Perrin’s experiment: the concept

Perrin developed the concept to analyse the vertical distribution of a volume of molecules of an ideal gas, with total mass of m_c, in a cylinder of cross-sectional area, A, and small height, h, at temperature, T. Firstly, he considered an infinitesimal layer of the gas of mass, dm_c, and thickness, dh, in the cylinder.

This layer of gas experiences an upward force, F_u= pA, from the pressure exerted by the gas below the layer, and a downward force, F_d = gdm_c + (p+dp)A, as a result of the gravitational force and the pressure exerted by the gas above the layer. At equilibrium, F_u= F_dand we get:

$dp=-\frac{g}{A}dm_{c}\; \; \; \; \; \; \; \; (3)$

At this juncture, Perrin, with reference to the works of previous scientists (Avogadro, Dalton, etc.) on relative mass of molecules, introduced an important definition:

A gram-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 gram-molecule is somewhat similar to the 1967 definition of the mole. This definition, together with Avogadro’s law, implies that a gram-molecule of gas X and a gram-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 gram-molecule M:

$p(Adh)=\frac{dm_{c}}{M}RT\; \; \; \; \; \; \; (4)$

Substituting eq4 in eq3 by eliminating dm_c/A :

$\frac{1}{p}dp=-\frac{Mg}{RT}dh\; \; \; \; \; \; \; (5)$

The expression for the distribution of the gas in the entire cylinder can be obtained by integrating both sides of eq5 (see above diagram):

$\int_{p}^{p'}\frac{1}{p}\: dp=-\frac{Mg}{RT}\int_{0}^{h}dh$

$\frac{p'}{p}=e^{-\frac{Mgh}{RT}}\; \; \; \; \; \; \; (6)$

Using eq2 again where p’ = n’RT/V’ and p = nRT/V and substituting them in eq6, we have:

$\frac{N'}{N}=e^{-\frac{Mgh}{RT}}\; \; \; \; \; \; \; (7)$

where N’ = n’/V’ and N = n/V, i.e. the number densities of the gas at the upper and lower levels of the cylinder.

Furthermore, the gram-molecule of the gas, M, is equal to the number of molecules in a gram-molecule, N_A, multiplied by the mass of a molecule of the gas, m:

$M=N_{A}m\; \; \; \; \; \; \; (8)$

Substituting eq8 in eq7, we have

$\frac{N'}{N}=e^{-\frac{N_{A}mgh}{RT}}\; \; \; \; \; \; \; (9)$

If Avogadro’s hypothesis that ‘equal volumes of all gases at the same temperature and pressure have the same number of molecules’ is true, N_Ain eq9 must be constant for different ideal gases. To determine the value of N_A, Perrin in principle needed to perhaps 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 and repeat the count for other gases with different m, and calculate the average value of N_A. However, a problem arose.

Perrin’s experiment: the concept

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