Why a cell's voltage depends on concentration
A table of standard reduction potentials gives you one number per half-reaction, measured with every dissolved species at unit activity — roughly 1 M — and every gas at 1 bar. Combine two of them and you get E°, the standard cell potential. Your actual cell is almost certainly not at those concentrations, and its voltage is not E°.
The Nernst equation supplies the correction. It says the potential falls below E° in proportion to the logarithm of the reaction quotient Q, which measures how far the mixture has already moved towards products. A cell rich in products has less driving force left and a lower voltage; a cell starved of products has more.
This is not a small effect. At 25 °C a one-electron reaction loses 59.16 mV for every tenfold rise in Q. Six orders of magnitude — entirely realistic in a trace analysis — is 355 mV, a third of a volt, which would swamp any attempt to identify a species from its standard potential alone.
The same logarithmic response is what makes potentiometric sensors work. A pH electrode, a fluoride electrode, a calcium electrode and a dissolved-oxygen probe all report a voltage that varies linearly with the logarithm of the analyte activity, at 59.16 mV per decade divided by the ion's charge. That is why pH is defined logarithmically in the first place, and why an electrode reading is calibrated with two standard buffers to establish a slope and an intercept.
The Nernst equation also explains why a battery's voltage sags as it discharges. Consuming reactants and building up products raises Q, and the terminal voltage falls with it. When Q finally reaches K, E is zero and the battery is flat.
The four inputs, and where each one bites
E° is a difference, not a property of the cell. Compute it as the standard reduction potential of the cathode minus that of the anode, both taken from a table as written for reduction. For the Daniell cell, Cu²⁺/Cu is +0.34 V and Zn²⁺/Zn is −0.76 V, giving E° = 0.34 − (−0.76) = +1.10 V. Do not reverse the sign of the anode entry and then subtract; that double-counts the reversal.
n is the electron count in the balanced cell reaction as written. Getting it wrong scales the whole correction term. For Zn + Cu²⁺ → Zn²⁺ + Cu, two electrons move, so n = 2 and the slope is 29.58 mV per decade rather than 59.16. A crucial subtlety: E° itself does not scale with n. Doubling every coefficient doubles n and doubles ΔG, but leaves E° unchanged, because potential is energy per unit charge and both numerator and denominator double.
Q follows the same rules as any reaction quotient. Products over reactants, each raised to its coefficient, with pure solids and pure liquids omitted because their activity is 1. In the Daniell cell the metals drop out and Q = [Zn²⁺]/[Cu²⁺]. The two-concentration mode above assumes exactly this form — one species on top, one underneath, both to the first power. When your expression has exponents, or gas partial pressures, compute Q separately with the reaction quotient calculator and enter it directly.
T sets the slope. The factor 2.303RT/F is 59.16 mV per decade at 25 °C, 54.20 mV at 0 °C and 61.54 mV at 37 °C — a 4% change over the range a benchtop measurement might span. A pH meter's automatic temperature compensation is doing precisely this correction, and a meter calibrated at one temperature and used at another without it will be wrong by a predictable amount that grows with distance from pH 7.
Two constants tie the electrochemistry to the thermodynamics. ΔG = −nFE converts a voltage into a free energy, which is why a fuel cell's theoretical efficiency is calculable from a table of potentials. And setting E to zero — the condition for equilibrium — gives ln K = nFE°/RT, the bridge between a measured voltage and an equilibrium constant. That relation is how many otherwise inaccessible constants, including solubility products, are determined; see the solubility product calculator for what is done with them.
Worked example: a Daniell cell with 0.0010 M Zn²⁺ and 1.00 M Cu²⁺
The cell reaction is Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s), with E° = +1.100 V and n = 2. The zinc half-cell is at 0.0010 M and the copper half-cell at 1.00 M, both at 25 °C.
- Write Q. The two metals are pure solids, so they drop out: Q = [Zn²⁺] ÷ [Cu²⁺] = 0.0010 ÷ 1.00 = 1.0 × 10⁻³.
- Take the logarithm. log₁₀(1.0 × 10⁻³) = −3.000.
- Find the slope. 2.303RT/(nF) = (2.303 × 8.3145 × 298.15) ÷ (2 × 96485.33) = 0.05916 ÷ 2 = 0.029580 V per decade, or 29.58 mV.
