Engineering & Science
Graham’s Law of Gas Effusion & Knudsen Isotope Separation Factor
Calculate molecular pinhole effusion rate ratios r_1 / r_2 = SQRT(M_2 / M_1) from molar masses—comparing rapid H2 vs. O2 vacuum leaks (3.984x) against razor-thin single-stage 235UF6 / 238UF6 gaseous diffusion enrichment (alpha = 1.004303).
Why hydrogen leaks nearly 4x faster than oxygen—while enriching Uranium-235 takes thousands of stages!
At identical temperature T, kinetic theory requires all ideal gas molecules to carry the exact same average translational kinetic energy (3/2) * k_B * T = (1/2) * m * v_rms^2—so lighter molecules zip around faster by the inverse square root of molar mass! Thomas Graham’s Law of Effusion (1848) states that through a microscopic Knudsen pinhole (diameter << mean free path), molar effusion rate ratio equals r_1 / r_2 = SQRT(M_2 / M_1)!
Compare two engineering extremes: (1) Hydrogen leak safety—molecular H2 (M_1 = 2.016 g/mol) effuses through micro-pores SQRT(32.00 / 2.016) = 3.984 times faster than O2 (M_2 = 32.00 g/mol), multiplying volumetric leak rate nearly fourfold! (2) Gaseous diffusion uranium enrichment—volatile 235UF6 (M_1 = 349.03 g/mol) vs. 238UF6 (M_2 = 352.04 g/mol) has an ideal single-stage separation factor of just alpha = SQRT(352.04 / 349.03) = 1.004303, requiring > 1,000 cascaded porous barriers to enrich reactor fuel from 0.71% to 3.5–5.0%!
| Binary Gas Effusion Pair | Molar Masses M1 & M2 (g/mol) | Graham Effusion Ratio SQRT(M2 / M1) | Engineering Consequence |
|---|---|---|---|
| Hydrogen H2 vs. Oxygen O2 | 2.016 vs. 32.00 g/mol | 3.984095 (3.984x Faster) | Rapid H2 seal & flange permeation |
| Helium He vs. Nitrogen N2 | 4.0026 vs. 28.014 g/mol | 2.645554 (2.646x Faster) | Sensitive He mass-spec leak testing |
| 235UF6 vs. 238UF6 Isotope Pair | 349.03 vs. 352.04 g/mol | 1.004303 (+0.43% per stage) | Requires ~1,400 gaseous-diffusion stages |
Graham’s Law Effusion Rate & Ideal Knudsen Separation Factor SQRT(M_2 / M_1)
Computes r_1 / r_2 = SQRT(M_2 / M_1) to 6 decimal places so tiny isotopic enrichments like UF6 (1.004303) retain full precision.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Molar Mass", ROUND(SQRT(B3 / B2), 6))Predicted Gas 1 Effusion Flux From Measured Reference Gas 2 Leak Rate
Scales a measured N2/O2/He reference leak rate B4 (sccm) into predicted Gas 1 Knudsen leak rate B4 * SQRT(M_2 / M_1).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Molar Mass", ROUND(B4 * SQRT(B3 / B2), 3))Try the example
Use the original values or change them to test your understanding.