SCIENCE · INVESTIGATE AND EXPLAIN

Test integrated kinetic models with withheld raw concentrations

Any flexible two-parameter description can look convincing when judged only on the points used to set it.

Learning outcome: For each generating order, compare candidate endpoint fits and raw-scale held-out residuals. Perturb only the final reading by+.02 M and retain how the inferred constants and residual patterns change.

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

A first-order source with k=.05 s⁻¹ gives A8=exp(−.4)=.670320 M. Zero-, first- and second-order candidates can each fit endpoints0 and8s, but their predictions at2,4,6s differ.

Open the experiment

Predict

Can matching an initial and final concentration alone identify the reaction order?

Your prediction and reason

Experiment

For each generating order, compare candidate endpoint fits and raw-scale held-out residuals. Perturb only the final reading by+.02 M and retain how the inferred constants and residual patterns change.

Fit two endpoints and test withheld concentration evidence

Worked teaching model · synthetic values · Synthetic A0=1.00 M; true k=.05 in order-appropriate units. Exact observations t0,2,4,6,8 s except selected additive offset at 8 s only. Each candidate law fits endpoints 0/8, then is tested at held-out 2/4/6 on the same raw concentration scale. Not a probabilistic model-selection test.

Generating order:0,1,2: 1 order Added final concentration reading offset: 0 mol/L

Reset keeps your prediction, captures and practice history.

Fitted order 0: solid
Fitted order 1: long dashes
Fitted order 2: short dashes
Supplied observations: points

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

Fitted order 0
time (s)	concentration (mol/L)
0	1
0.08	0.9967032
0.16	0.9934064
0.24	0.9901096
0.32	0.9868128
0.4	0.983516
0.48	0.9802192
0.56	0.9769224
0.64	0.9736256
0.72	0.9703288
0.8	0.967032
0.88	0.9637352
0.96	0.9604384
1.04	0.9571416
1.12	0.9538448
1.2	0.950548
1.28	0.9472512
1.36	0.9439544
1.44	0.9406576
1.52	0.9373608
1.6	0.934064
1.68	0.9307672
1.76	0.9274704
1.84	0.9241736
1.92	0.9208768
2	0.91758
2.08	0.9142832
2.16	0.9109864
2.24	0.9076896
2.32	0.9043928
2.4	0.901096
2.48	0.8977992
2.56	0.8945024
2.64	0.8912056
2.72	0.8879088
2.8	0.884612
2.88	0.8813152
2.96	0.8780184
3.04	0.8747216
3.12	0.8714248
3.2	0.868128
3.28	0.8648312
3.36	0.8615344
3.44	0.8582376
3.52	0.8549408
3.6	0.851644
3.68	0.8483472
3.76	0.8450504
3.84	0.8417536
3.92	0.8384568
4	0.83516
4.08	0.8318632
4.16	0.8285664
4.24	0.8252696
4.32	0.8219728
4.4	0.818676
4.48	0.8153792
4.56	0.8120824
4.64	0.8087856
4.72	0.8054888
4.8	0.802192
4.88	0.7988952
4.96	0.7955984
5.04	0.7923016
5.12	0.7890048
5.2	0.785708
5.28	0.7824112
5.36	0.7791144
5.44	0.7758176
5.52	0.7725208
5.6	0.769224
5.68	0.7659272
5.76	0.7626304
5.84	0.7593336
5.92	0.7560368
6	0.75274
6.08	0.7494432
6.16	0.7461464
6.24	0.7428496
6.32	0.7395528
6.4	0.736256
6.48	0.7329592
6.56	0.7296624
6.64	0.7263656
6.72	0.7230688
6.8	0.719772
6.88	0.7164752
6.96	0.7131784
7.04	0.7098816
7.12	0.7065848
7.2	0.703288
7.28	0.6999912
7.36	0.6966944
7.44	0.6933976
7.52	0.6901008
7.6	0.686804
7.68	0.6835072
7.76	0.6802104
7.84	0.6769136
7.92	0.6736168
8	0.67032