- Apply the correction. E = E° − slope × log Q = 1.100 − 0.029580 × (−3.000) = 1.100 + 0.08874 = 1.1887 V.
- Sanity-check the direction. Products are scarce relative to the standard state, so the reaction has further to run and the voltage should exceed E°. It does, by 89 mV.
- Convert to free energy. ΔG = −nFE = −2 × 96485.33 × 1.1887 = −229 400 J/mol = −229.4 kJ/mol.
Now let the cell discharge. As zinc dissolves and copper plates out, [Zn²⁺] rises and [Cu²⁺] falls, so Q climbs and E falls. The cell dies when Q reaches K, which follows from ln K = nFE°/RT = (2 × 96485.33 × 1.100) ÷ (8.3145 × 298.15) = 85.63, giving K ≈ 1.5 × 10³⁷. The copper ion concentration at that point is unmeasurably small: the reaction is effectively complete, which is why zinc-copper cells discharge fully rather than reaching a visible equilibrium.
A second case shows the extreme. Build a cell with copper electrodes in both compartments — identical chemistry, so E° = 0 — but with 0.0010 M Cu²⁺ on one side and 1.00 M on the other. With n = 2, E = 0 − 0.029580 × (−3.000) = +0.0887 V. A concentration cell generates a voltage from nothing but a concentration difference, and the same principle drives the electrical potential across a nerve cell membrane.
Reading the sign, the slope and the limits
The sign of E gives the direction. Positive means the reaction as written runs spontaneously and the cell delivers current; negative means the reverse reaction is the spontaneous one and driving the written reaction requires an external supply exceeding |E|. That is exactly the calculation behind electroplating and electrolysis voltages, before overpotential is added.
The magnitude of E is a free energy in disguise. One volt at n = 1 is 96.5 kJ/mol. Ordinary cells span roughly 0.5 to 2 V per cell for this reason: chemical reactions release energies of tens to hundreds of kilojoules per mole, and dividing by nF lands in that range.
The slope tells you what a measurement can resolve. At 59.16 mV per decade for a monovalent ion, a millivolt of measurement error corresponds to about 4% in concentration. For a divalent ion the slope is halved to 29.58 mV per decade, so the same millivolt is nearly 8%. This is the fundamental reason potentiometric methods are more precise for monovalent species, and why they are best for order-of-magnitude work rather than for high-accuracy assay.
Three limitations matter in practice. Concentration is not activity. The Nernst equation is exact in activities; substituting molar concentrations is accurate only in dilute solution, and by 0.1 M the activity coefficient of a divalent ion can be well below 0.5. Practical electrode work handles this by calibrating with standards at the same ionic strength as the samples, often with an added ionic-strength adjustment buffer.
Nernst says nothing about rate. A thermodynamically favourable cell can deliver negligible current if the electrode kinetics are slow. The gap between the calculated potential and the working potential under current is the overpotential, described by the Butler-Volmer and Tafel relations, and it is the reason a water electrolyser needs well above the 1.23 V that thermodynamics demands.
Junction potentials add an offset. A real cell contains a salt bridge or a porous frit, and the unequal mobilities of ions across it generate a small extra voltage, typically a few millivolts, which the equation does not include. It is one reason electrode calibration is empirical rather than computed.
Standard reduction potentials at 25 °C, against the standard hydrogen electrode
| Half-reaction (reduction) | E° (V) |
|---|---|
| F₂ + 2e⁻ → 2F⁻ | +2.87 |
| MnO₄⁻ + 8H⁺ + 5e⁻ → Mn²⁺ + 4H₂O | +1.51 |
| Cl₂ + 2e⁻ → 2Cl⁻ | +1.36 |
| Cr₂O₇²⁻ + 14H⁺ + 6e⁻ → 2Cr³⁺ + 7H₂O | +1.33 |
| O₂ + 4H⁺ + 4e⁻ → 2H₂O | +1.23 |
| Br₂ + 2e⁻ → 2Br⁻ | +1.07 |
| Ag⁺ + e⁻ → Ag | +0.80 |
| Fe³⁺ + e⁻ → Fe²⁺ | +0.77 |
| I₂ + 2e⁻ → 2I⁻ | +0.54 |
| Cu²⁺ + 2e⁻ → Cu | +0.34 |
| 2H⁺ + 2e⁻ → H₂ (reference) | 0.00 |
| Pb²⁺ + 2e⁻ → Pb | −0.13 |
| Ni²⁺ + 2e⁻ → Ni | −0.26 |
| Fe²⁺ + 2e⁻ → Fe | −0.44 |
| Zn²⁺ + 2e⁻ → Zn | −0.76 |
| Al³⁺ + 3e⁻ → Al | −1.66 |
| Mg²⁺ + 2e⁻ → Mg | −2.37 |
| Na⁺ + e⁻ → Na | −2.71 |
| Li⁺ + e⁻ → Li | −3.04 |
Values are for unit activity of every dissolved species and 1 bar for gases; the copper and zinc rows give the Daniell cell's 1.10 V directly.
