SCIENCE · INVESTIGATE AND EXPLAIN

Validate an approximation using depletion and output error

A small equilibrium constant alone does not justify neglecting a concentration change.

Learning outcome: Vary analytical concentration at fixed Kd. Retain exact and approximate roots, x/C and relative output error; identify a case where the approximation creates more product than the supply permits.

Starting recap

Refresh dimensionless activities, component balances, admissible roots, normalized partial pressures and independent versus shared measurement errors.

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Try

For C=.100 M and concentration-form Kd=.001 M, exact x≈.00951249 M, while √(KdC)=.0100 M overestimates it by about 5.12%.

Open the experiment

Predict

Does a small numeric Kd guarantee a small fraction dissociated for every starting concentration?

Your prediction and reason

Experiment

Vary analytical concentration at fixed Kd. Retain exact and approximate roots, x/C and relative output error; identify a case where the approximation creates more product than the supply permits.

Audit a small-change approximation against an exact root

Worked teaching model · synthetic values · Fictional ideal AB⇌A+B, initial AB=C and products zero. Kd=x²/(C−x) has concentration units in this explicitly concentration-form convention; dimensionless K=Kd/c°. No acid/base, charge or solvent-ionization interpretation is assumed.

Analytical AB concentration: 0.1 mol/L Dissociation quotient Kd: 0.001 mol/L

Reset keeps your prediction, captures and practice history.

Exact root versus analytical concentration: solid
Square-root approximation: long dashes

Axes rescale. A quotient comparison, inventory intersection and relaxation trajectory have different horizontal meanings; inspect labels rather than treating every curve as elapsed time.

All calculated plot coordinates

Exact root versus analytical concentration
fraction of analytical concentration range .01 to .10 M	dissociation concentration (mol/L)
0	0.002701562
0.01666667	0.002927827
0.03333333	0.003140055
0.05	0.003340573
0.06666667	0.003531129
0.08333333	0.003713075
0.1	0.003887482
0.1166667	0.004055217
0.1333333	0.004216991
0.15	0.004373397
0.1666667	0.004524938
0.1833333	0.00467204
0.2	0.004815073
0.2166667	0.004954356
0.2333333	0.00509017
0.25	0.005222762
0.2666667	0.00535235
0.2833333	0.00547913
0.3	0.005603278
0.3166667	0.00572495
0.3333333	0.005844289
0.35	0.005961424
0.3666667	0.006076473
0.3833333	0.006189544
0.4	0.006300735
0.4166667	0.006410137
0.4333333	0.006517834
0.45	0.006623903
0.4666667	0.006728416
0.4833333	0.006831439
0.5	0.006933034
0.5166667	0.00703326
0.5333333	0.007132169
0.55	0.007229812
0.5666667	0.007326238
0.5833333	0.00742149
0.6	0.00751561
0.6166667	0.007608637
0.6333333	0.00770061
0.65	0.007791562
0.6666667	0.007881527
0.6833333	0.007970537
0.7	0.008058621
0.7166667	0.008145808
0.7333333	0.008232125
0.75	0.008317596
0.7666667	0.008402247
0.7833333	0.0084861
0.8	0.008569179
0.8166667	0.008651503
0.8333333	0.008733093
0.85	0.008813968
0.8666667	0.008894147
0.8833333	0.008973648
0.9	0.009052487
0.9166667	0.00913068
0.9333333	0.009208244
0.95	0.009285193
0.9666667	0.009361541
0.9833333	0.009437303
1	0.009512492

