Maxwell-Boltzmann Speed Distribution

Maxwell-Boltzmann Speed Distribution

In an ideal gas at temperature $T$, molecules move with a wide range of speeds. The Maxwell-Boltzmann distribution describes how those speeds are distributed. Rather than all molecules moving at the same speed, the distribution is a broad bell-shaped curve with a characteristic spread determined by $T$ and the molecular mass $m$.

Three characteristic speeds summarise the distribution:

Most Probable Speed

The speed at the peak of the distribution — the speed most molecules travel near:

$v_{\text{mp}} = \sqrt{\frac{2k_B T}{m}} = \sqrt{\frac{2RT}{M}}$

Mean Speed

The arithmetic average speed:

$\bar{v} = \sqrt{\frac{8k_B T}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$

Root-Mean-Square Speed

The square root of the average of the squared speeds. This is directly related to the average kinetic energy $\langle KE \rangle = \tfrac{3}{2} k_B T$:

$v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3RT}{M}}$

Here $R = 8.314\,\text{J/(mol·K)}$ is the gas constant and $M$ is the molar mass in kg/mol.

Speed Ordering

The three speeds always satisfy:

$v_{\text{mp}} < \bar{v} < v_{\text{rms}}$

This ordering reflects the asymmetry of the distribution: the high-speed tail pulls the mean and RMS above the peak.

Example Molar Masses

Higher molar mass → slower speeds at the same temperature.