SCIENCE · INVESTIGATE AND EXPLAIN

Test an intermediate profile without singular equal-rate arithmetic

The candidate sequence A→B→C forms B in one step and consumes it in the next.

Learning outcome: Compare(.2,.2),(.2,.4) and(.4,.2). Retain A,B,C profiles, the peak time and conserved total; explain how formation and consumption compete at the peak.

Starting recap

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

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

For equal constants k=.20 s⁻¹, B=.20t exp(−.20t) reaches its maximum at 5 s, where A=B=exp(−1)≈.367879 and C≈.264241.

Open the experiment

Predict

Does the intermediate peak mean the entire irreversible network has reached chemical equilibrium?

Your prediction and reason

Experiment

Compare(.2,.2),(.2,.4) and(.4,.2). Retain A,B,C profiles, the peak time and conserved total; explain how formation and consumption compete at the peak.

Test an intermediate profile and equal-rate limit

Worked teaching model · synthetic values · Fictional isomers A→B→C, initial A = 1.00 M and B,C = 0; irreversible constant-volume first-order steps. B=k1/(k2−k1)[exp(−k1t)−exp(−k2t)]; equal-rate limit B=kt exp(−kt). Analytical supplied solution, not proof that a real species is an obligatory intermediate.

A→B first-order constant: 0.2 s⁻¹ B→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	0.9607894
0.4	0.9231163
0.6	0.8869204
0.8	0.8521438
1	0.8187308
1.2	0.7866279
1.4	0.7557837
1.6	0.726149
1.8	0.6976763
2	0.67032
2.2	0.6440364
2.4	0.6187834
2.6	0.5945205
2.8	0.5712091
3	0.5488116
3.2	0.5272924
3.4	0.506617
3.6	0.4867523
3.8	0.4676664
4	0.449329
4.2	0.4317105
4.4	0.4147829
4.6	0.398519
4.8	0.3828929
5	0.3678794
5.2	0.3534547
5.4	0.3395955
5.6	0.3262798
5.8	0.3134862
6	0.3011942
6.2	0.2893842
6.4	0.2780373
6.6	0.2671353
6.8	0.2566608
7	0.246597
7.2	0.2369278
7.4	0.2276377
7.6	0.2187119
7.8	0.2101361
8	0.2018965
8.2	0.19398
8.4	0.186374
8.6	0.1790661
8.8	0.1720449
9	0.1652989
9.2	0.1588174
9.4	0.1525901
9.6	0.146607
9.8	0.1408584
10	0.1353353
10.2	0.1300287
10.4	0.1249302
10.6	0.1200316
10.8	0.1153251
11	0.1108032
11.2	0.1064585
11.4	0.1022842
11.6	0.09827359
11.8	0.09442022
12	0.09071795
12.2	0.08716085
12.4	0.08374323
12.6	0.08045961
12.8	0.07730474
13	0.07427358
13.2	0.07136127
13.4	0.06856315
13.6	0.06587475
13.8	0.06329177
14	0.06081006
14.2	0.05842567
14.4	0.05613476
14.6	0.05393369
14.8	0.05181892
15	0.04978707
15.2	0.04783489
15.4	0.04595926
15.6	0.04415717
15.8	0.04242574
16	0.0407622
16.2	0.0391639
16.4	0.03762826
16.6	0.03615283
16.8	0.03473526
17	0.03337327
17.2	0.03206469
17.4	0.03080741
17.6	0.02959944
17.8	0.02843882
18	0.02732372
18.2	0.02625234
18.4	0.02522297
18.6	0.02423397
18.8	0.02328374
19	0.02237077
19.2	0.0214936
19.4	0.02065083
19.6	0.01984109
19.8	0.01906311
20	0.01831564

B
time (s)	concentration (mol/L)
0	0
0.2	0.03804981
0.4	0.07239182
0.6	0.1033005
0.8	0.1310319
1	0.1558251
1.2	0.1779031
1.4	0.1974738
1.6	0.2147313
1.8	0.2298561
2	0.2430168
2.2	0.2543702
2.4	0.2640623
2.6	0.2722291
2.8	0.2789971
3	0.284484
3.2	0.2887991
3.4	0.2920441
3.6	0.2943135
3.8	0.2956948
4	0.2962695
4.2	0.296113
4.4	0.2952952
4.6	0.293881
4.8	0.2919303
5	0.2894986
5.2	0.2866372
5.4	0.2833937
5.6	0.2798116
5.8	0.2759316
6	0.2717906
6.2	0.2674232
6.4	0.2628607
6.6	0.2581321
6.8	0.2532641
7	0.2482811
7.2	0.2432053
7.4	0.2380572
7.6	0.2328554
7.8	0.2276169
8	0.2223571
8.2	0.2170902
8.4	0.2118287
8.6	0.2065843
8.8	0.2013672
9	0.1961868
9.2	0.1910513
9.4	0.1859683
9.6	0.1809444
9.8	0.1759854
10	0.1710964
10.2	0.166282
10.4	0.1615461
10.6	0.1568919
10.8	0.1523225
11	0.14784
11.2	0.1434465
11.4	0.1391435
11.6	0.1349323
11.8	0.1308138
12	0.1267885
12.2	0.1228567
12.4	0.1190185
12.6	0.1152738
12.8	0.1116223
13	0.1080633
13.2	0.1045963
13.4	0.1012204
13.6	0.09793458
13.8	0.09473783
14	0.09162897
14.2	0.08860673
14.4	0.08566976
14.6	0.08281666
14.8	0.08004596
15	0.07735614
15.2	0.07474566
15.4	0.07221292
15.6	0.06975631
15.8	0.06737419
16	0.06506491
16.2	0.06282682
16.4	0.06065825
16.6	0.05855754
16.8	0.05652302
17	0.05455305
17.2	0.05264597
17.4	0.05080016
17.6	0.04901401
17.8	0.04728591
18	0.04561428
18.2	0.04399758
18.4	0.04243425
18.6	0.0409228
18.8	0.03946174
19	0.03804961
19.2	0.03668498
19.4	0.03536644
19.6	0.03409262
19.8	0.03286217
20	0.03167377

