Graham's Law

Open a bottle of perfume across the room and you smell it within seconds. Release a bottle of heavy cologne next to it and the lighter perfume molecules reach your nose first. Lighter molecules travel faster, so they spread through a room and escape through tiny openings more quickly. Graham’s law quantifies exactly how much faster.

Effusion vs. Diffusion

These terms are often confused but describe different processes:

Effusion: Gas molecules escape through a tiny hole (smaller than the mean free path of the gas) into a vacuum. One molecule at a time squeezes through. Think of a pinhole leak in a tire - the air slowly effuses out.

Diffusion: Gas molecules spread through another gas (or into empty space) by random motion. This is what happens when you open a bottle of ammonia and smell it across the room. Diffusion is slower than effusion because molecules collide with other gas molecules along the way.

Graham’s law applies strictly to effusion, but the same mathematical relationship approximately describes relative diffusion rates as well.

The Law

Why Graham’s Law Works

Graham’s law is a direct consequence of kinetic molecular theory. At the same temperature, all gas molecules have the same average kinetic energy:

KE = $\frac{1}{2}$ mv² = $\frac{3}{2}$ kT

Since KE is the same for both gases at the same temperature:

$\frac{1}{2}$ m₁v₁² = $\frac{1}{2}$ m₂v₂²

Solving for the velocity ratio:

v₁/v₂ = sqrt(m₂/m₁) = sqrt(M₂/M₁)

Since the rate of effusion is proportional to molecular speed (faster molecules reach the hole more quickly), Graham’s law follows directly.

Worked Examples

Example 1: How much faster does H2 (M = 2) effuse compared to O2 (M = 32)?

rate(H2)/rate(O2) = sqrt($\frac{32}{2}$) = sqrt(16) = 4

Hydrogen effuses 4 times faster than oxygen.

Example 2: Gas A effuses 3 times faster than Gas B (M = 36). What is the molar mass of Gas A?

rate(A)/rate(B) = sqrt(M(B)/M(A))

3 = sqrt(36/M(A))

9 = 36/M(A)

M(A) = $\frac{36}{9}$ = 4 g/mol (This is helium.)

Applications

Uranium enrichment: The original method for enriching uranium for nuclear fuel used Graham’s law. Natural uranium is mostly U-238, with only 0.7% U-235 (the fissile isotope). UF6 gas containing U-235 effuses slightly faster than UF6 containing U-238. By passing UF6 through thousands of porous barriers, the lighter isotope is gradually concentrated. The separation factor per stage is tiny (sqrt($\frac{352}{349}$) = 1.004), requiring thousands of stages.