Diffusion: Graham's law

Principle

Graham's law is also known as Graham's law of effusion. Effusion is the process where individual molecules flow through a hole without collisions. This will occur if the diameter of the hole is considerably smaller than the mean (collision-)free path of the molecules. The free path is at 1 bar of the oder of 70 nm. The law states that the rate of effusion of a gas is inversely proportional to the square root of the molecular mass of its particles. This formula can be written as:

v₁/v₂ = (m₁/m₂)^0.5, (1)

where:

v₁ is the rate of effusion of the first gas;

v₂ is the rate of effusion for the second gas;

m₁ is the molar mass of gas 1;

m₂ is the molar mass of gas 2.

Graham's law is most accurate for molecular effusion which involves the movement of one gas at a time through a hole. It is only approximate for diffusion of one gas in another or in air, as these processes involve the movement of more than one gas.

According to the kinetic theory of gases, the absolute temperature T (Kelvin) is directly proportional to the average kinetic energy of the gas molecules (0.5mv²). This can be derived from:

<v²> = 3RT/(N_Am),

where <v²> the mean of the squared velocity of the particles, R the molar gas constant (= 8315 J/kmol·K), N_Am the gas mass with N_A the Avogadro’s number (see Gas laws).

Thus, to have equal kinetic energies and so temperature, the velocities of two different molecules would have to be in inverse proportion to the square roots of their masses. Since the rate of diffusion is determined by the average molecular velocity, Graham's law for diffusion could be understood in terms of the molecular kinetic energies being equal at the same temperature.

Application

The law is of important for processes of diffusion of a gas into another gas or gas mixture (see Diffusion: general). For diffusion across a liquid membrane, gas concentration should be low and the membrane thin.

Graham's Law can also be used to find the approximate molecular weight of a gas if the rates are measured and the molecular weight of one of the gases is a known.

Graham's law was the basis for separating ²³⁵UF₆ from ²³⁸UF₆. Both isotopes of uranium, as element, are found in natural uranium ore, but the 235-isotope about 100 times less. By repeated diffusion through porous barriers the slightly lighter ²³⁵U isotope is enriched.