Graham's Law of Diffusion – From Intuition to Precision
Imagine you're in a room where someone opens a bottle of perfume at one end. You don't smell it instantly — it takes time for the perfume molecules to wander across the room. Now imagine the same experiment with a bottle of ammonia. You'd smell the ammonia much faster. Why? The ammonia molecules are lighter.
That's the core physical idea: lighter gas molecules move faster, on average, than heavier ones at the same temperature. Since diffusion and effusion are processes driven by molecular motion, a lighter gas will spread out (diffuse) or escape through a tiny hole (effuse) more quickly than a heavier gas.
Note
Diffusion is the mixing of gases due to random molecular motion. Effusion is the escape of a gas through a tiny hole into a vacuum. Graham's Law applies to both.
The Precise Statement
Graham's Law of Diffusion/Effusion states:
At constant temperature and pressure, the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass (or density).
Mathematically, for two gases A and B:
rBrA=MAMB=ρAρB
where:
r = rate of diffusion/effusion (volume or moles per unit time)
M = molar mass
ρ = density (at same T and P)
r2r1=M1M2
Why the Square Root? (The Physics)
The reason comes from kinetic molecular theory. At a given temperature, the average kinetic energy of gas molecules is the same for all gases:
21mv2=constant
Here m is the mass of one molecule and v is its speed. Rearranging:
v∝m1
Since molar mass M is proportional to molecular mass m, the average molecular speed is inversely proportional to M. And since the rate of diffusion/effusion is directly proportional to this average speed, you get Graham's Law.
Watch out
A common mistake is to invert the ratio. If gas A is lighter (MA<MB), then rA>rB. Check: MB/MA>1, so rA/rB>1 — correct. Always put the lighter gas in the numerator if you want a ratio > 1.
Worked Example
Problem: Hydrogen (M=2g/mol) and oxygen (M=32g/mol) are allowed to effuse through identical pinholes. How much faster does hydrogen effuse?
H2 is by far the lightest gas, so it moves fastest and is most prone to escaping Earth's gravity over time; the Moon's gravity is too weak to retain any atmosphere at all. …
Step 1. At a given temperature, lighter gas molecules move faster on average than heavier ones (this is the same molar-mass dependence behind Graham's law). Hydrogen (H2), being the very lightest gas, has the highest average molecular speed of any common gas.
Step 2. A significant fraction of H2 molecules, over long enough time, reach speeds high enough to exceed Earth's escape velocity and leave the atmosphere permanently -- so free hydrogen gas does not accumulate in Earth's atmosphere the way heavier gases like nitrogen and oxygen do. …