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Nernst Equation Calculator

Ready to calculate
E = E° − (RT/nF)·ln(Q).
16 Reference Reactions.
ΔG = −nFE Output.
100% Free.
No Data Stored.

How it Works

01Enter Standard Potential

E° from a reduction-potential table — or click a quick-fill from 16 standard half-reactions

02Enter T, n, Activities

Temperature (K, °C, °F), electrons transferred, and activities of reduced and oxidized forms

03Apply E = E° − (RT/nF)·ln(Q)

Cell potential at non-standard conditions — the Nernst equation in its full form

04Get E, ΔG, Spontaneity

Cell potential + Gibbs free energy + 5-band spontaneity classification + closest reference reaction

What is a Nernst Equation Calculator?

The Nernst equation is the foundation of electrochemistry — it tells you the actual cell potential of a reduction half-reaction at any temperature, electron count, and activity (effective concentration) of the reactants. Walther Nernst published it in 1889, and it remains the single most-used equation in batteries, sensors, biology (membrane potentials, the Hodgkin-Huxley model of nerve signaling), and corrosion analysis. Our Nernst Equation Calculator computes E = E° − (RT/nF) · ln(Q) in real time from five inputs: standard reduction potential E°, temperature, electrons transferred (n), and the activities of the reduced and oxidized forms.

A built-in quick-fill library of 16 standard reduction potentials covers the most-encountered half-reactions from active alkali metals (Li, Na, Mg) through the standard hydrogen electrode (SHE) up to the strong oxidizers (Cl₂, F₂). Click any reference reaction and the calculator pre-fills E° and n with the textbook values, so you only need to enter your activities and temperature. Output includes the cell potential E in volts and millivolts, the reaction quotient Q, the Nernst slope (mV per decade of Q — the famous "59 mV per electron" at 25°C), Gibbs free energy ΔG = −nFE, and a five-band spontaneity classification telling you whether the reduction is strongly favored, near equilibrium, or strongly disfavored under the conditions you specified.

Designed for general and analytical chemistry coursework, electrochemistry research, biochemistry (membrane potentials, redox biology), and battery engineering, the tool runs entirely in your browser — no data is stored or transmitted.

Pro Tip: Pair this with our pKa Calculator for related acid-base equilibrium work, or our Molarity Calculator to convert concentrations into activities.

How to Use the Nernst Equation Calculator?

Enter the Standard Reduction Potential (E°): In volts. Look up your half-reaction in a standard reduction potential table — Cu²⁺/Cu is +0.34 V, Zn²⁺/Zn is −0.76 V, etc. Or click a reaction in the quick-fill library to auto-populate E° and n.
Enter Temperature: In Celsius (°C), Fahrenheit (°F), or kelvins (K). Default convention in electrochemistry tables is 25°C (298.15 K). The tool auto-converts to absolute K.
Enter Electrons Transferred (n): The number of electrons gained per mole of reduction. Cu²⁺ + 2e⁻ → Cu has n = 2; Ag⁺ + e⁻ → Ag has n = 1; Au³⁺ + 3e⁻ → Au has n = 3. Auto-fills from the quick-fill library.
Enter Activities (Reduced and Oxidized Forms): Activity is the "effective concentration" — for dilute solutions, it equals concentration in mol/L. For solids and pure liquids, activity = 1. For standard conditions, set both to 1 (the calculator returns E = E°).
Press Calculate: The Nernst equation gives the cell potential E, reaction quotient Q = a_red/a_ox, Nernst slope, ΔG = −nFE, and the spontaneity classification. The calculation breakdown shows every step including (RT/nF) and ln(Q).

How do I calculate cell potential with the Nernst equation?

The Nernst equation generalizes the standard cell potential to non-standard conditions — it's a direct application of the Gibbs free energy relation ΔG = ΔG° + RT·ln(Q). Here's the complete derivation:


Think of E° as the "rated voltage" of a battery in fresh, factory conditions. As the battery discharges (concentrations of products build up, reactants drop), the actual voltage drifts away from the rated value. The Nernst equation tells you exactly how much.

