Problem — Why the atmosphere has no helium

Problem

Why the atmosphere has no helium. Earth’s atmosphere contains large amounts of nitrogen and oxygen, but very little helium. The reason is kinetic: the escape velocity from Earth is $v_f \approx 11\ \text{km/s}$.

• (a) Why does a helium molecule have a much higher probability than a nitrogen molecule of reaching $v_f$ at room temperature?
• (b) Why does no gas stay trapped for long on the Moon?

Answer with purely qualitative but quantitatively justified arguments.

Solution

(a) At the same temperature all molecules have the same mean kinetic energy $\langle E_\text{cin}\rangle = \tfrac{3}{2}k_B T$, but the speed depends on the mass: $v_\text{rms}\propto 1/\sqrt{m}$. Helium ($m=4$ u) is lighter than nitrogen ($m=28$ u), so it is faster by a factor $\frac{v_\text{He}}{v_{N_2}} = \sqrt{\frac{28}{4}} = \sqrt{7} \approx 2{,}6$ Since its typical speed is ~$2{,}6$ times greater, helium’s speed distribution is much more “shifted to the right”: its Maxwell–Boltzmann tail beyond the threshold $v_f$ contains an enormously larger fraction of molecules. Over geological eras, helium evaporates into space, while nitrogen, much slower, stays trapped.

(b) The escape velocity depends on the mass of the body: on the Moon it is much lower, $v_f \approx 2{,}4\ \text{km/s}$, against Earth’s $11$ km/s. With such a low threshold, even heavy molecules (nitrogen, oxygen, carbon dioxide) have a significant fraction in the tail beyond $v_f$ already at ordinary temperature. So any gas evaporates rapidly and the Moon retains no atmosphere. $\ev{\text{He} \sim 2{,}6\times\ \text{faster} \Rightarrow \text{escapes};\quad \text{Moon } v_f\approx 2{,}4\ \text{km/s} \Rightarrow \text{no gas remains}}$

Links

Topics: Kinetic theory of gases Concepts: Maxwell-Boltzmann distribution · Root-mean-square speed Skills: Micro-macro interpretation