💨 Graham's Law of Effusion

Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass: rate ∝ 1/√M. For two gases at the same temperature and pressure, rate1/rate2 = √(M2/M1).

From kinetic theory, the root-mean-square speed of gas molecules is v_rms = √(3RT/M). Since effusion rate through a tiny orifice is proportional to how often molecules strike and pass through the opening, which scales with average molecular speed, the effusion rate inherits the same 1/√M dependence.

Effusion is the escape of gas molecules one at a time through a hole much smaller than the mean free path, so molecules pass through without colliding with each other near the opening. Diffusion is the bulk mixing of gases through a medium, driven by concentration gradients and involving many intermolecular collisions. Both follow similar mass-dependent scaling but are physically distinct processes.

If the orifice is comparable to or larger than the mean free path, gas flows out as a collective hydrodynamic stream (bulk flow) rather than molecule-by-molecule. True effusion requires a small enough opening that molecules escape independently, each governed only by its own velocity.

Enlarging the orifice increases the chance that any molecule striking that region escapes, speeding up escapes for both gases proportionally. Since the extra flux depends only on geometry, not molecular mass, the ratio of escape rates stays fixed at √(M_B/M_A); only the absolute number of escapes per second changes.

Helium is a light molecule (M ≈ 4 g/mol), so it has a high average speed at any given temperature and passes through the microscopic pores in balloon rubber more quickly than heavier gases like nitrogen or oxygen, causing helium balloons to deflate noticeably faster.

Gaseous diffusion/effusion plants historically separated uranium-235 from uranium-238 by converting uranium to uranium hexafluoride (UF6) gas and passing it through porous barriers. Because 235UF6 is very slightly lighter than 238UF6, it effuses marginally faster, and repeating the process through thousands of stages gradually enriched the 235U fraction.

The same v_rms = √(3RT/M) formula that sets each gas's typical speed here also determines the width and peak of its Maxwell-Boltzmann speed distribution. Lighter gases have broader, faster distributions, which is exactly why their molecules strike the orifice more often and escape at a higher rate.

Gas chromatography separates volatile compounds partly based on how they diffuse through a stationary phase; industrial isotope separation historically used gaseous diffusion cascades; and the escape of light gases like hydrogen and helium from the top of planetary atmospheres (Jeans escape) follows the same mass-dependent speed argument.

As gas empties out of the box, fewer particles remain to strike the orifice, so the absolute number of escapes per second declines for both gases, even though the ratio of their rates stays governed by √(M_B/M_A).