The Nernst Equation


E = E° − (RT / nF) · ln(Q)


where E° is the standard reduction potential (V), R = 8.314 J/(mol·K) is the universal gas constant, T is absolute temperature (K), n is the number of electrons transferred per reduction event, F = 96 485 C/mol is the Faraday constant, and Q is the reaction quotient. For a reduction half-reaction Ox + ne⁻ → Red, Q = a_red / a_ox.

Standard Conditions


At standard conditions, all activities equal 1, so Q = 1, ln(Q) = 0, and E = E° exactly. Standard reduction potentials are tabulated for 25°C (298.15 K). Tables list values like Cu²⁺/Cu = +0.34 V, H⁺/H₂ = 0 V (the standard hydrogen electrode reference), Zn²⁺/Zn = −0.76 V.

The "59 mV per Electron" Rule


At T = 25°C, the prefactor 2.303 RT/F = 0.05916 V ≈ 59.2 mV. So the Nernst equation is often written:


E = E° − (0.0592 / n) · log₁₀(Q) (at 25°C)


For a 1-electron transfer (Ag⁺/Ag, Fe³⁺/Fe²⁺), each 10× change in Q shifts E by 59.2 mV. For 2-electron transfers (Cu²⁺/Cu, Zn²⁺/Zn), the slope is 29.6 mV per decade. This is the famous "Nernst slope" used to validate ion-selective electrodes (a properly working pH electrode shows 59 mV per pH unit at 25°C).

Gibbs Free Energy Connection


ΔG = −nFE


Cell potential and Gibbs free energy are directly related. Positive E ↔ negative ΔG ↔ spontaneous reduction. Negative E ↔ positive ΔG ↔ non-spontaneous (the reverse oxidation is favored). At equilibrium, E = 0 and ΔG = 0. The calculator reports both E and ΔG simultaneously.

Activity vs. Concentration


Strictly, the Nernst equation uses activities (effective concentrations corrected for non-ideality), not concentrations. Activity a_i = γ_i × c_i, where γ_i is the activity coefficient. In dilute solutions (≤0.01 M), γ ≈ 1 and you can substitute concentrations directly. In concentrated electrolytes, γ deviates from 1 — Debye-Hückel theory or experimental tables are needed. For solids and pure liquids, a = 1 by definition.

Real-World Example

Nernst Equation Calculator – Cell Potential In Practice

Consider a Daniell cell with non-standard concentrations: Cu²⁺/Cu half-cell with [Cu²⁺] = 0.01 M (rather than the standard 1 M). Compute the half-cell potential at 25°C:
  • Step 1: Identify the half-reaction. Cu²⁺ + 2e⁻ → Cu. Standard E° = +0.34 V. n = 2 (two electrons transferred).
  • Step 2: Identify activities. Activity of Cu (solid) = 1. Activity of Cu²⁺ (dilute solution) ≈ concentration = 0.01.
  • Step 3: Compute Q. Q = a_red / a_ox = 1 / 0.01 = 100.
  • Step 4: Compute (RT/nF) at 25°C. (8.314 × 298.15) / (2 × 96485) = 0.01285 V.
  • Step 5: Apply the Nernst equation. E = 0.34 − 0.01285 × ln(100) = 0.34 − 0.01285 × 4.605 = 0.34 − 0.0592 = 0.281 V.
  • Step 6: Interpret. The half-cell potential drops from +0.34 V (standard) to +0.281 V (at [Cu²⁺] = 0.01 M) — a 59 mV decrease per 100× dilution, exactly the "59 mV / 2 electrons / decade Q" Nernst slope. Reduction is still favored, but less so than at standard conditions.

