Graham’s Law of Effusion

What Graham’s Law actually says

In 1846 the Scottish chemist Thomas Graham pushed gases through a fine plaster-of-Paris plug and timed how quickly each escaped. He found a strikingly simple rule: the rate of effusion is inversely proportional to the square root of the gas’s density, and since density at fixed temperature and pressure is proportional to molar mass, this is equivalent to

Rate ∝ 1/√M

For two gases measured under identical conditions, the molar masses cancel into a tidy ratio:

Rate₁ / Rate₂ = √(M₂ / M₁)

The heavier mass sits on top of the square root for the faster gas. That single relation — no fitting constants, no empirical fudge — is why Graham’s Law is one of the cleanest demonstrations that gases are made of moving molecules whose speed depends on their mass.

Effusion versus diffusion

The two words get conflated, but they describe different physical situations, and Graham’s Law applies most exactly to only one of them.

• Effusion is the escape of molecules, one at a time, through an opening that is smaller than the mean free path of the gas. Molecules pass through without colliding with each other in the hole — it’s a pure "do you happen to be heading at the hole" counting problem. This is the regime Graham measured and the regime where Rate ∝ 1/√M is exact.
• Diffusion is the gradual interpenetration and mixing of one gas through another (or through a still gas). Here molecule-molecule collisions dominate; a molecule takes a random walk rather than a clean shot. Graham’s 1/√M dependence still describes the trend — lighter gases spread faster — but the net mixing rate is slowed and complicated by collision cross-sections, so the square-root law is approximate.

The classic ammonia–HCl ring experiment shows the diffusion side. Cotton soaked in concentrated ammonia (NH₃, M = 17) and concentrated hydrochloric acid (HCl, M = 36.5) are placed at opposite ends of a glass tube. A white ring of ammonium chloride forms where the vapors meet — and it forms closer to the HCl end, because NH₃ diffuses faster: the distance ratio comes out near √(36.5/17) ≈ 1.47, so ammonia travels roughly 1.5 times as far.

Why mass sets the speed

The mechanism is kinetic-molecular theory. At thermal equilibrium, every gas in a container has the same average translational kinetic energy, fixed only by temperature:

½ m v̄² = (3/2) k_B T

Solving for the root-mean-square molecular speed gives

v_rms = √(3RT / M)

So speed falls off as 1/√M. A light molecule must sprint to carry the same energy a heavy one carries while ambling. At 25 °C, hydrogen molecules average about 1920 m/s, while oxygen molecules average only about 480 m/s — a 4× difference, exactly √(32/2).

The effusion rate is the frequency at which molecules strike the aperture from the inside, which is proportional to (number density) × (mean speed) × (hole area). When you compare two pure gases at the same temperature, pressure, and hole, the number density and area are identical, so the rate ratio reduces to the speed ratio — and the speed ratio is 1/√M. That is the whole derivation. Notice that temperature cancels in a rate ratio: both gases speed up as √T together, so the comparison between two gases is temperature-independent, even though each gas individually effuses faster when hot.

Worked numbers

The square root compresses big mass differences into modest rate differences, which is exactly why isotope work is so slow.

Gas pair (same T, P)	Molar masses (g/mol)	Rate ratio √(M₂/M₁)	Faster gas
H₂ vs O₂	2 vs 32	4.00×	H₂
He vs CH₄	4 vs 16	2.00×	He
He vs N₂	4 vs 28	2.65×	He
CO₂ vs O₂	44 vs 32	0.85× (O₂ faster)	O₂
²³⁵UF₆ vs ²³⁸UF₆	349 vs 352	1.0043×	²³⁵UF₆

A handy companion relation: because rate and time are reciprocal, if two equal volumes effuse, the times are in the inverse ratio, t₁/t₂ = √(M₁/M₂). If 50 mL of an unknown gas effuses in 80 s while 50 mL of oxygen takes 40 s, the unknown is √-related: (80/40)² × 32 = 4 × 32 = 128 g/mol — which points to a heavy gas such as xenon or sulfur hexafluoride’s heavier cousins. Timing effusion is, in fact, a textbook method for estimating an unknown gas’s molar mass.

Isotope enrichment: the law that built the bomb

Graham’s Law sat as a tidy classroom result for almost a century until the Manhattan Project needed to separate fissile ²³⁵U (0.72% of natural uranium) from the dominant ²³⁸U. The trick was to make uranium gaseous as uranium hexafluoride (UF₆), which sublimes at 56 °C. The two isotopologues differ only by the three-mass-unit gap between the uranium isotopes, so:

α = √(352 / 349) ≈ 1.0043

That single-stage separation factor means each pass through a porous barrier enriches the ²³⁵U fraction by under half a percent. To climb from 0.72% to reactor-grade 3–5% takes well over a thousand cascade stages plumbed in series; weapons-grade (>90%) demands far more. The gaseous-diffusion plant at Oak Ridge (K-25) covered 44 acres under one roof for exactly this reason. The tiny separation factor — a direct consequence of the square root crushing a 0.86% mass difference into a 0.43% rate difference — is why enrichment is so capital-intensive, and why gas centrifuges (which exploit mass differently, via centrifugal force ∝ M) eventually displaced diffusion. The principle that lighter molecules move faster is the same one that lets a helium leak detector sniff out a vacuum-system pinhole long before a heavier tracer would register.

Where the law shows up

• Balloons. A helium balloon (He, M = 4) deflates noticeably faster than an air-filled one because helium effuses through the latex pores; hydrogen would be faster still.
• Isotope geochemistry. Mass-dependent effusion and diffusion fractionate light isotopes in volcanic gases, ice cores, and the slow escape of hydrogen from planetary atmospheres — Earth has lost much of its primordial H₂ this way.
• Leak detection. Helium’s high effusion rate makes it the standard tracer for finding micro-leaks in vacuum chambers and refrigeration lines.
• Analytical chemistry. Estimating an unknown gas’s molar mass by timing its effusion against a reference gas is a routine lab exercise grounded directly in the law.
• Nuclear fuel cycle. Gaseous-diffusion uranium enrichment, the historical workhorse, is Graham’s Law applied at industrial scale.

Common misconceptions

• Effusion and diffusion are the same. They’re not — effusion is collisionless escape through a small hole; diffusion is collision-limited mixing. The law is exact only for effusion.
• The rate scales with mass. It scales with the square root of mass — a 16× mass increase only quarters the rate.
• Hotter gas changes the ratio. Temperature cancels in a two-gas comparison; each gas individually speeds up as √T, but their ratio doesn’t move.
• Heavier always means slower in every context. True for effusion at equal T and P, but real diffusion also depends on collision cross-section and pressure gradients.
• You can enrich uranium in one pass. The √M law makes the per-stage factor ~1.0043, so thousands of cascade stages are needed.
• Graham’s Law needs ideal gases only. It’s a kinetic-theory result; real-gas attractions matter only at high pressure, far from the low-density effusion regime.