Square-root approximation
fraction of analytical concentration range .01 to .10 M	dissociation concentration (mol/L)
0	0.003162278
0.01666667	0.003391165
0.03333333	0.003605551
0.05	0.003807887
0.06666667	0.004
0.08333333	0.0041833
0.1	0.004358899
0.1166667	0.004527693
0.1333333	0.004690416
0.15	0.00484768
0.1666667	0.005
0.1833333	0.005147815
0.2	0.005291503
0.2166667	0.00543139
0.2333333	0.005567764
0.25	0.005700877
0.2666667	0.005830952
0.2833333	0.005958188
0.3	0.006082763
0.3166667	0.006204837
0.3333333	0.006324555
0.35	0.006442049
0.3666667	0.006557439
0.3833333	0.006670832
0.4	0.00678233
0.4166667	0.006892024
0.4333333	0.007
0.45	0.007106335
0.4666667	0.007211103
0.4833333	0.007314369
0.5	0.007416198
0.5166667	0.007516648
0.5333333	0.007615773
0.55	0.007713624
0.5666667	0.00781025
0.5833333	0.007905694
0.6	0.008
0.6166667	0.008093207
0.6333333	0.008185353
0.65	0.008276473
0.6666667	0.0083666
0.6833333	0.008455767
0.7	0.008544004
0.7166667	0.008631338
0.7333333	0.008717798
0.75	0.008803408
0.7666667	0.008888194
0.7833333	0.008972179
0.8	0.009055385
0.8166667	0.009137833
0.8333333	0.009219544
0.85	0.009300538
0.8666667	0.009380832
0.8833333	0.009460444
0.9	0.009539392
0.9166667	0.009617692
0.9333333	0.00969536
0.95	0.00977241
0.9666667	0.009848858
0.9833333	0.009924717
1	0.01

Calculated under the declared model
Quantity	Value	Unit
Exact dissociation concentration	0.009512492	mol/L
Small-change approximation	0.01	mol/L
Dissociated fraction	0.09512492	fraction
Relative approximation error	0.05124922	fraction
Recovered concentration quotient	0.001	mol/L

Replacing C−x by C must be checked after solving, not justified by calling a constant small.

The approximation always overestimates x in this model because it neglects depletion. Inspect x/C and the actual output error; a chosen 5% depletion convention is not a guarantee that every downstream derived quantity has less than 5% error. The stable exact root avoids cancellation and does not require that approximation.

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

Replacing C−x by C must be checked after solving, not justified by calling a constant small.

Observation to attach to a capture

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

Understand

Declare the concentration-form convention

The fictional dissociation AB⇌A+B starts with AB=C and no products. Define Kd=x²/(C−x), with units mol/L. The corresponding thermodynamic constant is dimensionless K=Kd/c° for ideal concentration activities. This is not an acid-ionization calculation; charge balances and solvent ions are absent by design.

Derive and test the shortcut

Replacing C−x by C gives x_approx=√(KdC). The exact positive root solves x²+Kd x−KdC=0. A numerically stable expression is 2KdC/[√(Kd²+4KdC)+Kd], which avoids subtracting nearly equal numbers.

At C=.100 and Kd=.001 M, exact x≈.00951249 M versus .0100 M approximate. Dissociation consumes about 9.51% of the original AB; the approximation overstates x by about 5.12%. The chosen acceptable error must come from the intended inference, not from calling .001 a small number.

Inspect the failure region

At lower C, the same constant can produce a large fractional depletion. An approximation can even request an amount above the starting inventory. Compare both curves and identify where a five-percent depletion rule would reject the shortcut. Such a rule is a screening convention, not a universal guarantee for nonlinear quantities calculated from x.

Transfer and check

For C=.0100 M and Kd=.0100 M, exact x=(√5−1)×.005≈.00618034 M, while the shortcut gives .0100 M. Its relative error is about 61.8%. Reinsert each candidate into the original quotient; the approximate candidate exhausts AB and makes that quotient undefined. Reject it rather than reporting a plausible-looking rounded value.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. These original hypothetical packets use explicitly defined activities, inventories and response assumptions. No material is mixed, heated, pressurized or measured. Numerical compatibility does not certify practical technique or uniquely establish a real chemical mechanism. Retain your prediction, feasible inventory, quotient check and changed-case interpretation.

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Why should a depletion convention and an error bound on a downstream observable not be treated as identical?

Your explanation and limitations

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Previous: Close a shared inventory across two equilibrium relations Next: Check phase existence before imposing saturation

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