Graham's Law of Effusion: Why Light Gases Escape Faster

Effusion is not diffusion

Effusion is the escape of gas molecules, one at a time, through a hole so small that a molecule is far more likely to sail straight through it than to collide with another molecule while passing. That condition — the orifice being much smaller than the gas's mean free path — puts effusion in a different regime from ordinary diffusion, which is bulk mixing driven by huge numbers of intermolecular collisions and described instead by Fick's laws. Effusion cares only about how fast individual molecules arrive at the hole; diffusion cares about how a whole crowd jostles its way through a medium.

Where the square root comes from

Kinetic theory says every gas at temperature T carries the same average translational kinetic energy per molecule, regardless of what the molecule is. Set that equal to (1/2)mv² and solve for speed, and mass drops out of the energy but not out of the speed — lighter molecules must move faster to carry the same energy.

Equal kinetic energy per molecule at temperature T:
  (1/2) m v_rms^2 = (3/2) k_B T   =>   v_rms = sqrt(3 R T / M)

Effusion flux through a small orifice (Knudsen regime):
  d (orifice diameter) << λ (mean free path)
  Rate ∝ n · v_avg / 4  ∝  1 / sqrt(M)   (T, P held constant)

The effusion flux through a small hole is proportional to the number density of molecules times their average speed toward that hole — and since v_avg itself scales as 1/√M at fixed temperature, the whole rate inherits that square-root dependence on molar mass.

Graham's law itself

Thomas Graham stated the empirical rule in the 1830s, decades before kinetic theory could explain it: the effusion rate of a gas is inversely proportional to the square root of its molar mass. Comparing two gases at the same temperature and pressure collapses to a single ratio, rate₁/rate₂ = √(M₂/M₁) — no calculus needed once you trust the kinetic-theory derivation above.

Real-world use: isotope separation

The historic gaseous-diffusion route to enriched uranium pumped uranium hexafluoride (UF6) gas through porous membrane stages, relying on the tiny effusion-rate difference between molecules built around U-235 versus U-238. Because the mass ratio is only about 1.0043, a single stage separates the isotopes by barely 0.2%, which is why real plants cascaded thousands of stages to reach usable enrichment — a direct, industrial-scale consequence of a square-root law.

Limits of the law

Graham's law only holds in the Knudsen regime described above — a true pinhole into vacuum or a much lower-pressure region, with the hole smaller than the mean free path. Push gas through a porous plug, a long tube, or any path where molecules collide with each other on the way out, and you have left the effusion regime for viscous or diffusive flow, which follows different scaling entirely. The law also assumes ideal-gas behaviour, so it drifts at very high pressure or very low temperature where real intermolecular forces start to matter.

Frequently asked questions

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