SCIENCE · INVESTIGATE AND EXPLAIN

Use product selectivity to separate indistinguishable decay traces

Fictional isomer A independently forms B and C through irreversible first-order branches with constants kB and kC.

Learning outcome: Compare(.1,.4),(.2,.3) and(.4,.1). Retain the identical total-decay cases and distinct product curves, then check A+B+C conservation at several times.

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

Branches(.1,.4) and(.2,.3)s⁻¹ both deplete A with total constant.5 s⁻¹, but their B fractions are.20 and.40 of reacted material.

Open the experiment

Predict

Can an A-disappearance trace alone identify two parallel branch constants separately?

Your prediction and reason

Experiment

Compare(.1,.4),(.2,.3) and(.4,.1). Retain the identical total-decay cases and distinct product curves, then check A+B+C conservation at several times.

Separate overall disappearance from product selectivity

Worked teaching model · synthetic values · Fictional isomers A,B,C, initial A = 1.00 M and products 0, independent irreversible first-order branches at fixed volume. A=exp[−(kB+kC)t], products partition reacted amount by branch rate fractions. Total disappearance alone identifies only the sum of constants.

A→B first-order constant: 0.2 s⁻¹ A→C first-order constant: 0.3 s⁻¹

Reset keeps your prediction, captures and practice history.

A: solid
B: long dashes
C: short 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
0.2	0.9048374
0.4	0.8187308
0.6	0.7408182
0.8	0.67032
1	0.6065307
1.2	0.5488116
1.4	0.4965853
1.6	0.449329
1.8	0.4065697
2	0.3678794
2.2	0.3328711
2.4	0.3011942
2.6	0.2725318
2.8	0.246597
3	0.2231302
3.2	0.2018965
3.4	0.1826835
3.6	0.1652989
3.8	0.1495686
4	0.1353353
4.2	0.1224564
4.4	0.1108032
4.6	0.1002588
4.8	0.09071795
5	0.082085
5.2	0.07427358
5.4	0.06720551
5.6	0.06081006
5.8	0.05502322
6	0.04978707
6.2	0.0450492
6.4	0.0407622
6.6	0.03688317
6.8	0.03337327
7	0.03019738
7.2	0.02732372
7.4	0.02472353
7.6	0.02237077
7.8	0.02024191
8	0.01831564
8.2	0.01657268
8.4	0.01499558
8.6	0.01356856
8.8	0.01227734
9	0.011109
9.2	0.01005184
9.4	0.009095277
9.6	0.008229747
9.8	0.007446583
10	0.006737947
10.2	0.006096747
10.4	0.005516564
10.6	0.004991594
10.8	0.004516581
11	0.004086771
11.2	0.003697864
11.4	0.003345965
11.6	0.003027555
11.8	0.002739445
12	0.002478752
12.2	0.002242868
12.4	0.002029431
12.6	0.001836305
12.8	0.001661557
13	0.001503439
13.2	0.001360368
13.4	0.001230912
13.6	0.001113775
13.8	0.001007785
14	0.000911882
14.2	0.0008251049
14.4	0.0007465858
14.6	0.0006755388
14.8	0.0006112528
15	0.0005530844
15.2	0.0005004514
15.4	0.0004528272
15.6	0.000409735
15.8	0.0003707435
16	0.0003354626
16.2	0.0003035391
16.4	0.0002746536
16.6	0.0002485168
16.8	0.0002248673
17	0.0002034684
17.2	0.0001841058
17.4	0.0001665858
17.6	0.0001507331
17.8	0.0001363889
18	0.0001234098
18.2	0.0001116658
18.4	0.0001010394
18.6	9.142423e-5
18.8	8.272407e-5
19	7.485183e-5
19.2	6.772874e-5
19.4	6.128350e-5
19.6	5.545160e-5
19.8	5.017468e-5
20	4.539993e-5

B
time (s)	concentration (mol/L)
0	0
0.2	0.03806503
0.4	0.0725077
0.6	0.1036727
0.8	0.131872
1	0.1573877
1.2	0.1804753
1.4	0.2013659
1.6	0.2202684
1.8	0.2373721
2	0.2528482
2.2	0.2668516
2.4	0.2795223
2.6	0.2909873
2.8	0.3013612
3	0.3107479
3.2	0.3192414
3.4	0.3269266
3.6	0.3338804
3.8	0.3401726
4	0.3458659
4.2	0.3510174
4.4	0.3556787
4.6	0.3598965
4.8	0.3637128
5	0.367166
5.2	0.3702906
5.4	0.3731178
5.6	0.375676
5.8	0.3779907
6	0.3800852
6.2	0.3819803
6.4	0.3836951
6.6	0.3852467
6.8	0.3866507
7	0.387921
7.2	0.3890705
7.4	0.3901106
7.6	0.3910517
7.8	0.3919032
8	0.3926737
8.2	0.3933709
8.4	0.3940018
8.6	0.3945726
8.8	0.3950891
9	0.3955564
9.2	0.3959793
9.4	0.3963619
9.6	0.3967081
9.8	0.3970214
10	0.3973048
10.2	0.3975613
10.4	0.3977934
10.6	0.3980034
10.8	0.3981934
11	0.3983653
11.2	0.3985209
11.4	0.3986616
11.6	0.398789
11.8	0.3989042
12	0.3990085
12.2	0.3991029
12.4	0.3991882
12.6	0.3992655
12.8	0.3993354
13	0.3993986
13.2	0.3994559
13.4	0.3995076
13.6	0.3995545
13.8	0.3995969
14	0.3996352
14.2	0.39967
14.4	0.3997014
14.6	0.3997298
14.8	0.3997555
15	0.3997788
15.2	0.3997998
15.4	0.3998189
15.6	0.3998361
15.8	0.3998517
16	0.3998658
16.2	0.3998786
16.4	0.3998901
16.6	0.3999006
16.8	0.3999101
17	0.3999186
17.2	0.3999264
17.4	0.3999334
17.6	0.3999397
17.8	0.3999454
18	0.3999506
18.2	0.3999553
18.4	0.3999596
18.6	0.3999634
18.8	0.3999669
19	0.3999701
19.2	0.3999729
19.4	0.3999755
19.6	0.3999778
19.8	0.3999799
20	0.3999818

