Nernst equation
/ NAIRNST /
Standard reduction potentials are quoted for a tidy, idealized world: every concentration exactly 1 molar, every gas at 1 bar. Real solutions are messier — half-used batteries, dilute ions, changing acidity. The Nernst equation is the correction that tells you the actual electrode or cell potential away from those neat standard conditions. It is, in effect, the dial that shows how a battery's voltage sags as it drains.
It reads E = E° - (RT / nF) ln Q, where E° is the standard potential, R the gas constant, T the temperature, n the number of electrons transferred, F the Faraday constant, and Q the reaction quotient (the same concentration ratio as the equilibrium expression, but with the present, not-yet-equilibrium values). At 298 K the clumsy constants collapse into the handy form E = E° - (0.0592 / n) log10 Q. The message is intuitive: pile up reactants (small Q) and the potential rises; pile up products (large Q) and it falls. When the reaction reaches equilibrium, Q equals K, E drops to zero, and the battery is flat.
This equation is what makes electrochemistry quantitative in the real world. It underlies pH meters and ion-selective electrodes (which literally read concentration as a voltage), it explains why dilute cells give less than their textbook voltage, and it is the engine behind a Pourbaix diagram, since changing pH changes Q for any couple that involves H+. The main honest caveat: strictly the equation uses activities, not concentrations, so in concentrated or strongly interacting solutions the simple concentration form drifts from reality.
For MnO4- + 8 H+ + 5 e- -> Mn2+ + 4 H2O, Q contains [H+]^8 in the denominator, so the potential drops by about 0.0592 x 8/5 V (roughly 0.095 V) for every unit rise in pH — permanganate is a far weaker oxidant in base.
Because H+ appears in the half-reaction, the potential is strongly pH-dependent.
The 0.0592/n shortcut is only valid at 298 K; at other temperatures you must use the full RT/nF form. And it is built from activities, not concentrations — the difference matters once solutions are concentrated.