SCIENCE · INVESTIGATE AND EXPLAIN

Compare integrated laws without confusing units or depletion

The single-reactant irreversible models use disappearance rate k[A]^n at fixed volume.

Learning outcome: Compare three orders at A0=1 M, then double A0 for each. Predict half-life scaling and explicitly locate the zero-order depletion boundary.

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 A0=1 M and numerical k=.10 in each order-specific unit, half-lives are 5 s, 6.931 s and 10 s for zero, first and second order.

Open the experiment

Predict

Does a common numerical value of k mean these three physical rate constants are the same quantity?

Your prediction and reason

Experiment

Compare three orders at A0=1 M, then double A0 for each. Predict half-life scaling and explicitly locate the zero-order depletion boundary.

Compare integrated laws with their own dimensions

Worked teaching model · synthetic values · Single irreversible A→P at fixed volume. The numerical constant is .10 with order-specific units M/s,s⁻¹,M⁻¹s⁻¹. These constants cannot be compared as the same physical quantity across orders. Zero-order law ceases at depletion; no negative concentration is extended beyond it.

Order:0,1,2: 1 order Initial A concentration: 1 mol/L

Reset keeps your prediction, captures and practice history.

A: 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

A
time (s)	concentration (mol/L)
0	1
0.2	0.9801987
0.4	0.9607894
0.6	0.9417645
0.8	0.9231163
1	0.9048374
1.2	0.8869204
1.4	0.8693582
1.6	0.8521438
1.8	0.8352702
2	0.8187308
2.2	0.8025188
2.4	0.7866279
2.6	0.7710516
2.8	0.7557837
3	0.7408182
3.2	0.726149
3.4	0.7117703
3.6	0.6976763
3.8	0.6838614
4	0.67032
4.2	0.6570468
4.4	0.6440364
4.6	0.6312836
4.8	0.6187834
5	0.6065307
5.2	0.5945205
5.4	0.5827483
5.6	0.5712091
5.8	0.5598984
6	0.5488116
6.2	0.5379444
6.4	0.5272924
6.6	0.5168513
6.8	0.506617
7	0.4965853
7.2	0.4867523
7.4	0.4771139
7.6	0.4676664
7.8	0.458406
8	0.449329
8.2	0.4404317
8.4	0.4317105
8.6	0.4231621
8.8	0.4147829
9	0.4065697
9.2	0.398519
9.4	0.3906278
9.6	0.3828929
9.8	0.3753111
10	0.3678794
10.2	0.3605949
10.4	0.3534547
10.6	0.3464558
10.8	0.3395955
11	0.3328711
11.2	0.3262798
11.4	0.319819
11.6	0.3134862
11.8	0.3072787
12	0.3011942
12.2	0.2952302
12.4	0.2893842
12.6	0.283654
12.8	0.2780373
13	0.2725318
13.2	0.2671353
13.4	0.2618457
13.6	0.2566608
13.8	0.2515786
14	0.246597
14.2	0.241714
14.4	0.2369278
14.6	0.2322363
14.8	0.2276377
15	0.2231302
15.2	0.2187119
15.4	0.2143811
15.6	0.2101361
15.8	0.2059751
16	0.2018965
16.2	0.1978987
16.4	0.19398
16.6	0.190139
16.8	0.186374
17	0.1826835
17.2	0.1790661
17.4	0.1755204
17.6	0.1720449
17.8	0.1686381
18	0.1652989
18.2	0.1620258
18.4	0.1588174
18.6	0.1556726
18.8	0.1525901
19	0.1495686
19.2	0.146607
19.4	0.1437039
19.6	0.1408584
19.8	0.1380692
20	0.1353353

Calculated under the declared model
Quantity	Value	Unit
Half-life from original concentration	6.931472	s
Initial disappearance rate	0.1	mol/(L s)
Remaining concentration at 10 s	0.3678794	mol/L
Rate-constant numerical value	0.1	s⁻¹

A first-order amount halves in equal time intervals.

The three numerical constants have different units. Zero-order kinetics is ended at zero amount rather than continued into an unphysical negative concentration. First- and second-order formulas here assume one reactant and fixed volume; a multireactant law requires additional constraints.

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

A first-order amount halves in equal time intervals.

Observation to attach to a capture

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

Understand

Connect a local law with an amount trajectory

The single-reactant irreversible models use disappearance rate k[A]^n at fixed volume. Their integrated forms are A0−kt for zero order, A0exp(−kt) for first order and A0/(1+kA0t) for second order. These formulas encode different concentration dependence, not three interchangeable fitting styles.

Preserve dimensions

A zero-order k has concentration/time units, a first-order k inverse-time units and a second-order k inverse-concentration/time units. The activity deliberately uses numerical.10 in each appropriate unit. It does not assert that those constants are one identical physical quantity.

Derive and compare half-lives

Set the remaining concentration to A0/2. The half-lives are A0/(2k),ln2/k and1/(kA0). For A0=1 M, the displayed values are5,6.931 and10s. Doubling A0 doubles the zero-order half-life, leaves first-order half-life fixed and halves the second-order half-life. These contrasting predictions can discriminate models over a suitable range.

Respect the zero-order domain

The zero-order segment reaches A=0 at t=A0/k. Extending the line would produce negative concentration, so the model ends that reaction segment at depletion and holds amount at zero. That end condition is not a claim the same zero-order differential law remains valid with no reactant. First- and second-order traces approach zero asymptotically in this idealization.

Transfer and check

For A0=2 M and second-order k=.10 M⁻¹s⁻¹, half-life is5s. At10s, A=2/(1+2)=.666667 M, not zero. Contrast this with a zero-order.10 M/s packet, which retains1 M at 10 s and depletes only at20s.

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.

Why must a constant-rate law stop being extrapolated when no reactant remains?

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Previous: Infer rate exponents through separate concentration interventions Next: Test integrated kinetic models with withheld raw concentrations

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