Graham's law

Graham's law of effusion (sometimes called Graham's law of diffusion) was formulated by Scottish physical chemist Thomas Graham in 1848. Graham found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. This formula can be written as:

where:

Graham's law states that the rate of diffusion or of effusion of a gas is inversely proportional to the square root of its molecular weight. Thus, if the molecular weight of one gas is four times that of another, it would diffuse through a porous plug or escape through a small pinhole in a vessel at half the rate of the other (heavier gases diffuse more slowly). A complete theoretical explanation of Graham's law was provided years later by the kinetic theory of gases. Graham's law provides a basis for separating isotopes by diffusion—a method that came to play a crucial role in the development of the atomic bomb.

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.

In the same conditions of temperature and pressure, the molar mass is proportional to the mass density. Therefore the rate of diffusion of different gases is inversely proportional to the square root of their mass densities.

Let gas 1 be H₂ and gas 2 be O₂.

Therefore, hydrogen molecules effuse four times faster than those of oxygen.

Graham's Law can also be used to find the approximate molecular weight of a gas if one gas is a known species, and if there is a specific ratio between the rates of two gases (such as in the previous example). The equation can be solved for the unknown molecular weight.

Graham's law was the basis for separating ²³⁵U from ²³⁸U found in natural uraninite (uranium ore) during the Manhattan Project to build the first atomic bomb. The United States government built a gaseous diffusion plant in Clinton, Tennessee at the cost of $100 million (~$7.7 billion in 2014 dollars). In this plant, uranium from uranium ore was first converted to uranium hexafluoride and then forced repeatedly to diffuse through porous barriers, each time becoming a little more enriched in the slightly lighter ²³⁵U isotope.

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