The three characteristic speeds

From the Maxwell-Boltzmann curve three “characteristic” speeds can be derived, which summarise the whole distribution in three numbers:

• most probable speed $v_p$ (the peak of $f(v)$, the speed possessed by the largest number of molecules): $v_p = \sqrt{2 k_B T / m}$.
• average arithmetic speed $\langle v\rangle$: $\langle v\rangle = \sqrt{8 k_B T / (\pi m)}$.
• root-mean-square speed $v_\text{rms}$: $v_\text{rms} = \sqrt{3 k_B T / m}$.

The three numbers are always in the order $v_p < \langle v\rangle < v_\text{rms}$, with fixed ratios, independent of $T$ and $m$:

$\frac{\langle v\rangle}{v_p} = \frac{2}{\sqrt\pi} \approx 1{,}128 \qquad \frac{v_\text{rms}}{v_p} = \sqrt{\frac{3}{2}} \approx 1{,}225$

Key formula — Maxwell-Boltzmann

$v_p = \sqrt{\frac{2k_BT}{m}}\qquad \langle v\rangle = \sqrt{\frac{8 k_BT}{\pi m}}\qquad v_\text{rms} = \sqrt{\frac{3 k_BT}{m}}$ $v_p : \langle v\rangle : v_\text{rms} \approx 1{,}00 : 1{,}13 : 1{,}22$

Example — Nitrogen at room temperature

For N$_2$ at $T = 300$ K, with mass per molecule $m = 4{,}65\cdot 10^{-26}$ kg: $v_p = \sqrt{\frac{2\cdot 1{,}38\cdot 10^{-23}\cdot 300}{4{,}65\cdot 10^{-26}}} \approx 421\;\text{m/s}$ and also $\langle v\rangle \approx 475$ m/s, $v_\text{rms} \approx 517$ m/s. The fastest molecules, however, have tails that extend well beyond $1000$ m/s: it is thanks to these tails (exponentially small but non-zero) that some molecules can exceed the escape velocity from the atmosphere or cross the Coulomb barrier for nuclear fusion in the Sun.

Historical context

Maxwell derived the distribution in 1860 starting from symmetry assumptions (isotropy and statistical independence of the components $v_x, v_y, v_z$). Boltzmann in 1872 derived it dynamically, showing that it is the steady state of binary collisions: it was the first step towards modern statistical mechanics. It is the same distribution that governs the high-energy tail for fusion in stellar cores (Atkins 2010).

Collegamenti

Argomenti: Teoria cinetica dei gas Concetti: Distribuzione di Maxwell-Boltzmann · Velocità quadratica media

Esercizi collegati: Problem — Why the atmosphere has no helium · Problem — Speed distribution: hydrogen or oxygen · Problem — Escape temperature of hydrogen