The velocity distribution

$v_\text{rms}$ is an average speed (in the quadratic sense), but in reality molecules have very different speeds from one another: some almost still, others extremely fast. The distribution of all the speeds of a gas in thermal equilibrium was found independently by James Clerk Maxwell (1860) and Ludwig Boltzmann (1872).

Principle — Maxwell-Boltzmann distribution

For a gas of molecules of mass $m$ in thermal equilibrium at temperature $T$, the fraction of molecules with speed (in magnitude) between $v$ and $v + dv$ is $\ev{f(v) = 4\pi\,n\,\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2\,\exp\!\left(-\frac{m v^2}{2 k_B T}\right)}$ The area under the curve $f(v)$ between $0$ and $\infty$ is the total number of molecules $N$.

The shape of the curve arises from two competing factors: the $v^2$ term makes it grow from zero (few molecules have almost zero speed), while the exponential term $\exp(-mv^2/2k_BT)$ then makes it decay (very few molecules have enormous speeds). The result is an asymmetric bell shape, with a peak and a long tail towards high speeds.

Speed distribution at two temperatures. As $T$ increases, the peak shifts to the right and lowers, while the area underneath (the total number of molecules $N$) remains unchanged.

In summary

Molecules do NOT all have the same speed: there is a distribution. Peak at $v_p$, exponential tail towards high speeds. As $T$ increases, the peak shifts right and lowers, but the area under the curve is fixed.

Try it — interactive simulation

In the simulation below two hard-disk gases — a hot one (red, on top) and a cold one (blue, at the bottom) — are separated by a lid: each settles into its own Maxwell-Boltzmann distribution. Remove the lid and the two gases mix and, exchanging energy through collisions, both distributions converge onto a single common Maxwell-Boltzmann curve at the shared temperature (in 2D, Rayleigh distributions).

Two gases split by a lid (hot red on top, cold blue at the bottom): remove the lid and watch the two speed distributions merge into one common Maxwell-Boltzmann distribution.

Collegamenti

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

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