Last updated: 2026-06-16

Understanding Graham's law of effusion

Effusion is the escape of gas molecules through an opening so small that they slip out one at a time without colliding with one another. Graham's law says the rate of effusion depends on the molar mass of the gas:

The molar mass appears under a square root, so the effect of mass is gentle. A gas that is four times heavier than another effuses only half as fast, not four times slower.

Why mass controls the rate

At a fixed temperature every gas has the same average kinetic energy, $\tfrac{1}{2} m v^2$. To carry that same energy a lighter molecule has to travel faster, so it reaches the opening more often. Because kinetic energy scales with the square of speed, the speed itself — and therefore the rate of effusion — scales with $1/\sqrt{M}$. Take the ratio for two gases and the temperature cancels, leaving the clean square-root relationship above.

Worked example

Hydrogen (H₂, M₁ = 2.016 g/mol) and oxygen (O₂, M₂ = 32.00 g/mol) sit on opposite sides of a pinhole at the same temperature and pressure. How much faster does hydrogen effuse?

Hydrogen effuses almost four times faster than oxygen. Running the relationship backwards is just as useful: if a known gas of 2.016 g/mol is measured to effuse exactly four times faster than an unknown gas, the unknown molar mass is

which points to oxygen.

Comparing common gases

The table below lists how fast each gas effuses relative to hydrogen, the lightest. Each figure is $\sqrt{M_{\text{gas}} / 2.016}$ inverted — that is, $\sqrt{2.016 / M_{\text{gas}}}$.

Effusion versus diffusion

Effusion is escape through a single small hole; diffusion is the mixing of gases as their molecules spread through one another, colliding constantly along the way. Graham's law was measured for effusion, but because both processes are driven by molecular speed, the same $\sqrt{M_2/M_1}$ ratio is widely used as an approximation for relative rates of diffusion.

Where the law applies and its limits

Graham's law holds for ideal-gas effusion: the hole must be small compared with the average distance molecules travel between collisions, and the two gases must be at the same temperature and pressure. It is the principle behind gaseous separation of uranium isotopes, where the tiny mass difference between UF₆ molecules is exploited over thousands of stages. The law assumes ideal behaviour, so it loses accuracy at high pressures, near condensation, or when the opening is large enough that ordinary flow rather than molecule-by-molecule escape takes over.