How many molecules are in a glass of water? Count them yourself.

What is Brownian motion?

It is the endless trembling of anything small enough in a liquid or a gas. The beads in the footage are one micrometre across, a thousandth of a millimetre. In one second a bead typically wanders 0.93 micrometres along any one direction, then turns back, then sets off elsewhere, with no visible cause.

It is not a current in the water: two neighbouring beads go different ways. It is not life: the beads are latex. And it does not stop: water at room temperature keeps pushing them for ever.

Why does the bead tremble?

Because water comes in pieces. Its molecules, some three thousand times smaller than the bead, move in every direction and strike it from every side. If the blows were countless and infinitely small, they would cancel exactly and the bead would sit still. Because they are a number, however large, at one instant a few more arrive from the left, at the next a few more from below, and the bead wobbles.

Albert Einstein worked out the law in 1905. Behind it is a simple idea: a bead, however large, gets on average the same energy of motion as a single water molecule. The energy of motion of a whole mole of molecules follows from the temperature and the gas constant, R, measured long before anyone could count molecules. Divided by the number of molecules, it gives the energy of one. The more molecules in a mole, the less each one has, and the more gently the bead trembles.

How you counted the molecules

Each bead you catch brings its pairs of positions a sixth of a second and one second apart. The difference between the two averages gives the rate of spreading, D, free of the footage's constant error (each frame is a short exposure during which the bead moves). All 712 beads followed for at least two seconds give D = 0.43 µm²/s, from 80,334 pairs of positions a sixth of a second apart and 67,343 one second apart.

Whoever filmed it did not record the temperature. Water at 20 degrees is more viscous than at 25, and the result moves from 6.0 to 6.9 × 10²³ molecules in a mole. A few degrees in the room shift the result by 14%, about as much as Perrin's entire error. The bead size is nominal too: a radius 5% larger would lower the result by almost 5%. Both today's value, fixed by definition at 6.022 140 76 × 10²³, and Perrin's, 6.85 × 10²³, fall inside that range.

How the number settles, for one arbitrary order of beads (at 22 degrees): one bead, 6.4; five, 6.8; twenty, 6.0; a hundred, 6.2; all of them, 6.3 × 10²³.

Who discovered Brownian motion?

In the summer of 1827 the Scottish botanist Robert Brown was looking through his microscope at the pollen of a flower, Clarkia pulchella. What he saw moving in the water was not the pollen grains. It was the particles inside them, a few micrometres long, and smaller ones of a micrometre or so.

To find out whether the motion belonged to living matter, he tried dust from things that had never lived: window glass, rocks, among them “a fragment of the Sphinx”. They moved just the same. Brown published what he had seen, credited earlier observers, Needham and Gleichen, and would not “hazard any conjectures whatever” about these particles; a year later he still called their motions ones “for which I am unable to account”.

Is dust in a sunbeam Brownian motion?

Some two thousand years ago the Roman poet Lucretius asked his readers to watch the motes in a sunbeam: they change course and are beaten back, “impelled by viewless blows”, in W. E. Leonard's translation. It was his argument that the world is made of atoms, though his atoms were not today's molecules. And the example did not hold. Those specks are carried by faint air currents, as Perrin himself pointed out in 1909. The motion driven by molecules shows only in particles of a few micrometres or smaller, and only under a microscope.

Did Einstein prove that atoms exist?

Not quite. In 1905 Einstein calculated how far small particles suspended in water should spread if the molecular theory were true, and wrote that this motion might be identical with Brownian motion; the data of the day were too vague to decide. He gave an example: for a particle a thousandth of a millimetre across, exactly the size of the beads in the footage, with the values he used (water at 17 degrees, 6 × 10²³ molecules in a mole), about 0.8 micrometres in one second and 6 micrometres in one minute. The beads in the footage, at room temperature, typically wander 0.93 micrometres in a second.

The paper ends with a wish that some researcher would soon settle the question. Three years later, one did.

How did Perrin count the atoms?

The French physicist Jean Perrin needed grains all of one size. He made them from gamboge, a yellow resin that, rubbed under water like a bar of soap, gives a yellow emulsion. He sorted the grains by centrifuging at 2,500 revolutions a minute: in his most careful run, a kilogram of resin gave, after several months, a few tenths of a gram of grains of the size he wanted.

With his collaborators he followed the grains under the microscope and marked their positions every 30 seconds. By 1909 he had some 3,000 displacements and a number: 71.5 × 10²² molecules in a mole. He also used other methods, among them the way the grains thin out with height, like the air on a mountain. The 1926 Nobel Prize was awarded for his work on the discontinuous structure of matter, especially for this “sedimentation equilibrium”.

The decisive argument was that numbers reached by unrelated routes came out close: all between 6.0 and 7.5 × 10²³. In his 1913 book Les Atomes, Perrin put 13 determinations into one table: from the viscosity of gases to the blue of the sky and radioactivity.

In November 1908 the chemist Wilhelm Ostwald wrote, in the preface to a textbook, that J. J. Thomson's counting of gas ions, together with the agreement of the Brownian movements with the kinetic hypothesis, “established by many investigators and most conclusively by J. Perrin”, justified “the most cautious scientist in now speaking of the experimental proof of the atomic nature of matter” (in W. W. Taylor's authorised English translation).

How many molecules are in a glass of water?

A mole of water weighs 18 grams. A 250 millilitre glass therefore holds almost 14 moles, or about 8.4 × 10²⁴ molecules: eight septillion and more.

Catch at least one bead in the experiment above, and the glass you measured appears here.

Next time you see dust drifting in a sunbeam, it is the air that is moving. The water molecules in the glass on your table are striking everything in it right now, just as they strike the beads in the footage. All it takes to see their blows is something small enough and a microscope.