Now consider biological membrane potential: a typical resting nerve cell has [K⁺]_inside ≈ 140 mM, [K⁺]_outside ≈ 5 mM, T = 37°C (310 K). Treating this as a K⁺ concentration cell with E° = 0 (reference): E_K = (8.314 × 310 / (1 × 96485)) × ln(5/140) = 0.0267 × ln(0.0357) = 0.0267 × (−3.33) ≈ −89 mV. This is the famous Nernst (equilibrium) potential for potassium across a neural membrane — the foundation of action-potential physiology.

Who Should Use the Nernst Equation Calculator?

1
Chemistry Students: Solve electrochemistry problems on cell potential at non-standard conditions — the most-tested non-trivial topic in general chemistry.
2
Battery Engineers: Predict open-circuit voltage as state-of-charge changes (concentrations of redox species shift during discharge).
3
Corrosion Engineers: Use Nernst-corrected potentials to assess when metals will corrode in a given electrolyte composition.
4
Analytical Chemists: Calibrate ion-selective electrodes (pH, F⁻, Ca²⁺, etc.). Nernst slope = 59 mV/decade is the validation criterion.
5
Biochemists & Physiologists: Compute equilibrium potentials for ions across membranes — foundation of nerve signaling, action potentials, mitochondrial bioenergetics.
6
Electrochemistry Researchers: Build calibration curves, design redox sensors, validate cyclic voltammetry data against thermodynamic predictions.

Technical Reference

Origin (Nernst, 1889). Walther Nernst derived the equation while working on the thermodynamics of galvanic cells. He won the 1920 Nobel Prize in Chemistry partially for this work. The equation links Gibbs free energy (ΔG = ΔG° + RT·ln Q) to electrochemical potential via ΔG = −nFE.

Constants.

  • R = 8.314 J/(mol·K)
  • F = 96 485 C/mol (Faraday's constant — charge of one mole of electrons)
  • RT/F at 25°C = 0.02569 V (the "thermal voltage" used in semiconductor physics)
  • 2.303 RT/F at 25°C = 0.05916 V ≈ 59.2 mV (the Nernst slope per decade per electron)

Selected Standard Reduction Potentials (25°C, 1 atm, 1 M, vs SHE):

  • F₂ + 2e⁻ → 2F⁻: +2.87 V (strongest common oxidizer)
  • O₂ + 4H⁺ + 4e⁻ → 2H₂O: +1.23 V (oxygen reduction; basis of fuel cells)
  • Cl₂ + 2e⁻ → 2Cl⁻: +1.36 V
  • Ag⁺ + e⁻ → Ag: +0.80 V
  • Fe³⁺ + e⁻ → Fe²⁺: +0.77 V
  • Cu²⁺ + 2e⁻ → Cu: +0.34 V
  • 2H⁺ + 2e⁻ → H₂: 0.00 V (Standard Hydrogen Electrode — the reference)
  • Pb²⁺ + 2e⁻ → Pb: −0.13 V
  • Fe²⁺ + 2e⁻ → Fe: −0.44 V
  • Zn²⁺ + 2e⁻ → Zn: −0.76 V
  • Al³⁺ + 3e⁻ → Al: −1.66 V
  • Mg²⁺ + 2e⁻ → Mg: −2.37 V
  • Li⁺ + e⁻ → Li: −3.04 V (most active common metal)

Cell EMF. For a full electrochemical cell, EMF = E_cathode − E_anode (both as reduction potentials). The cathode is the electrode where reduction happens (positive lead in a galvanic cell); the anode is where oxidation happens (negative lead). Spontaneous galvanic cells have EMF > 0.

Activity Coefficients. Strictly, Nernst uses activities (a = γ × c). For dilute solutions (c < 0.01 M), γ ≈ 1 — concentration ≈ activity. For higher ionic strengths, use Debye-Hückel: log γ = −A·z²·√I / (1 + B·a·√I), where I is ionic strength, z is charge, a is ion size. For routine work, the activity ≈ concentration approximation is sufficient.

