Lesson 4 · 30 min

Brownian motion: counting molecules under a microscope

Why do particles from pollen never stop jiggling? Einstein tied their random walk to molecular motion, and Perrin, tracking grains one by one and counting them layer by layer under the microscope, obtained the Avogadro constant. Retrace his route with his own data, and meet an “atmosphere” inside a drop of water.

In this lesson

Explain where Brownian motion comes from and describe a random walk with ⟨x²⟩ = 2Dt: displacement grows as the square root of time
Estimate a particle’s diffusion coefficient with the Stokes–Einstein equation D = kT/(6πηa), and obtain the Avogadro constant from measured displacements
Treat sedimentation equilibrium as an “atmosphere” in a drop and compare the effective weight of a grain with that of an air molecule using the Boltzmann distribution

In the summer of 1827 the botanist Robert Brown watched particles released from pollen grains under his microscope and saw them jiggling ceaselessly in water. At first he suspected life, but powdered glass and rock jiggled just the same, and for more than seventy years nobody could say why. In 1905 Einstein pointed out that if a liquid is made of molecules in constant motion, a small suspended particle is struck unevenly from all sides and must wander in a random walk — one whose reach can be predicted exactly.

From 1908 the French physicist Jean Perrin followed uniform spheres of gamboge and mastic under the microscope, counting and measuring grain by grain; he confirmed Einstein’s predictions and obtained the Avogadro constant from them. Molecules stopped being a hypothesis, and Perrin received the 1926 Nobel Prize in Physics.

Predict first

In water at 20 °C a grain of radius 0.212 μm moves, root-mean-square, about 7.8 μm along x in 30 s. After 120 s, about how far has it moved?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

About 7.8 μm — no further About 15.6 μm About 31 μm About 125 μm

The random walk and Einstein’s relation

A grain is struck by water molecules an enormous number of times every second. Each blow does next to nothing, but the blows from different sides never cancel exactly, and the fluctuations add up to push the grain about. Einstein linked two things: moving through a liquid, the grain feels viscous drag, with the drag coefficient 6πηa from Stokes’ law; and thermal motion gives every degree of freedom an average energy of ½kT. Together they give the diffusion coefficient D = kT/(6πηa) — larger grains and more viscous liquids mean slower wandering, higher temperatures faster.

Hidden in the formula is Boltzmann’s constant, k = R/N_A. Measure a grain’s diffusion coefficient, know its size and the viscosity of the liquid, and you have the Avogadro constant — which is exactly what Perrin did.

\langle x^2 \rangle = 2Dt,\qquad D = \frac{kT}{6\pi\eta a} = \frac{RT}{6\pi\eta a\,N_\mathrm{A}}

The mean-square displacement along one direction grows linearly with time; in two dimensions ⟨r²⟩ = 4Dt.

$D$: diffusion coefficient
$\eta$: viscosity of the liquid
$a$: grain radius
$k$: Boltzmann constant

Random walks from deterministic random numbers · compared with Perrin’s measurements of 1908–1911

Counting molecules under a microscope

Grain radius a = 0.367 μm Record a position every 30 s Follow 400 grains

Microscope field (water at 20 °C)

Mean-square displacement against time

Einstein: ⟨x²⟩ = 2Dt
simulated grains

Perrin’s seven series

D = kT/(6πηa) = 0.584 μm²/s; from 400 grains the measured value is 0.597 μm²/s, so N = RT/(6πηaD) = 5.90 × 10²³ mol⁻¹ (−2.1 % from the exact value).

On the left, the tracks of three grains with a point every 30 s — just as Perrin drew them. Top right, the mean-square displacement of many grains: the points fall on a straight line whose slope is 2D, and the N it gives scatters around the exact value; raise the number of grains from 100 to 1600 and the scatter shrinks. Perrin’s seven series gave 5.5–8.0 × 10²³; the three with the most recorded displacements — VII, IV and II, with 1500, 1000 and 900 — gave 6.88, 7.25 and 6.95 × 10²³.

