Q.Rate of diffusion of a gas is

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.

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:

where:

• $r$ = rate of diffusion/effusion (volume or moles per unit time)
• $M$ = molar mass
• $\rho$ = density (at same T and P)

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:

Here $m$ is the mass of one molecule and $v$ is its speed. Rearranging:

Since molar mass $M$ is proportional to molecular mass $m$, the average molecular speed is inversely proportional to $\sqrt{M}$. And since the rate of diffusion/effusion is directly proportional to this average speed, you get Graham's Law.

Worked Example

Problem: Hydrogen ($M = 2\ \text{g/mol}$) and oxygen ($M = 32\ \text{g/mol}$) are allowed to effuse through identical pinholes. How much faster does hydrogen effuse?

Solution:

Hydrogen effuses 4 times faster than oxygen.

Key Points for Exams

1. Conditions matter: Graham's Law holds strictly only at constant temperature and pressure, and for gases behaving ideally.
2. Rate can be measured as volume per unit time, moles per unit time, or distance travelled per unit time in a diffusion tube.
3. Density form: Since density $\rho \propto M$ at fixed T and P, you can use $\sqrt{\rho_2/\rho_1}$ directly.
4. For mixtures: Graham's Law is used in uranium enrichment — separating $^{235}\text{UF}_6$ from $^{238}\text{UF}_6$ by effusion, though the mass difference is tiny.

Graham's law of diffusion is a standard NCERT/CBSE Class 11 Chemistry topic, and "Graham's law of diffusion formula and numericals" is a commonly searched query during board exam preparation. It's also a frequent JEE Main and NEET important-question type, often used to compare effusion rates of two gases.