What is the Nernst equation?

E = E° − (RT/nF) × ln(Q), where E° is the standard cell potential, R is the gas constant (8.314 J/(mol·K)), T is temperature in Kelvin, n is electrons transferred, F is Faraday's constant (96,485 C/mol), and Q is the reaction quotient. It gives the cell potential under non-standard concentrations.

What is the simplified Nernst equation at 25°C?

At 25 °C (298.15 K): E = E° − (0.05916/n) × log₁₀(Q). The 0.05916 V is the "Nernst slope" — a 10-fold change in Q changes E by 59.16/n mV. This form is easier for mental math with logarithm tables.

Why does battery voltage drop as it discharges?

As the battery delivers current, reactants get consumed and products accumulate. This increases the reaction quotient Q. The Nernst equation says E = E° − (RT/nF) × ln(Q) — as Q grows, E decreases. When the cell reaches equilibrium (Q = K), E = 0 and the battery is "dead."

How does a pH meter use the Nernst equation?

A pH electrode generates a voltage proportional to log[H⁺]. By the Nernst equation, the voltage changes 59.16 mV per pH unit at 25 °C (n=1, 10× change in [H⁺]). The meter converts measured voltage to pH using this calibrated relationship.

What is n in the Nernst equation?

n is the number of electrons transferred in the balanced overall redox reaction. For Cu²⁺ + 2e⁻ → Cu paired with Zn → Zn²⁺ + 2e⁻, the overall reaction has n = 2. Always balance both half-reactions so electrons cancel; the cancelled count is n.

How is the Nernst equation related to thermodynamics?

ΔG = −nFE relates Gibbs free energy to cell potential. For E > 0: ΔG 0, non-spontaneous (electrolysis requires external power). Each volt corresponds to about 96.5 kJ/mol of free energy per electron.

What's the difference between E and E°?

E° is the standard cell potential — voltage at 1 M, 1 atm, 25 °C standard conditions. E is the actual potential under non-standard conditions. They're related by the Nernst equation. Real batteries operate at non-standard conditions, so E (not E°) is what you measure at the terminals.

Nernst Equation Calculator

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.

Inputs

Standard Cell Potential (E0) V

Electrons Transferred (n)

Temperature (K)

Reaction Quotient (Q)

Results

Cell Potential

1.1592 V

ΔG

-223.68 kJ/mol

Result

Spontaneous (galvanic)

Nernst Equation Results

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)

Last updated: May 28, 2026

Formula

**Nernst equation (general form):** E = E° − (RT / nF) × ln(Q) Where: - **E**: cell potential under given conditions (V) - **E°**: standard cell potential (V) at 25 °C, 1 M / 1 atm - **R**: gas constant = 8.314 J/(mol·K) - **T**: absolute temperature (K) - **n**: number of electrons transferred in the balanced redox reaction - **F**: Faraday constant = 96,485 C/mol - **Q**: reaction quotient ([products] / [reactants], each raised to stoichiometric power) - **ln**: natural logarithm **At 25 °C (298.15 K), simplified form using log₁₀:** E = E° − (0.05916 / n) × log₁₀(Q) The 0.05916 V is the famous "Nernst slope" — a 10-fold change in Q changes E by 59.16/n mV. **For 1-electron reactions (n=1)**: - Q goes from 1 to 10: E drops by 59 mV - Q goes from 1 to 100: E drops by 118 mV - Q goes from 1 to 1000: E drops by 177 mV **For 2-electron reactions (n=2)**: - Each 10× change in Q: 29.6 mV. **Example: Daniell cell (Zn/Cu²⁺)** Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s); n = 2 E° = +1.10 V (standard) If [Cu²⁺] = 0.001 M and [Zn²⁺] = 1.0 M: Q = [Zn²⁺] / [Cu²⁺] = 1.0 / 0.001 = 1000 E = 1.10 − (0.05916/2) × log(1000) = 1.10 − 0.0296 × 3 = 1.10 − 0.089 = **1.011 V** The voltage drops 89 mV from the standard value due to depleted Cu²⁺ and built-up Zn²⁺. **At equilibrium, E = 0 and Q = K:** 0 = E° − (RT/nF) × ln(K) → E° = (RT/nF) × ln(K) At 25 °C: E° = (0.0592/n) × log₁₀(K) So E° = 0.30 V (n=1) → K = 10^(0.30/0.0592) = 10⁵. **pH measurement with a Nernst-type electrode:** E_glass = E_constant + 0.0592 × pH (at 25 °C) A pH-sensitive electrode is essentially a Nernst-equation device. Each pH unit corresponds to 59.16 mV of electrode response. Modern pH meters convert measured voltage directly to pH. **Biological membrane potentials (Nernst form for ions):** For Na⁺, K⁺, Cl⁻ across a cell membrane: equilibrium potential = (RT/zF) × ln([ion_out] / [ion_in]) At 37 °C (body temp) for K⁺ (z = 1, [K]out = 4 mM, [K]in = 140 mM): E_K = 0.0615 × log(4/140) = 0.0615 × (−1.544) = −95 mV This is the resting potential contributed by K⁺ in neurons.

How to use this calculator

Identify the balanced overall redox reaction. Number of electrons transferred (n) comes from the balanced equation.
Look up the standard cell potential E° from a table of reduction potentials. E°_cell = E°_cathode − E°_anode.
Enter the temperature (K). Use 298.15 for 25 °C; physiological temperature is 310 K.
Calculate Q from concentrations: products / reactants, each to the stoichiometric power.
The calculator returns the actual cell potential E under these conditions.
For a "what if" battery design, vary Q to see how voltage drops as the cell discharges.

Worked examples

Daniell cell with diluted electrolyte

**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.

pH electrode calibration

**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.

Resting potential of a neuron

**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."

When to use this calculator

**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.

Common mistakes to avoid

Using temperature in Celsius instead of Kelvin. The RT/nF factor requires absolute temperature.
Mixing up Q and K. Q is for current concentrations (use in Nernst eqn). K is the equilibrium constant (use to compute E° → K relation).
Forgetting the stoichiometric powers in Q. For aA + bB → cC + dD, Q = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ.
Confusing single half-cell potentials with overall cell potential. E_cell = E_cathode − E_anode, both reported as reductions.
Using the wrong sign convention. By IUPAC, all half-reactions are reductions; the one running backward (oxidation) gets a negative sign when added.
Treating the Nernst equation as describing rate. It predicts equilibrium voltage; kinetics (overpotential) can make actual battery voltage even lower under current draw.
Forgetting that pure solids and pure liquids are activity = 1 (not in Q). Same convention as equilibrium constants.

Frequently Asked Questions

Sources & further reading

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