From grams to counts

A chemical equation is a statement about numbers of particles. A balance reads grams. The mole is the unit that converts between them — and in 2019 its definition was turned inside out, so that the count is now exact by decree and the mass of twelve grams of carbon is the thing being measured.

1811 – 201916 min

By the end you should be able to

Explain why chemistry needs a unit that is neither a mass nor a pure number
Convert between grams, moles and numbers of particles, and say which step needs a measured constant
Describe how the atoms in a silicon sphere were counted, and why that made the 2019 redefinition possible
State what the 2019 redefinition changed, and the one familiar sentence it quietly stopped being exactly true

A chemical equation is a statement about numbers. 2H₂ + O₂ → 2H₂O says that molecules of hydrogen are consumed two at a time for every one of oxygen. Not two grams. Two things. Every reaction in chemistry works like this, because a bond forms between particular atoms and does not care what they weigh. And there is no instrument that counts molecules. There never has been, there is no prospect of one, and the numbers involved would defeat it anyway. The only operations a chemist can actually perform on a quantity of matter are weighing it and measuring its volume. So there is a gap in the middle of the subject: the theory speaks in counts and the laboratory speaks in grams. Everything in this lesson is about the unit built to bridge that gap, and about what it takes to fix the size of the bridge.

The bridge is two steps, and it is worth separating them, because they are different kinds of thing. Grams to moles. Divide by the molar mass. This is a property of the substance in front of you: 18.02 g/mol for water, 63.55 for copper, 342.30 for sugar. It comes ultimately from Cannizzaro — the relative weights are exactly what Karlsruhe settled. Moles to particles. Multiply by Avogadro’s number, 6.02214076 × 10²³ per mole. This is the same for everything in the universe. It is not a property of water or copper or anything else; it is the size of the unit itself. That is the whole apparatus. grams ÷ molar mass × N_A = particles. The interesting question is not how to use it — it is arithmetic — but where the second number came from, and what kind of statement it is. That turns out to have a surprising answer, and the answer changed in 2019.

A teaspoon of water on the pan. Read the count, then find it on the axis below. Now drag the mass down as far as it will go, and watch how little difference it makes.

Loading the mole scale…

The pan is on the left, the count is on the right, and the axis between them is logarithmic — every tick is a factor of a thousand. Five grams of water is 1.671 × 10²³ molecules: far past the sand, far past the oceans, and level with the stars in the observable universe. Drag the mass down to a quarter of a gram, a drop you would not notice on a sleeve, and it is still past every teacup of water in every ocean. That is why chemistry is done on a balance.

The mole is a unit, not a number

The usual gloss is that a mole is a chemist’s dozen: a big number with a name. That is close enough for most work and it hides the part worth knowing. A dozen is a pure number. Twelve eggs, twelve atoms, twelve galaxies — the twelve is the same twelve, and it is dimensionless. The mole is an SI base unit, for a base quantity called amount of substance, standing alongside the metre, the kilogram and the second. Avogadro’s constant is correspondingly not a number but a constant with a unit: N_A = 6.02214076 × 10²³ mol⁻¹. That is why molar masses come out in grams per mole rather than being dimensionless ratios, and why the gas constant, every equilibrium constant and every rate constant in the subject carry the units they do. Take the mole out of the base units and the dimensional bookkeeping of physical chemistry has to be rewritten. And this is genuinely contested. A serious body of metrologists argues that amount of substance is not an independent quantity at all — that a mole is a count, N_A is a pure number, and the SI is carrying a historical accident. The 2019 redefinition made that position more tenable rather than less, because it defines the mole by a count and says nothing about mass. The argument has not been settled; it has been made polite.