Fitted order 1
time (s)	concentration (mol/L)
0	1
0.08	0.996008
0.16	0.9920319
0.24	0.9880717
0.32	0.9841273
0.4	0.9801987
0.48	0.9762857
0.56	0.9723884
0.64	0.9685066
0.72	0.9646403
0.8	0.9607894
0.88	0.956954
0.96	0.9531338
1.04	0.9493289
1.12	0.9455391
1.2	0.9417645
1.28	0.938005
1.36	0.9342605
1.44	0.9305309
1.52	0.9268162
1.6	0.9231163
1.68	0.9194313
1.76	0.9157609
1.84	0.9121051
1.92	0.908464
2	0.9048374
2.08	0.9012253
2.16	0.8976276
2.24	0.8940443
2.32	0.8904752
2.4	0.8869204
2.48	0.8833798
2.56	0.8798534
2.64	0.876341
2.72	0.8728426
2.8	0.8693582
2.88	0.8658877
2.96	0.8624311
3.04	0.8589883
3.12	0.8555592
3.2	0.8521438
3.28	0.848742
3.36	0.8453538
3.44	0.8419792
3.52	0.838618
3.6	0.8352702
3.68	0.8319358
3.76	0.8286147
3.84	0.8253069
3.92	0.8220122
4	0.8187308
4.08	0.8154624
4.16	0.812207
4.24	0.8089647
4.32	0.8057353
4.4	0.8025188
4.48	0.7993151
4.56	0.7961243
4.64	0.7929461
4.72	0.7897807
4.8	0.7866279
4.88	0.7834876
4.96	0.7803599
5.04	0.7772447
5.12	0.774142
5.2	0.7710516
5.28	0.7679735
5.36	0.7649078
5.44	0.7618543
5.52	0.7588129
5.6	0.7557837
5.68	0.7527666
5.76	0.7497616
5.84	0.7467685
5.92	0.7437874
6	0.7408182
6.08	0.7378609
6.16	0.7349153
6.24	0.7319815
6.32	0.7290595
6.4	0.726149
6.48	0.7232502
6.56	0.720363
6.64	0.7174873
6.72	0.7146231
6.8	0.7117703
6.88	0.7089289
6.96	0.7060989
7.04	0.7032801
7.12	0.7004726
7.2	0.6976763
7.28	0.6948912
7.36	0.6921172
7.44	0.6893542
7.52	0.6866023
7.6	0.6838614
7.68	0.6811314
7.76	0.6784123
7.84	0.6757041
7.92	0.6730067
8	0.67032

Fitted order 2
time (s)	concentration (mol/L)
0	1
0.08	0.9951058
0.16	0.9902593
0.24	0.9854598
0.32	0.9807066
0.4	0.975999
0.48	0.9713364
0.56	0.9667181
0.64	0.9621435
0.72	0.957612
0.8	0.9531231
0.88	0.948676
0.96	0.9442702
1.04	0.9399051
1.12	0.9355802
1.2	0.9312949
1.28	0.9270487
1.36	0.9228411
1.44	0.9186714
1.52	0.9145393
1.6	0.9104442
1.68	0.9063856
1.76	0.902363
1.84	0.898376
1.92	0.894424
2	0.8905067
2.08	0.8866235
2.16	0.8827741
2.24	0.8789579
2.32	0.8751746
2.4	0.8714237
2.48	0.8677048
2.56	0.8640175
2.64	0.8603615
2.72	0.8567362
2.8	0.8531414
2.88	0.8495766
2.96	0.8460415
3.04	0.8425357
3.12	0.8390588
3.2	0.8356105
3.28	0.8321904
3.36	0.8287982
3.44	0.8254335
3.52	0.8220961
3.6	0.8187855
3.68	0.8155015
3.76	0.8122437
3.84	0.8090118
3.92	0.8058056
4	0.8026247
4.08	0.7994688
4.16	0.7963376
4.24	0.7932308
4.32	0.7901482
4.4	0.7870895
4.48	0.7840543
4.56	0.7810425
4.64	0.7780537
4.72	0.7750877
4.8	0.7721442
4.88	0.769223
4.96	0.7663239
5.04	0.7634465
5.12	0.7605906
5.2	0.757756
5.28	0.7549424
5.36	0.7521497
5.44	0.7493776
5.52	0.7466258
5.6	0.7438941
5.68	0.7411824
5.76	0.7384904
5.84	0.7358178
5.92	0.7331646
6	0.7305303
6.08	0.727915
6.16	0.7253183
6.24	0.7227401
6.32	0.7201801
6.4	0.7176382
6.48	0.7151142
6.56	0.7126079
6.64	0.7101191
6.72	0.7076476
6.8	0.7051933
6.88	0.7027559
6.96	0.7003353
7.04	0.6979313
7.12	0.6955438
7.2	0.6931726
7.28	0.6908174
7.36	0.6884782
7.44	0.6861549
7.52	0.6838471
7.6	0.6815548
7.68	0.6792778
7.76	0.677016
7.84	0.6747692
7.92	0.6725373
8	0.67032

