• word_count: 200

Three characteristic speeds describe molecular motion, each derived from the Maxwell-Boltzmann distribution:

RMS speed: $v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{3k_BT/m} = \sqrt{3RT/M}$. Appears in pressure and energy calculations. Always the largest of the three.

Average speed: $v_{\text{avg}} = \sqrt{8k_BT/(\pi m)} = \sqrt{8RT/(\pi M)}$. Used in mean free path and collision rate calculations.

Most probable speed: $v_{\text{mp}} = \sqrt{2k_BT/m} = \sqrt{2RT/M}$. The peak of the Maxwell distribution curve. Always the smallest.

The universal ratio is $v_{\text{mp}} : v_{\text{avg}} : v_{\text{rms}} = 1 : 1.128 : 1.224 = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$. All three scale as $\sqrt{T}$ and $1/\sqrt{M}$: lighter and hotter gases have faster molecules.

Key applications: the speed of sound $v_s = \sqrt{\gamma RT/M}$ relates to $v_{\text{rms}}$ by $v_s = v_{\text{rms}}\sqrt{\gamma/3}$. The escape velocity comparison determines which gases an atmosphere retains — if $v_{\text{rms}} > v_{\text{escape}}/6$, the gas gradually escapes (explaining why Earth retains N$_2$ and O$_2$ but not H$_2$ or He).