Worked example

How far a 1 μm sphere diffuses

A sphere 1.0 μm across is suspended in water at 25 °C (η = 0.890 mPa s). Find its diffusion coefficient and its root-mean-square displacement along x after 1 s, 1 min and 1 h.

An atmosphere in a drop

Perrin’s other experiment is even more direct. Leave an emulsion to stand: gravity pulls the grains down, thermal motion kicks them back up, and they settle into sedimentation equilibrium — densest at the bottom and thinner above, just like the atmosphere. Both follow the same Boltzmann distribution: each equal step in height reduces the concentration by the same factor, set by the ratio of the effective weight to kT.

Air has to rise about 6 km for its concentration to halve; Perrin’s mastic grains needed only about 6 μm. That factor of about 10⁹ means a grain weighs, in water, about 10⁹ times as much as an air molecule. The grain’s weight can be measured, so the mass of an air molecule — and with it the Avogadro constant — follows. Varying the grain size, the material, the viscosity of the liquid and the temperature, Perrin always found N between 6.5 × 10²³ and 7.2 × 10²³.

\frac{n(h)}{n_0} = \exp\!\left(-\frac{m^\prime g h}{kT}\right),\qquad m^\prime = \tfrac{4}{3}\pi a^3(\rho - \rho_0)

The effective mass m′ allows for buoyancy; for a gas molecule m′ is simply its mass.

$n(h)$: concentration of grains at height h
$m^\prime$: effective (buoyancy-corrected) mass
$\rho - \rho_0$: density difference between grain and liquid

Grains counted by Perrin (Atoms, 1916) · the line is a least-squares fit of the logarithm

An atmosphere in a drop

Grains at sedimentation equilibrium (sketch)

Grains counted at each height (log scale)

The count halves every 6.9 μm (r² = 0.9970). Air at 15 °C halves only every 5.85 km — 8.5 × 10⁸ times farther — so a grain weighs, in water, about 8.5 × 10⁸ times as much as an air molecule.

On a log scale Perrin’s counts fall on a straight line. The mastic grains halve every 6.9 μm or so, the smaller gamboge grains about every 29 μm — the lighter the grains, the taller their “atmosphere”.

Concept check

Double the grain radius and keep everything else the same. What happens to the diffusion coefficient D and to the halving height at sedimentation equilibrium?

D halves; the halving height drops to one-eighth D halves; the halving height halves too D drops to one-eighth; the halving height is unchanged Neither changes

Try the calculation

Treat the atmosphere as isothermal: at 15 °C, how many kilometres must air (M = 28.97 g/mol) rise for its concentration to halve? h½ = RT ln 2/(Mg).

Key takeaways

Brownian motion is a random walk driven by fluctuations in molecular blows: ⟨x²⟩ = 2Dt, so displacement grows only as √t.
The Stokes–Einstein equation D = kT/(6πηa) turns a measured D, the grain radius and the viscosity into k, and hence N_A = R/k.
Sedimentation equilibrium follows the same Boltzmann distribution as the atmosphere: air halves in about 6 km, mastic grains in about 6.9 μm; from this and from Brownian displacements Perrin found N ≈ 6.5–7.2 × 10²³ and established the reality of molecules.

Progress is stored only in this browser and never uploaded.

Data and references

R. Brown, “A brief account of microscopical observations … on the particles contained in the pollen of plants”, Philos. Mag. 4, 161–173 (1828).
A. Einstein, “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen”, Ann. Phys. 17 (322), 549–560 (1905).
J. Perrin, Atoms, authorised translation by D. Ll. Hammick, Constable, London (1916), chapter IV (pp. 103–105, 123).
M. L. Huber et al., “New international formulation for the viscosity of H₂O”, J. Phys. Chem. Ref. Data 38, 101–125 (2009).