Nernst Equation Calculator
Compute electrochemical cell potential E and Gibbs free energy ΔG under any non-standard conditions. Enter E°, temperature, electron count, and Q — or build Q species-by-species — for an instant, fully explained result.
Trusted by chemistry students and educators at He Loves Math, with worked examples for the Daniell cell, silver electrode, and more.
The Nernst Equation
At 25 °C (298.15 K) the simplified form using log10 is:
Both forms are equivalent; this calculator uses the exact general form valid at any temperature.
Nernst Equation Calculator
Select a preset example or enter your own values. Choose whether to supply Q directly or build it from species concentrations and stoichiometric coefficients.
Products — Q numerator
Reactants — Q denominator
Pure solids and pure liquids have activity = 1 (omit or enter activity 1, coefficient 1).
Constants used: R = 8.314 J mol⁻¹ K⁻¹, F = 96 485 C mol⁻¹. Results are rounded to 4 decimal places.
What Is the Nernst Equation?
The Nernst equation is one of the most powerful tools in physical chemistry and electrochemistry. Named after German physicist and chemist Walther Hermann Nernst (1864–1941), who derived it in 1889, the equation describes how the electrical potential of an electrochemical cell changes when the concentrations, partial pressures, or temperature of the participating species deviate from the idealised standard conditions.
Under standard conditions — all dissolved species at 1 M activity, all gases at 1 atm (or 1 bar) partial pressure, and temperature fixed at 25 °C (298.15 K) — the cell potential is the standard cell potential E°. In real experiments, however, such perfect concentrations rarely exist. A battery discharges and ion concentrations shift; a nerve cell maintains different ionic concentrations inside and outside the membrane; a corroding steel structure contacts solutions of varying composition. The Nernst equation translates these real conditions into a precise prediction of the actual (non-standard) cell potential E.
Beyond batteries and corrosion, the Nernst equation underpins potentiometry (pH meters, ion-selective electrodes), biological electrochemistry (Nernst potential of neuronal membranes), and the thermodynamics of chemical equilibrium (the Nernst equation at equilibrium yields the direct relationship between E° and the equilibrium constant K).
Variable definitions
- E — Cell potential under actual conditions (V)
- E° — Standard cell potential (V)
- R — Gas constant = 8.314 J mol⁻¹ K⁻¹
- T — Temperature in Kelvin (K)
- n — Moles of electrons transferred
- F — Faraday constant = 96 485 C mol⁻¹
- Q — Reaction quotient (dimensionless)
History — Walther Nernst (1864–1941)
Nernst derived his equation from the thermodynamic relationship between Gibbs free energy and chemical potential. His work on electrochemistry and the Nernst heat theorem (third law of thermodynamics) earned him the 1920 Nobel Prize in Chemistry. The equation was a cornerstone in unifying classical thermodynamics with electrochemistry and remains a standard tool in every chemistry curriculum worldwide.
Thermodynamic Derivation
The Nernst equation is not an empirical formula — it emerges directly from classical thermodynamics. The derivation requires just two fundamental relationships:
1. Gibbs energy and cell potential:
2. Gibbs energy under non-standard conditions:
Substituting the first pair into the second:
Dividing through by \(-nF\):
The term \(RT/nF\) is the thermal voltage factor. Its value at 298.15 K per electron (n = 1) is:
The Reaction Quotient Q
The reaction quotient Q is central to the Nernst equation. For a general balanced redox reaction:
Where [X] represents the activity of species X:
- Dissolved solutes: activity ≈ molar concentration (in mol/L = M), referenced to 1 M
- Gases: activity = partial pressure in bar (or atm), referenced to 1 bar (or 1 atm)
- Pure solids and pure liquids: activity = 1 (exact), omit from Q expression
- Solvent (water in dilute solutions): activity ≈ 1, omit from Q
The At-25 °C Simplification: 0.05916 V
In most university textbooks, the Nernst equation appears in a slightly different form that uses the common (base-10) logarithm instead of the natural logarithm:
At exactly 25 °C (T = 298.15 K), the factor \(2.303 \times RT/F\) evaluates to:
Giving the widely used shorthand:
This form is only valid at exactly 25 °C (298.15 K). For any other temperature — blood temperature, industrial smelting, environmental measurements at 10 °C — you must use the full \(RT/nF\) expression. This calculator handles all temperatures correctly.
Gibbs Free Energy and Spontaneity
The relationship between cell potential E and Gibbs free energy ΔG is the conceptual bridge linking thermodynamics and electrochemistry:
E > 0 → Spontaneous
When E is positive, ΔG = −nFE is negative. The reaction releases electrical energy and proceeds spontaneously — a galvanic cell (battery) operating in forward mode. The maximum work the cell can do per mole of reaction is |ΔG| joules.
