Graham's Law of Diffusion Calculator

About the Graham's Law of Diffusion Calculator

Graham's law of effusion, formulated by Scottish chemist Thomas Graham in 1848, states that the rate of effusion (or diffusion) of a gas is inversely proportional to the square root of its molar mass. Mathematically, for two gases: Rate₁/Rate₂ = √(M₂/M₁). This elegant relationship is a direct consequence of kinetic molecular theory — lighter molecules move faster at the same temperature because they have the same average kinetic energy as heavier ones.

This law has profound practical applications: it explains why hydrogen gas escapes from containers faster than any other gas, why helium balloons deflate faster than air-filled ones, and why uranium isotope enrichment by gaseous diffusion through porous membranes is possible (albeit inefficient). The uranium enrichment process exploits the tiny mass difference between UF₆ containing ²³⁵U versus ²³⁸U isotopes.

This calculator computes relative rates, absolute velocities (root-mean-square and most probable speeds), effusion times, and helps identify unknown gases from experimental effusion data. It includes presets for common gas pairs, isotope systems, and a comprehensive reference table of molecular speeds.

When This Page Helps

This calculator quickly compares gas effusion rates, calculates molecular speeds at any temperature, and solves for unknown molar masses from experimental effusion data — useful for chemistry students, researchers, and engineers working with gas systems.

How to Use the Inputs

1. Enter the molar masses of the two gases you want to compare.
2. Optionally enter the temperature to calculate absolute molecular speeds.
3. Use presets to load common gas pairs (H₂ vs O₂, He vs Ar, UF₆ isotopes, etc.).
4. If you measured an effusion time, enter it to calculate the unknown gas's molar mass.
5. Review the rate ratio, velocity comparison, and effusion time ratio.
6. Check the reference table for molecular speeds of common gases at your temperature.
7. Use the isotope separation data to understand enrichment factors.

Example Calculation

Tips & Best Practices

• Graham's law is exact for ideal gases at low pressures where intermolecular forces are negligible.
• For real gases at high pressures, deviations from Graham's law occur due to intermolecular interactions.
• To identify an unknown gas, measure its effusion time relative to a known gas under identical conditions.
• The separation factor for isotopes is very small — practical enrichment requires hundreds of cascade stages.
• RMS speed at room temperature ranges from ~200 m/s (heavy gases) to ~1900 m/s (H₂).
• Graham's law also approximately predicts the relative rates of gas leaks through small holes in containers.

Historical Context and Thomas Graham

Thomas Graham (1805–1869) was a Scottish chemist who studied gas movement through porous plugs and small orifices. His 1848 publication established that lighter gases pass through porous barriers faster, with rates inversely proportional to the square root of their densities. This empirical finding preceded kinetic molecular theory by over a decade and was later explained by Maxwell and Boltzmann's statistical mechanics.

Kinetic Molecular Theory Connection

Graham's law follows directly from the equipartition theorem: at temperature T, every gas molecule has average translational kinetic energy E = 3/2 kT, regardless of mass. Setting ½m₁v₁² = ½m₂v₂² gives v₁/v₂ = √(m₂/m₁). Since effusion rate is proportional to molecular speed (faster molecules hit the orifice more often), the rate ratio equals the speed ratio.

Modern Applications

Beyond the historical uranium enrichment application, Graham's law is relevant in modern technology: helium leak detection for vacuum systems, natural gas pipeline leak estimation, semiconductor fabrication (CVD gas delivery), and respiratory physiology (understanding gas exchange rates in the lungs). It's also fundamental to understanding why certain gases permeate through polymer membranes faster than others.

Frequently Asked Questions

• What is the difference between effusion and diffusion?

  Effusion is gas escape through a tiny hole (smaller than the mean free path); diffusion is spreading through another gas. Graham's law applies exactly to effusion and approximately to diffusion.

• Why does Graham's law work?

  At the same temperature, all gas molecules have the same average kinetic energy (½mv² = 3/2 kT). Lighter molecules must therefore move faster, with speed inversely proportional to √M.

• How was Graham's law used for uranium enrichment?

  UF₆ gas containing ²³⁵U (M = 349) effuses slightly faster than ²³⁸UF₆ (M = 352). The separation factor is √(352/349) = 1.0043 per stage, requiring ~1400 stages for weapons-grade enrichment.

• Can I identify an unknown gas using effusion?

  Yes. If gas A (known M) effuses in time t₁ and the unknown effuses in time t₂, then M_unknown = M_A × (t₂/t₁)². This is a classic general chemistry experiment.

• Does Graham's law apply to gas mixtures?

  It applies to individual components. In a mixture, each gas effuses independently based on its own molar mass, which is how gaseous diffusion plants separate isotopes.

• What is root-mean-square speed?

  The v_rms is the square root of the average of the squared molecular speeds: v_rms = √(3RT/M). It's slightly higher than the mean speed and corresponds to the average kinetic energy.