Last updated: May 8, 2026
Nernst Equation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Nernst equation calculator computes the actual electrochemical cell potential E using the formula 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 the number of electrons transferred, F is Faraday's constant (96485 C/mol), and Q is the reaction quotient. At 25 °C this simplifies to E = E° − (0.05916/n)·log₁₀(Q). The calculator also returns the thermal voltage term RT/nF and a spontaneity classification.
To calculate the cell potential with the Nernst equation, use E equals E-standard minus RT over nF times the natural log of Q. At 25 degrees Celsius, this simplifies to E equals E-standard minus 0.05916 divided by n, times log base 10 of Q. Plug in the standard potential in volts, the number of electrons transferred, the reaction quotient, and the temperature in Kelvin.
Key Takeaways
- The Nernst equation E = E° − (RT/nF)·ln(Q) gives the actual cell potential under non-standard concentrations, pressures, and temperatures.
- At 25 °C, the simplified form E = E° − (0.05916/n)·log₁₀(Q) holds; the 0.05916 V factor equals RT·ln(10)/F.
- When Q = K (equilibrium constant), the cell potential equals zero — the battery is dead and no electrical work is possible.
- Temperature increases the Nernst correction term: at 37 °C the prefactor RT/F ≈ 26.71 mV compared to 25.69 mV at 25 °C.
- The Nernst equation underpins pH meters, ion-selective electrodes, biological membrane potentials, and modern battery management systems.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
E = E° − (R·T)/(n·F) · ln(Q)
Where:
- E=Cell potential(V)
- E°=Standard cell potential(V)
- R=Universal gas constant(J mol⁻¹ K⁻¹)
- T=Temperature(K)
- n=Number of electrons transferred(dimensionless)
- F=Faraday's constant(C mol⁻¹)
- Q=Reaction quotient(dimensionless)
Worked Examples
Daniell Cell at Standard Conditions
The classic zinc-copper Daniell cell (Zn²⁺/Cu²⁺) at standard conditions (Q = 1, 25 °C). The cell emf equals the standard potential.
- 1Identify the half-reactions: Zn → Zn²⁺ + 2e⁻ (E°anode = −0.76 V) and Cu²⁺ + 2e⁻ → Cu (E°cathode = +0.34 V).
- 2Calculate E°cell = E°cathode − E°anode = 0.34 − (−0.76) = 1.10 V.
- 3At Q = 1, ln(Q) = 0, so the Nernst correction term = 0.
- 4E = 1.10 − (8.314 × 298.15) / (2 × 96485) × 0 = 1.10 V.
Daniell Cell with Q = 100 (product-rich solution)
The same Daniell cell when the reaction quotient has risen to 100, representing higher product (Zn²⁺) or lower reactant (Cu²⁺) concentrations.
- 1E°cell = 1.10 V, n = 2, Q = 100, T = 298.15 K.
- 2Thermal term = RT/nF = (8.314 × 298.15) / (2 × 96485) ≈ 0.012847 V.
- 3Nernst correction = 0.012847 × ln(100) = 0.012847 × 4.60517 ≈ 0.05916 V.
- 4E = 1.10 − 0.05916 ≈ 1.04084 V.
Standard Hydrogen Electrode at Physiological Temperature
The standard hydrogen electrode (E° = 0 V, n = 1) operating at 37 °C (310 K) with Q = 0.1 (proton concentration = 0.1 M, pH = 1).
- 1E°cell = 0 V, n = 1, Q = 0.1, T = 310 K.
- 2Thermal term = RT/nF = (8.314 × 310) / (1 × 96485) ≈ 0.026709 V.
- 3Nernst correction = 0.026709 × ln(0.1) = 0.026709 × (−2.302585) ≈ −0.06150 V.
- 4E = 0 − (−0.06150) = +0.06150 V.
Introduction
The Nernst equation is one of the most fundamental relationships in electrochemistry, describing how the electromotive force (EMF) of an electrochemical cell changes as reactant and product concentrations deviate from standard conditions. Formulated by German physical chemist Walther Nernst in 1889, it bridges thermodynamics and electrochemistry: E = E° − (RT/nF) · ln(Q). Under standard conditions (all activities = 1, so Q = 1), the natural log term vanishes and E = E°. At any other composition, the actual potential differs — sometimes dramatically — from the standard value. This calculator computes the real cell potential for any combination of standard potential, electron stoichiometry, reaction quotient, and temperature, making it indispensable for battery design, corrosion science, biological membrane potential analysis, and analytical electrochemistry.
