Graham's Law of Effusion
Learn Graham's law formula relating gas effusion rates to molar masses, with step-by-step examples and applications.
The Formula
Graham's law of effusion states that the rate at which a gas escapes through a tiny hole is inversely proportional to the square root of its molar mass. In simpler terms, lighter gas molecules move faster and escape more quickly than heavier ones. This relationship was discovered experimentally by the Scottish chemist Thomas Graham in 1848 and is a direct consequence of the kinetic molecular theory of gases.
The law applies specifically to effusion, which is the process of gas molecules passing through a hole so small that they go through one at a time without colliding with each other. This is slightly different from diffusion, which is the mixing of gases through a larger space. However, Graham's law gives a good approximation for diffusion rates as well, since both processes depend on molecular speed.
At any given temperature, all gas molecules have the same average kinetic energy regardless of their mass. Since kinetic energy equals ½mv², lighter molecules must move at higher velocities to have the same energy as heavier ones. A hydrogen molecule (molar mass 2 g/mol) moves about four times faster than an oxygen molecule (molar mass 32 g/mol) at the same temperature, because √(32/2) = 4.
One of the most historically significant applications of Graham's law was during the Manhattan Project in the 1940s. Scientists at Oak Ridge, Tennessee, used gaseous diffusion to separate uranium-235 from uranium-238 by converting uranium to uranium hexafluoride gas. The slightly lighter U-235 hexafluoride (349 g/mol) effused marginally faster than U-238 hexafluoride (352 g/mol). Thousands of diffusion stages were needed because the mass difference is so small, but the method worked and remains a landmark application of this simple formula. Today, Graham's law is applied in industrial gas separation, leak detection, atmospheric science, and understanding how gases behave in vacuum systems.
Variables
| Symbol | Meaning |
|---|---|
| Rate₁ | Effusion rate of gas 1 (in mol/s, mL/s, or relative units) |
| Rate₂ | Effusion rate of gas 2 (same units as Rate₁) |
| M₁ | Molar mass of gas 1 (in g/mol) |
| M₂ | Molar mass of gas 2 (in g/mol) |
| √ | Square root — the key mathematical relationship in the formula |
Example 1: Hydrogen vs Oxygen
Problem: How much faster does hydrogen gas (H₂, M = 2 g/mol) effuse compared to oxygen gas (O₂, M = 32 g/mol)?
Assign gas 1 = H₂ (M₁ = 2) and gas 2 = O₂ (M₂ = 32).
Rate_H₂ / Rate_O₂ = √(M_O₂ / M_H₂) = √(32 / 2) = √16
Hydrogen effuses 4 times faster than oxygen at the same temperature.
Example 2: Identifying an Unknown Gas
Problem: An unknown gas effuses at 0.354 times the rate of helium (M = 4 g/mol). What is the molar mass of the unknown gas?
Rate_He / Rate_unknown = √(M_unknown / M_He)
1 / 0.354 = √(M_unknown / 4), so 2.825 = √(M_unknown / 4)
Square both sides: 7.98 = M_unknown / 4, so M_unknown = 31.9 g/mol
The unknown gas has a molar mass of approximately 32 g/mol, suggesting it is oxygen (O₂).
When to Use It
Graham's law is useful whenever you need to compare how fast different gases move or escape through small openings.
- Comparing effusion or diffusion rates of different gases
- Identifying unknown gases by measuring their effusion rate relative to a known gas
- Industrial gas separation processes (such as isotope enrichment)
- Understanding why helium balloons deflate faster than air-filled ones
- Predicting how quickly gas leaks will spread in safety engineering