Table of Contents
Maxwell-Boltzmann Distribution
The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in an ideal gas at thermal equilibrium. Not all molecules move at the same speed; instead there is a distribution with a characteristic peak and tail. Three important characteristic speeds are defined: the most probable velocity (peak of distribution), mean velocity (average), and root-mean-square velocity (related to kinetic energy).
The distribution depends on temperature and molecular mass. Higher temperatures broaden the distribution and shift it to higher speeds. Lighter molecules move faster than heavier ones at the same temperature, which is why hydrogen escapes Earth's atmosphere but nitrogen does not.
Velocity Formulas
Where R = 8.314 J/(mol·K), T is temperature in Kelvin, and M is molar mass in kg/mol. Note: vmp < vmean < vrms always.
Gas Velocities at 300 K
| Gas | M (g/mol) | vrms (m/s) |
|---|---|---|
| H₂ | 2 | 1920 |
| He | 4 | 1370 |
| N₂ | 28 | 517 |
| O₂ | 32 | 484 |
| CO₂ | 44 | 412 |
Frequently Asked Questions
Why are there three different velocity measures?
Each captures a different aspect of the speed distribution. The most probable velocity is where you'd find the most molecules. The mean velocity is the arithmetic average. The RMS velocity is connected to kinetic energy (KE = ½mv²rms). For a Maxwell-Boltzmann distribution, they are always in the ratio vmp : vmean : vrms = 1 : 1.128 : 1.225.
How fast do air molecules move?
At room temperature (300 K), nitrogen molecules (the main component of air) have an RMS speed of about 517 m/s, or roughly 1860 km/h. Despite this incredible speed, molecules travel only about 68 nm between collisions due to the extremely high number density of air.