SCIENCE · INVESTIGATE AND EXPLAIN

Normalize species changes before comparing reaction rates

Different species can change at different numerical rates even while describing one chemical transformation.

Learning outcome: Compare durations 1, 2 and 4 s from the same start. Normalize A disappearance by 2, compare B appearance, and distinguish interval average from the start and midpoint derivatives.

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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Try

For 2A→B with [A]=exp(−.10t) M, interval 2→4 s has normalized rate [exp(−.2)−exp(−.4)]/4≈.037103 M/s.

Open the experiment

Predict

Will the numerical disappearance rate of A equal the appearance rate of B when the net coefficient of A is 2?

Your prediction and reason

Experiment

Compare durations 1, 2 and 4 s from the same start. Normalize A disappearance by 2, compare B appearance, and distinguish interval average from the start and midpoint derivatives.

Normalize species changes to one reaction rate

Worked teaching model · synthetic values · Hypothetical 2A→B at fixed volume: [A]=1.00exp(−.10t) M and [B]=(1−[A])/2 M. Reaction rate is −(1/2)d[A]/dt=d[B]/dt. Constant temperature, irreversible supplied law; finite interval rates are not instantaneous derivatives.

Interval start: 2 s Interval duration: 2 s

Reset keeps your prediction, captures and practice history.

A: solid
B: long dashes

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

A
time (s)	concentration (mol/L)
0	1
0.12	0.9880717
0.24	0.9762857
0.36	0.9646403
0.48	0.9531338
0.6	0.9417645
0.72	0.9305309
0.84	0.9194313
0.96	0.908464
1.08	0.8976276
1.2	0.8869204
1.32	0.876341
1.44	0.8658877
1.56	0.8555592
1.68	0.8453538
1.8	0.8352702
1.92	0.8253069
2.04	0.8154624
2.16	0.8057353
2.28	0.7961243
2.4	0.7866279
2.52	0.7772447
2.64	0.7679735
2.76	0.7588129
2.88	0.7497616
3	0.7408182
3.12	0.7319815
3.24	0.7232502
3.36	0.7146231
3.48	0.7060989
3.6	0.6976763
3.72	0.6893542
3.84	0.6811314
3.96	0.6730067
4.08	0.6649789
4.2	0.6570468
4.32	0.6492094
4.44	0.6414654
4.56	0.6338138
4.68	0.6262535
4.8	0.6187834
4.92	0.6114024
5.04	0.6041094
5.16	0.5969034
5.28	0.5897834
5.4	0.5827483
5.52	0.5757971
5.64	0.5689288
5.76	0.5621424
5.88	0.555437
6	0.5488116
6.12	0.5422653
6.24	0.535797
6.36	0.5294058
6.48	0.5230909
6.6	0.5168513
6.72	0.5106862
6.84	0.5045946
6.96	0.4985756
7.08	0.4926285
7.2	0.4867523
7.32	0.4809461
7.44	0.4752093
7.56	0.4695408
7.68	0.46394
7.8	0.458406
7.92	0.452938
8.04	0.4475352
8.16	0.4421969
8.28	0.4369223
8.4	0.4317105
8.52	0.426561
8.64	0.4214728
8.76	0.4164454
8.88	0.4114779
9	0.4065697
9.12	0.40172
9.24	0.3969281
9.36	0.3921935
9.48	0.3875153
9.6	0.3828929
9.72	0.3783256
9.84	0.3738129
9.96	0.3693539
10.08	0.3649481
10.2	0.3605949
10.32	0.3562937
10.44	0.3520437
10.56	0.3478444
10.68	0.3436952
10.8	0.3395955
10.92	0.3355447
11.04	0.3315423
11.16	0.3275875
11.28	0.32368
11.4	0.319819
11.52	0.3160041
11.64	0.3122347
11.76	0.3085103
11.88	0.3048303
12	0.3011942

