Effusion is what happens when gas molecules sneak out through a hole that’s smaller than the distance they typically travel between collisions — a true microscopic leak, not a bulk flow. Thomas Graham figured out in 1846 that lighter gases escape faster than heavier ones, and the relationship is so clean it’s still the basis of uranium isotope enrichment two centuries later. The Manhattan Project’s gaseous-diffusion plants at Oak Ridge separated ²³⁵UF₆ from ²³⁸UF₆ by exploiting a rate ratio of about 1.0043 — a tiny edge, but compounded thousands of times across cascades of porous barriers.

The equation

rate₁ / rate₂ = √(M₂ / M₁)

Light gas on top of the rate ratio, heavy gas on top of the molar mass ratio (i.e., the molar masses swap sides). It comes from kinetic theory: average kinetic energy is the same for any gas at the same temperature, and KE = ½mv² implies v ∝ 1/√m. Faster molecules hit the hole more often, so they effuse faster.

The same √(M₂/M₁) relationship applies to: average molecular speed, RMS speed, and effusion time (with the ratio inverted, since slower means longer).

Worked example: hydrogen vs oxygen

Compare H₂ (M = 2.016) and O₂ (M = 32.00):

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

Hydrogen effuses about four times faster than oxygen. That’s why a hydrogen-filled balloon goes flat overnight while a helium balloon takes a few days and an air-filled one stays inflated for weeks — molecular weight controls the leak rate.

Identifying an unknown from effusion data

An unknown gas effuses 0.355 times as fast as helium (M = 4.003). What is it?

rate(unknown) / rate(He) = √(M(He) / M(unknown))

0.355 = √(4.003 / M)

Square both sides: 0.126 = 4.003 / M

M = 4.003 / 0.126 = 31.8 g/mol

That’s essentially O₂ (32.00 g/mol). This is the lab technique behind effusion-based molar mass determination — measure two effusion rates against a known reference, do one square root, get the molecular weight of an unknown.

Effusion time

Time and rate are inverse, so the ratio flips:

time₁ / time₂ = √(M₁ / M₂)

It takes 4.00 minutes for 1.0 L of helium to effuse through a pinhole. How long for the same volume of methane (M = 16.04)?

time(CH₄) / time(He) = √(16.04 / 4.003) = √4.007 = 2.002

time(CH₄) = 4.00 × 2.002 = 8.01 minutes

CH₄ is four times heavier than He, so it moves at half the average speed (√4 = 2), so it takes twice as long to leak through. Internally consistent.

Why uranium enrichment is hard

²³⁵UF₆ has M = 349.03 and ²³⁸UF₆ has M = 352.04. The effusion rate ratio:

rate(²³⁵) / rate(²³⁸) = √(352.04 / 349.03) = √1.00863 = 1.00430

A 0.43% advantage per stage. Enrichment to weapons-grade (~90% ²³⁵U) from natural uranium (0.72% ²³⁵U) requires thousands of stages cascaded together. The math is trivial; the engineering is staggering. Modern centrifuges have replaced gaseous diffusion because they can achieve much larger separation factors per stage, but Graham’s law still describes the underlying phenomenon.

Where the calculation goes sideways

Inverting the ratio. The light gas should always be the faster one. If your math says hydrogen effuses slower than xenon, you’ve flipped the molar masses. Sanity check by asking which gas should be faster, then verify your ratio agrees.

Atomic vs molar mass for diatomics. Use the molar mass of the molecule. H₂ is 2.016, not 1.008. O₂ is 32.00, not 16.00. Forgetting the diatomic factor of 2 is the single most common Graham’s law error on exams.

Effusion vs diffusion. Graham’s law is rigorously derived for effusion through a small orifice into vacuum. Diffusion (mixing of gases in normal pressure) follows the same √M dependence approximately, but with corrections for collision frequency. The textbook problem usually doesn’t care about the distinction; lab work sometimes does.

Forgetting that temperature affects both equally. Graham’s law assumes both gases are at the same temperature. Compare gases at different temperatures and you need the full kinetic-theory treatment, not just √(M₂/M₁).

Practice

Verify against the Graham’s Law Calculator:

How much faster does He effuse than Ar (M = 39.95)?
An unknown gas effuses 0.500 times as fast as H₂. Molar mass?
Rank by effusion speed: CO₂ (44.01), H₂ (2.016), Ne (20.18), Cl₂ (70.90).
N₂ effuses through a hole in 15.0 s. How long for the same amount of SO₂ (M = 64.07)?
Ratio of average molecular speeds, ²³⁵UF₆ (349.03) to ²³⁸UF₆ (352.04)?

How to Calculate Effusion Rate