Nernst Equation Calculator
The Nernst equation converts a standard electrode potential into the actual voltage under real concentration conditions. Choose what you want to solve for, enter the known values, and the calculator returns the cell potential E, the Gibbs free energy change deltaG, the equilibrium constant K, and a full step-by-step breakdown.
Formula
Worked example
Daniell cell: E0 = 1.10 V, n = 2, Q = 0.10 at 25 C. RT/nF = (8.314 x 298.15) / (2 x 96485) = 0.01285 V. Shift = -0.01285 x ln(0.10) = +0.02958 V. E = 1.10 + 0.02958 = 1.1296 V. deltaG = -2 x 96485 x 1.1296 = -218.0 kJ/mol. K = exp(2 x 96485 x 1.10 / (8.314 x 298.15)) = 1.51 x 10^37.
What the Nernst equation tells you
Standard reduction potentials are tabulated under one specific set of reference conditions: all dissolved species at unit activity (effectively 1 M), all gases at 1 bar, and temperature at 25 °C. Real cells almost never meet those conditions, so the measured voltage drifts from the textbook number. The Nernst equation E = E0 - (RT/nF) x ln Q is the bridge between the idealized standard potential and what a voltmeter would actually read, accounting for the concentrations and partial pressures of all reacting species through the reaction quotient Q. A related quantity, Gibbs free energy deltaG = -nFE, tells you how much useful electrical work the cell can deliver: when deltaG is negative, the reaction proceeds spontaneously and can power a device; when it is positive, external work must be supplied.
Reading the reaction quotient Q and equilibrium constant K
The reaction quotient Q has the same algebraic form as the equilibrium constant K, but uses instantaneous concentrations rather than equilibrium values. You write it as products over reactants, each raised to its stoichiometric coefficient, with pure solids, pure liquids and (usually) water omitted. In half-cell mode this simplifies to the ratio [Ox]/[Red]. When Q is less than one, reactants dominate, the logarithm is negative, and E rises above E0. When Q is greater than one, products dominate and E drops below E0. As the cell discharges, Q climbs toward K, the driving voltage shrinks, and the cell finally reaches equilibrium when E is exactly zero and deltaG is zero. The equilibrium constant itself follows from the standard potential: K = exp(nFE0/RT), which at 25 C and n = 1 gives roughly a factor of 10^(E0/0.0592).
The Nernst factor RT/nF and temperature effects
The coefficient RT/nF determines how sensitive the cell potential is to concentration changes. At 25 °C, RT/F equals 0.02569 V in natural-log form or 0.05916 V per decade in base-ten log form (the familiar "0.0592/n" rule). Increasing the temperature raises the factor, making potential more sensitive to changes in Q; increasing n flattens the response and stabilizes potential. The Nernst equation is also central to pH electrodes, ion-selective electrodes and biological membrane potentials, where n corresponds to the ion valence z and Q reduces to a single concentration ratio of intra- to extracellular ion.
Reverse-solve modes: finding E0 or Q from a measured voltage
Rearranging E = E0 - (RT/nF) x ln Q yields two additional solve modes. To find E0 from a measured voltage: E0 = E + (RT/nF) x ln Q. This is how standard potentials are experimentally determined - the cell is prepared at a known Q and the voltage is measured. To find Q from a measured voltage: ln Q = (E0 - E) / (RT/nF), so Q = exp((E0 - E) x nF / RT). This is useful when you want to back-calculate concentration ratios in a solution, for example in a potentiometric titration or when troubleshooting a battery at a known state of charge.
How Q, deltaG and K relate to E (n = 1, 25 °C, E0 = 0.800 V)
| Q | ln Q | Shift (V) | E (V) | deltaG (kJ/mol) | Spontaneous? |
|---|---|---|---|---|---|
| 0.001 | -6.91 | +0.178 | 0.978 | -94.4 | Yes |
| 0.01 | -4.61 | +0.118 | 0.918 | -88.6 | Yes |
| 0.1 | -2.30 | +0.059 | 0.859 | -82.9 | Yes |
| 1 | 0.00 | 0.000 | 0.800 | -77.2 | Yes |
| 10 | +2.30 | -0.059 | 0.741 | -71.5 | Yes |
| 100 | +4.61 | -0.118 | 0.682 | -65.8 | Yes |
| 1000 | +6.91 | -0.178 | 0.622 | -60.0 | Yes |
Each tenfold change in Q shifts E by about 59 mV for a one-electron reaction. All three quantities are directly connected through E.
Frequently asked questions
Why does the factor change between 0.0257 and 0.0592?
They describe the same thing in different logarithm bases. RT/F at 25 °C equals 0.02569 V when you use the natural log (ln Q). Multiplying by ln(10) = 2.3026 converts it to 0.05916 V for use with base-ten log (log Q). Use 0.02569 with ln Q, or 0.0592 with log Q, never mix them in the same equation.
What is the reaction quotient Q and how do I calculate it?
Q is the ratio of product activities to reactant activities at any instant, each species raised to its stoichiometric coefficient, with pure solids and liquids omitted. For a simple redox half-reaction Ox + ne- -> Red, Q = [Red]/[Ox] (or its reciprocal depending on how you write the half-reaction). The calculator's half-cell mode lets you enter [Ox] and [Red] directly so you do not have to compute the ratio yourself.
What does a negative cell potential mean?
A negative E means the reaction as written is non-spontaneous under those conditions. The reverse reaction would proceed spontaneously instead, and the cell would need an external voltage (like a battery charger) to drive it forward. The corresponding deltaG is positive. A galvanic cell delivers useful work only while E is positive, and it stops once E reaches zero at equilibrium.
How do I find the equilibrium constant K from the Nernst equation?
At equilibrium, E = 0 and Q = K. Setting E = 0 in the Nernst equation gives E0 = (RT/nF) x ln K, so K = exp(nFE0/RT). At 25 °C with n = 1, this simplifies to K = 10^(E0/0.05916). A standard potential of just +0.2 V gives K roughly 2400, showing that even modest voltages correspond to strongly product-favored equilibria.
How is the Nernst equation used in biology and pH electrodes?
Ion-selective electrodes (including the glass pH electrode) rely on the Nernst equation. For a single ion with charge z, the potential across the membrane is V = (RT/zF) x ln([X]out/[X]in). For a proton (z = 1) this becomes 59 mV per decade of concentration ratio at 25 °C, which is exactly the Nernst slope of a pH electrode. Biologists also use it to calculate reversal potentials for ion channels (the Goldman-Hodgkin-Katz potential is the multi-ion generalization).
What is Gibbs free energy deltaG and how does it connect to the cell potential?
Gibbs free energy change deltaG = -nFE is the maximum electrical work the cell can perform per mole of reaction. A negative deltaG (spontaneous) pairs with positive E, confirming the cell delivers energy. At standard conditions deltaG0 = -nFE0, and the relationship to the equilibrium constant is deltaG0 = -RT x ln K, tying all three key electrochemistry quantities together.