The short answer
โMore favourableโ and โfasterโ are different statements. A defined barrier model allows us to test how they can diverge without fitting an experimental curve.
Selected record and original analysis
Specify the quantity being varied
Read the stated model and observation separately.
| Question | Record or stipulated input | Interpretation |
|---|---|---|
| Driving energy | Five invented ฮGยฐ values | ฮป, kBT and A are fixed. |
| Barrier | A squared dependence | Negative driving energy does not mean a negative barrier. |
| Rate | k/A peaks at 1 here | Not a measured molecular rate. |
Chronology
A timeline of the turning points
Selected lecture
Discussed electron-transfer theory and experiment.
Use the chemical-transfer model
Marcusโs lecture gives ฮG*=(ฮป/4)(1+ฮGยฐ/ฮป)ยฒ and k=A exp(โฮG*/kBT). The reorganisation energy ฮป includes environmental and vibrational contributions. The normal-to-inverted comparison holds other relevant model conditions fixed.
Here ฮGยฐ is reaction free energy, ฮG* the activation free energy, ฮป reorganisation energy, A the prefactor, and kBT the thermal energy at temperature T.
This study uses that classical chemical-transfer expression. It does not transfer the lectureโs separately qualified electrode expression into an invented high-driving-force calculation.
Sources for this section: [1]
Original fixed-parameter question
Stipulate ฮป=1 electronvolt, kBT=0.025 electronvolt, and the same positive prefactor A for every case. Let ฮGยฐ take 0, โ0.5, โ1, โ1.5, โ2 electronvolts. Calculate ฮG* and k/A at each value. Which case is fastest within the stipulated model?
Use the barrier and exponential-rate expressions above.
All parameters and cases here are invented. They do not represent measured rates or a fit to a particular molecule.
Sources for this section: [1]
Reference answer: keep the square
The barriers, in the listed order, are 0.25, 0.0625, 0, 0.0625, 0.25 electronvolt. Dividing by 0.025 gives 10, 2.5, 0, 2.5, 10. Thus k/A is exp(โ10), exp(โ2.5), 1, exp(โ2.5), exp(โ10), approximately 0.0000454, 0.0821, 1, 0.0821, 0.0000454.
The fastest listed case is ฮGยฐ=โ1 electronvolt. Increasing the magnitude of the negative driving energy beyond it raises the squared barrier again. The two equally spaced cases on opposite sides of โ1 have the same barrier in this fixture.
This is a model-dependent result. If reorganisation energy, prefactor, temperature or the physical transfer regime changed between cases, the comparison would require fresh justification. The arithmetic is not evidence that every chemical reaction follows the same trend.
Sources for this section: [1]
What to compare with experiment
The selected lecture discusses theory and experimental investigation rather than treating a plotted prediction as its own verification. A real comparison would identify the chemical system and which quantities were controlled or inferred. Here the useful conclusion is conditional and exact: the supplied equation and fixed parameters produce a nonmonotonic rate.
Sources for this section: [1]
Continue the chronology
Related history guides
Biography context
Explore United States through connected lives
Connected biographies
The working life behind the record
Quick answers
Common questions
- Is the rate prefactor inferred from data?
- No. The exercise compares k/A and stipulates a common prefactor.
- Is this an electrode-rate calculation?
- No. It uses the explicitly selected chemical-transfer model.
Primary and institutional sources
How this history was checked
These selected records support the historical account. Invented inputs and their original reference answers are identified explicitly. This article was reviewed on 31 October 2026.