E < 0 → Non-Spontaneous
When E is negative, ΔG > 0. The reaction requires external electrical energy input to proceed — an electrolytic cell. For example, recharging a battery forces the non-spontaneous reverse reaction using external current.
At electrochemical equilibrium, no net current flows and E = 0. Setting E = 0 in the Nernst equation gives the fundamental link between E° and the equilibrium constant K:
Real-World Applications of the Nernst Equation
1. pH Measurement — Glass Electrode
The glass electrode used in every laboratory pH meter is a direct application of the Nernst equation. The half-reaction at the glass membrane involves hydrogen ions (H⁺), n = 1, E° = 0 V (vs SHE). The Nernst potential becomes:
The Nernst slope of −59.16 mV per pH unit at 25 °C is the theoretical ideal. Temperature affects this slope (at 37 °C it becomes approximately −61.54 mV/pH), which is why pH meters require temperature compensation.
2. Nerve and Muscle Cell Potentials
The resting membrane potential of neurons (approximately −70 mV) arises because different ions (K⁺, Na⁺, Cl⁻) are maintained at very different concentrations inside and outside the cell by ion pumps. The Nernst potential for each ion gives the equilibrium potential at which there is no net ionic flux:
For K⁺ in a typical neuron at 37 °C: EK ≈ −90 mV. For Na⁺: ENa ≈ +60 mV. The actual resting potential is set by the Goldman equation, a weighted combination of multiple Nernst potentials.
3. Electrochemical Corrosion
Steel corrodes because iron dissolution (Fe → Fe²⁺ + 2e⁻) and oxygen reduction (O₂ + 4H⁺ + 4e⁻ → 2H₂O) form a galvanic couple. The Nernst equation predicts how the corrosion driving force changes with O₂ partial pressure, pH, and Fe²⁺ concentration, informing protective coating and cathodic protection design in marine and infrastructure engineering.
4. Batteries and Fuel Cells
As a battery discharges, reactants are consumed and products accumulate. Q increases with discharge, reducing E towards zero (the battery "dies"). The Nernst equation accurately predicts this voltage drop, which is observed as the terminal voltage of a lithium-ion cell decreasing from ~4.2 V fully charged to ~3.0 V near depletion. Engineers use it to design battery management systems (BMS) that estimate state-of-charge from measured voltage.
5. Concentration Cells
A concentration cell has identical electrodes but different electrolyte concentrations. Since E° = 0, the potential is driven purely by the concentration gradient:
This principle is used in electromotive force measurements, lead-acid battery internal diagnostics, and understanding biological ion channels.
Worked Examples
Example 1 — Daniell Cell Under Non-Standard Conditions
Setup: Zn(s) | Zn²⁺(0.10 M) ‖ Cu²⁺(1.50 M) | Cu(s). Find E at 25 °C.
Half-reactions (reduction potentials):
Cu²⁺ + 2e⁻ → Cu(s), E° = +0.34 V (cathode)
Zn²⁺ + 2e⁻ → Zn(s), E° = −0.76 V (reversed at anode)
E°cell = 0.34 − (−0.76) = +1.10 V, n = 2
Reaction quotient: The net reaction is Zn(s) + Cu²⁺ → Zn²⁺ + Cu(s).
Since Q < 1olean, ln Q < 0, so E > E°. The lower Zn²⁺ and higher Cu²⁺ concentration make the cell more powerful than at standard conditions.
Example 2 — Silver Electrode Half-Cell
Setup: Ag⁺(0.01 M) | Ag(s). Half-reaction: Ag⁺ + e⁻ → Ag(s), E° = +0.80 V, n = 1, T = 25 °C.
Example 3 — Iron Half-Cell at Elevated Temperature
Setup: Fe²⁺(0.001 M) + 2e⁻ → Fe(s), E° = −0.44 V, n = 2, T = 40 °C = 313.15 K, Q = 1/[Fe²⁺] = 1000.
E is more negative than E° because Q > 1. The cell remains non-spontaneous for iron reduction, but the magnitude indicates more energy is needed to drive iron plating at these conditions.
Standard Reduction Potentials (25 °C, vs SHE)
Use these values to calculate E°cell = E°cathode − E°anode. More positive = stronger oxidising agent; more negative = stronger reducing agent.
