Graham's Formula of Diffusion and Effusion

Chemist Thomas Graham
Thomas Graham. Wikipedia/Public Domain

Graham's law expresses the relationship between the rate of effusion or diffusion of a gas and that gas's molar mass. Diffusion describes the spreading of a gas throughout a volume or second gas and effusion describes the movement of a gas through a tiny hole into an open chamber.

In 1829, Scottish chemist Thomas Graham determined through experimentation that a gas's rate of effusion is inversely proportional to the square root of the gas particle's density. In 1848, he showed that the rate of effusion of a gas is also inversely proportional to the square root of its molar mass. Graham's law also shows that the kinetic energies of gases are equal at the same temperature.

Graham's Law Formula

Graham's law states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. See this law in equation form below.

r ∝ 1/(M)½


r(M)½ = constant

In these equations, r = rate of diffusion or effusion and M = molar mass.

Generally, this law is used to compare the difference in diffusion and effusion rates between gases, often denoted as Gas A and Gas B. It assumes that temperature and pressure are constant and equivalent between the two gases. When Graham's law is used for such a comparison, the formula is written as follows:

rGas A/rGas B = (MGas B)½/(MGas A)½

Example Problems

One application of Graham's law is to determine how quickly a gas will effuse in relation to another and quantify the difference in rate. For example, if you want to compare the effusion rates of hydrogen (H2) and oxygen gas (O2), you can use their molar masses (hydrogen = 2 and oxygen = 32) and relate them inversely.

Equation for comparing effusion rates: rate H2/rate O2 = 321/2 / 21/2 = 161/2 / 11/2 = 4/1

This equation shows that hydrogen molecules effuse four times faster than oxygen molecules.

Another type of Graham's law problem may ask you to find the molecular weight of a gas if you know its identity and the effusion ratio between two different gases.

Equation for finding molecular weight: M2 = M1Rate12 / Rate22

Uranium Enrichment

Another practical application of Graham's law is uranium enrichment. Natural uranium consists of a mixture of isotopes with slightly different masses. In gaseous effusion, uranium ore is first made into uranium hexafluoride gas, then repeatedly effused through a porous substance. Through each effusion, the material passing through the pores becomes more concentrated in U-235 (the isotope used to generate nuclear energy) because this isotope diffuses at a faster rate than the heavier U-238.