C
time (s)	concentration (mol/L)
0	0
0.2	0.00116075
0.4	0.004491834
0.6	0.009779113
0.8	0.01682436
1	0.02544418
1.2	0.03546907
1.4	0.04674242
1.6	0.05911967
1.8	0.07246753
2	0.08666313
2.2	0.1015934
2.4	0.1171543
2.6	0.1332504
2.8	0.1497939
3	0.1667044
3.2	0.1839085
3.4	0.2013389
3.6	0.2189343
3.8	0.2366388
4	0.2544015
4.2	0.2721765
4.4	0.2899219
4.6	0.3076
4.8	0.3251769
5	0.342622
5.2	0.3599081
5.4	0.3770108
5.6	0.3939086
5.8	0.4105823
6	0.4270151
6.2	0.4431926
6.4	0.459102
6.6	0.4747326
6.8	0.4900751
7	0.505122
7.2	0.519867
7.4	0.5343052
7.6	0.5484328
7.8	0.5622471
8	0.5757464
8.2	0.5889298
8.4	0.6017973
8.6	0.6143496
8.8	0.6265879
9	0.6385144
9.2	0.6501313
9.4	0.6614416
9.6	0.6724486
9.8	0.6831562
10	0.6935683
10.2	0.7036893
10.4	0.7135237
10.6	0.7230764
10.8	0.7323524
11	0.7413569
11.2	0.750095
11.4	0.7585722
11.6	0.7667941
11.8	0.774766
12	0.7824936
12.2	0.7899825
12.4	0.7972383
12.6	0.8042666
12.8	0.811073
13	0.8176631
13.2	0.8240424
13.4	0.8302165
13.6	0.8361907
13.8	0.8419704
14	0.847561
14.2	0.8529676
14.4	0.8581955
14.6	0.8632497
14.8	0.8681351
15	0.8728568
15.2	0.8774194
15.4	0.8818278
15.6	0.8860865
15.8	0.8902001
16	0.8941729
16.2	0.8980093
16.4	0.9017135
16.6	0.9052896
16.8	0.9087417
17	0.9120737
17.2	0.9152893
17.4	0.9183924
17.6	0.9213866
17.8	0.9242753
18	0.927062
18.2	0.9297501
18.4	0.9323428
18.6	0.9348432
18.8	0.9372545
19	0.9395796
19.2	0.9418214
19.4	0.9439827
19.6	0.9460663
19.8	0.9480747
20	0.9500106

Calculated under the declared model
Quantity	Value	Unit
A at 5 s	0.3678794	mol/L
Intermediate B at 5 s	0.2894986	mol/L
Final C at 5 s	0.342622	mol/L
Conserved total at 5 s	1	mol/L
Intermediate maximum time	4.054651	s

A rising and falling intermediate requires both a formation and a consumption account.

At equal constants use the finite limit B=kt exp(−kt), not division by zero. The intermediate peak occurs when its formation and consumption rates match; it is not chemical equilibrium of the whole irreversible network. A transient detected species is not automatically an obligatory intermediate without discriminating kinetic or structural evidence.

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

A rising and falling intermediate requires both a formation and a consumption account.

Observation to attach to a capture

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

Understand

Separate formation and consumption

The candidate sequence A→B→C forms B in one step and consumes it in the next. A rising then falling B profile reflects the competition between those rates. With initial A1 M and no products, the irreversible first-order assumptions determine all three trajectories; an observed transient alone does not establish that every real molecule follows this path.

Use the supplied analytic solution

A=exp(−k1t). For unequal constants, B=k1[exp(−k1t)−exp(−k2t)]/(k2−k1), and C=1−A−B. The numerator and denominator change signs together when the constants swap, preserving nonnegative amounts. The expression is supplied as a solved model; deriving the coupled differential equations is optional deeper work.

Handle equal constants physically

When k1=k2=k, the unequal-rate expression has a zero-over-zero form, not zero concentration or an infinite concentration. Its finite limit is B=kt exp(−kt). At k=.20 and t=5s, A and B both equal exp(−1), while C=1−2exp(−1). All amounts remain finite and sum to1.

Interpret the intermediate maximum

At the B peak, k1A=k2B so its instantaneous formation and consumption rates match. This does not mean the complete irreversible network is at equilibrium: A is still being consumed and C produced. For unequal constants the peak time is ln(k2/k1)/(k2−k1); the equal-rate limit is1/k.

Transfer and propose discriminating evidence

For k1=.10 and k2=.20 s⁻¹, predict peak time ln2/.10≈6.931s. At that time A=.50,B=.25,C=.25 M. Independent identity, calibrated response and perturbation evidence would be needed to establish B as an obligatory intermediate rather than a parallel transient species.

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 would seeing a transient species not by itself prove it is an obligatory intermediate?

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.

Previous: Use product selectivity to separate indistinguishable decay traces Next: Build a dimensionless quotient and separate direction from speed

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.