Membrane Potentials. For ion concentration gradients across membranes, the Nernst equation gives the equilibrium ('reversal') potential for each ion: E_ion = (RT/zF) × ln(c_outside/c_inside). The Goldman-Hodgkin-Katz equation generalizes this to multiple permeable ions and gives the resting membrane potential for nerve and muscle cells.

Key Takeaways

The Nernst equation bridges thermodynamics and electrochemistry — given a standard potential and any set of concentrations, it tells you the actual cell potential under your conditions. The key insight is the 59 mV per electron per decade rule at 25°C: every 10× change in the activity ratio shifts the potential by 59 mV / n. Use the ToolsACE Nernst Equation Calculator to compute E for any half-reaction with full transparency on the math (RT/nF coefficient, ln(Q), the closing E = E° − slope × ln(Q) step), get ΔG = −nFE alongside, and validate against the 16-reaction quick-fill library of standard reduction potentials. Bookmark it for general chemistry homework, biochemistry membrane-potential calculations, ion-selective-electrode validation, and battery engineering.

Frequently Asked Questions

What is the Nernst Equation Calculator?
The Nernst equation tells you the actual cell potential of a reduction half-reaction at non-standard conditions: E = E° − (RT/nF)·ln(Q), where Q = a_red/a_ox is the reaction quotient. Our calculator computes E from the standard reduction potential (E°), temperature, electron count, and activities. A built-in 16-reaction quick-fill library covers the major standard reduction potentials from Li (most reducing) through F₂ (most oxidizing).

Output includes the cell potential E in volts and millivolts, the reaction quotient Q, the Nernst slope (mV per decade of Q — 59 mV per electron at 25°C), Gibbs free energy ΔG = −nFE, and a 5-band spontaneity classification. Designed for general and analytical chemistry coursework, electrochemistry research, biochemistry (membrane potentials), and battery engineering.

Pro Tip: For more chemistry tools, try our pKa Calculator.

