A battery's voltage is a concentration ratio, read out loud. Move an ion from a crowded compartment to an empty one and the pull is a potential you can measure in volts. Nernst wrote the exchange rate: E = E₀ − (RT/nF) ln Q — every tenfold change in the quotient Q costs 59 mV / n. Down the center the quantities go in, the engine computes the cell potential, the volts come out. The blue team builds and defends it; the red team tries to break it.
source Nernst, Die elektromotorische Wirksamkeit der Jonen (1889), Z. phys. Chem. 4, 129–181 — doi.org/10.1515/zpch-1889-0412. Rendered, not quoted.
The cell potential is the standard potential minus a logarithmic correction for how far the mixture sits from 1:1:
E = E₀ − (RT/nF)·ln Q. Here E₀ is the standard cell potential, n the electrons transferred, F the Faraday constant, R the gas constant, T the temperature, and Q the reaction quotient (the same ratio that becomes K at equilibrium).
Convert to base-10 and one decade of Q costs 2.303·RT/nF. The thermal voltage RT/F is the whole scale — live below for this T:
| n | mV / decade of Q |
|---|
RT/F at this T = … mV. The 59 is not a law of nature; it is 2.303·RT/F and it scales with temperature.
Run the reaction until it stops: current dies, E = 0, and Q has become the equilibrium constant K. Set E to zero and the equation collapses to
E₀ = (RT/nF)·ln K — the standard voltage is just the equilibrium ratio, read as a potential. A voltmeter is an equilibrium-constant meter.
That is the seam to the-equilibrium-constant: Q and K are one object, and Nernst turns the ratio into 59 millivolts to the decade. Each sphere is the next one's premise.
The blue team's live check: recompute the standard case, the per-n slope, the concentration cell and the equilibrium tie, and confirm them against the known physics. If red drops the n, this badge is where it shows.
Six numbers feed the engine. Three are constants of nature; three you set for the cell:
| symbol | quantity | value / unit |
|---|---|---|
| R | gas constant | 8.314463 J mol⁻¹ K⁻¹ |
| F | Faraday constant | 96485.33 C mol⁻¹ |
| T | temperature | 298.15 K |
| E₀ | standard potential | volts (you set) |
| n | electrons moved | 1, 2, 3 (you set) |
| Q | reaction quotient | dimensionless (you set) |
RT/F = 25.693 mV is the thermal voltage — the natural unit of the correction. That is what you feed the panel below.
General cell: Q is the product/reactant ratio. Q = 1 sits at standard; E = E₀ exactly.
E is a straight line in log₁₀ Q, slope −59.16/n mV per decade — computed live from R, T, n, F, never looked up.
What the machine proves: at Q = 1 the answer is E₀ exactly; each decade of Q shifts it by −59.16/n mV; a concentration cell (E₀=0) reads a strictly positive voltage down its own gradient; and at Q = K the potential is 0. The current cell's answer is above.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
It also writes concentrations where thermodynamics demands activities — true only in the dilute limit. At seawater or physiological ionic strength the activity coefficients drift from 1 and the naive ratio is wrong AMBER. Liquid-junction potentials add an uncontrolled term at the salt bridge, and as Q→0 or ∞ it predicts unbounded E that no real cell delivers.
"The slope is 59 mV per decade, always." Cut. It is 59.16/n mV — a 2-electron couple gives 29.6, a 3-electron 19.7. Dropping the n is exactly the planted tamper.
"59 mV is a universal constant." Kept, corrected. It is 2.303·RT/F and scales with T: ≈54.2 mV at 0°C, 59.2 at 25°C, 61.5 at 37°C.
"Concentration equals activity." Cut. Q is an activity ratio; concentration only approximates it when the solution is dilute.
"A loaded battery reads its Nernst voltage." Cut. Nernst is the zero-current EMF; under load, overpotentials subtract.
The red team's move: drop the n — use RT/F regardless of how many electrons the couple transfers. A 2-electron cell then reports twice its true voltage. The blue team's witness (window 7) is watching.
Divide by F instead of nF and every multi-electron slope inflates — n=2 jumps from 29.6 to 59.2 mV/decade. The witness recomputes, disagrees with the known physics, and turns red. Nothing is faked; the attack is real and it is caught.