Most-probable speed
v_mp = √(2k_BT/m) — the speed at the peak of the Maxwell–Boltzmann distribution, the slowest of its three characteristic speeds.
Definition
The most-probable speed is the speed at which the Maxwell–Boltzmann distribution f(v) reaches its maximum, found by setting df/dv = 0: v_mp = √(2k_BT/m). It is the single speed most molecules are travelling near, and it is the smallest of the three characteristic speeds, below the mean and the root-mean-square: v_mp < ⟨v⟩ < v_rms, in the fixed ratio √2 : √(8/π) : √3.
The three speeds differ because the distribution is asymmetric, with a long high-speed tail that pulls the average above the peak. The most-probable speed is the natural width parameter of the distribution; the Gaussian factor can be written purely in terms of v/v_mp, so v_mp sets the scale on which the whole curve lives.
Like the other characteristic speeds it scales as √(T/m), so heating a gas or using lighter molecules pushes the peak to higher speed. It is the convenient reference speed for laboratory work, including the interpretation of molecular-beam and Doppler-broadening measurements.
History
Identified as part of Maxwell's 1859 analysis of the speed distribution, alongside the mean and rms speeds.