Problem — Hydrogen or oxygen faster

Problem

Two identical containers hold hydrogen ($H_2$) and oxygen ($O_2$) respectively, at the same temperature and pressure. In which of the two do the molecules move faster on average? By how much (ratio of the $v_\text{rms}$)? Is the average kinetic energy per molecule the same or different?

Solution

The root-mean-square speed depends on mass: $v_\text{rms} = \sqrt{3k_B T/m}$. At equal temperature, the lighter molecule is the faster one. Hydrogen ($m_{H_2}=2$ u) is much lighter than oxygen ($m_{O_2}=32$ u), so $H_2$ is faster.

The ratio of speeds (same $T$, the factor $3k_B T$ cancels): $\frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{m_{O_2}}{m_{H_2}}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4$ Hydrogen moves on average 4 times faster.

The average translational kinetic energy per molecule depends only on temperature: $\langle E_\text{kin}\rangle = \tfrac{3}{2}k_B T$ Being at the same $T$, it is equal for both gases: the higher speed of $H_2$ exactly compensates for its lower mass. $\ev{H_2\ \text{faster};\ \ v_{H_2}/v_{O_2}=4;\ \ \langle E_\text{kin}\rangle\ \text{equal}}$

Connections

Topics: Teoria cinetica dei gas Concepts: Velocità quadratica media · Energia cinetica Skills: Impostazione simbolica