Chemistry 1889

The Electromotive Activity of Ions

Walther Nernst

A difference in concentration is, quietly, a voltage.

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In depth · the introduction

Take the same salty water at two different strengths, dip a metal in each, and a voltmeter twitches — a difference in concentration is, quietly, a voltage.

The big idea

A battery makes electricity because chemicals “want” to move from where they are crowded to where they are sparse. Nernst's achievement was to say exactly how much voltage that wanting is worth. His equation links the voltage of a cell to the concentrations of the ions inside it.

The rule of thumb that falls out is beautifully simple: at room temperature, every tenfold difference in concentration of a singly-charged ion is worth about 59 millivolts. Ten times more concentrated on one side: 59 mV. A hundred times: about 118 mV. The voltage grows with the logarithm of the ratio, not the ratio itself.

How it came about

In the late 1880s the brand-new science of physical chemistry was being built in Wilhelm Ostwald's laboratory in Leipzig. Two ideas had just arrived there: Jacobus van 't Hoff had shown that dissolved particles press outward like a gas (osmotic pressure), and Svante Arrhenius had argued that a salt in water is already broken into charged ions. Walther Nernst, a sharp and ambitious young assistant, saw how to fuse them.

If an ion is like a gas under pressure, he reasoned, then a metal electrode feels two opposing pushes — its own tendency to dissolve into ions, and the pressure of the ions already in the water pushing back. Where they balance sets the voltage. In his 1889 habilitation he turned that picture into an equation. (He would win a Nobel Prize in 1920, but for something else entirely — his heat theorem, the third law of thermodynamics.)

Why it mattered

Before Nernst, you could rank metals by how vigorously they reacted, but you could not predict a voltage. Afterwards you could calculate it. That single ability underlies the design of batteries, the measurement of acidity with a pH meter, the sensors that read sodium or calcium in a blood sample, and the science of why metals corrode — and, remarkably, the tiny voltage that every nerve cell in your body maintains across its membrane.

A way to picture it

Imagine two adjoining rooms connected by a single doorway, one packed with people and one nearly empty. People naturally drift from the crowded room toward the empty one, and if you put a turnstile in the doorway, that drift can turn it and do work. The bigger the crowd difference, the harder it turns. An ion concentration cell is exactly this: ions “want” to move from the concentrated side to the dilute side, and that wanting shows up as a voltage. Nernst's equation is the precise exchange rate between crowdedness and volts.

Where it sits

Volta had built the first battery in 1800 and Faraday had quantified electrolysis in the 1830s, but neither could say why a given cell produced a particular voltage. Nernst supplied that link, resting it on van 't Hoff's and Arrhenius's fresh ideas about solutions. From here a line runs straight to Hodgkin and Huxley's account of the nerve impulse — built on the very same equation — and to the batteries in the device you are reading this on.

The original document

Original source text

Walther Nernst · Zeitschrift für physikalische Chemie 4 (1889): 129–181 · Leipzig

Written as Nernst's habilitation in Wilhelm Ostwald's Leipzig laboratory — the same rooms where van 't Hoff's osmotic theory and Arrhenius's ions were turning physical chemistry into a quantitative science — the memoir asks a deceptively simple question: how large is the electrical force an ion can exert, and on what does it depend?

1 · The osmotic analogy

Nernst takes over van 't Hoff's result that dissolved particles behave like a gas, exerting an osmotic pressure proportional to their concentration, and Arrhenius's claim that in solution a salt is already split into free ions. An ion in solution is therefore like a gas under pressure; differences in that pressure are differences in a tendency to move.

2 · Electrolytic solution pressure (Lösungstension)

To this he adds one new quantity: every metal is supposed to have an “electrolytic solution pressure” P, an intrinsic tendency to shed ions into the solution, leaving electrons behind on the metal. Against it pushes the osmotic pressure p of the ions already dissolved, which tends to drive them back onto the metal. The electrode comes to rest where the two are balanced, and the charge separation built up in reaching that balance is the electrode potential.

3 · The resulting law

Equating the electrical work of moving the ions against the osmotic work of compressing them from one pressure to the other gives a logarithmic law: the potential depends on the logarithm of the concentration (strictly, the ratio of solution pressure to osmotic pressure). In the form used ever since, the electromotive force of a cell is E = E° − (RT/nF) ln Q.

E = (RT / nF) · ln(P / p) → E = E° − (RT / nF) · ln Q ; 2.303 RT/F ≈ 59.2 mV per tenfold concentration ratio at 25 °C.

4 · Concentration cells and consequences

The clearest test is a cell with the same metal in the same salt at two concentrations: the standard term cancels and a voltage appears from the concentration difference alone. From the same law Nernst reads off how cell voltages, solubilities and equilibria depend on concentration — the working equations of electrochemistry.

[ … ]

The literal picture of a “solution pressure” was later dropped in favour of Gibbs's chemical potential, and dilute concentrations were replaced by activities — but the logarithmic law itself, and the 59 mV-per-decade slope, are exactly as Nernst left them.

Leipzig · 1889