Derivation from Gibbs Free Energy
The Nernst equation emerges naturally from classical thermodynamics. The Gibbs free energy change for a reaction at arbitrary composition is:
ΔG = ΔG° + RT·ln(Q)
Because electrical work is related to cell potential by ΔG = −nFE and ΔG° = −nFE°, substituting gives:
−nFE = −nFE° + RT·ln(Q)
Dividing through by −nF yields the Nernst equation:
E = E° − (RT/nF)·ln(Q)
At equilibrium, E = 0 and Q = K (the equilibrium constant), giving E° = (RT/nF)·ln(K) — a powerful link between standard potential and equilibrium constant. You can explore thermodynamic driving forces further with our reaction quotient calculator and equilibrium constant calculator.
The 25 °C Simplified Form and the 0.05916 V Factor
At 25 °C (298.15 K), the prefactor RT/F evaluates to exactly 0.025693 V (the thermal voltage). Converting from natural log to base-10 log (multiply by ln(10) ≈ 2.302585) gives:
E = E° − (0.05916 / n) · log₁₀(Q)
The value 0.05916 V (often rounded to 0.0592 V or 59.16 mV) is the workhorse of analytical electrochemistry — it appears in pH meter calibration, ion-selective electrode response (the Nernst slope), and the Nernst equation for concentration cells. Each decade change in Q shifts E by 59.16/n millivolts. At temperatures other than 25 °C, the full formula E = E° − (RT/nF)·ln(Q) must be used; this calculator applies the general form automatically. For a detailed treatment of the temperature dependence, refer to Atkins and de Paula's *Physical Chemistry* (Oxford University Press, 2014).
Understanding the Reaction Quotient Q
The reaction quotient Q measures how far a system has progressed toward equilibrium relative to its standard state: - Q < K: Reaction proceeds forward (more product forms) → E > E_eq - Q = 1: Standard conditions → E = E° (by definition) - Q = K: System at equilibrium → E = 0 (cell is dead) - Q > K: Reaction runs in reverse → E can be negative For the Daniell cell Zn | Zn²⁺(aq) ‖ Cu²⁺(aq) | Cu, Q = Zn²⁺]/[Cu²⁺]. Doubling the Cu²⁺ concentration halves Q and increases E by (0.05916/2)·log₂ ≈ 8.9 mV. This sensitivity is exploited in concentration cells, which generate EMF purely from concentration differences (E° = 0). Ionic strength corrections to activity coefficients become important in concentrated solutions — see our [ionic strength calculator. External reference: NIST Chemistry WebBook on electrochemical data.
Temperature Effects and Biological Applications
Temperature modifies both the thermal prefactor RT/nF and (indirectly) the standard potential E° through the Gibbs-Helmholtz equation. The direct effect through the prefactor is straightforward: at 37 °C (310 K, physiological temperature), the Nernst factor rises to 26.71 mV, versus 25.69 mV at 25 °C — a modest ~4% increase that nonetheless shifts potentials measurably in sensitive biosensors. Biological systems exploit the Nernst equation constantly: - Nernst potential (equilibrium potential): The membrane voltage at which the electrical gradient balances the concentration gradient for an ion. For K⁺ with [K⁺]out/[K⁺]in ≈ 1/30, E_K = (RT/F)·ln(1/30) ≈ −90 mV at 37 °C. - Goldman-Hodgkin-Katz equation: Extends the Nernst equation to multiple ions to predict the resting membrane potential of neurons. - Mitochondrial proton-motive force: The chemiosmotic gradient driving ATP synthase is a direct application of electrochemical potential differences. In all these contexts, this Nernst equation calculator provides the quantitative backbone for understanding how ion concentration ratios translate into electrical potentials.
Practical Applications: Batteries, Corrosion, and Sensors
The Nernst equation governs the real-world performance of virtually every electrochemical device: Batteries: The open-circuit voltage of a lithium-ion cell depends on the lithium chemical potential in both electrodes — a Nernst-type relationship. As the battery discharges, Q changes and the voltage drops. State-of-charge algorithms in battery management systems numerically integrate the Nernst equation. For related work, see our cell EMF calculator and electrolysis calculator. pH meters: The glass electrode obeys the Nernst equation: E = const − (0.05916/1)·pH at 25 °C. This is why pH meters must be temperature-compensated — the Nernst slope changes with T. Corrosion (Pourbaix diagrams): Metal dissolution or passivation depends on Nernst-corrected potentials as a function of pH and potential. Iron corrodes when E_Fe²⁺/Fe, corrected for local [Fe²⁺], is below the corrosion potential. Ion-selective electrodes (ISE): These sensors — for Na⁺, K⁺, Ca²⁺, F⁻ — exploit the Nernst response directly. The theoretical sensitivity (Nernst slope) is 59.16/z mV per decade concentration change, where z is the ion charge. For additional electrochemical reference data, see LibreTexts Electrochemistry).
