= Most probable speed in the Maxwell distribution
{title2=$v_{\rm mp}=\sqrt{2k_BT/m}$}
= Most probable speed
{synonym}
The most probable speed is the <mode>, or density maximum, of the <Maxwell speed distribution>:
$$
v_{\rm mp}=\sqrt{\frac{2k_BT}{m}}=\sqrt2\,s.
$$
For small equal-width <speed> bins, the bin near $v_{\rm mp}$ contains the largest expected fraction of particles. This does not mean that any exact <speed> has positive <probability>, or that every particle has this <speed>. The three common summaries obey
$$
\frac{v_{\rm mp}}s=\sqrt2\approx1.414,
\qquad \frac{\langle v\rangle}s=\sqrt{\frac8\pi}\approx1.596,
\qquad \frac{v_{\rm rms}}s=\sqrt3\approx1.732,
$$
so $v_{\rm mp}<\langle v\rangle<v_{\rm rms}$. The first describes a density peak, the second the <mean> <speed>, and the third the <root mean square> associated with <kinetic energy>. https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds[OpenStax, characteristic molecular speeds]{ref}
For a concrete scale $s=300\,\mathrm{m\,s^{-1}}$, these are approximately $424$, $479$, and $520\,\mathrm{m\,s^{-1}}$. The most probable <velocity> vector and each most probable component remain zero in the gas rest frame. A zero vector labels one small region of <velocity> space, whereas a positive <speed> labels an entire <spherical shell>.
A short check uses the <derivative> of the <logarithm> of the <Maxwell speed distribution> for $v>0$:
$$
\frac{d}{dv}\log g(v)=\frac2v-\frac{v}{s^2}.
$$
It is positive below $\sqrt2\,s$ and negative above it, locating the unique peak. The factor $v^2$ from the <spherical shell> supplies the term $2/v$; omitting it would incorrectly move the <speed> peak to zero.
Back to article page
