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Avogadro’s number from Brownian motion

Follow suspended grains under the microscope as Perrin did, get the diffusion coefficient and the Avogadro constant from the mean-square displacement, and see how the number of grains decides how far you can trust the result.

Difficulty ●●○ ≈ 25 min

Aim Determine N from ⟨x²⟩ = 2Dt and D = RT/(6πηaN), and compare statistical scatter with the systematic errors in Perrin’s experiments.
Random walks from deterministic random numbers · compared with Perrin’s measurements of 1908–1911

Counting molecules under a microscope

Microscope field (water at 20 °C)
10 μm
Mean-square displacement against time
050100150200250300time / s0100200300400⟨x²⟩ / μm²
  • Einstein: ⟨x²⟩ = 2Dt
  • simulated grains
Perrin’s seven series
today’s exact value IIIIIIIVVVIVIIseries5678N / 10²³ mol⁻¹

D = kT/(6πηa) = 0.584 μm²/s; from 100 grains the measured value is 0.528 μm²/s, so N = RT/(6πηaD) = 6.67 × 10²³ mol⁻¹ (10.7 % from the exact value).

Each step is drawn from a Gaussian with variance 2DΔt, with D from the Stokes–Einstein equation and the viscosity of water at 20 °C (1.0016 mPa s, IAPWS 2008); the random numbers come from a deterministic hash, so “new batch” gives a different but repeatable set. The more grains you follow, the closer N comes to the exact value — the error shrinks roughly as 1/√n. Perrin’s data are from Atoms (English translation, 1916), p. 123: his seven series varied the grain size (radius 0.212–5.5 μm), the material and the viscosity of the liquid (1 to 125 times that of water); in series III the temperature was only roughly known.

Procedure

  1. With a grain radius of 0.367 μm, a position every 30 s and 100 grains, record the measured D and N; then press “new batch of grains” twice more, recording each time.
  2. Raise the number of grains to 400 and then 1600, recording N each time.
  3. Keeping 400 grains, change the radius to 0.212 μm and then 0.8 μm, and watch what D and N do.
  4. Finally compare with Perrin’s seven series in the chart.

What to notice

  • With 100 grains the three batches give N = 6.67, 5.48 and 5.94 × 10²³ — about ±10 % apart. The theoretical diffusion coefficient is 0.584 μm²/s.
  • 400 grains give 5.90 × 10²³ and 1600 give 5.85 × 10²³, both within 3 % of the exact 6.022 × 10²³: the statistical error shrinks roughly as 1/√n.
  • Grains of radius 0.212, 0.367 and 0.8 μm diffuse with D = 1.011, 0.584 and 0.268 μm²/s, inversely proportional to the radius, while the N obtained stays the same.
  • Perrin’s seven series gave 5.5–8.0 × 10²³, a wider spread than the simulation: errors in the grain radius, temperature and viscosity are systematic, and counting more grains cannot remove them.

In a real lab

  • Uniform grains are essential: Perrin centrifuged a kilogram of gamboge for several months to obtain a few decigrams of grains of the size he wanted; today monodisperse polystyrene beads are used.
  • Seal the cell and keep it at constant temperature to avoid evaporation and convection — drift carries all the grains together and swamps the Brownian displacements.
  • Record positions frame by frame with a camera and track the grains with software; use only grains far from the walls, where the drag is larger.

Related lessons

All experiments