Graham's law expresses the relationship between the rate of effusion or diffusion of a gas and the gas's molar mass. Diffusion describes the spreading of a gas throughout a volume or a second gas, while 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 the rate of effusion of a gas is inversely proportional to the square root of the gas particle mass and to its density. In 1848, he showed that the rate of effusion of a gas is also inversely proportional to the square root of the molar mass of the gas. 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 the molar mass of the gas:

r ∝ 1/(M)^{½}

or

r(M)^{½} = constant

****where *r* = rate of diffusion or effusion and *M* = molar mass.

Generally, this law is used to compare the difference in rates between between two different gases: Gas A and Gas B. The law assumes that the temperature and pressure are the same for the two gases. When Graham's law is used for such a comparison, the formula is written:

r_{Gas A}/r_{Gas B} = (M_{Gas B})^{½}/(M_{Gas A})^{½}

## Example Problems

One way to apply Graham's law is to determine whether one gas will effuse more quickly or slowly than another and to quantify the difference in rate. For example, if you want to compare the rates of effusion of hydrogen gas (H_{2}) and oxygen gas (O_{2}), you use the molar masses of the gases (two for hydrogen and 32 for oxygen) and relate them inversely:

rate H_{2}/rate O_{2} = 32^{1/2} / 2^{1/2} = 16^{1/2} / 1^{1/2} = 4/1

The equation shows that hydrogen gas molecules effuse four times more quickly than oxygen molecules.

Another type of Graham's law problem may ask you to find the molecular weight of a gas if you know the identity of one gas and the ratio between the rates of effusion of two different gases. This problem can be expressed as:

M_{2} = M_{1}Rate_{1}^{2} / Rate_{2}^{2}

A practical application of Graham's law is uranium enrichment. Natural uranium consists of a mixture of isotopes, each of which has a slightly different mass. In gaseous diffusion, uranium ore is made into uranium hexafluoride gas, which is repeatedly diffused through a porous substance. Each time, the material that passes through the pores becomes more concentrated in U-235 (the isotope used to generate nuclear energy). This is because the isotope diffuses at a faster rate than U-238, which is heavier.