Calculator
Formula Used
- Nernst: E = E° − (RT / nF) ln(Q)
- Link to energy: ΔG = −n F E and ΔG° = −n F E°
- Equilibrium: K = exp(−ΔG° / RT)
- Marcus rate: k_ET = (2π/ħ)|V|²(1/√(4πλk_BT)) exp(−(ΔG°+λ)²/(4λk_BT))
How to Use This Calculator
- Select a calculation mode matching your experiment or model.
- Enter temperature in kelvin, then fill required fields.
- For Nernst, provide E°, n, and a positive Q value.
- For energy conversion, enter either E or ΔG, not both.
- For Marcus, enter ΔG°, λ, V, and consistent energy units.
- Press Calculate to view results above the form instantly.
- Use the export buttons to save results as CSV or PDF.
Example Data Table
| Scenario | Inputs | Typical Output |
|---|---|---|
| Nernst correction | E°=1.10 V, n=2, Q=0.10, T=298.15 K | E≈1.13 V, ΔG≈−218 kJ/mol |
| Energy from potential | E=0.50 V, n=1, T=298.15 K | ΔG≈−48.2 kJ/mol, K≫1 |
| Marcus rate | ΔG°=−20 kJ/mol, λ=60 kJ/mol, V=10 meV, T=298.15 K | k_ET≈10⁹–10¹¹ s⁻¹ (order-of-magnitude) |
Professional Article
1) What electron transfer describes
Electron transfer is the movement of charge between a donor and an acceptor. In electrochemistry it appears as a redox half‑reaction at an electrode; in molecular systems it can occur through space or along bonds. The direction is governed by energy differences, while the speed depends on coupling and reorganization effects.
2) Connecting potential to energy
Electrical potential is an energy-per-charge measure. For a reaction that transfers n electrons, the free-energy change is ΔG = −nFE, where F is Faraday’s constant. A positive cell potential implies negative ΔG, meaning the process is thermodynamically favorable under the specified conditions.
3) Why the reaction quotient matters
Real systems rarely remain at standard state. The reaction quotient Q summarizes how far the mixture is from standard conditions using activities or effective concentrations. The Nernst correction shifts the potential by (RT/nF) ln(Q). Small changes in Q can noticeably change E, especially at higher temperature or smaller n.
4) Temperature sensitivity and scaling
Temperature enters both equilibrium and kinetics. In the Nernst equation, higher T increases the magnitude of the correction term, making E more responsive to composition. In thermodynamics, the factor exp(−ΔG/RT) controls the equilibrium constant, so moderate ΔG values can translate into very large or very small K.
5) Equilibrium constant interpretation
The calculator reports K using K = exp(−ΔG°/RT). Values much greater than one indicate products dominate at equilibrium; values much less than one indicate reactants dominate. Because K is exponential, reporting results in scientific notation is practical and helps prevent misreading extremely large magnitudes.
6) Marcus framework for rates
Thermodynamics does not guarantee speed. The Marcus model estimates the electron‑transfer rate constant by combining electronic coupling V, temperature, and the reorganization energy λ. The activation term depends on (ΔG° + λ)², predicting an “inverted region” where overly strong driving force can slow the transfer.
7) Choosing realistic inputs
For electrochemical work, use measured or tabulated E° values, correct n, and compute Q from activities when possible. For Marcus estimates, ΔG° and λ are commonly reported in kJ/mol or eV, while couplings are often in the meV range. Keep sign conventions consistent and verify temperature in kelvin.
8) Practical workflow and validation
A practical workflow is to compute an operating potential via Nernst, convert to ΔG for thermodynamic insight, then explore Marcus parameters to assess likely rates. Compare outputs against known benchmarks: typical electrochemical ΔG values are tens to hundreds of kJ/mol, while many molecular rates fall between 106 and 1012 s⁻¹ depending on coupling and λ.
FAQs
1) What does Q represent in the Nernst mode?
Q is the reaction quotient built from activities (or effective concentrations) of products divided by reactants, each raised to stoichiometric powers. It must be positive and dimensionless for the logarithm to be valid.
2) Why does the calculator ask for temperature in kelvin?
The constants and exponential relations use absolute temperature. Using kelvin keeps the Nernst correction, ΔG conversions, and equilibrium calculations consistent and prevents negative or shifted temperature scales from producing invalid results.
3) When should I use the potential ↔ free energy mode?
Use it when you already know either the potential E or the free energy change ΔG and want the other quantity plus the equilibrium constant. Enter only one of them so the conversion stays unambiguous.
4) What sign should ΔG° have for a favorable transfer?
A negative ΔG° indicates an exergonic process and corresponds to an equilibrium constant greater than one at the specified temperature. A positive ΔG° indicates the reverse direction is favored at equilibrium.
5) What is the reorganization energy λ in Marcus theory?
λ is the energy required to reorganize the solvent and internal molecular coordinates so the electron can transfer. Larger λ generally increases the activation requirement, reducing the rate when other parameters remain fixed.
6) Why can a larger driving force reduce the Marcus rate?
Marcus theory predicts an “inverted region.” If −ΔG° becomes much larger than λ, the activation term increases again because it depends on (ΔG°+λ)², which can decrease the predicted rate constant.
7) Are the CSV and PDF exports reproducible?
Yes. Exports capture the displayed results and key calculation metadata. For reproducible records, keep your input values and units consistent, and export immediately after computing so the saved file matches the on‑screen results.
Accurate outputs help you interpret electron transfer confidently always.