Grade 12 Chemistry · Nernst, off standard
The correction that keeps a cell honest
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The correction that keeps a cell honest

EE^{\circ} is a promise made under laboratory conditions: every solute at 1 mol/L1\ \mathrm{mol/L}, every gas at one bar, 25 °C. A real cell leaves those conditions the instant it starts working — it eats its reactants. The Nernst equation is the correction that follows it out: E=ERTnFlnQE = E^{\circ} - \dfrac{RT}{nF}\ln Q.

Every letter, before any of them works. EE is the cell's ACTUAL potential right now, in volts — the thing a voltmeter reads. EE^{\circ} is its standard potential in volts, the number you built last lesson. nn is the moles of electrons the balanced cell reaction moves — a pure count, no unit. QQ is the reaction quotient: products over reactants at this instant, each concentration raised to its coefficient, and also unitless. R=8.314 J/(molK)R = 8.314\ \mathrm{J/(mol \cdot K)} is the gas constant, TT the absolute temperature in kelvin, and F=96500 C/molF = 96\,500\ \mathrm{C/mol} the Faraday constant. You are solving for EE.

At 25 °C the constants collapse into one number every exam hall knows: RTFln10=0.0592 V\dfrac{RT}{F}\ln 10 = 0.0592\ \mathrm{V}, which turns the natural log into the log you can read off a scale — E=E0.0592nlogQE = E^{\circ} - \dfrac{0.0592}{n}\log Q. Read aloud: E equals E-naught minus zero-point-oh-five-nine-two over n, log Q. One decade of QQ is worth 59.2 mV59.2\ \mathrm{mV} to a one-electron cell — and that is not trivia. It is exactly why a pH meter reads about 59 millivolts per pH unit.

Three sanity rails, free with every calculation. Q<1Q < 1 means reactant-rich, the cell is eager, and EE sits ABOVE EE^{\circ}. Q=1Q = 1 is standard, and E=EE = E^{\circ} exactly. And Q=KQ = K is equilibrium, where E=0E = 0 — which is all a dead battery ever was.