The Distribution of Molecular Speeds
In any gas at thermal equilibrium, not all molecules move at the same speed. Some crawl, others race — their speeds follow a precise statistical distribution first derived by James Clerk Maxwell in 1860 and later placed on rigorous statistical-mechanical foundations by Ludwig Boltzmann. The Maxwell–Boltzmann distribution is one of the cornerstones of kinetic theory and statistical mechanics.
Three Characteristic Speeds
The distribution defines three important speeds: v_mp (most probable speed, where the curve peaks), v_mean (average speed), and v_rms (root-mean-square speed, connected to kinetic energy via E = ½mv²_rms). These three are always ordered: v_mp < v_mean < v_rms. The differences between them arise from the asymmetric shape of the distribution — it has a long tail toward high speeds.
Temperature and Mass Effects
Move the temperature slider and watch the curve flatten and shift right. Higher temperature pumps more kinetic energy into the gas, spreading speeds over a wider range. Now try changing the molecular mass: heavier molecules (like Xe at 131 amu) have a sharper, slower distribution than light ones (like He at 4 amu) at the same temperature. This mass dependence has real consequences — it determines which gases a planet can retain in its atmosphere over geological timescales.
Atmospheric Escape
Enable the escape velocity overlay to see Earth's escape speed (11.2 km/s) on the distribution. For heavy molecules like N₂, the tail beyond 11.2 km/s is negligibly small — Earth holds onto its nitrogen easily. But for hydrogen (mass 2 amu), a meaningful fraction of molecules exceed escape speed at any moment. Over 4.5 billion years, this thermal leakage (Jeans escape) has stripped Earth of nearly all its primordial hydrogen. This is why massive planets like Jupiter, with their much higher escape velocities, retain thick hydrogen-helium atmospheres while Earth cannot.