SCIENCE · INVESTIGATE AND EXPLAIN

Interpret temperature sensitivity without confusing equilibrium

Reactants must encounter one another in suitable configurations with sufficient energy to access a reaction pathway.

Learning outcome: At fixed barrier compare280,300 and320 K. Then compare barriers40 and60 kJ/mol at300 K with A fixed. Derive the inverse-temperature slope and name the assumptions that allow a two-temperature inference.

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 Ea50 kJ/mol and A=1×10⁸ s⁻¹, k300≈.1967 s⁻¹. The ratio k320/k300 is exp[(50000/8.314)(1/300−1/320)]≈3.50.

Open the experiment

Predict

Does a faster forward reaction at a higher temperature by itself prove a larger equilibrium product fraction?

Your prediction and reason

Experiment

At fixed barrier compare280,300 and320 K. Then compare barriers40 and60 kJ/mol at300 K with A fixed. Derive the inverse-temperature slope and name the assumptions that allow a two-temperature inference.

Infer temperature sensitivity from an explicit barrier model

Worked teaching model · synthetic values · Supplied first-order Arrhenius relation k=Aexp(−Ea/RT), A=1.0×10⁸ s⁻¹ and R = 8.314 J/(mol K). Constant A and mechanism in the selected range. Activation energy is not reaction enthalpy; this forward kinetic illustration alone does not determine an equilibrium constant.

Temperature: 300 K Activation energy: 50 kJ/mol

Reset keeps your prediction, captures and practice history.

Supplied Arrhenius line: 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

Supplied Arrhenius line
inverse temperature (K⁻¹)	ln(k / 1 s⁻¹)
0.003571429	-3.057721
0.003558719	-2.981285
0.003546099	-2.905391
0.003533569	-2.830034
0.003521127	-2.755208
0.003508772	-2.680907
0.003496503	-2.607125
0.003484321	-2.533857
0.003472222	-2.461098
0.003460208	-2.388843
0.003448276	-2.317086
0.003436426	-2.245822
0.003424658	-2.175047
0.003412969	-2.104754
0.003401361	-2.03494
0.003389831	-1.965598
0.003378378	-1.896726
0.003367003	-1.828317
0.003355705	-1.760367
0.003344482	-1.692872
0.003333333	-1.625827
0.003322259	-1.559227
0.003311258	-1.493069
0.00330033	-1.427347
0.003289474	-1.362057
0.003278689	-1.297196
0.003267974	-1.232758
0.003257329	-1.168741
0.003246753	-1.105139
0.003236246	-1.041948
0.003225806	-0.9791656
0.003215434	-0.9167867
0.003205128	-0.8548076
0.003194888	-0.7932246
0.003184713	-0.7320338
0.003174603	-0.6712315
0.003164557	-0.6108141
0.003154574	-0.5507778
0.003144654	-0.4911192
0.003134796	-0.4318345
0.003125	-0.3729204
0.003115265	-0.3143734
0.00310559	-0.25619
0.003095975	-0.1983668
0.00308642	-0.1409006
0.003076923	-0.08378809
0.003067485	-0.02702591
0.003058104	0.02938909
0.00304878	0.08546011
0.003039514	0.1411903
0.003030303	0.1965827
0.003021148	0.2516404
0.003012048	0.3063664
0.003003003	0.3607637
0.002994012	0.4148353
0.002985075	0.4685841
0.00297619	0.522013
0.002967359	0.5751248
0.00295858	0.6279223
0.002949853	0.6804083
0.002941176	0.7325855

Calculated under the declared model
Quantity	Value	Unit
Selected first-order rate constant	0.1967489	s⁻¹
Rate constant at 300 K	0.1967489	s⁻¹
Selected/300K constant ratio	1	ratio
Arrhenius slope	-6013.952	K
Activation energy	50	kJ/mol

The slope in inverse temperature reflects the supplied activation-energy scale, not the reaction enthalpy.

Temperature changes the fraction of collision configurations able to cross a barrier; orientation and encounter factors also matter. The fixed pre-exponential and mechanism are assumptions. A catalyst supplies a different path and changes rates without changing equilibrium at the same temperature; this one forward relation alone cannot compute K.

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

The slope in inverse temperature reflects the supplied activation-energy scale, not the reaction enthalpy.

Observation to attach to a capture

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

Understand

Connect collisions to an effective barrier

Reactants must encounter one another in suitable configurations with sufficient energy to access a reaction pathway. The Arrhenius model packages temperature dependence into k=Aexp(−Ea/RT). The pre-exponential represents additional encounter and configuration factors; a lower barrier alone is not a complete account if A or the mechanism changes too.

Keep energy and temperature units consistent

Use Ea in joules per mole with R=8.314 J/(mol K), and use absolute temperature. With Ea=50000 J/mol,A=1×10⁸ s⁻¹ and T=300 K, k is about.1967 s⁻¹. Replacing50000 by50 while retaining that R would create a thousand-fold exponent error.

Derive a ratio and a linearized test

At fixed A and Ea, ln(k2/k1)=−Ea/R(1/T2−1/T1). Thus320 K compared with300 K gives a ratio about3.50. The graph of ln(k divided by1 s⁻¹) versus1/T has slope−Ea/R and a dimensionless logarithm argument. Its straightness tests the constant-parameter approximation, not every possible mechanism.

Do not confuse rate and equilibrium

Ea describes an activation scale; reaction enthalpy compares endpoint states. A catalyst changes pathways and can speed approach to equilibrium without changing the equilibrium constant at the same temperature. This single forward model contains no reverse-rate or Gibbs-energy information, so it cannot determine an equilibrium product fraction.

Transfer and check

A different first-order packet doubles k between300 and310 K while the same Arrhenius form applies. Infer Ea=R ln2/(1/300−1/310)≈53.6 kJ/mol. A new mechanism or temperature-dependent pre-exponential could invalidate that interpretation; report the inference with its assumptions rather than as a unique microscopic barrier.

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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Why is activation energy not interchangeable with reaction enthalpy or a complete equilibrium prediction?

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Previous: Separate optical background from a transformed kinetic slope Next: Derive a saturating rate from bound-catalyst conservation

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