B
time (s)	concentration (mol/L)
0	0
0.12	0.005964144
0.24	0.01185715
0.36	0.01767985
0.48	0.02343311
0.6	0.02911773
0.72	0.03473455
0.84	0.04028437
0.96	0.04576799
1.08	0.0511862
1.2	0.05653978
1.32	0.0618295
1.44	0.06705613
1.56	0.0722204
1.68	0.07732308
1.8	0.08236489
1.92	0.08734657
2.04	0.09226881
2.16	0.09713235
2.28	0.1019379
2.4	0.1066861
2.52	0.1113776
2.64	0.1160132
2.76	0.1205935
2.88	0.1251192
3	0.1295909
3.12	0.1340092
3.24	0.1383749
3.36	0.1426884
3.48	0.1469506
3.6	0.1511618
3.72	0.1553229
3.84	0.1594343
3.96	0.1634967
4.08	0.1675106
4.2	0.1714766
4.32	0.1753953
4.44	0.1792673
4.56	0.1830931
4.68	0.1868732
4.8	0.1906083
4.92	0.1942988
5.04	0.1979453
5.16	0.2015483
5.28	0.2051083
5.4	0.2086259
5.52	0.2121015
5.64	0.2155356
5.76	0.2189288
5.88	0.2222815
6	0.2255942
6.12	0.2288674
6.24	0.2321015
6.36	0.2352971
6.48	0.2384545
6.6	0.2415743
6.72	0.2446569
6.84	0.2477027
6.96	0.2507122
7.08	0.2536858
7.2	0.2566239
7.32	0.2595269
7.44	0.2623954
7.56	0.2652296
7.68	0.26803
7.8	0.270797
7.92	0.273531
8.04	0.2762324
8.16	0.2789015
8.28	0.2815389
8.4	0.2841447
8.52	0.2867195
8.64	0.2892636
8.76	0.2917773
8.88	0.2942611
9	0.2967152
9.12	0.29914
9.24	0.3015359
9.36	0.3039033
9.48	0.3062424
9.6	0.3085536
9.72	0.3108372
9.84	0.3130936
9.96	0.315323
10.08	0.3175259
10.2	0.3197025
10.32	0.3218532
10.44	0.3239782
10.56	0.3260778
10.68	0.3281524
10.8	0.3302022
10.92	0.3322276
11.04	0.3342289
11.16	0.3362062
11.28	0.33816
11.4	0.3400905
11.52	0.3419979
11.64	0.3438826
11.76	0.3457448
11.88	0.3475848
12	0.3494029

Calculated under the declared model
Quantity	Value	Unit
A interval disappearance rate	0.07420535	mol/(L s)
Normalized interval reaction rate	0.03710268	mol/(L s)
B interval appearance rate	0.03710268	mol/(L s)
Instantaneous normalized rate at interval start	0.04093654	mol/(L s)
Instantaneous normalized rate at interval midpoint	0.03704091	mol/(L s)

Stoichiometric normalization makes species-based estimates of the same reaction rate agree.

The finite-interval average generally differs from either endpoint derivative; the midpoint derivative is only an approximation to that average. This supplied exponential decrease is first order in A disappearance even though the net coefficient is 2; a net equation does not determine an empirical rate exponent.

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

Stoichiometric normalization makes species-based estimates of the same reaction rate agree.

Observation to attach to a capture

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

Understand

Define one reaction extent rate

Different species can change at different numerical rates even while describing one chemical transformation. For 2A→B at fixed volume, normalize disappearance of A by 2 and appearance of B by1. Both then report the same reaction-progress rate. A negative concentration change is not itself a negative forward reaction rate.

Reconcile a finite interval

The supplied irreversible packet has [A]=exp(−.10t) M and [B]=(1−[A])/2. From2 to4 s, A falls from.818731 to.670320 M. Its disappearance average is.074205 M/s, while B appears at.037103 M/s. Dividing the former by 2 reconciles the two accounts.

Distinguish averages from derivatives

The instantaneous normalized rate is.05exp(−.10t) M/s. At the start2s it is.040937 M/s; at midpoint3s it is.037041 M/s. The interval average is close to, but not exactly, the midpoint derivative. Shrinking a time interval can approach a local rate in a smooth model, while real reading noise complicates very short differences.

Separate stoichiometry from rate order

Here A disappearance is first order even though its net coefficient is 2. A net reaction sums a pathway and does not generally encode an elementary molecular collision. Only an explicitly elementary step licenses direct molecularity reasoning. This distinction will matter when multiple mechanisms fit the same empirical dependence.

Transfer and check

For a hypothetical3X→2Y, X disappears at.090 M/s. Predict the normalized progress rate.030 M/s and Y appearance.060 M/s. Check both against stoichiometric ratios before comparing them with any independently inferred kinetic law.

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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Why does a coefficient 2 in the net equation not establish a second-order empirical law?

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Previous: Bound an apparent molar mass without assuming its identity Next: Infer rate exponents through separate concentration interventions

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