Problem — Escape temperature of hydrogen

Problem

At what temperature would the root-mean-square speed of a hydrogen molecule ($m_\text{mol} = 2\cdot 10^{-3}$ kg/mol) reach the escape velocity from Earth’s surface, $v_\text{fuga}\approx 1{,}12\cdot 10^4$ m/s? Comment on the result in relation to the scarcity of hydrogen in Earth’s atmosphere.

Solution

Set the root-mean-square speed equal to the escape velocity: $v_\text{rms} = \sqrt{\frac{3RT}{m_\text{mol}}} = v_\text{fuga}$ Square both sides and solve for $T$: $T = \frac{m_\text{mol}\,v_\text{fuga}^2}{3R}$ Substitute the numerical values: $T = \frac{2\cdot 10^{-3}\cdot (1{,}12\cdot 10^{4})^2}{3\cdot 8{,}31}$ $T = \frac{2\cdot 10^{-3}\cdot 1{,}25\cdot 10^{8}}{24{,}9} \approx 1{,}0\cdot 10^{4}\ \text{K}$ Comment. About $10\,000$ K would be needed for the average hydrogen molecule to reach the escape velocity: the atmosphere is much colder (~$300$ K). However, the Maxwell–Boltzmann distribution has a high-speed tail: since hydrogen is extremely light, a non-negligible fraction of its molecules nonetheless exceed $v_\text{fuga}$ and escape into space over geological ages. This is why free hydrogen is rare in Earth’s atmosphere, while nitrogen and oxygen, much heavier, remain trapped. $\ev{T \approx 1{,}0\cdot 10^{4}\ \text{K}}$

Collegamenti

Argomenti: Teoria cinetica dei gas Concetti: Velocità quadratica media · Temperatura Competenze: Impostazione simbolica