Article — Nernst Equation Calculator
Nernst Equation Calculator: E = E° − (RT/nF) ln Q at Any Temperature
The Nernst equation is E = E° − (RT / nF) ln Q. It gives the electrode or cell potential at any temperature and any ion concentration, with E° the standard potential, R the gas constant, T the absolute temperature, n the number of electrons transferred, F = 96485 C/mol Faraday's constant, and Q the reaction quotient. At 25 C, the convenient form is E = E° − (0.0592 / n) log10(Q).
Walther Nernst published the equation in 1889 and won the Nobel Prize in 1920 for related thermodynamic work. The equation links electrochemistry to thermodynamics and is the workhorse formula for batteries, ion-selective electrodes, corrosion calculations, and neural membrane potentials.
What the Nernst equation does
The Nernst equation answers the question: "How does the voltage of an electrochemical cell change when concentrations are not at the standard 1 mol/L?" Standard electrode potentials in tables (E°) assume every aqueous species sits at unit activity (~1 M) and every gas at 1 atm. Real cells almost never meet those conditions.
Plugging in actual concentrations of products and reactants (through the quotient Q) gives the cell potential under operating conditions. The equation is also what determines when a battery is "dead" (E = 0, Q = K), what concentration a pH meter is sensing (it is a Nernstian device), and how iron rusts faster at low pH (the H+ concentration enters Q).
Where the Nernst equation comes from
The starting point is the relationship between cell potential and Gibbs free energy: ΔG = − n F E. Free energy under non-standard conditions is ΔG = ΔG° + RT ln Q. Combining these and dividing by − nF gives the Nernst equation directly.
E = E° − (RT/nF) ln Q general formE = E° − (0.0592/n) log10(Q) at 25 CE = 0 → Q = K equilibriumΔG = −nFE energy linkE° = (RT/nF) ln K from KThe derivation is short but the implication is enormous: the simple act of measuring a voltage with a voltmeter gives access to the thermodynamic equilibrium constant K, with no calorimetry required.
The Nernst equation at 25 C
At 298.15 K, the factor (RT/F) ln 10 equals exactly 0.0592 V. So the equation becomes E = E° − (0.0592/n) log10(Q). The slope of 59.2 mV per decade of Q (for n = 1) is famous enough to have a name: the Nernstian slope. It is what every pH electrode and ion-selective electrode is designed to deliver.
For a 2-electron transfer like Cu²+ + 2e− → Cu, the slope is 0.0592 / 2 = 0.0296 V per decade. For 3-electron iron(III) reduction, 0.0197 V per decade. The smaller the slope, the harder it is to use the electrode as a precise concentration sensor.
The pH electrode is literally a Nernst-equation device. A glass membrane permits H+ ions to exchange between two solutions; the resulting potential follows E = E° − 0.0592 pH at 25 C. A modern pH meter is a Nernst equation translator with a built-in temperature sensor that corrects the 0.0592 to whatever T the sample is at.
The reaction quotient Q in detail
Q is the ratio of products to reactants, each raised to its stoichiometric coefficient, at the moment you compute it. For Cu²+ + 2 e− → Cu, Q = [Cu (solid)] / [Cu²+]. Solid copper is a pure phase with activity = 1, so Q reduces to 1 / [Cu²+]. Dropping the copper from Q is a common abbreviation; technically you are using its unit activity.
For gas-phase reactions, use partial pressures in atm. For dilute aqueous, use molar concentration in mol/L. For concentrated electrolytes, strict thermodynamics calls for activities (concentration times an activity coefficient γ), but for most undergraduate work the concentration form is good enough.
Nernst equation temperature dependence
The coefficient RT/nF is proportional to temperature. At 0 C (273 K), the slope at 25 C scales by 273/298 = 0.916, giving 0.0542 V/decade for n = 1. At body temperature (310 K) it rises to 0.0615 V/decade. At a furnace at 1000 K, 0.198 V/decade.
| Temperature | Slope at n = 1 (V/decade) | Use |
|---|---|---|
| 0 C (273 K) | 0.0542 | Refrigerated electrochemistry |
| 25 C (298 K) | 0.0592 | Standard lab |
| 37 C (310 K) | 0.0615 | Body temperature, neurons |
| 100 C (373 K) | 0.0740 | High-temperature fuel cells (lower end) |
Nernst equation in biology
The resting potential of a neuron is roughly −70 mV. The driver is the K+ concentration gradient across the cell membrane: about 140 mM inside, 5 mM outside. Applied to the K+ Nernst potential at 37 C:
E_K = (RT/F) ln(K_out / K_in) = 0.0615 V × log10(5/140) = −0.089 V = −89 mV.
The actual resting potential is −70 mV because Na+, Cl−, and Ca²+ also contribute, weighted by their membrane permeabilities. The Goldman-Hodgkin-Katz equation generalizes the Nernst equation to multiple ions and is the central equation of computational neuroscience.
Nernst equation applications
Battery cell voltage. A Li-ion cell's open-circuit voltage at any state of charge follows the Nernst equation applied to the lithium intercalation chemistry. Battery management systems estimate state-of-charge from open-circuit voltage by inverting the Nernst form.
Corrosion. Iron rusts faster in acidic environments because the H+ concentration shifts the Nernst potential of the Fe / Fe²+ couple, making oxidation more thermodynamically favorable.
Ion-selective electrodes. Fluoride, calcium, chloride, ammonia, sodium, nitrate sensors. All work by establishing a selective membrane potential that obeys the Nernst equation, with a calibration curve giving concentration from measured voltage.
Nernst equation mistakes
RT is the absolute temperature times the gas constant. Plug in Celsius and you get nonsense. The toggle on this calculator handles the conversion; double-check the unit button matches the value you typed. The leading source of Nernst errors is feeding Celsius into a Kelvin slot.
Other regular slips: writing Q upside down (products go on top, reactants on bottom; pure phases drop out), forgetting to use a balanced half-reaction so the n is wrong, mixing up oxidation and reduction directions (the standard table is written for reductions; reverse the sign of E° if you treat a species as the oxidized form being formed), and using activity coefficients in dilute solutions where concentrations would do.
At 25 C and n = 1, every factor of 10 in Q changes E by 59.2 mV. For a quick mental check of any electrochemistry problem, see whether the calculated ΔE matches the order of magnitude of (decades of Q) times 59 mV. If you are off by 100x, you probably have a unit slip.