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Derive a saturating rate from bound-catalyst conservation

The supplied candidate has rapid reversible binding C+A⇌CA followed by slower CA→C+P.

Learning outcome: Compare A=.1,.5,1.0 M at Kd=.5 and then Kd=.1. Derive free/bound catalyst amounts and compare the exact rate with the low-A approximation and saturation ceiling.

Starting recap

Distinguish a net reaction coefficient, empirical exponent and elementary step. Refresh logarithmic ratios, signed residuals, concentration/time dimensions and parameter identifiability.

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At free A=.50 M,Kd=.50 M,total catalyst .0100 M and kcat2 s⁻¹, bound catalyst.00500 M gives rate.0100 M/s.

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Predict

Can a catalyst-binding model predict indefinitely linear rate growth as free A becomes much larger than Kd?

Your prediction and reason

Experiment

Compare A=.1,.5,1.0 M at Kd=.5 and then Kd=.1. Derive free/bound catalyst amounts and compare the exact rate with the low-A approximation and saturation ceiling.

Derive a bound-intermediate rate using catalyst conservation

Worked teaching model · synthetic values · Supplied rapid pre-equilibrium C+A⇌CA followed by slow CA→C+P. Total catalyst .0100 M, kcat = 2.00 s⁻¹, free A controlled externally. Kd=[C][A]/[CA]; conservation gives [CA]=Ctotal[A]/(Kd+[A]). Approximation assumes binding equilibrates faster than turnover; no enzyme/clinical or actual catalyst experiment.

Free A concentration: 0.5 mol/L Dissociation constant Kd: 0.5 mol/L

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Rapid-pre-equilibrium prediction: solid

Axes rescale. A concentration trajectory, initial-rate comparison and inverse-temperature plot have different horizontal meanings; use the labels rather than assuming every curve shows elapsed time.

All calculated plot coordinates

Rapid-pre-equilibrium prediction
free A concentration (mol/L)	initial product rate (mol/(L s))
0	0
0.02	0.0007692308
0.04	0.001481481
0.06	0.002142857
0.08	0.002758621
0.1	0.003333333
0.12	0.003870968
0.14	0.004375
0.16	0.004848485
0.18	0.005294118
0.2	0.005714286
0.22	0.006111111
0.24	0.006486486
0.26	0.006842105
0.28	0.007179487
0.3	0.0075
0.32	0.007804878
0.34	0.008095238
0.36	0.008372093
0.38	0.008636364
0.4	0.008888889
0.42	0.009130435
0.44	0.009361702
0.46	0.009583333
0.48	0.009795918
0.5	0.01
0.52	0.01019608
0.54	0.01038462
0.56	0.01056604
0.58	0.01074074
0.6	0.01090909
0.62	0.01107143
0.64	0.01122807
0.66	0.01137931
0.68	0.01152542
0.7	0.01166667
0.72	0.01180328
0.74	0.01193548
0.76	0.01206349
0.78	0.0121875
0.8	0.01230769
0.82	0.01242424
0.84	0.01253731
0.86	0.01264706
0.88	0.01275362
0.9	0.01285714
0.92	0.01295775
0.94	0.01305556
0.96	0.01315068
0.98	0.01324324
1	0.01333333
1.02	0.01342105
1.04	0.01350649
1.06	0.01358974
1.08	0.01367089
1.1	0.01375
1.12	0.01382716
1.14	0.01390244
1.16	0.0139759
1.18	0.01404762
1.2	0.01411765
1.22	0.01418605
1.24	0.01425287
1.26	0.01431818
1.28	0.01438202
1.3	0.01444444
1.32	0.01450549
1.34	0.01456522
1.36	0.01462366
1.38	0.01468085
1.4	0.01473684
1.42	0.01479167
1.44	0.01484536
1.46	0.01489796
1.48	0.01494949
1.5	0.015
1.52	0.0150495
1.54	0.01509804
1.56	0.01514563
1.58	0.01519231
1.6	0.0152381
1.62	0.01528302
1.64	0.0153271
1.66	0.01537037
1.68	0.01541284
1.7	0.01545455
1.72	0.0154955
1.74	0.01553571
1.76	0.01557522
1.78	0.01561404
1.8	0.01565217
1.82	0.01568966
1.84	0.0157265
1.86	0.01576271
1.88	0.01579832
1.9	0.01583333
1.92	0.01586777
1.94	0.01590164
1.96	0.01593496
1.98	0.01596774
2	0.016

Calculated under the declared model
Quantity	Value	Unit
Bound catalyst CA	0.005	mol/L
Free catalyst C	0.005	mol/L
Total catalyst inventory	0.01	mol/L
Product formation rate	0.01	mol/(L s)
Low-A linear approximation	0.02	mol/(L s)
Saturation-rate ceiling	0.02	mol/(L s)

Catalyst conservation produces saturation instead of unlimited linear rate growth.

Solve Kd=[C][A]/[CA] with [C]+[CA]=Ctotal before substituting r=kcat[CA]. At A much smaller than Kd the rate is approximately first order in A; at A much larger it approaches zero-order dependence. A matching curve supports a candidate model, not a unique microscopic mechanism; the pre-equilibrium time-scale assumption needs independent evidence.

These calculated packets are not performed chemical or instrument measurements. Keep the hypothesis, fitting observations, withheld checks and exposure record distinct.

Catalyst conservation produces saturation instead of unlimited linear rate growth.

Observation to attach to a capture

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

Understand

Specify the proposed mechanism

The supplied candidate has rapid reversible binding C+A⇌CA followed by slower CA→C+P. C is regenerated rather than consumed overall. CA is a bound intermediate. The free reactant concentration is externally controlled in this initial-rate illustration; it is not silently equated with every possible analytical total.

Use conservation before substitution

The dissociation relation is Kd=[C][A]/[CA], and catalyst conservation gives [C]+[CA]=Ctotal. Eliminating free catalyst produces [CA]=Ctotal[A]/(Kd+[A]). Product rate is kcat[CA]. Ignoring conservation and setting every catalyst molecule free would miss saturation.

Calculate a half-occupied packet

At[A]=Kd=.50 M, half of the.0100 M catalyst is bound, so[CA]=.00500 M. With kcat=2.00 s⁻¹, rate is.0100 M/s. The saturation ceiling is kcat Ctotal=.0200 M/s, not an unlimited rate as A rises.

Derive limiting orders and check error

When A is much smaller than Kd, replace Kd+A approximately by Kd and obtain first-order dependence on A. When A is much larger, the ratio approaches one and rate becomes approximately independent of A. At A=Kd the low-A approximation overpredicts by a factor2, so naming an approximation is not enough; test its domain quantitatively.

Transfer and avoid mechanism overclaim

For Ctotal=.0200 M,kcat=3.00 s⁻¹,Kd=.20 M and free A=.60 M, bound fraction is.75 and rate.0450 M/s. A matching saturation curve is consistent with this candidate but does not prove it uniquely. The rapid-binding assumption needs evidence that equilibration is fast compared with turnover, and other pathways can produce similar macroscopic shapes.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. All numerical packets, prose and diagrams are original. Models are supplied/simulated, not measured rate constants or observed chemical mechanisms. No reactive substances, heating, optical instrument, catalytic preparation or physical chemical procedure is requested. Retain predictions, contrasting evidence, residuals and a changed-case explanation.

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

Which independent time-scale or species evidence would be needed before calling the rapid pre-equilibrium mechanism established?

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