SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish a van’t Hoff inference from an activation barrier

Equilibrium temperature dependence and rate temperature dependence encode different information.

Learning outcome: Compare endothermic, zero and exothermic enthalpies; infer the inverse-temperature slope and test the stated300 K reference.

Starting recap

Refresh state-function boundaries, absolute temperature, dimensionless logarithms, written reaction/electron equivalents, conserved charge and formal orbital/identity bookkeeping.

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

Try

With K300=1 and ΔrH°+30 kJ/mol, warming to320 K gives lnK≈.7517 and K≈2.12 under constant enthalpy.

Open the experiment

Predict

Would an Arrhenius activation energy be a valid substitute for reaction enthalpy in this equilibrium relation?

Your prediction and reason

Experiment

Compare endothermic, zero and exothermic enthalpies; infer the inverse-temperature slope and test the stated300 K reference.

Infer a conditional equilibrium temperature response

Worked teaching model · synthetic values · Fictional reaction with K=1 at300 K and constant ΔrH° across the range. ln[K(T)/K(300)]=−ΔrH°/R(1/T−1/300). No phase changes or heat-capacity correction. Equilibrium K changes with temperature; a catalyst changes approach rates, not this fixed-temperature K.

Temperature: 320 K Standard reaction enthalpy: 30 kJ/mol reaction

Reset keeps your prediction, captures and practice history.

ln K versus inverse temperature: solid

Axes rescale. A multiplicity distribution, inverse-temperature relation and charge-budget graph have different meanings. Read each axis and do not treat a computed curve as additional measured evidence.

All calculated plot coordinates

ln K versus inverse temperature
inverse temperature (K⁻¹)	ln dimensionless K
0.002777778	2.004539
0.002791005	1.956812
0.002804233	1.909085
0.00281746	1.861358
0.002830688	1.813631
0.002843915	1.765904
0.002857143	1.718177
0.00287037	1.670449
0.002883598	1.622722
0.002896825	1.574995
0.002910053	1.527268
0.00292328	1.479541
0.002936508	1.431814
0.002949735	1.384087
0.002962963	1.33636
0.00297619	1.288632
0.002989418	1.240905
0.003002646	1.193178
0.003015873	1.145451
0.003029101	1.097724
0.003042328	1.049997
0.003055556	1.00227
0.003068783	0.9545425
0.003082011	0.9068154
0.003095238	0.8590883
0.003108466	0.8113611
0.003121693	0.763634
0.003134921	0.7159069
0.003148148	0.6681798
0.003161376	0.6204526
0.003174603	0.5727255
0.003187831	0.5249984
0.003201058	0.4772713
0.003214286	0.4295441
0.003227513	0.381817
0.003240741	0.3340899
0.003253968	0.2863628
0.003267196	0.2386356
0.003280423	0.1909085
0.003293651	0.1431814
0.003306878	0.09545425
0.003320106	0.04772713
0.003333333	1.564795e-15
0.003346561	-0.04772713
0.003359788	-0.09545425
0.003373016	-0.1431814
0.003386243	-0.1909085
0.003399471	-0.2386356
0.003412698	-0.2863628
0.003425926	-0.3340899
0.003439153	-0.381817
0.003452381	-0.4295441
0.003465608	-0.4772713
0.003478836	-0.5249984
0.003492063	-0.5727255
0.003505291	-0.6204526
0.003518519	-0.6681798
0.003531746	-0.7159069
0.003544974	-0.763634
0.003558201	-0.8113611
0.003571429	-0.8590883

Calculated under the declared model
Quantity	Value	Unit
Equilibrium constant at selected temperature	2.120607	dimensionless
Natural log K	0.7517022	dimensionless
Reference K at300 K	1	dimensionless
Inverse-temperature slope	-3608.171	K

An endothermic forward reaction gains equilibrium favorability on warming in this model.

The slope reflects a reaction enthalpy under the stated constant-enthalpy approximation, not an Arrhenius activation barrier. Changing temperature changes K; changing concentrations at fixed temperature changes Q. A catalyst can change equilibration time but does not alter the same fixed-temperature thermodynamic K.

These calculated packets are not performed chemical measurements or separation procedures. Keep feasible inventories, activity conventions, fitting observations and independent checks distinct.

An endothermic forward reaction gains equilibrium favorability on warming in this model.

Observation to attach to a capture

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

Understand

Keep the reference and approximation visible

For a reaction with K=1 at300 K and constant reaction enthalpy, ln[K(T)/K(300)]=−ΔrH°/R(1/T−1/300). The logarithm acts on a dimensionless ratio. The supplied enthalpy is the difference between reaction product and reactant states, not a barrier height.

For ΔrH°+30 kJ/mol and T320 K, lnK is about.7517 and K about2.12. The endothermic forward reaction becomes more favorable at equilibrium on warming. A−30 kJ/mol reaction gives the reciprocal K under the same comparison. Zero enthalpy gives no temperature dependence in this approximation.

Separate two temperature plots

An Arrhenius ln k versus1/T slope concerns a kinetic activation-energy model and assumes an appropriate pre-exponential behavior. A van’t Hoff ln K versus1/T slope concerns equilibrium reaction enthalpy. Both may appear as straight lines over a limited interval, but their slopes cannot be interchanged without additional mechanistic relationships.

Changing concentrations at fixed temperature changes Q, not K. A catalyst changes the approach to equilibrium, not the same thermodynamic equilibrium constant. Changing temperature can change both rates and K, but those are distinct claims.

Transfer and check

For an endothermic reaction with ΔrH°15 kJ/mol and K300=1, compare T360 K: lnK=15000/R(1/300−1/360), about1.0023, so K≈2.724. State why a phase transition or appreciable reaction heat-capacity difference would invalidate an unqualified extrapolation of this line.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. All prose, cases, graphs and questions are original. These supplied thermodynamic, electrical, nuclear or orbital models are not actual measurements, apparatus instructions, synthesis protocols or practical certification. Retain conventions, independent constraints, changed-case reasoning and limits. No heating, wiring, electrolysis, source handling or chemical manipulation is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which evidence would challenge the constant-enthalpy assumption over a wider temperature range?

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: Separate standard free energy from the current driving force Next: Preserve reaction direction and electron equivalents in a cell potential

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.