Graham's Law Calculator

How to Use the Calculator

1. Open What to Calculate and choose a mode: Rate ratio only (just the ratio), Rate of gas 1 (you know the rate of gas 2), Rate of gas 2 (you know the rate of gas 1), or Compare two measured rates (checks how well your data match the law).
2. Enter the Molar Mass of Gas 1 in g/mol — this is typically the lighter, faster gas.
3. Enter the Molar Mass of Gas 2 in g/mol.
4. If your mode needs it, enter the known Rate of Gas 1 or Rate of Gas 2 in any unit you like; the answer comes back in the same unit.
5. Read the results: the rate ratio \(r_1/r_2\), the inverse ratio \(r_2/r_1\), and a line telling you which gas is faster and by how much.

What Is Graham's Law?

Graham's law, formulated by the Scottish chemist Thomas Graham in 1848, states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. In practical terms: lighter gases spread out and escape through small openings faster than heavier gases.

Diffusion is the spontaneous mixing of a gas with another gas through random molecular motion — for example, the smell of perfume spreading through a room. Effusion is the escape of a gas through a tiny hole into a vacuum, such as air slowly leaking from a balloon. Graham's law applies to both processes because both are governed by how fast the molecules travel.

The law follows directly from the kinetic theory of gases. At a given temperature, all gases have the same average kinetic energy, so the average molecular speed is \(v \propto 1/\sqrt{M}\). A molecule of hydrogen (2.016 g/mol) therefore moves about four times faster than a molecule of oxygen (32 g/mol) at the same temperature — exactly what the example above shows.

Interpreting the result is simple: if \(r_1/r_2 = 3.98\), gas 1 covers distance or escapes through an opening 3.98 times faster than gas 2. If the ratio is below 1, gas 1 is the slower one, and the inverse ratio \(r_2/r_1\) tells you the speed advantage of gas 2. One classic application was the Manhattan Project, where Graham's law of effusion guided the separation of uranium isotopes by effusing gaseous UF₆ through porous membranes.

Molar Masses of Common Gases

Assumptions and Limitations

• Both gases must be at the same temperature and pressure. The rates only depend on molar mass when temperature is identical, since speed scales with \(\sqrt{T}\).
• The law assumes ideal-gas behaviour. Real gases deviate at high pressure, low temperature, or when molecules interact strongly (polar gases, hydrogen bonding). Deviations of a few percent are normal.
• The hole must be tiny for effusion. If the opening is large compared to the mean free path, gas flows as a bulk fluid and the simple law no longer holds.
• Use molar mass, not density or molecular diameter. If you compare a gas with air, use the effective molar mass of air, about 28.96 g/mol.
• For gas-law problems involving pressure and volume, see the Dalton's Law calculator, the Combined Gas Law calculator, and the Charles' Law calculator.