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The Distribution of Molecular Speeds

Source lecture(s): SC133 Lec 29

Intuition

At 300 K, air molecules average ~500 m/s — but that's an average over anarchy. Any instant, some molecules crawl and a lucky few streak at several kilometres per second, their speeds constantly reshuffled by ~10⁹ collisions per second each. Maxwell's great insight (1860): despite the chaos, the distribution of speeds is fixed, universal, and calculable — statistical order emerging from molecular mayhem. It was physics' first probability distribution, and the door to statistical mechanics.

The Maxwell–Boltzmann distribution

The fraction of molecules with speed near \(v\):

\[f(v) = 4\pi N\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2\, e^{-mv^2/2k_BT}\]

Two competing factors sculpt the famous skewed bell:

  • \(v^2\) — more directions available at higher speed (phase-space volume) — kills the distribution at \(v = 0\)
  • \(e^{-mv^2/2k_BT}\) — the Boltzmann energy penalty — kills it at high \(v\)

Heat the gas and the peak shifts right and flattens; the area (total count) stays fixed. The exponential tail never quite dies — and much of nature lives in that tail.

Three landmark speeds

\[v_p = \sqrt{\frac{2k_BT}{m}} \;<\; \bar v = \sqrt{\frac{8k_BT}{\pi m}} \;<\; v_\text{rms} = \sqrt{\frac{3k_BT}{m}}\]

(most probable : mean : root-mean-square = \(\sqrt2 : \sqrt{8/\pi} : \sqrt3\) ≈ 1 : 1.13 : 1.22). The rms speed is the one tied to temperature and pressure (\(\overline{K} = \tfrac32 k_BT\)); the distribution's skew is why they differ.

The tail runs the world

  • Evaporative cooling: only the fastest molecules escape a liquid's surface; the remainder is cooler — sweating, and the trick behind Bose–Einstein-condensate experiments.
  • Atmospheric escape: a tiny hydrogen/helium tail exceeds Earth's escape velocity (11.2 km/s); over eons those gases leaked away — while the Moon (2.4 km/s) lost everything. N₂ and O₂, 14–16× heavier, have effectively no escaping tail: that's why Earth kept this atmosphere.
  • Chemical reactions & fusion: reactions need collisions above an activation energy — rates track the tail population, hence the exponential temperature sensitivity (Arrhenius), and why fusion plasmas fixate on temperature.

Worked example: hydrogen vs nitrogen at 300 K

\[v_\text{rms}^{H_2} = \sqrt{\frac{3k_BT}{m_{H_2}}} \approx 1930\,\text{m/s} \qquad v_\text{rms}^{N_2} \approx 517\,\text{m/s}\]

Same temperature, same average energy — but H₂ is 14× lighter, hence \(\sqrt{14} \approx 3.7×\) faster, with a far fatter high-speed tail relative to escape velocity. Geology by statistics.

Common mistakes

  • "All molecules move at \(v_\text{rms}\)." The spread is comparable to the mean — the distribution is the physics.
  • Peak = mean. The skew puts \(\bar v\) and \(v_\text{rms}\) to the right of the peak \(v_p\).
  • Doubling \(T\) doubles speeds. Speeds scale as \(\sqrt T\) — doubling 300 K → 600 K raises speeds by 41%.
  • Using Celsius anywhere near these formulas. Kelvin only.

Knowledge graph position

Prerequisites: Ideal gas. Leads to: Second law (statistics → entropy), kinetic theory, Vlasov theory.

Quiz

Q1 (computational). At what temperature does H₂'s rms speed equal N₂'s at 300 K?

Answer

\(v_\text{rms} \propto \sqrt{T/m}\): need \(T_{H_2}/m_{H_2} = 300/m_{N_2}\), so \(T = 300\times(2/28) \approx 21\,\text{K}\) — hydrogen at liquid-helium-ish temperatures still moves like room-temperature nitrogen.

Q2 (conceptual). Why does a puddle evaporate at 25 °C when water "boils at 100 °C"?

Answer

The Maxwell tail: even at 25 °C some surface molecules carry enough KE to break free. Boiling is bulk vapor formation; evaporation is the tail deserting one molecule at a time — cooling what remains.

Q3 (multiple choice). For a gas at temperature \(T\), the ordering is: (a) \(v_p < \bar v < v_\text{rms}\) (b) \(v_\text{rms} < \bar v < v_p\) (c) all equal

Answer

(a) — the high-speed skew drags the averages (especially the squared one) above the peak.