Supplied observations
time (s)	concentration (mol/L)
0	1
2	0.9048374
4	0.8187308
6	0.7408182
8	0.67032

Calculated under the declared model
Quantity	Value	Unit
Order 0 held-out raw-scale squared error	0.0005744242	(mol/L)²
Order 0 fitted constant	0.04120999	mol/(L s)
Order 1 held-out raw-scale squared error	0	(mol/L)²
Order 1 fitted constant	0.05	s⁻¹
Order 2 held-out raw-scale squared error	0.0005706153	(mol/L)²
Order 2 fitted constant	0.06147809	L/(mol s)

Observed minus predicted, excluding fitted endpoints
Test	Residual	Unit
Order 0: residual at 2 s	-0.01274259	mol/L
Order 0: residual at 4 s	-0.01642927	mol/L
Order 0: residual at 6 s	-0.01192181	mol/L
Order 1: residual at 2 s	0	mol/L
Order 1: residual at 4 s	0	mol/L
Order 1: residual at 6 s	0	mol/L
Order 2: residual at 2 s	0.01433072	mol/L
Order 2: residual at 4 s	0.01610607	mol/L
Order 2: residual at 6 s	0.01028787	mol/L

All candidates match their fitted endpoints; held-out raw concentrations test what that fit cannot.

Compare residuals on one measurement scale, not R² values from differently transformed axes. A small error favors compatibility for this packet but is not a probability, mechanism proof or guarantee under another concentration range. The deliberately biased last reading can distort all fitted constants.

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

All candidates match their fitted endpoints; held-out raw concentrations test what that fit cannot.

Observation to attach to a capture

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

Understand

Separate fitting information from testing information

Any flexible two-parameter description can look convincing when judged only on the points used to set it. Here initial concentration is fixed at1 M and each candidate order uses the final8s concentration to infer its own k. All candidates therefore match both fitted endpoints by construction.

Derive three endpoint inferences

For final concentration Af, the constants are(1−Af)/8 for zero order,ln(1/Af)/8 for first order and(1/Af−1)/8 for second order, with their respective units. An exact first-order generator with k=.05 gives Af=.670320. Substitute that same value into all three formulas rather than assuming the generator label was experimentally known.

Challenge the predictions on one scale

Now reveal concentrations at2,4 and6s that were not used in fitting. Compute observed-minus-predicted residuals in mol/L and sum their squares on that same scale. For the exact packet, the true generating model has zero residual up to floating arithmetic. The other two can fit endpoints while missing intermediate curvature.

Introduce one controlled reading fault

The second control adds an offset to the last reading only. That distorts every endpoint-derived constant and can make the correct physical family fail to pass through withheld observations. The residual pattern is evidence to inspect calibration and model assumptions, not an automatic proof that the reaction changed mechanism.

Transfer and critique a shortcut

A new packet gives A0=1,A4=.50,A8=.25 M. Predict equal first-order half-lives and k=ln2/4≈.173287 s⁻¹. A zero-order endpoint fit would predict.625 M at4 s, contrary to.50. Explain why comparing linear-fit R² for A,ln A and1/A uses different error scales and does not substitute for a stated measurement-error model.

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.

Apply

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

Why are R² values from different transformed response axes not a direct common-error comparison?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

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