Avogadro’s Number: How Chemists Learned to Count Invisible Things

A chemist cannot pick up an atom. What a chemist can do is weigh things and measure volumes, and the whole of quantitative chemistry rests on a bridge between those two operations and the invisible particles underneath them.

The bridge was proposed in 1811 by a Piedmontese lawyer turned physics professor, ignored for nearly fifty years, and rescued at a conference by a Sicilian who handed out pamphlets. The number that carries his name was not measured until eighty years after he wrote, and he never calculated it himself.

The Problem With Gay-Lussac’s Volumes

In 1808 Joseph Louis Gay-Lussac published a result that was clean and awkward. Gases combine in simple whole number ratios by volume. Two volumes of hydrogen and one of oxygen give two volumes of water vapour. One volume of hydrogen and one of chlorine give two volumes of hydrogen chloride.

Whole numbers are a strong hint that something discrete is being counted, and Dalton’s atomic theory had been published five years earlier. But the numbers would not fit. If one volume of hydrogen and one of chlorine each contain the same number of particles, and each particle of hydrogen joins one of chlorine, the result should occupy one volume, not two.

Dalton’s response was to reject Gay-Lussac’s measurements. He had a reason: he assumed that atoms of the same kind repel one another, so a molecule of two identical atoms was impossible.

Avogadro’s Two Sentences

Amedeo Avogadro, professor of physics at Turin, published a paper in a French journal in 1811 containing the fix. It has two parts, and the second is the one that took decades to swallow.

First, equal volumes of any gas at the same temperature and pressure contain equal numbers of particles. This follows from gases behaving in much the same way regardless of what they are made of, and it means that a volume ratio is a particle ratio.

Second, the particles of an elementary gas need not be single atoms. They can be molecules made of two identical atoms, and Avogadro drew the distinction explicitly.

Allow hydrogen to be H₂ and chlorine to be Cl₂, and the arithmetic works instantly: one molecule of each splits and recombines into two molecules of HCl, so one volume plus one volume gives two volumes. Water becomes H₂O rather than HO, and the whole table of relative atomic weights shifts onto its modern footing.

Half a Century of Silence

Almost nobody took it up. The reasons are worth knowing because they are not about stubbornness.

• It contradicted a respected theory. Berzelius held that compounds are held together by electrical attraction between opposite charges, which makes two identical atoms bonding to each other incomprehensible
• The vocabulary did not exist. Atom, molecule and equivalent were used interchangeably and inconsistently by different chemists, so two people could agree on a formula and mean different things by it
• Avogadro was isolated. He worked in Turin, published in journals few read, wrote densely, and made no effort to promote the idea. André Marie Ampère reached a similar conclusion independently in 1814 and it was similarly ignored

The cost was chaos. By the 1850s the same compound might be written in half a dozen ways in different laboratories, atomic weights differed by factors of two, and organic chemistry in particular was drowning in incompatible formulae.

Karlsruhe, 1860

The first international chemical congress met at Karlsruhe in September 1860, called specifically to settle the mess. Around 140 chemists attended, including Mendeleev, Meyer, Bunsen and Kekule.

The argument was not resolved on the floor. It was resolved by Stanislao Cannizzaro, who had published a pamphlet in 1858 setting out how Avogadro’s hypothesis, applied consistently, produces a single unambiguous set of atomic weights, and who arranged for copies to be handed out to delegates as they left.

Lothar Meyer wrote afterwards that reading it on the journey home felt as though scales fell from his eyes, that doubts disappeared and a feeling of quiet certainty took their place. Mendeleev, then twenty six, took the same reasoning back to Russia. The periodic table of 1869 is built on atomic weights that are only correct because Cannizzaro had made Avogadro’s rule stick nine years earlier.

Avogadro had died in 1856, four years too early to see any of it.

Nobody Knew How Many

The hypothesis says equal volumes hold equal numbers. It does not say what the number is, and for most of the nineteenth century nobody could say.

The first serious estimate came in 1865 from Josef Loschmidt in Vienna, who combined the kinetic theory of gases with a measurement of the viscosity of air to work out the size of a molecule and, from that, how many are packed into a given volume. His figure was within a factor of a few of the modern one, which for a first attempt at counting the uncountable is remarkable. In German the constant is still called the Loschmidt number.

Better values came from several independent directions, which is what made them convincing.

• Brownian motion. Einstein’s 1905 analysis showed that the random jiggling of a pollen grain depends quantitatively on molecular bombardment, and Jean Perrin measured the motion of suspended particles under a microscope and extracted the number. His work, published in 1909, is generally regarded as the point at which the physical reality of atoms stopped being arguable, and it is Perrin who proposed naming the constant after Avogadro
• Electrolysis. The charge required to deposit a mole of a metal, divided by the charge on one electron once Millikan had measured it, gives the count directly
• X ray crystallography. Measure the spacing of atoms in a crystal, measure the density, and the number falls out of geometry

All of them agree. Boltzmann, who spent his career defending the reality of atoms against colleagues who regarded them as a convenient fiction, died in 1906, just before Perrin’s results settled the question in his favour.

What the Number Is For

Avogadro’s number is 6.02214076 × 10²³, and since 2019 that value is exact, because the mole is now defined as that many entities rather than being tied to a physical sample of carbon.

Its purpose is bookkeeping between two scales. Chemistry happens between individual particles, one atom joining another, and laboratory work happens in grams. Choose a number large enough that a mole of atoms weighs a convenient amount, and the relative atomic mass in atomic units becomes the mass in grams of a mole. Carbon 12 weighs 12 units per atom and 12 grams per mole. That correspondence is the entire point.

The size of it is genuinely difficult to hold in mind. A mole of water is 18 grams, about a tablespoon. A mole of sand grains would bury the surface of the Earth. If you could count a million particles a second, a mole would take about twenty billion years.

The Pattern

The chain here is worth stating plainly, because it is one of the neatest in the history of science. A measurement of gas volumes leads to a hypothesis about particle counts, which resolves a dispute about formulae, which fixes the atomic weights, which orders the elements into a table, which predicts elements nobody had found.

Every step in it was made by someone working on a problem that looked narrow at the time. Avogadro was trying to reconcile two published results. He proposed the right answer in a journal nobody read, in a vocabulary that did not yet exist, and was dead before it mattered.