• id: JTHERM-02-N08
• title: Most Probable, Mean, and RMS Speeds
• tags: vp, vavg, vrms, speed-comparison

Three speeds characterise molecular motion: (1) Most probable speed $v_p = \sqrt{2k_BT/m} = \sqrt{2RT/M}$ — the speed at the peak of the Maxwell-Boltzmann distribution. (2) Mean speed $v_{\text{avg}} = \sqrt{8k_BT/(\pi m)} = \sqrt{8RT/(\pi M)}$ — the arithmetic average of all speeds. (3) RMS speed $v_{\text{rms}} = \sqrt{3k_BT/m} = \sqrt{3RT/M}$ — the square root of the mean of squared speeds. The ratio $v_p : v_{\text{avg}} : v_{\text{rms}} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1 : 1.128 : 1.224$. Always $v_p < v_{\text{avg}} < v_{\text{rms}}$.