Coulometry: counting electrons, counting water

In March 1975 V. E. Bower and R. S. Davis of the US National Bureau of Standards (NBS) hung a weighed bar of pure silver as the anode in perchloric acid, switched on a constant current of 0.203 838 18 A and let the silver dissolve for 21 939 seconds. Weighed again, the bar had lost 4.999 830 4 g. Divide mass by charge: each coulomb carried off 1.118 mg of silver.

That is the idea of coulometry. Electrons cannot be counted one by one, but a current and a time can both be measured extremely well, and the charge Q = It is the number of electrons times the elementary charge. As long as every electron goes into the same reaction, counting electrons means counting atoms. Scale the same trick down a millionfold and it measures a few micrograms of water in a lithium-battery electrolyte or a transformer oil: generate iodine in the solution with a current, let it react with the water, and read the answer in coulombs.

Each electron carries e = 1.602 176 634 × 10⁻¹⁹ C, and a mole of electrons carries F = N_A e = 96 485.332 12 C/mol — the Faraday constant. Since 2019 the SI has fixed both the Avogadro constant and the elementary charge exactly, so F is exact too.

If the electrode reaction needs z electrons per particle, a charge Q converts n = Q/(zF) moles. Coulometry measures amounts through charge, in two ways: in controlled-potential coulometry the electrode reaction is driven to completion and the decaying current is integrated over time; in constant-current coulometry the current is held steady and only the time has to be measured. Both need 100 % current efficiency: every electron must go into the wanted reaction, none into electrolysing water or some impurity. When that holds, the measurement rests only on a current, a time and a fundamental constant, with no standard solution at all — a so-called absolute method.

Why the corrections? As the anode dissolves, some silver always falls off as a fine powder instead of becoming ions — from a few hundred micrograms to over a milligram per run, an error of 50 to 230 ppm. The authors redissolved this “anode sludge”, counted its silver by coulometric titration, and calibrated the background of that titration with an aliquot of known silver mass. Oxygen was another threat: any Ag₂O in the silver dissolves in perchloric acid without taking electrons, and the authors estimated that 1 ppm of oxygen would make F 14.6 ppm too low, so the anodes were annealed hot under vacuum to drive it out.

The eight runs average 1.117 962 7 mg/C with a standard deviation of the mean of only 0.85 ppm; the impurity corrections bring it to 1.117 964 8 mg/C. Mass spectrometry gave the atomic mass of their silver as 107.868 33, so F = 107.868 33/1.117 964 8 × 10⁻³ = 96 486.33 C/mol (in the electrical units NBS maintained in 1975) — about one part in 100 000 from today’s exact value.

An ordinary titration adds a standard solution of known concentration from a burette; a coulometric titration makes the titrant on the spot by electrolysis in the titration cell: Br₂ from Br⁻ or I₂ from I⁻ at a platinum anode, OH⁻ from water at a cathode, or Ag⁺ from a dissolving silver anode to precipitate Cl⁻. With 100 % current efficiency the titrant generated is It/(zF), and together with the end point found by an indicator electrode that gives the result.

This has several advantages: no standard solution has to be made up, standardised or stored, and unstable titrants such as Br₂ or Cl₂ can be generated as they are needed; the current can be made very small — 1 μA for 1 s is only about 10 pmol of electrons — so micrograms or even nanograms can be titrated; and the whole process only needs a controlled current and a clock, which makes it easy to automate.

In 1935 the German chemist Karl Fischer found that iodine reacts quantitatively with water in methanol when SO₂ and a base are present. Today’s reagents use bases such as imidazole (RN) in place of the original pyridine. Methanol first forms a methyl sulfite salt with SO₂ and the base, and oxidising that salt with iodine uses up exactly one water molecule:

In 1959 Meyer and Boyd generated the iodine at an anode from iodide (2I⁻ → I₂ + 2e⁻): coulometric Karl Fischer titration. One water molecule uses one I₂, that is two electrons, so every milligram of water corresponds to a fixed charge.

How is the end point found? Two platinum wires dip into the solution with a small voltage between them (the “dead-stop” end point of Foulk and Bawden, 1926). For a current to flow, something must be oxidised at one wire and something reduced at the other. I⁻ is always plentiful and can be oxidised; but as long as water remains, iodine is consumed the moment it forms, and there is no I₂ to reduce — no current. When the water runs out, a trace of I₂ appears, the I₂/I⁻ couple is present in both forms, and a current flows in proportion to the I₂ concentration.

Drift is the other key idea. Water vapour from the air creeps into the cell through seals and the injection septum, so even before the injection the instrument keeps generating a little iodine to consume it and hold the cell at the end point; that current, converted to μg/min, is the drift. The drift carries on during the titration, so “drift × time” has to be subtracted from the result. The less water in the sample and the slower the titration, the more this correction matters.