How big 10²³ actually is

Large numbers stop meaning anything past about a million, so here are three that can be checked rather than felt. A mole of pennies. Not atoms — ordinary British pennies, 20.3 mm across and 1.52 mm thick. Pour a mole of them evenly over the entire surface of the Earth, oceans included, and the layer would be 581 metres deep. They would weigh 2.14 × 10²¹ kg, which is about 2.9% of the mass of the Moon. Counting them. At one per second, without stopping, a mole takes 1.91 × 10¹⁶ years — around 1.4 million times the age of the universe. Start at the Big Bang and you are still on the first millionth of it. The teaspoon. Five grams of water holds 1.671 × 10²³ molecules. The observable universe holds something like 10²³ stars — the published estimates run from 10²² to 10²⁴, so treat that as an order of magnitude and nothing more. The two numbers are the same size. A teaspoon of water and the entire visible universe are, on this one count, a draw. None of this is decoration. It is the reason the mole exists: no procedure that touches particles one at a time can ever be a chemical procedure.

The experiment

Counting the atoms in a silicon sphere

The International Avogadro Coordination — PTB Braunschweig, NMIJ, NMI Australia, BIPM, INRIM and others · 1990s onwards; the decisive results 2011 and 2015 – 2017 · Silicon-28 crystal grown in Nizhny Novgorod and Berlin; spheres polished in Sydney; measured across five continents

The question: How many atoms are there in a macroscopic object? Not an estimate from a theory — an actual count, of an object you can hold.
The apparatus: A boule of silicon enriched to better than 99.99% ²⁸Si, ground into a sphere about 93.7 mm across and one kilogram in mass, and polished until it is the roundest object ever manufactured. Its diameter is measured in hundreds of directions by optical interferometry; the spacing of its crystal lattice is measured by X-ray interferometry; its mass is compared against the kilogram; and its molar mass is determined from the isotope ratios by mass spectrometry. The surface oxide layer, a few nanometres thick, is measured separately and subtracted, because at this precision it is not negligible.

Theory predicted

Nothing was in dispute about the answer. What was in dispute was whether the count could be made accurate enough — to a few parts in a hundred million — to serve as a definition, and whether it would agree with the entirely independent watt-balance route to the same constants.

They measured

Silicon’s lattice is cubic with eight atoms to a cell, so the count is a division: the volume of the sphere over the volume of one cell, times eight. It comes to about 2.151 × 10²⁵ atoms. The best determinations gave N_A = 6.02214076 × 10²³ mol⁻¹ with a relative uncertainty around 2 × 10⁻⁸, and they agreed with the entirely independent watt-balance route.

How sure could they be? A few parts in a hundred million, which is the whole point — it is the level at which a count can replace a mass as a definition. The diameter has to be known to a fraction of a nanometre and the lattice spacing to about a part in a thousand million. Almost all the remaining uncertainty is in the surface layer and the isotopic composition, not in the geometry.

Why it mattered

Avogadro’s number stopped being a thing to measure. Once two independent routes agreed to a few parts in a hundred million, the CGPM could fix the value by definition and let the laboratories go on measuring what they had previously defined. That is the 2019 redefinition in one sentence, and this sphere is the reason it was possible.

A mirror-polished silicon sphere held in a mount, reflecting the room around it. — About 93.7 mm across, a kilogram in mass, and round to within a few tens of nanometres — scaled up to the size of the Earth, its mountains would be a few metres high. The mirror finish is not for show: the diameter is measured by bouncing light off it in hundreds of directions, and any roughness is an error in the volume, and therefore an error in the count.

On 20 May 2019 that measurement was used to abolish itself. Before. One mole is the amount of substance containing as many elementary entities as there are atoms in 0.012 kg of carbon-12. On this definition, 12 g of carbon-12 is exactly one mole — it is true by construction, and no experiment could ever disturb it. Avogadro’s number was the unknown: a quantity to be measured, with an uncertainty, revised every few years as the measurements improved. After. One mole contains exactly 6.02214076 × 10²³ elementary entities. The count is now the definition. And the mass of a mole of carbon-12 has become the measured quantity, with an uncertainty of its own. Read those two paragraphs again in order. The definition was turned inside out. The thing that used to be exact is now measured; the thing that used to be measured is now exact. Not one number in any laboratory changed, because the value was chosen so that nothing would — but which sentence is a decree and which is a result swapped places, and that is a real change even though nothing moved.

You might think

Twelve grams of carbon-12 contains exactly Avogadro’s number of atoms.

