.5176721
0.6666667	0.5462709
0.7083333	0.5737366
0.75	0.6
0.7916667	0.625
0.8333333	0.6486869
0.875	0.6710253
0.9166667	0.6919958
0.9583333	0.7115967
1	0.7298438
1.041667	0.7467695
1.083333	0.7624209
1.125	0.776857
1.166667	0.7901457
1.208333	0.8023608
1.25	0.8135792
1.291667	0.8238781
1.333333	0.8333333
1.375	0.8420177
1.416667	0.85
1.458333	0.8573444
1.5	0.8641101
1.541667	0.8703513
1.583333	0.8761174
1.625	0.881453
1.666667	0.8863983
1.708333	0.8909896
1.75	0.8952595
1.791667	0.8992372
1.833333	0.9029488
1.875	0.906418
1.916667	0.9096658
1.958333	0.9127112
2	0.9155711
2.041667	0.918261
2.083333	0.9207945
2.125	0.9231842
2.166667	0.9254413
2.208333	0.927576
2.25	0.9295974
2.291667	0.931514
2.333333	0.9333333
2.375	0.9350623
2.416667	0.9367072
2.458333	0.9382739
2.5	0.9397675
2.541667	0.9411929
2.583333	0.9425545
2.625	0.9438563
2.666667	0.9451021
2.708333	0.9462954
2.75	0.9474393
2.791667	0.9485366
2.833333	0.9495902
2.875	0.9506026
2.916667	0.951576
2.958333	0.9525126
3	0.9534144

Calculated under the declared model
Quantity	Value	Unit
Complex ML concentration	0.0007298438	mol/L
Unbound metal concentration	0.0002701562	mol/L
Unbound ligand concentration	0.0002701562	mol/L
Metal-bound fraction	0.7298438	fraction
Recovered concentration formation constant	10000	L/mol
Excess-ligand approximation to bound metal	0.0009090909	mol/L

A strong formation constant does not eliminate a finite ligand shortage.

Solve x=beta(Mtotal−x)(Ltotal−x) within both inventories. Replacing free ligand by total ligand assumes ligand depletion is negligible; it may overpredict bound metal near equal supplies. The ligand donates an electron pair and the metal accepts it; a coordinate bond origin is not a separate weaker bond class once formed. Other stoichiometries and competing ligands require extra equations.

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

A strong formation constant does not eliminate a finite ligand shortage.

Observation to attach to a capture

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

Understand

Make the Lewis interaction quantitative

In fictional M++L⇌ML+, a neutral monodentate ligand donates an electron pair and the metal accepts it. A coordinate-bond description records where the bonding pair originated; it does not establish a separate inherently weaker bond class after formation. Other ligand denticities or complex stoichiometries need different ledgers.

Let complex concentration be x. Free metal is Mtotal−x and free ligand is Ltotal−x. The concentration-form formation constant beta has units M⁻¹ in x=beta(Mtotal−x)(Ltotal−x). The normalized thermodynamic constant is beta c°, dimensionless under the declared standard.

Solve both inventories together

With both totals .001 M and beta10⁴ M⁻¹, the feasible root is .000729844 M. Each free concentration is about.000270156 M. Replacing free ligand by its .001 M total gives the excess-ligand estimate .000909091 M, which is too high because ligand depletion is substantial.

Even an extremely favorable constant cannot bind more metal than the available1:1 ligand amount. The feasible interval for x is bounded by both totals. This constraint is more fundamental than a rule that a large constant means complete reaction.

Transfer and check

Suppose total metal is .00200 M and ligand .00100 M. Before any equation, bound metal cannot exceed .00100 M. Explain why an apparent predicted .00180 M complex is impossible even if beta is very large. Then identify the independent observation needed to distinguish unbound ligand from a total-ligand assay when checking the formation quotient.

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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Which additional equations would a second ligand or a protonated ligand require?

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Previous: Test a quantitative selectivity target against the second onset Next: Increase dissolved total while decreasing free metal

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