Common Mistakes and How to Avoid Them
Even experienced chemists encounter pitfalls when applying the Nernst equation: 1. Sign of E°: Always compute E°cell = E°cathode − E°anode using reduction potentials. A common error is using oxidation potentials for the anode without sign reversal. 2. Units of Q: Q must be dimensionless. Express concentrations relative to the standard state (1 mol/L), pressures relative to 1 bar, and pure solids/liquids have activity = 1 (they don't appear in Q). 3. ln vs. log₁₀: The thermodynamically correct form uses the natural logarithm. The 0.05916 V shorthand already incorporates the ln(10) conversion factor and applies only at 25 °C with log base 10. 4. n is stoichiometric, not ionic charge: For Cr₂O₇²⁻ + 14H⁺ + 6e⁻ → 2Cr³⁺ + 7H₂O, n = 6, not 2 or 3. 5. Q ≤ 0 is undefined: The logarithm of a non-positive number is undefined. Ensure all concentrations/activities are strictly positive. This calculator returns zero and flags invalid inputs if Q ≤ 0.
Quick Reference Card
Nernst Equation Quick Reference
Quick reference • Nernst Equation Calculator
E = E° − (RT/nF)·ln(Q) | At 25°C: E = E° − (0.05916/n)·log₁₀(Q)Valid range: T > 0 K; n ≥ 1 (integer); Q > 0; E° can be any real number
Common Values
⚠ Watch Out
- •Q must be strictly greater than zero — ln(0) and ln(negative) are undefined.
- •Always use reduction potentials for both half-reactions: E°cell = E°cathode − E°anode.
- •The 0.05916 V shorthand applies ONLY at 25 °C (298.15 K). Use full RT/nF at other temperatures.
- •n is the stoichiometric electron count from the balanced equation, not the ion charge.
- •Activities, not concentrations, are thermodynamically rigorous — use activity coefficients (γ) in concentrated solutions.
Pro Tips
- →To find E° from half-reactions: look up standard reduction potentials in an IUPAC or NIST table, then E°cell = E°(more positive) − E°(more negative).
- →A quick sanity check: when Q = 1, the Nernst correction is zero and E = E°. Verify your calculator returns E° for Q = 1.
- →For pH-sensitive reactions, remember that H⁺ appears in Q. Each pH unit change (×10 in [H⁺]) shifts E by 0.05916 × (n_H/n) volts at 25 °C.
- →The equilibrium constant K can be obtained from K = exp(nFE°/RT) — use this to cross-check thermodynamic data with electrochemical measurements.
FAQs
What is the Nernst equation and when do I use it?
The Nernst equation — E = E° − (RT/nF)·ln(Q) — gives the actual cell potential when reactant/product concentrations (or gas pressures) differ from the standard state (1 M, 1 bar, 25 °C). Use it whenever you know the standard potential but the cell operates under non-standard conditions: real batteries, dilute or concentrated electrolytes, biological membranes, or any time you need an accurate EMF rather than a textbook E° value.
What are the values of R, F, and the 25 °C Nernst factor?
R (universal gas constant) = 8.314 J mol⁻¹ K⁻¹; F (Faraday's constant) = 96485 C mol⁻¹. At exactly 25 °C (298.15 K), RT/F = 25.693 mV, and RT·ln(10)/F = 59.16 mV (the '0.05916 V' Nernst factor used in the base-10 shorthand E = E° − (0.05916/n)·log₁₀Q).
How does temperature affect the Nernst equation?
Temperature appears linearly in the thermal prefactor RT/nF. A higher T amplifies the Q-dependent correction: at 37 °C the prefactor increases to 26.71 mV (≈4% more than at 25 °C). Additionally, E° itself varies slightly with temperature through the Gibbs-Helmholtz equation, but this calculator focuses on the direct RT/nF effect — which dominates for small temperature changes — using the user-supplied E° as the baseline.
What is the reaction quotient Q and how do I calculate it?
Q is computed from the balanced half-cell or overall cell reaction using activities (approximately equal to molar concentrations for dilute solutions). For Zn | Zn²⁺(c₁) ‖ Cu²⁺(c₂) | Cu, Q = [Zn²⁺]/[Cu²⁺]. Solid electrodes and pure solvents have activity = 1. Gas-phase species use partial pressure in bars divided by 1 bar. Q = 1 under standard conditions; Q = K at equilibrium.
What happens when Q equals K (the equilibrium constant)?
When Q = K, the reaction is at equilibrium and E = 0 — the cell can do no net electrical work. This is the condition of a 'dead battery'. Setting E = 0 in the Nernst equation recovers the thermodynamic relation E° = (RT/nF)·ln(K), which is used to calculate equilibrium constants from electrochemical data (and vice versa).
Can the cell potential be negative?
Yes. A negative E means the overall cell reaction as written is non-spontaneous under those conditions (ΔG > 0). This occurs when Q > K — the system has already exceeded equilibrium and would need to run in reverse to reach it. Negative E is also the basis of electrolysis: applying a voltage greater than |E| drives a thermodynamically unfavourable reaction. This calculator flags negative results as 'non-spontaneous'.