Actually

Not since 2019. It is now a measured statement rather than a definition, and the measurement does not come out exactly right: the molar mass of carbon-12 is 11.9999999958 g/mol, short of 12 by about three parts in ten thousand million. The same applies to the rule you were taught alongside it — that the molar mass in grams per mole equals the relative atomic mass in daltons. That is now an approximation too, accurate to the same three parts in ten thousand million. Both remain true far beyond any precision a chemist will ever work at; a balance reading to a microgram is eleven orders of magnitude coarser. But "exactly" is a strong word, and it is no longer the right one. Notice also which way round this is: the redefinition did not make chemistry less certain. It moved the uncertainty out of the constant, where it was a nuisance to everybody, and into a mass measurement, where it belongs.

What the redefinition actually bought

The headline is a fixed number. The consequence is independence. Before 2019, the mole depended on the kilogram. The definition ran through 0.012 kg of carbon-12, so any drift in the kilogram was a drift in the mole. And the kilogram was a cylinder of platinum-iridium in a vault at Sèvres, which over a century appeared to lose around 50 micrograms relative to its own official copies — with no way to tell whether the prototype was shedding mass or the copies were gaining it, because the prototype was the definition and could not be wrong. After 2019, nothing depends on an object. The mole is a count. The kilogram is defined through the Planck constant. The two are decoupled, and both are reproducible from first principles by any laboratory with the right apparatus. That is the real prize, and it is the same one Lavoisier was after in a cruder form. A measurement is only as trustworthy as the thing it is compared against. Comparing against a lump of metal in a French basement means trusting the basement. Comparing against a defined constant of nature means trusting only that physics is the same everywhere — which is, in the end, the assumption this entire site is built on.

Problem

A teaspoon of water

A teaspoon holds 5.00 g of water. How many water molecules is that? (These are the simulation’s starting values, so you can check yourself against it.)

Mass of water: 5.00 g
Molar mass of water: M(H₂O) = 18.02 g/mol
Avogadro constant: N_A = 6.02214076 × 10²³ mol⁻¹

The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 × 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, N_A, when expressed in the unit mol⁻¹ and is called the Avogadro number.
26th General Conference on Weights and MeasuresResolution 1, November 2018; in force from 20 May 2019. The second sentence of the definition is the whole change: it names a count, and it never mentions a mass.

1811Avogadro: equal volumes of gas hold equal numbers of particles. Nobody can say how many.
1865Loschmidt makes the first real estimate of the number of molecules in a volume of gas, from kinetic theory. He is within a factor of ten.
1900Planck, fitting the blackbody curve, gets a value for the number within a per cent — from a problem that has nothing to do with chemistry.
1909Perrin measures it four independent ways from Brownian motion and gets the same answer each time. Ostwald concedes that atoms are real.
1971The mole becomes an SI base unit, defined as the number of atoms in 0.012 kg of carbon-12.
2011The Avogadro Project’s silicon-28 spheres reach a relative uncertainty of 3 × 10⁻⁸.
2018The CGPM votes to fix N_A, h, e and k by definition. The vote is unanimous.
2019On 20 May the new definitions take effect. The mole is a count; the kilogram is a constant; the cylinder at Sèvres is a museum piece.

Where this leaves us

The bridge is finished. Grams to moles to particles, in two multiplications, with one constant fixed by decree and one that is a property of whatever is on the balance. Chemistry can now do exact arithmetic on objects nobody will ever see one of, and it can do it to more decimal places than any experiment needs. It took two centuries to build. Lavoisier supplied the conserved quantity, Dalton the whole-number ratios, Avogadro the hypothesis nobody would accept, Cannizzaro the procedure that made the weights consistent, Perrin the demonstration that the particles are real, and a sphere of silicon the count that let the unit be defined without reference to any object at all. And now put that finished scale of atomic weights to work, in the most obvious way imaginable: line the elements up in order. Something very strange happens when you do. The properties come back round. Lithium, sodium and potassium behave alike and sit at intervals; fluorine, chlorine and bromine do the same. It happens again and again, at spacings that are regular but not equal, and for fifty years nobody could say why — or whether the pattern was a fact about matter or a coincidence of arithmetic.