Graham's law of effusion (also 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 molar mass of its particles. This formula is stated as
rate 1 rate 2 = M 2 M 1 , {\displaystyle {\frac {{\text{rate}}_{1}}{{\text{rate}}_{2}}}={\sqrt {\frac {M_{2}}{M_{1}}}},}
where
rate1 is the rate of effusion for the first gas (volume or number of moles per unit time), rate2 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 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 rates of diffusion of different gases are inversely proportional to the square roots of their mass densities:
rate ∝ 1 ρ , {\displaystyle {\text{rate}}\propto {\frac {1}{\sqrt {\rho }}},}
where ρ is the mass density.
Examples First example: Let gas 1 be H2, and gas 2 be O2. This example is solving for the ratio between the rates of the two gases:
rate ( H 2 ) rate ( O 2 ) = M ( O 2 ) M ( H 2 ) = 32 2 = 16 = 4. {\displaystyle {\frac {{\text{rate}}({\ce {H2}})}{{\text{rate}}({\ce {O2}})}}={\sqrt {\frac {M({\ce {O2}})}{M({\ce {H2}})}}}={\frac {\sqrt {32}}{\sqrt {2}}}={\sqrt {16}}=4.}
Therefore, hydrogen molecules effuse four times faster than oxygen molecules. Graham's law can also be used to approximately determine the molecular weight of a gas if one gas is a known species and there is a specific ratio between the rates of two gases (as in the previous example). The equation can be solved for the unknown molecular weight:
M 2 = ( rate 1 rate 2 ) 2 M 1 . {\displaystyle M_{2}=\left({\frac {{\text{rate}}_{1}}{{\text{rate}}_{2}}}\right)^{2}M_{1}.}
Graham's law was the basis for separating uranium-235 from uranium-238 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 at the Clinton Engineer Works in Oak Ridge, Tennessee, at the cost of $479 million (equivalent to $6.61 billion in 2024). In this plant, yellowcake, uranium concentrate recovered from the leaching of uranium ore, was first converted to volatile uranium hexafluoride. UF6 was then heated in the vapor state and repeatedly forced to diffuse through porous barriers, each time becoming a little more enriched in the slightly lighter uranium-235 isotope. Second example: An unknown gas diffuses 0.25 times as fast as He. What is the molar mass of the unknown gas? Using the formula of gaseous diffusion, we can set up the equation
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