Avogadro's Number

6.02214076 × 10²³. Since 20 May 2019 that figure is exact by definition, not by measurement — it is one of seven fixed constants the modern SI is built on.

Two things about it confuse people in different ways. The first is that the man it is named after never calculated it, never attempted to, and died more than fifty years before anyone managed it. The second is that it is not really a number about chemistry at all: it is a conversion factor between a human-sized unit of mass and the actual size of atoms, and it is ugly because both of those were chosen for unrelated reasons.

If what you need is how to use it in a calculation, that belongs to the mole — the unit is where the arithmetic lives. This page is about the constant itself.

Avogadro never knew the number named after him

Amedeo Avogadro published his hypothesis in 1811: equal volumes of different gases, at the same temperature and pressure, contain equal numbers of particles. That is a statement about ratios. It says nothing whatsoever about how many particles are in any particular volume, and Avogadro had no way of finding out.

His idea was also largely ignored for half a century, partly because it required accepting that elemental gases were made of two-atom molecules rather than single atoms — a claim Dalton rejected outright. It took Stanislao Cannizzaro, presenting Avogadro's reasoning at the Karlsruhe Congress of 1860, to make the chemistry community take it seriously. Avogadro had died in 1856.

The name is therefore an honorific rather than an attribution. It was attached by Jean Perrin in 1909, who was measuring the quantity and chose to name it after the person whose hypothesis made the measurement conceptually possible.

Loschmidt counted first

The first real attempt came in 1865 from Josef Loschmidt in Vienna. He combined two things that were separately measurable — the mean free path of molecules in a gas, from viscosity data, and the volume a gas occupies when condensed to a liquid — to extract an estimate of molecular diameter, and from that the number of molecules in a given volume of gas.

His answer is not the Avogadro constant. It is the number of particles per unit volume of an ideal gas at a stated temperature and pressure, a distinct quantity now called the Loschmidt constant, about 2.687 × 10²⁵ per cubic metre at 0 °C and one atmosphere. Loschmidt's own figure was off by roughly an order of magnitude, which in 1865, working from indirect evidence about entities most physicists doubted existed, was a remarkable result.

This is why German-language sources for many decades called the Avogadro constant the Loschmidt number, and why the symbol L still appears alongside N_A in some textbooks. It is a genuine naming disagreement rather than an error, and if you meet L in a European text it almost certainly means what your book calls N_A.

Perrin's converging roads

What settled the question — and, along the way, settled whether atoms were real — was Jean Perrin's work in the first decade of the twentieth century. Perrin studied Brownian motion, the jittering of pollen-sized particles suspended in a fluid, which Einstein had shown in 1905 should follow a statistical law containing the number of molecules per mole.

The persuasive part was not any single measurement. It was that Perrin assembled roughly a dozen completely independent routes to the same quantity — Brownian displacement, sedimentation equilibrium under gravity, gas viscosity, the blackbody radiation law, the blue of the sky, the charge measured in electrolysis divided by the electron's charge, the counting of alpha particles from radium — and they all landed on the same figure, near 6 × 10²³.

No one of those methods was above suspicion. All of them agreeing, when there was no reason for their errors to conspire, was an argument that even committed sceptics of atomic theory could not answer. Perrin published the case in Les Atomes in 1913 and received the Nobel Prize in Physics in 1926.

A silicon sphere, and why the kilogram needed one

The modern determination is a very different exercise. The X-ray crystal density method takes an almost perfect single crystal of silicon, enriched to better than 99.99% silicon-28 so that its molar mass is known with extreme precision, and works out how many atoms are in it.

Silicon's atoms sit on a diamond-cubic lattice with eight atoms per unit cell. Measure the lattice spacing by X-ray interferometry, measure the sphere's volume and mass, and the count follows:

N_A = 8 × M / (ρ × a³)

where M is molar mass, ρ density and a the lattice parameter. Getting this to a relative uncertainty of a few parts in 10⁸ required a sphere round to within tens of nanometres — if it were scaled up to the size of the Earth, its highest mountain would be a few metres.

That precision was not pursued for chemistry's sake. It was one of the two routes to redefining the kilogram without a physical artefact, the other being the Kibble balance's determination of the Planck constant. Because the two constants are tightly linked by relations that are known far better than either is individually, agreement between the silicon spheres and the balances was the condition for redefining both units at once. When the numbers finally converged, the kilogram, the mole, the ampere and the kelvin were all redefined together.

How big it actually is

The figure is nearly impossible to hold in your head, and comparisons help more than digits.

• Stack a mole of sheets of ordinary printer paper, each a tenth of a millimetre thick, and the pile is about 6,400 light years tall — roughly a quarter of the way from here to the centre of the galaxy.
• If every person alive counted one atom per second, without stopping, the eight billion of us would need about 2.4 million years to count one mole.
• Estimates of the number of grains of sand on all the world's beaches sit around 10¹⁹. A mole is tens of thousands of times more than that.

And yet a mole of water molecules is 18 grams — a bit more than a tablespoon. That contrast is the honest lesson of the number: it is astronomically large only because atoms are astronomically small, and the two cancel out at the scale of a spoon.

Constant or number, and the units that decide

Examiners do care about this distinction, and it is a cheap mark.

Avogadro's number is the dimensionless quantity 6.02214076 × 10²³ — just a count, like a dozen being twelve.

The Avogadro constant, N_A, is 6.02214076 × 10²³ mol⁻¹. The units are what make it a conversion factor rather than a count, and they are what cause the moles to cancel when you multiply an amount in mol by N_A and get a pure number of particles.

Write the units, and the dimensional check will tell you whether you have multiplied or divided correctly without your having to think about it.

Why such an awkward value

There is nothing fundamental about 6.022. Two arbitrary human decisions fix it: the size of the gram, which descends from a revolutionary-era definition based on water, and the choice of carbon-12 as the mass reference. Had the mole been pegged to the kilogram instead, the constant would be a thousand times larger and no less arbitrary.

That arbitrariness sits behind a live disagreement among metrologists that most courses never mention. One camp holds that amount of substance is a genuine base quantity and the mole a genuine base unit; another argues that since the mole is now defined as a fixed count, it is dimensionally no different from "dozen" and should not be an SI base unit at all. The 2019 redefinition sharpened that argument rather than resolving it, because it removed the last link between the mole and any physical object.