What is the Nernst equation, exactly?
E = E° − (RT/nF) · ln(Q). It's a thermodynamic statement that the cell potential at any non-standard condition equals the standard potential minus a correction term that depends on temperature, electron count, and the reaction quotient. The equation comes directly from the Gibbs energy relation ΔG = ΔG° + RT·ln(Q) combined with ΔG = −nFE. Walther Nernst derived it in 1889; he won the 1920 Nobel Prize in chemistry partly for this work.
What's the '59 mV per electron' rule?
At T = 25°C (298.15 K), the prefactor 2.303 RT/F equals exactly 0.05916 V, or about 59.2 mV. So at 25°C the Nernst equation in log₁₀ form is E = E° − (0.0592/n) · log₁₀(Q). Each 10× change in Q shifts E by 59.2 mV / n. For 1-electron transfers (Ag⁺/Ag, Fe³⁺/Fe²⁺) the slope is 59 mV per decade; for 2-electron transfers (Cu²⁺/Cu) it's 29.6 mV per decade. This is the validation criterion for ion-selective electrodes — a working pH electrode shows ~59 mV per pH unit at 25°C.
What's the difference between activity and concentration?
Activity is the effective concentration accounting for non-ideal behavior — interactions between solute particles in real solutions. Activity a = γ × c, where γ is the activity coefficient. In dilute solutions (≤ 0.01 M), γ ≈ 1 and you can substitute concentration directly. In concentrated electrolytes, γ deviates significantly from 1 — Debye-Hückel theory or experimental tables are needed for precision. For solids and pure liquids, activity = 1 by definition. For routine chemistry, treating concentration as activity is fine.
How do I find the standard reduction potential E°?
Look it up in a standard reduction potential table. Most general chemistry textbooks have one. Common values (at 25°C, vs Standard Hydrogen Electrode): F₂/F⁻ = +2.87, Cl₂/Cl⁻ = +1.36, Ag⁺/Ag = +0.80, Cu²⁺/Cu = +0.34, H⁺/H₂ = 0.00 (the reference), Zn²⁺/Zn = −0.76, Mg²⁺/Mg = −2.37, Li⁺/Li = −3.04. The 16-reaction quick-fill library in the calculator covers the most-needed values.
How do I count electrons (n)?
n = the number of electrons transferred in the balanced reduction half-reaction. Cu²⁺ + 2e⁻ → Cu: n = 2. Ag⁺ + e⁻ → Ag: n = 1. Au³⁺ + 3e⁻ → Au: n = 3. O₂ + 4H⁺ + 4e⁻ → 2H₂O: n = 4. The number is what you'd use to balance the charge on each side of the half-reaction. Quick-fill auto-populates the right value for each library reaction.
What does positive vs. negative E mean?
E > 0 (positive cell potential): reduction is spontaneous, ΔG = −nFE is negative. The reaction proceeds in the reduction direction as written. E < 0: reduction is non-spontaneous, ΔG > 0. The reverse oxidation reaction is favored instead. E = 0: equilibrium — no net driving force in either direction. Strong reducers (Li, Na, Mg) have very negative E° because their reduction is highly disfavored — they want to oxidize. Strong oxidizers (F₂, Cl₂) have very positive E°.
What's a 'concentration cell'?
An electrochemical cell where both half-cells use the same electrode and electroactive species but at different concentrations. Since E° is the same on both sides, the cell potential comes entirely from the Nernst equation: E_cell = (RT/nF) · ln(c_high/c_low). The biological membrane potential is a special case — the [K⁺] gradient across a nerve cell membrane creates ~−90 mV via the Nernst equation.
Why does the Nernst slope change with temperature?
Because RT/F scales linearly with absolute T. At 25°C the slope is 59.2 mV per decade per electron; at 100°C (373 K) it's 74 mV/decade. Higher temperature → larger Nernst slope → more sensitive to concentration changes. For high-temperature work (e.g., molten-salt electrochemistry), always use the actual T, not the 25°C value.
How does the Nernst equation relate to ΔG?
Directly: ΔG = −nFE. Cell potential and Gibbs free energy are two views of the same thermodynamic quantity. The calculator reports both. Standard ΔG° = −nFE°; non-standard ΔG = −nFE. Equilibrium constant K relates to standard cell potential by E° = (RT/nF) · ln(K) — large positive E° corresponds to large K (products favored), large negative E° to small K.
What's the difference between EMF and cell potential?
EMF (electromotive force) is the open-circuit cell potential — the voltage measured when no current flows. Under load, the actual measured voltage drops below EMF due to internal resistance and overpotential. The Nernst equation gives EMF, the reversible thermodynamic potential. Real-world battery voltage at 1 A discharge is lower than the Nernst EMF.

Author Spotlight

The ToolsACE Team - ToolsACE.io Team

The ToolsACE Team

Our chemistry tools team implements the Nernst equation E = E° − (RT/nF)·ln(Q) — the foundation of electrochemistry. The calculator handles temperature in K/°C/°F (auto-converted to absolute), arbitrary electron-transfer counts, and any combination of activities. A 16-reaction quick-fill library covers the major standard reduction potentials from the active alkali metals (Li, Na, Mg) through the noble metals and strong oxidizers (Cl₂, F₂). Output includes the cell potential, reaction quotient Q, Nernst slope (mV per decade of Q), Gibbs free energy ΔG = −nFE, and a 5-band spontaneity classification.

ElectrochemistryNernst Equation (1889)Software Engineering Team

Disclaimer

The Nernst equation assumes thermodynamic equilibrium and uses activities (effective concentrations). For real solutions, activity = γ × concentration; γ ≈ 1 in dilute solutions. For concentrated electrolytes, use Debye-Hückel theory or experimental activity-coefficient tables. The equation predicts the reversible cell potential — real-world voltages under load are lower due to overpotential and internal resistance.