| Half-Reaction (Reduction) | E° (V) | Common role |
|---|---|---|
| F₂ + 2e⁻ → 2F⁻ | +2.87 | Strongest common oxidising agent |
| Cl₂ + 2e⁻ → 2Cl⁻ | +1.36 | Chlorine in swimming pools & bleach |
| O₂ + 4H⁺ + 4e⁻ → 2H₂O | +1.23 | Oxygen reduction in fuel cells & corrosion |
| Ag⁺ + e⁻ → Ag(s) | +0.80 | Silverware tarnishing, silver plating |
| Cu²⁺ + 2e⁻ → Cu(s) | +0.34 | Copper plating, Daniell cell cathode |
| 2H⁺ + 2e⁻ → H₂(g) | 0.000 | Standard Hydrogen Electrode (reference) |
| Ni²⁺ + 2e⁻ → Ni(s) | −0.25 | Nickelcadmium batteries, plating |
| Fe²⁺ + 2e⁻ → Fe(s) | −0.44 | Iron corrosion (rusting) |
| Zn²⁺ + 2e⁻ → Zn(s) | −0.76 | Daniell cell anode, galvanisation |
| Al³⁺ + 3e⁻ → Al(s) | −1.66 | Aluminium smelting (Hall–Héroult) |
| Na⁺ + e⁻ → Na(s) | −2.71 | Sodium production, Na-ion batteries |
| Li⁺ + e⁻ → Li(s) | −3.04 | Lithium-ion batteries (most negative common) |
Frequently Asked Questions
What is the Nernst equation?
The Nernst equation (\(E = E^\circ - \frac{RT}{nF}\ln Q\)) relates the cell potential E under actual (non-standard) conditions to the standard potential E°, temperature T, number of electrons n, and the reaction quotient Q. It was derived in 1889 by Walther Nernst from the identity ΔG = ΔG° + RT ln Q combined with ΔG = −nFE. It quantifies how concentrations, pressures, and temperature shift the driving force of a redox reaction away from standard conditions.
Why is 0.0592 V used in the Nernst equation at 25 °C?
At 25 °C (T = 298.15 K), the factor 2.303 × R × T / F = 2.303 × 8.314 × 298.15 / 96 485 = 0.05916 V. This converts the natural log (ln) form to the common log (log₁₀) form: \(E = E^\circ - \frac{0.05916}{n}\log_{10}Q\). The value 0.0592 is a further rounded approximation. This simplification only holds at exactly 25 °C — for other temperatures, the full RT/nF must be used.
What is the reaction quotient Q?
Q is the ratio of the product of activities of products (raised to stoichiometric coefficients) to the product of activities of reactants (raised to stoichiometric coefficients), evaluated at the current (not equilibrium) state. For solutes, activity ≈ molar concentration (M); for gases, activity ≈ partial pressure (bar or atm). Pure solids and liquids have activity = 1 and do not appear in Q. When Q equals the equilibrium constant K, the system is at equilibrium and E = 0.
How are cell potential and Gibbs free energy related?
\(\Delta G = -nFE\). Under standard conditions: \(\Delta G^\circ = -nFE^\circ\). Positive E → negative ΔG → spontaneous. Negative E → positive ΔG → non-spontaneous. The maximum electrical work a cell can perform equals |ΔG| in joules per mole of reaction. At equilibrium E = 0, ΔG = 0, and \(E^\circ = \frac{RT}{nF}\ln K\), linking standard potential directly to equilibrium constant K.
What does E° (standard electrode potential) mean?
E° is the electrode half-cell potential measured under standard conditions: all dissolved species at unit activity (≈ 1 M), all gases at 1 atm or 1 bar, temperature = 25 °C (298.15 K). All E° values are reported relative to the Standard Hydrogen Electrode (SHE), which is assigned E° = 0.000 V by IUPAC convention. For a full cell: E°_cell = E°_cathode − E°_anode, using the standard reduction potential table.
What is a concentration cell?
A concentration cell uses two identical electrode materials in electrolytes of different concentrations. Since E° = 0 (same material), \(E = -\frac{RT}{nF}\ln Q = \frac{RT}{nF}\ln\frac{[\text{C}]_\text{high}}{[\text{C}]_\text{low}}\). The cell potential is driven entirely by the concentration difference. As ions transfer through the salt bridge until concentrations equalise, E approaches zero. Biological examples include ionic gradients across cell membranes.
How does the Nernst equation apply to pH measurements?
A pH glass electrode exploits the Nernst equation for H⁺: \(E = -\frac{0.05916\,\text{V}}{1}\times\text{pH}\) at 25 °C, giving a Nernst slope of −59.16 mV per pH unit. The glass membrane develops a potential proportional to log[H⁺] across the membrane. At 37 °C (body temperature), the slope becomes −61.54 mV/pH, which is why clinical pH meters apply automatic temperature correction. Electrode aging causes the slope to decrease below the theoretical Nernst slope, reducing electrode accuracy.
How does temperature affect cell potential?
Temperature appears directly in the thermal voltage factor RT/nF. At 25 °C, RT/F = 25.69 mV; at 37 °C = 26.96 mV; at 60 °C = 28.77 mV. Higher T amplifies the Nernst correction term. If Q < 1 (ln Q negative), raising T makes E larger. If Q > 1 (ln Q positive), raising T reduces E. The standard potential E° also shifts with temperature via the Gibbs–Helmholtz relation: (∂E°/∂T)_P = ΔS°/nF, meaning entropy-driven cells may gain or lose E° with heating.
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