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The Maxwell-Boltzmann Speed Distribution

A gas at a single temperature is not a crowd of molecules all moving alike; it is a riot of speeds, some crawling, some tearing along, most somewhere in the middle. The spread is not arbitrary. Because each component of a molecule's velocity is an independent Gaussian set by the temperature, the speed, which is the length of that velocity vector, follows the Maxwell-Boltzmann distribution $f(v) \propto v^2 e^{-mv^2/2kT}$. The $v^2$ from the growing surface area of faster-and-faster velocity shells pushes the curve up from zero, and the exponential from the Boltzmann factor pulls it back down, leaving a lopsided bell with a long high-speed tail. Three speeds pin it down, always in the same order and the same ratio: the most probable speed at the peak, the slightly larger mean, and the larger still root-mean-square that sets the pressure and the average kinetic energy. The scene is a box of molecules drawn from exactly this distribution, coloured from blue for the slow ones to red for the fast; the panel below samples speeds and stacks them into a histogram that fills in the smooth $f(v)$ curve as the count grows, with the three characteristic speeds marked. Turn the temperature up and the whole distribution slides to higher speeds and flattens out, keeping its area at one; make the molecules heavier and it shrinks back toward the slow end, since at the same temperature a heavier molecule moves more sedately.

Figure 1. The Maxwell-Boltzmann speed distribution. Top: a box of molecules with MB-distributed speeds, coloured blue (slow) to red (fast). Bottom: sampled speeds stacking into a histogram (blue) that converges to f(v) (gold), with the most probable (green), mean (orange), and rms (red) speeds marked. Method: speeds sampled as the magnitude of a Gaussian velocity. Source: Reif, Fundamentals of Statistical and Thermal Physics, Sec. 7.9-7.10.
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WHAT TO TRY

  • Watch the histogram fill in the smooth $f(v)$ curve as more speeds are sampled, peaking at the most probable speed.
  • Raise the temperature: the distribution slides right and flattens, the molecules in the box turning redder, but the area stays one.
  • Increase the mass: at the same temperature the distribution shrinks back toward slow speeds, $v_p \propto 1/\sqrt{m}$.
  • Note the fixed order $v_p \lt v_\text{avg} \lt v_\text{rms}$ and that the tail always reaches far past the peak.