The Nernst slope 2.303RT/F against temperature
| Temperature | T (K) | Slope (mV/decade) |
|---|---|---|
| 0 °C | 273.15 | 54.20 |
| 20 °C | 293.15 | 58.17 |
| 25 °C | 298.15 | 59.16 |
| 30 °C | 303.15 | 60.15 |
| 37 °C | 310.15 | 61.54 |
| 50 °C | 323.15 | 64.12 |
| 100 °C | 373.15 | 74.04 |
A pH meter's temperature compensation applies exactly this scaling. Uncompensated, a meter calibrated at 25 °C and used at 50 °C errs by about 8% of the deviation from the isopotential point.
Mistakes that give the wrong potential
- Reversing the anode potential and then subtracting it. E° = E°(cathode) − E°(anode), both taken as reduction potentials straight from the table. Flipping the sign first and then subtracting double-counts.
- Multiplying E° by a stoichiometric factor. Potential is energy per unit charge, so it does not scale when you double the equation. ΔG does; E° does not.
- Including pure solids or liquids in Q. Their activity is 1 by definition. In the Daniell cell only the two ion concentrations appear.
- Using the wrong n. It is the electron count in the balanced cell reaction, and it sets the slope. A factor-of-two error here is a factor-of-two error in the entire correction.
- Substituting concentration for activity above about 0.1 M. Activity coefficients drop well below 1 in concentrated or high-ionic-strength solutions. Calibrate against standards of matching ionic strength.
- Assuming the calculated potential is what you will measure under load. The Nernst value is the open-circuit, zero-current potential. Drawing current adds overpotential and IR drop, both of which reduce the terminal voltage.
- Ignoring temperature. The slope is proportional to absolute temperature, so a 25 °C calibration used at 5 °C is 7% off in slope.
- Forgetting the junction potential. A salt bridge or frit contributes a few millivolts that no calculation predicts; empirical calibration absorbs it.
How the Nernst equation connects to the rest of the chemistry
Electrochemistry is thermodynamics measured with a voltmeter. The bridge is ΔG = −nFE, which means everything the Gibbs free energy calculator tells you has an electrochemical equivalent. A spontaneous reaction has negative ΔG and positive E; the crossover where ΔG changes sign is the same crossover where the cell voltage changes sign. And ΔG = ΔG° + RT ln Q, divided through by −nF, is the Nernst equation — the two are the same statement in different units.
Setting E to zero gives the equilibrium condition and hence ln K = nFE°/RT. This is one of the most precise routes to an equilibrium constant, because voltages are easy to measure to a fraction of a millivolt while concentrations of a trace species are not. Solubility products of sparingly soluble salts, formation constants of complexes and acid dissociation constants have all been determined this way; the resulting constants feed the equilibrium constant calculator and the solubility product calculator.
The most familiar application is the pH electrode, whose glass membrane responds to hydrogen ion activity with a near-ideal 59.16 mV per pH unit at 25 °C. Because pH is itself −log₁₀ of an activity, the Nernst response is linear in pH by construction. The pH calculator covers the solution chemistry side, and the buffer calculator the standards you calibrate with.
What the Nernst equation deliberately excludes is kinetics. It gives the potential at zero current, the thermodynamic ceiling. Real electrodes need extra voltage to drive a useful current, and how much extra depends on the exchange current density and the electrode material — which is why platinum catalyses hydrogen evolution far better than lead does at the same thermodynamic potential. That temperature-dependent rate behaviour is described by the Arrhenius equation and its electrochemical analogues.