SCIENCE · INVESTIGATE AND EXPLAIN

Infer rate exponents through separate concentration interventions

Write r=k[A]^m[B]^n as a hypothesis at fixed temperature and controlled conditions.

Learning outcome: Compare base (1,1), A-only (2,1), B-only (1,2) and combined (2,2) factors. Derive exponents from rate ratios and check the combined held-out prediction.

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

At [A]=.10 M and [B]=.20 M, r=.50[A]²[B]=.00100 M/s. Doubling A alone gives .00400; doubling B alone gives .00200.

Open the experiment

Predict

Can one experiment doubling both concentrations identify both exponents separately?

Your prediction and reason

Experiment

Compare base (1,1), A-only (2,1), B-only (1,2) and combined (2,2) factors. Derive exponents from rate ratios and check the combined held-out prediction.

Separate independent concentration interventions

Worked teaching model · synthetic values · Original empirical packet r=.50[A]²[B], with concentrations .10*a and .20*b M. Rate is M/s and k has units M⁻²s⁻¹. This is a measured-law illustration, not elementary molecularity or a mechanism proof. Initial rates only, fixed temperature.

A concentration factor: 2 ratio B concentration factor: 1 ratio

Reset keeps your prediction, captures and practice history.

At selected B: 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

At selected B
initial A concentration (mol/L)	initial reaction rate (mol/(L s))
0	0
0.004	1.600000e-6
0.008	6.400000e-6
0.012	1.440000e-5
0.016	2.560000e-5
0.02	4.000000e-5
0.024	5.760000e-5
0.028	7.840000e-5
0.032	0.0001024
0.036	0.0001296
0.04	0.00016
0.044	0.0001936
0.048	0.0002304
0.052	0.0002704
0.056	0.0003136
0.06	0.00036
0.064	0.0004096
0.068	0.0004624
0.072	0.0005184
0.076	0.0005776
0.08	0.00064
0.084	0.0007056
0.088	0.0007744
0.092	0.0008464
0.096	0.0009216
0.1	0.001
0.104	0.0010816
0.108	0.0011664
0.112	0.0012544
0.116	0.0013456
0.12	0.00144
0.124	0.0015376
0.128	0.0016384
0.132	0.0017424
0.136	0.0018496
0.14	0.00196
0.144	0.0020736
0.148	0.0021904
0.152	0.0023104
0.156	0.0024336
0.16	0.00256
0.164	0.0026896
0.168	0.0028224
0.172	0.0029584
0.176	0.0030976
0.18	0.00324
0.184	0.0033856
0.188	0.0035344
0.192	0.0036864
0.196	0.0038416
0.2	0.004
0.204	0.0041616
0.208	0.0043264
0.212	0.0044944
0.216	0.0046656
0.22	0.00484
0.224	0.0050176
0.228	0.0051984
0.232	0.0053824
0.236	0.0055696
0.24	0.00576
0.244	0.0059536
0.248	0.0061504
0.252	0.0063504
0.256	0.0065536
0.26	0.00676
0.264	0.0069696
0.268	0.0071824
0.272	0.0073984
0.276	0.0076176
0.28	0.00784
0.284	0.0080656
0.288	0.0082944
0.292	0.0085264
0.296	0.0087616
0.3	0.009
0.304	0.0092416
0.308	0.0094864
0.312	0.0097344
0.316	0.0099856
0.32	0.01024
0.324	0.0104976
0.328	0.0107584
0.332	0.0110224
0.336	0.0112896
0.34	0.01156
0.344	0.0118336
0.348	0.0121104
0.352	0.0123904
0.356	0.0126736
0.36	0.01296
0.364	0.0132496
0.368	0.0135424
0.372	0.0138384
0.376	0.0141376
0.38	0.01444
0.384	0.0147456
0.388	0.0150544
0.392	0.0153664
0.396	0.0156816
0.4	0.016

Calculated under the declared model
Quantity	Value	Unit
A initial concentration	0.2	mol/L
B initial concentration	0.2	mol/L
Initial normalized reaction rate	0.004	mol/(L s)
Rate relative to base packet	4	ratio
Second-order-in-A exponent	2	order
First-order-in-B exponent	1	order

Changing one concentration at a time distinguishes squared A dependence from linear B dependence.

If both controls change together, the single rate ratio combines both exponents and is insufficient to identify them separately. Overall order 3 sets k dimensions M⁻²s⁻¹, not a claim that three particles collide in an elementary event. Temperature and all hidden conditions are held fixed.

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

Changing one concentration at a time distinguishes squared A dependence from linear B dependence.

Observation to attach to a capture

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

Understand

Start with a conditional empirical law

Write r=k[A]^m[B]^n as a hypothesis at fixed temperature and controlled conditions. The exponents describe sensitivity to concentrations in that range. They are not automatically the coefficients in a balanced overall reaction, and overall order is not necessarily the number of colliding particles.

Change one variable at a time

Our original packet has r=.50[A]²[B], with k in M⁻²s⁻¹. At.10 M A and.20 M B, rate is.00100 M/s. Doubling A while holding B fixed multiplies rate by4; doubling B while holding A fixed multiplies it by 2. Taking the corresponding concentration-factor powers identifies m=2 and n=1.

Use a combined change as a prediction

Doubling both concentrations predicts a factor8, hence.00800 M/s. That combined observation alone would constrain m+n=3 but would not tell whether the separate exponents were2 and1,1 and2 or another pair. Controlled single-variable packets add the needed independent information.

Check dimensions and omitted influences

Rate has concentration/time units. For total order 3, k must have M⁻²s⁻¹ units so the dimensions close. Temperature, solvent activity, mixing or catalyst amount changes could confound a nominal concentration intervention. A good design states what stayed fixed and verifies that initial-rate intervals are short enough to limit concentration drift.

Transfer and check

A new hypothetical power law gives rate factors2 when A doubles alone and9 when B triples alone. Predict orders1 and2, then a factor18 when both changes occur. Explain why those values still do not uniquely identify an elementary mechanism.

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.

What hidden change could make a concentration intervention appear to change an exponent?

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: Normalize species changes before comparing reaction rates Next: Compare integrated laws without confusing units or depletion

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