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In Class 11 you write 6.022 × 10²³ without a second thought. Around 1900 many scientists still argued over whether atoms and molecules really existed or were just a handy way of doing sums. Jean Perrin settled the question by counting tiny grains under a microscope. This post explains how his experiment worked, takes you through a worked example of the calculation, and ends with what you can borrow from his way of working.
For how the idea of Avogadro's number began, see our post on Avogadro's hypothesis. For the jiggling motion Perrin also measured, see Robert Brown and Brownian motion. Here we stay with Perrin's own experiment.
Who was Jean Perrin?
Perrin was born in Lille on 30 September 1870, according to his Nobel Prize facts page. His Nobel biography says his first research showed that cathode rays are negatively charged particles. He joined the Sorbonne in 1897 and was a professor there from 1910 to 1940. When Germany invaded France in 1940 he escaped to the USA, and he died in New York on 17 April 1942. He was also the creator of France's national research organisation, the CNRS. His popular book Les Atomes (1913) had sold 30,000 copies by 1936.
He won the 1926 Nobel Prize in Physics "for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium".
Why were atoms still in doubt?
The facts page puts it plainly: in the 1880s atoms and molecules were important ideas, but whether they physically existed "was still a matter of dispute". In his Nobel lecture Perrin noted that the kinetic theory of gases gave values of Avogadro's number anywhere from 40 × 10²² to 120 × 10²², an uncertainty he called "largely 100 per cent". The order of magnitude was known; a convincing measurement was not.
What is sedimentation equilibrium?
Air thins as you climb. Perrin's lecture works out that in oxygen at 0 °C the density halves for every 5 kilometres of height, and in hydrogen, whose molecules are 16 times lighter, you must rise 80 kilometres.
His idea was bold. If grains suspended in water behave like very heavy molecules, they should thin out with height by the same law, only over microns instead of kilometres. And the halving height should tell you the grain's weight in "molecule units".
The hard part was making grains that were all the same size. Perrin dissolved resin in alcohol, poured in water so the resin came out as tiny spheres, then sorted them by repeated centrifuging. He wrote that he treated one kilogram of gamboge (a yellow resin) and, after several months of daily work, obtained a few decigrams of grains about three-quarters of a thousandth of a millimetre across.
He then sealed a drop under a microscope, focused on a layer only about 2 microns thick, and photographed it at different heights to count the grains at each level.
What did he find?
In his most careful measurements the number of grains halved for every 6 microns of rise, and the result was N = 68 × 10²². He then changed almost everything he could: grain mass (by a factor of 50), grain material (gamboge and mastic), the liquid (water, sugared water, glycerol) and the temperature (from −9 °C to +60 °C). The answers stayed close to one another.
Other methods agreed too. In the lecture he lists 55 × 10²² to 72 × 10²² from Brownian motion, 61 × 10²² from measurements of the electron's charge, and 62 × 10²² to 70 × 10²² from counting helium atoms produced by radioactivity. Today the CODATA value from NIST is exactly 6.022 140 76 × 10²³ per mole, so Perrin's best emulsion result was about 13% high.
Worked example: Avogadro's number from a halving height
First, Perrin's own ratio. Oxygen halves every 5 km and his emulsion every 6 µm, so each grain's effective weight was about 5,000 m ÷ 0.000006 m ≈ 8.3 × 10⁸ times that of an oxygen molecule.
Now a full calculation with sample numbers chosen to resemble his grains (these are not his recorded data):
Grain diameter 0.75 µm, so radius r = 0.375 × 10⁻⁶ m
Grain density 1,200 kg/m³ in water of 1,000 kg/m³
Temperature 293 K; measured halving height h = 6.5 µm
For a gas the halving height is h = RT ln 2 ÷ (Mg), where M is the molar mass. For grains, replace Mg with N × w, where w is one grain's weight minus its buoyancy.
Volume: V = (4/3)πr³ = 2.21 × 10⁻¹⁹ m³
Effective mass: V × (1,200 − 1,000) = 4.42 × 10⁻¹⁷ kg, so w = 4.42 × 10⁻¹⁷ × 9.81 = 4.33 × 10⁻¹⁶ N
N = RT ln 2 ÷ (w h) = (8.314 × 293 × 0.693) ÷ (4.33 × 10⁻¹⁶ × 6.5 × 10⁻⁶) = 1,688 ÷ 2.82 × 10⁻²¹ ≈ 6.0 × 10²³
Common slips to avoid:
Forgetting buoyancy. Using 1,200 kg/m³ instead of 200 makes w six times too big and N six times too small.
Using the diameter as the radius. That makes the volume eight times too large.
Mixing log bases. Using log₁₀ 2 (0.301) in place of ln 2 (0.693) makes N about 2.3 times too small.
What can a student take from Perrin's way of working?
Make the invisible countable. Perrin could not see molecules, so he built something that obeyed the same law at a size he could see.
Change everything you can. Agreement across different grains, liquids and temperatures was his real evidence, and agreement with other methods sealed it.
Do the patient part. Months of centrifuging came before a single useful count.
If it is hard to picture how random molecular collisions add up to a steady pattern, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read. For the gas laws behind the halving height, see our guide to kinetic theory of gases.
Next step: redo the worked example with a grain diameter of 1.0 µm and work out the halving height you would expect at 293 K.
