Graham's law states that the rate of diffusion 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. 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.
At the time Graham did his work the concept of molecular weight was being established, in large part through measurements of gases. Italian physicist Amadeo Avogadro had suggested in 1811 that equal volumes of different gases contain equal numbers of molecules. Thus, the relative molecular weights of two gases are equal to the ratio of weights of equal volumes of the gases. Avogadro's insight together with other studies of gas behavior provided a basis for theoretical work by Scottish physicist James Clerk Maxwell who was trying to explain the properties of gases as collections of small particles moving through largely empty space.
Perhaps the greatest success of the kinetic theory of gases, as it came to be called, was the discovery that for gases, the temperature as measured on the Kelvin (absolute) temperature scale is directly proportional to the average kinetic energy of the gas molecules, where the kinetic energy of any object is equal to one-half its mass times the square of its velocity. Thus, to have equal kinetic energies, 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.
Therefore, hydrogen molecules effuse four times as fast as 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 either one of the molecular weights provided the subscripts are consistent.
Graham's law was the basis for separating 235U from 238U found in natural uraninite (uranium ore). The United States government built a gaseous diffusion plant at the then phenomenal cost of $100 million in Clinton, Tennessee. 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.