Graham's Law of Effusion Calculator

Graham’s Law of Effusion (1848) states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

Effusion is the process of gas molecules passing through a tiny hole into a vacuum (one molecule at a time). Diffusion is mixing of gases throughout a space (both processes follow similar laws).

Rate ratio formula:

r₁/r₂ = √(M₂/M₁)

Time ratio (to effuse the same amount):

t₁/t₂ = √(M₁/M₂) = r₂/r₁

So the lighter gas is faster — a heavier gas takes longer.

Derivation: From kinetic theory, the average speed of a gas molecule:

v_avg = √(8RT/πM)

Since effusion rate is proportional to average molecular speed:

r₁/r₂ = v₁/v₂ = √(M₂/M₁)

Common molar masses (g/mol):

Uranium enrichment: The uranium isotope separation process uses UF₆ gas. ²³⁵UF₆ (M = 349) effuses slightly faster than ²³⁸UF₆ (M = 352). The rate ratio = √(352/349) ≈ 1.0043 — only 0.43% faster. Thousands of stages of gaseous diffusion were needed in the Manhattan Project!

Worked example

Compare hydrogen (2.016 g/mol) and oxygen (32.00 g/mol):

rate(H₂) / rate(O₂) = √(32.00 / 2.016) = √15.87 = 3.98

Hydrogen escapes through a pinhole almost four times faster than oxygen. Note which way round the molar masses go: the ratio of rates carries the inverse ratio of masses, because heavier molecules move more slowly at the same temperature. Writing √(M₁/M₂) instead is the single most common mistake with this equation, and it inverts the answer.

Why this ever mattered

The Manhattan Project’s gaseous diffusion plant at Oak Ridge existed because of this equation. Uranium-235 and uranium-238 are chemically identical, so no ordinary separation works. Converted to uranium hexafluoride, though, the two isotopes have molar masses of 349 and 352, and their effusion rates differ by a factor of √(352/349) = 1.0043.

That is a separation of four parts in a thousand per stage. Reaching weapons-grade enrichment took thousands of stages in series, a building over a kilometre long, and at its peak a significant fraction of the entire electrical output of the United States. Modern plants use gas centrifuges instead, which exploit the same tiny mass difference far more efficiently, but the underlying physics is unchanged.

Effusion is not diffusion

Graham’s law describes effusion, gas escaping through an opening small compared with the mean free path. Diffusion, gas spreading through another gas, follows the same square-root dependence on mass but is much slower and depends on collisions with the other gas as well.