Perrin needed to prove that NA is a constant in the following equation that he derived:
The challenge was that N’, N and m could not be measured directly. He circumvented this issue by replacing the gas in the cylinder with visible particles suspended in a liquid. Drawing from the principles of Brownian motion, Perrin assumed that collisions between liquid molecules and the visible particles would result in a distribution similar to that of gas molecules in the cylinder. He also assumed that the motion of the particles obeyed the ideal gas law.
Perrin used a monodisperse colloid of a gum called gamboge, consisting of thousands of gamboge spheres in a water cylinder. He studied the distribution of the spheres with a microscope and adjusted eq9 to account for the upthrust of water on the gamboge spheres. For a single gamboge sphere (“particle”),
upthrust = weight of liquid displaced = ml g = dVl g (10)
where ml is the mass of the liquid displaced, d is the density of the liquid and Vl is the volume of the liquid displaced.
Furthermore, the volume of liquid displaced is equal to the volume of the particle, Vp :
Vl = Vp (11)
Substituting eq11 in eq10 yields
upthrust = dVp g (12)
Since Vp = m/D, where m is the mass of a particle and D is the density of the particle, eq 12 becomes:
The effective weight of the particle is therefore the difference between the weight of the particle and the upthrust on the particle:
Replacing the weight mg of a gas molecule in eq9 with the effective weight of a suspended particle gives
The suspended particles must be heavier than the liquid molecules for the particles to exert a downward pressure on the liquid, producing an upthrust that results in a lower effective weight for the particles. This means that d < D for eq14 to be valid. Hence, the choice of particle material is important.
Perrin meticulously prepared emulsions containing particles that were equal in size. He calculated the average mass of a particle by weighing a specified number of particles, determined its density using various methods (including the specific gravity bottle method), and counted the number of suspended particles per unit volume at different heights using a microscope.
After repeating the experiment with different particle materials (e.g. mastic), sizes, masses, liquids and temperatures, he found that the value of NA remained fairly constant, reporting numbers ranging from 6.5 x 1023 to 7.2 x 1023. He further conducted experiments using methods based on radioactivity, blackbody radiation and the motion of ions in liquids, obtaining very similar results for the value of NA. Perrin concluded that the results justified the hypotheses that had guided him, including Avogadro’s law, and named the constant the Avogadro constant, in honour of Avogadro.
The accuracy of the value of the Avogadro constant was subsequently improved by other scientists, one of whom was Robert Millikan.