Perrin developed the concept to analyse the vertical distribution of a volume of molecules of an ideal gas, with total mass of mc, in a cylinder with a cross-sectional area A and small height h at temperature T. Firstly, he considered an infinitesimal layer of the gas with mass dmc and thickness dh in the cylinder.
This layer of gas experiences an upward force Fu = pA from the pressure exerted by the gas it, and a downward force Fd = gdmc + (p+dp)A due to the gravitational force and the pressure exerted by the gas above it. At equilibrium, Fu = Fd and we get:
At this juncture, Perrin, with reference to the works of previous scientists (Avogadro, Dalton, etc.) on relative mass of molecules and introduced an important definition:
A gramme-molecule, M, is the mass of a gas that occupies the same volume as 2g of hydrogen gas at the same temperature and pressure*
* Perrin’s definition of the gramme-molecule is somewhat similar to the 1967 definition of the mole. This definition, together with Avogadro’s law, implies that a gramme-molecule of gas X and a gramme-molecule of gas Y have the same number of molecules.
With Perrin’s definition, the physical state of the gas in the cylinder can be described using the ideal gas law (eq2), where the amount of gas n is expressed in multiples of gramme-molecule M:
Substituting eq4 in eq3 by eliminating dmc/A yields
The expression for the distribution of the gas in the entire cylinder can be obtained by integrating both sides of eq5 (see above diagram):
From eq2, we have p’ = n’RT/V’ and p = nRT/V. Substituting these two equations in eq6 gives
where N’ = n’/V’ and N = n/V; that is, the number densities of the gas at the upper and lower levels of the cylinder.
Furthermore, the gramme-molecule of the gas, M, is equal to the number of molecules in a gramme-molecule, NA, multiplied by the mass of a molecule of the gas, m:
Substituting eq8 in eq7 results in
If Avogadro’s hypothesis – that ‘equal volumes of all gases at the same temperature and pressure have the same number of molecules’ – is true, NA in eq9 must be constant for different ideal gases. To determine the value of NA, Perrin would need to (i) count the number of molecules per unit volume at the upper and lower levels of the cylinder (of fixed height, h, at a constant temperature, T) for a particular gas, (ii) repeat the count for other gases with different m, and (iii) calculate the average value of NA. However, a problem arose.