Determine the electrochemical cell potential under non-standard conditions using the Nernst equation. Enter standard cell potential, number of electrons transferred, temperature, and reaction quotient.
The Nernst equation predicts how a battery's voltage changes as it's used. Standard cell potentials (E°) describe the voltage of a galvanic cell with all species at 1 M, 1 atm, and 25 °C — neat lab conditions. Real batteries operate at non-standard concentrations: a discharging battery has decreasing reactant concentrations and increasing product concentrations, which shifts the cell voltage according to the Nernst equation E = E° − (RT/nF) ln(Q). As Q grows (more product, less reactant), the voltage drops. When voltage hits zero, the battery is at equilibrium — "dead" from an electrical standpoint, even though some unreacted starting material may remain.
The same equation governs all electrochemistry beyond the standard state: pH measurement (a pH electrode is a Nernst-equation device), redox titrations, corrosion potentials, biological membrane potentials in nerve cells, and the standard reduction potentials used to predict whether a given redox reaction will spontaneously occur. Once you can use the Nernst equation, electrochemistry becomes quantitative rather than just "list of standard potentials."
This calculator handles the general form. Enter the standard cell potential E°, the number of electrons transferred in the balanced redox reaction (n), the temperature (in K), and the reaction quotient Q. The output is E — the actual cell potential under your specific conditions. For "Nernst slope" intuition at 25 °C, the common approximation is E = E° − (0.0592/n) × log₁₀(Q) — easier for back-of-envelope estimates.
**Scenario:** Daniell cell (Zn/Cu²⁺) with [Cu²⁺] = 0.10 M and [Zn²⁺] = 1.0 M, at 25 °C. Find the actual voltage. **Calculation:** E° = 1.10 V (standard); n = 2; Q = [Zn²⁺]/[Cu²⁺] = 1.0/0.10 = 10. E = 1.10 − (0.0592/2) × log(10) = 1.10 − 0.0296 = 1.070 V. **Result:** The cell delivers 1.07 V instead of 1.10 V — small loss from 10× diluted Cu²⁺. If you let the battery discharge until [Cu²⁺] = 10⁻⁶ M: Q = 10⁶, voltage drops by 0.0296 × 6 = 0.18 V to ~0.92 V. The battery keeps going but each tenfold drop in Cu²⁺ costs another 29.6 mV.
**Scenario:** A pH electrode reads 415 mV in pH 4 buffer and 60 mV in pH 7 buffer at 25 °C. Is it functioning correctly? **Calculation:** Theoretical slope: 59.16 mV/pH unit. Measured slope: (415 − 60)/(7 − 4) = 355/3 = 118.3 mV/pH unit. That's 2× the Nernstian slope — sign of trouble. **Result:** The electrode is malfunctioning (real Nernstian behavior is 59.16 mV/pH unit at 25 °C; this electrode reads 118 mV/pH). Possibly a broken reference junction or contaminated bulb. Replace electrode and recalibrate. Properly functioning pH electrodes give measured slope within 95–105% of theoretical.
**Scenario:** A neuron has [K⁺]_out = 4 mM, [K⁺]_in = 140 mM at body T (310 K). Estimate the K⁺ equilibrium potential. **Calculation:** Nernst form for ion equilibrium: E_K = (RT/zF) × ln([K⁺]_out / [K⁺]_in) = (8.314 × 310 / (1 × 96485)) × ln(4/140) = 0.0267 × (−3.555) = −0.0950 V = −95 mV. **Result:** K⁺ equilibrium potential is ~−95 mV. Neurons rest at about −70 mV (not at the K⁺ equilibrium) because Na⁺ leak and Na/K ATPase contribute. When K⁺ channels open during repolarization after an action potential, the membrane voltage moves toward this −95 mV — the basis of "after-hyperpolarization."
**Use the Nernst equation in:**
- **Battery design and analysis**: predict voltage decline as cells discharge, calculate effective capacity. - **pH measurement**: every pH meter is fundamentally a Nernst device; calibration is checking that slope = 59.16 mV/pH unit. - **Ion-selective electrodes**: F⁻, Ca²⁺, K⁺ electrodes work the same way; one ion drives a potential. - **Corrosion analysis**: predicting potential of iron in different environments, sacrificial anode sizing. - **Electroplating**: knowing voltage at non-standard plating bath compositions. - **Bioelectrochemistry**: action potentials, mitochondrial proton gradient, ATP synthase. - **Redox titrations**: equivalence-point voltage prediction for indicators. - **Electrochemical synthesis**: organic electrosynthesis, Faradaic efficiency.
**Standard reduction potentials worth knowing:**
| Half-reaction | E° (V) | |---|---| | F₂ + 2e⁻ → 2F⁻ | +2.87 (strongest oxidizer) | | O₃ + 2H⁺ + 2e⁻ → O₂ + H₂O | +2.07 | | Cl₂ + 2e⁻ → 2Cl⁻ | +1.36 | | O₂ + 4H⁺ + 4e⁻ → 2H₂O | +1.23 | | Ag⁺ + e⁻ → Ag | +0.80 | | Cu²⁺ + 2e⁻ → Cu | +0.34 | | 2H⁺ + 2e⁻ → H₂ | 0.00 (reference) | | Fe²⁺ + 2e⁻ → Fe | −0.44 | | Zn²⁺ + 2e⁻ → Zn | −0.76 | | Na⁺ + e⁻ → Na | −2.71 | | Li⁺ + e⁻ → Li | −3.04 (strongest reducer) |
To predict spontaneity: E°_cell = E°_cathode − E°_anode. Positive → spontaneous (battery delivers power); negative → non-spontaneous (needs external energy, like electrolysis).
**Common Nernst-slope facts to memorize:**
- 59.16 mV per decade (n=1) at 25 °C - 29.58 mV per decade (n=2) - 19.72 mV per decade (n=3) - At 37 °C (body T): 61.5 mV (n=1)
**Connection to thermodynamics:**
ΔG° = −n × F × E°
For n=1 reaction with E° = 1 V: ΔG° = −96,485 J/mol = −96.5 kJ/mol — very favorable.
Each volt of cell potential corresponds to about 96.5 kJ/mol of free-energy change (for n=1). This is the energy that drives the external circuit.
Calculate Gibbs free energy using ΔG = ΔH - TΔS.
Calculate the equilibrium constant (Keq) from product and reactant concentrations.
Calculate radioactive or exponential decay half-life, remaining amount, and decay constant.
Calculate osmotic pressure using the formula pi = MRT.
Solve PV = nRT for any variable — pressure, volume, moles, or temperature.
Calculate absorbance using A = εbc (molar absorptivity, path length, concentration).
Cell Potential
1.1592 V
ΔG
-223.68 kJ/mol
Result
Spontaneous (galvanic)
| Parameter | Value |
|---|---|
| Standard Potential (E0) | 1.1000 V |
| Electrons Transferred (n) | 2 |
| Temperature | 298.15 K |
| Reaction Quotient (Q) | 1.0000e-2 |
| ln(Q) | -4.605170 |
| RT/nF | 0.012846 V |
| (RT/nF)ln(Q) | -0.059156 V |
| Cell Potential (E) | 1.159156 V |
| ΔG | -223.6824 kJ/mol |
| Spontaneity | Spontaneous (galvanic) |