Problem — Two gases at the same temperature

Problem

Two gases at the same temperature. Hydrogen (H$_2$) and nitrogen (N$_2$) are kept at the same temperature $T$. Which microscopic quantities of the two gases are equal and which are not? Quantify the ratio of the root-mean-square speeds.

Solution

At the same temperature, what is fixed is the mean translational kinetic energy per molecule, which does not depend on mass: $\langle E_k\rangle=\tfrac{3}{2}k_B T \qquad \text{equal for H}_2 \text{ and N}_2$ What does change is the speed, because at equal energy a larger mass corresponds to a smaller speed: $v_\text{rms}=\sqrt{\frac{3k_B T}{m}}\propto \frac{1}{\sqrt{m}}$ The ratio of the speeds depends only on the ratio of the masses. The molecular mass of N$_2$ ($\approx 28$ u) is $14$ times that of H$_2$ ($\approx 2$ u): $\frac{v_\text{rms}^{H_2}}{v_\text{rms}^{N_2}}=\sqrt{\frac{m_{N_2}}{m_{H_2}}}=\sqrt{14}\approx 3{,}7$ $\ev{\langle E_k\rangle=\tfrac{3}{2}k_B T\ \text{equal};\ v_\text{rms}\propto 1/\sqrt{m}\ \text{different: H}_2\ \sqrt{14}\approx 3{,}7\times\ \text{faster than N}_2}$

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