The Avogadro constant

Q: How many atoms are in a spoonful, and how did anyone count them?

A glass of water has about 10^{25} molecules. The Avogadro constant, N_A, is how many are in one mole: about 602 thousand billion billion. It is the bridge between atoms and things you can weigh. For a long time some scientists doubted atoms were even real. Counting them, by watching specks jiggle in water, settled the argument.

Q: What is the value of the Avogadro constant?

6.02214076 × 10²³ per mole. Since 2019 N_A is exactly 6.02214076 × 10²³ per mole. The value was frozen from silicon-sphere counts and Kibble balances, which agreed to about 12 parts per billion.

How many atoms are in a spoonful, and how did anyone count them?

N_A = 6.022\,140\,76 \times 10^{23}\ \text{mol}^{-1}

Exact, by definition, since 2019 13 min read, with an experiment to run

A glass of water has about $10^{25}$ molecules. The Avogadro constant, $N_A$ , is how many are in one mole: about 602 thousand billion billion. It is the bridge between atoms and things you can weigh.

For a long time some scientists doubted atoms were even real. Counting them, by watching specks jiggle in water, settled the argument.

1 · Play first

Track jiggling beads under a microscope

In 1908 Jean Perrin watched tiny resin beads in water. Water molecules hit them from all sides, a little unevenly, so they wander at random: Brownian motion. Einstein had shown in 1905 how far they should wander if molecules were real.

Let the tracker follow the beads for a while, then measure how far they wander. Einstein's formula turns that into $N_A$ .

What to notice

Smaller beads and warmer water jiggle more.
Wandering grows with time, but the distance squared grows in a straight line.
A thicker liquid slows the beads down without changing the answer.

2 · The equation, piece by piece

Einstein linked the spread of the beads to the number of molecules. Tap any piece.

\htmlClass{t0}{N_A} = \frac{\htmlClass{t1}{R}\,\htmlClass{t2}{T}}{\htmlClass{t3}{6\pi\eta a}\,\htmlClass{t4}{D}}, \qquad \htmlClass{t5}{\langle r^2 \rangle} = 4\htmlClass{t4}{D}\,t

N_A

The Avogadro constant

Molecules per mole, what we want.

Perrin got about $6.8\times10^{23}$ , close enough that most doubters accepted atoms. He won the 1926 Nobel Prize.

3 · How we found it

1Gases by the box

In 1811 Amedeo Avogadro proposed that equal volumes of gas at the same temperature and pressure hold equal numbers of molecules. He had no idea how many.

In 1865 Josef Loschmidt made the first estimate from how gases flow, about $10^{25}$ per cubic metre.

2Are atoms real?

Robert Brown saw pollen particles jiggle in 1827 but could not explain it. In 1905 Einstein predicted exactly how far they would wander if molecules were real.

Perrin tested it with beads, counting them at different heights and tracking their paths. Several different methods all gave the same $N_A$ , and that agreement convinced the world.

3Counting a silicon sphere

The Avogadro Project made near-perfect spheres of silicon-28 and counted their atoms using X-ray spacing. In 2019 $N_A$ was fixed and the mole became exactly $6.02214076\times10^{23}$ things.

1811
Avogadro's hypothesis about gas volumes.
1827
Brown sees particles jiggle in water.
1865
Loschmidt's first estimate of molecule numbers.
1905
Einstein's theory of Brownian motion.
1908
Perrin measures $N_A$ with beads in water.
1909
Perrin names the number after Avogadro.
2019
$N_A$ is fixed exactly and the mole is redefined.

4 · How the value is pinned down

Since 2019 $N_A$ is exactly 6.02214076 × 10²³ per mole. The value was frozen from silicon-sphere counts and Kibble balances, which agreed to about 12 parts per billion.

A mole is now simply that many of anything, the way a dozen means twelve. $R = N_A k_B$ is exact too.

5 · What if it were different?

If atoms were much bigger

You would see grainy water. Perrin's beads would barely jiggle, because a few big hits would average out less.

If there were no atoms

Matter would be smooth all the way down, and Brownian motion would never happen.

Why it matters to you

Every medicine dose and recipe for chemicals is worked out in moles, so $N_A$ sits behind them.

6 · Try it at home

Measure a molecule with oil on water

A tiny drop of oil spreads on water until it is one molecule thick. Measure the patch and you can work out how long one molecule is.

You need

A wide tray or baking dish of water
Talcum powder or pepper
Cooking oil, and a toothpick or thin wire to pick up a tiny drop
A ruler

Do this

Dust the water surface lightly with powder.
Touch the toothpick tip into oil to get the smallest drop you can. Estimate its diameter against the ruler.
Touch the drop to the middle of the water. The oil pushes the powder back into a circle.
Measure the circle's diameter.

The maths

Drop volume is $\tfrac{\pi}{6} d^3$ . Spread as a thin film, its thickness is volume / patch area. For a 0.5 mm drop and a 20 cm patch that is about $2\times10^{-9}$ m: a couple of nanometres, the length of one oil molecule.

Stay safe: Wipe up spilled oil so nobody slips, and keep powder away from faces.

7 · Common mix-ups

MythA mole is a weight.

ActuallyIt is a count, like a dozen. A mole of hydrogen and a mole of gold weigh very different amounts.

MythAvogadro measured his number.

ActuallyHe never did. It was named after him by Perrin in 1909.

MythBrownian motion is caused by currents in the water.

ActuallyIt is caused by individual molecules hitting the particle unevenly.

8 · References

CODATA value: Avogadro constant, NIST, 2018
The International System of Units (SI Brochure, 9th edition), BIPM, 2019
On the motion of small particles suspended in liquids at rest, A. Einstein, Annalen der Physik 17, 1905
Brownian movement and molecular reality, J. Perrin (English translation), 1910
Determination of the Avogadro constant by counting the atoms in a 28Si crystal, B. Andreas et al., Physical Review Letters 106, 2011

Next constantThe speed of light: How fast is light, and how did anyone time something that quick?