Problem — Hydrogen or oxygen at the same temperature

Problem

Two different gases, hydrogen and oxygen, are kept at the same temperature. Which of the two has the faster molecules on average? Explain why.

Solution

The temperature fixes the mean translational kinetic energy per molecule, which is the same for both gases and does not depend on mass: $\bar E = \tfrac{3}{2}k_B T$ From $\tfrac{1}{2}m\,v_\text{rms}^2 = \tfrac{3}{2}k_B T$ it follows that the speed depends on the mass: $v_\text{rms}=\sqrt{\frac{3k_B T}{m}}\propto \frac{1}{\sqrt{m}}$ At equal kinetic energy, lighter molecules must move faster. Hydrogen ($m_{H_2}\approx 2$ u) is much lighter than oxygen ($m_{O_2}\approx 32$ u), so it is faster: precisely, $v_\text{rms}$ of hydrogen is $\sqrt{32/2}=4$ times that of oxygen. $\ev{\text{Hydrogen: same } \bar E=\tfrac{3}{2}k_B T,\ \text{but } v_\text{rms}\propto 1/\sqrt{m}\Rightarrow H_2 \text{ faster}}$

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