Perrin needed to prove that *N _{A }*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 = **m _{l }g*

*= d*

*V*_{l }g (10)where *m _{l}* is the mass of the liquid displaced,

*d*is the density of the liquid and

*is the volume of the liquid displaced.*

*V*_{l}Furthermore, the volume of liquid displaced is equal to the volume of the particle, * V_{p }*:

*V _{l }= V_{p} (11)*

Substituting eq11 in eq10 yields

*upthrust =* *d**V _{p }*

*g*

*(12)*

Since *V _{p} = 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 *N _{A}* remained fairly constant, reporting numbers ranging from 6.5 x 10

^{23 }to 7.2 x 10

^{23}. 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

*N*. Perrin concluded that the results justified the hypotheses that had guided him, including Avogadro’s law, and named the constant the

_{A}**, in honour of Avogadro.**

*Avogadro constant*The accuracy of the value of the Avogadro constant was subsequently improved by other scientists, one of whom was Robert Millikan.