C
time (s)	concentration (mol/L)
0	0
0.2	0.05709755
0.4	0.1087615
0.6	0.1555091
0.8	0.197808
1	0.2360816
1.2	0.270713
1.4	0.3020488
1.6	0.3304026
1.8	0.3560582
2	0.3792723
2.2	0.4002773
2.4	0.4192835
2.6	0.4364809
2.8	0.4520418
3	0.4661219
3.2	0.4788621
3.4	0.4903899
3.6	0.5008207
3.8	0.5102588
4	0.5187988
4.2	0.5265261
4.4	0.5335181
4.6	0.5398447
4.8	0.5455692
5	0.550749
5.2	0.5554359
5.4	0.5596767
5.6	0.563514
5.8	0.5669861
6	0.5701278
6.2	0.5729705
6.4	0.5755427
6.6	0.5778701
6.8	0.579976
7	0.5818816
7.2	0.5836058
7.4	0.5851659
7.6	0.5865775
7.8	0.5878549
8	0.5890106
8.2	0.5900564
8.4	0.5910027
8.6	0.5918589
8.8	0.5926336
9	0.5933346
9.2	0.5939689
9.4	0.5945428
9.6	0.5950622
9.8	0.5955321
10	0.5959572
10.2	0.596342
10.4	0.5966901
10.6	0.597005
10.8	0.5972901
11	0.5975479
11.2	0.5977813
11.4	0.5979924
11.6	0.5981835
11.8	0.5983563
12	0.5985127
12.2	0.5986543
12.4	0.5987823
12.6	0.5988982
12.8	0.5990031
13	0.5990979
13.2	0.5991838
13.4	0.5992615
13.6	0.5993317
13.8	0.5993953
14	0.5994529
14.2	0.5995049
14.4	0.599552
14.6	0.5995947
14.8	0.5996332
15	0.5996681
15.2	0.5996997
15.4	0.5997283
15.6	0.5997542
15.8	0.5997776
16	0.5997987
16.2	0.5998179
16.4	0.5998352
16.6	0.5998509
16.8	0.5998651
17	0.5998779
17.2	0.5998895
17.4	0.5999
17.6	0.5999096
17.8	0.5999182
18	0.599926
18.2	0.599933
18.4	0.5999394
18.6	0.5999451
18.8	0.5999504
19	0.5999551
19.2	0.5999594
19.4	0.5999632
19.6	0.5999667
19.8	0.5999699
20	0.5999728

Calculated under the declared model
Quantity	Value	Unit
Total disappearance constant	0.5	s⁻¹
B fraction of reacted material	0.4	fraction
C fraction of reacted material	0.6	fraction
A remaining at 5 s	0.082085	mol/L
B at 5 s	0.367166	mol/L
C at 5 s	0.550749	mol/L

The disappearance curve determines a sum of branch constants, while product composition adds another constraint.

Pairs (.1,.4) and (.2,.3) share the same total decay but different product selectivity. The same-formula isomer stipulation makes A+B+C an amount-conservation check. In a real chemical network stoichiometric factors and secondary product reactions must also be audited.

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

The disappearance curve determines a sum of branch constants, while product composition adds another constraint.

Observation to attach to a capture

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

Understand

Write the complete candidate network

Fictional isomer A independently forms B and C through irreversible first-order branches with constants kB and kC. The same-formula stipulation makes A+B+C a valid amount-conservation check at fixed volume. A real network could need different stoichiometric weights or further product reactions.

Derive what disappearance can identify

Both branches remove A, so−dA/dt=(kB+kC)A. Initial A=1 M gives A=exp[−(kB+kC)t]. Any two positive constants with the same sum produce the same disappearance trace. Even an exact, densely sampled A curve cannot separate parameters that enter only through their sum.

Use product composition as new evidence

Reacted amount1−A partitions as B=kB/(kB+kC)(1−A) and C=kC/(kB+kC)(1−A). For(.1,.4), B receives20% of reacted material; for(.2,.3), it receives40%. Both sums are.5 s⁻¹. A calibrated product ratio therefore adds a genuinely different constraint.

Check a finite-time inventory

At total constant.5 and t=5s, remaining A is exp(−2.5)≈.082085 M. With kB=.2 and kC=.3, B≈.367166 and C≈.550749 M. Their total with A is1 M. This is supplied mechanism arithmetic, not proof that actual measured signals are proportional to those concentrations.

Transfer and identify a failure mode

A fresh packet has total constant.60 s⁻¹ and independently calibrated B fraction.25 of reacted material. Predict kB=.15 and kC=.45 s⁻¹. If B later converts to C, an endpoint product ratio need not equal the original branching ratio. State the no-secondary-reaction and response-calibration assumptions before inferring constants.

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.

What changes if products undergo later reactions or their response factors differ?

Your explanation and limitations

Continue

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

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