Physics 9702 · Paper 4

r.m.s. Speed

“How fast do air molecules move?” needs care: the average velocity of a gas going nowhere is zero. The useful average comes from the energy equation.

Root-mean-square

The is $c_{\text{rms}} = \sqrt{\langle c^2 \rangle}$ : square every speed, take the mean, then the square root. It is exactly the speed that appears in the kinetic energy $\tfrac{1}{2}m\langle c^2 \rangle$ , which is why physics prefers it to a plain average.
From $\tfrac{1}{2}m\langle c^2 \rangle = \tfrac{3}{2}kT$ : $\langle c^2 \rangle = \dfrac{3kT}{m}$ , so $c_{\text{rms}} = \sqrt{3kT/m}$ — hotter means faster, lighter means faster.

Worked example

Air molecules at room temperature

Find the r.m.s. speed of nitrogen molecules (mass 28 u, $1\,\text{u} = 1.66 \times 10^{-27}$ kg) at 300 K.

$m = 28 \times 1.66\times10^{-27} = 4.6 \times 10^{-26}\ \text{kg}$ .
$\langle c^2 \rangle = \dfrac{3kT}{m} = \dfrac{3 \times 1.38\times10^{-23} \times 300}{4.6\times10^{-26}} = 2.7 \times 10^{5}\ \text{m}^2\text{s}^{-2}$ .
$c_{\text{rms}} = \sqrt{2.7 \times 10^{5}} = 520\ \text{m s}^{-1}$ .

Answer

c_{\text{rms}} \approx 520\ \text{m s}^{-1}

Faster than sound (340 m s−¹) — as it must be, since sound is carried by these same molecules.

Common mistake

Asked for

\langle c^2 \rangle

, stop at the mean square; asked for the r.m.s. speed, remember the final square root. Papers deliberately alternate between the two, and the factor-of-thousands difference makes the slip obvious to examiners but invisible to a hurried candidate.

How speed scales

r.m.s. speed against T, sampled from the real function c ∝ √T: through the origin, gradient always decreasing. Quadrupling T only doubles the r.m.s. speed.

$c_{\text{rms}} \propto \sqrt{T}$ : the sketch asked in recent papers is a curve through the origin with decreasing gradient — quadruple the kelvin temperature and the speed only doubles (9702/41/O/N/24 Q3(b)(ii)) (9702/42/M/J/25 Q4(c)).
Two different gases at the same temperature have the same average kinetic energy, so the lighter molecules move faster: $c_{\text{rms}} \propto 1/\sqrt{m}$ . Hydrogen (2 u) beats oxygen (32 u) by a factor of $\sqrt{16} = 4$ (9702/41/O/N/24 Q3(b)) (9702/42/M/J/24 Q2(b)).

Why the Moon keeps no air

At 400 K a hydrogen molecule's r.m.s. speed is about 2200 m s−¹. The Moon's escape speed is only about 2400 m s−¹ — and in any speed distribution plenty of molecules move faster than the r.m.s. value. Over time they leak away into space, which is why the Moon holds no atmosphere. A 2023 paper walked through exactly this argument (9702/42/O/N/23 Q2(d)(e)).

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