Root-mean-square Speed

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A sealed flask contains oxygen gas in thermal equilibrium at 300 K; determine the root-mean-square speed of its molecules.

RankGuru · Accepted Answer

The root-mean-square speed measures the typical molecular speed and follows $v_{rms}=\sqrt{\dfrac{3RT}{M}}$, where $R=8.314$ J/mol\cdotK, $T$ is absolute temperature, and $M$ is molar mass in kg/mol. The formula comes from equating the average translational kinetic energy $\frac{3}{2}k_BT$ per molecule to $\frac{1}{2}m\overline{v^2}$ and then scaling up to one mole. For oxygen, $M=32	imes10^{-3}$ kg/mol. Substituting $T=300$ K: $v_{rms}=\sqrt{\dfrac{3	imes8.314	imes300}{0.032}}=\sqrt{233{,}800}\approx484$ m/s. Heavier molecules move more slowly at the same temperature, which is why oxygen is slower than a light gas such as hydrogen. The value 419 m/s would correspond to nitrogen ($M=28$) rather than oxygen. The value 681 m/s incorrectly drops a factor and overestimates the speed. The value 322 m/s mistakenly applies $\sqrt{2RT/M}$, the most probable speed expression, instead of the rms formula. This calculation reflects the NCERT kinetic theory treatment relating molecular speed to temperature and molar mass. As a sanity check, molecular speeds in common gases at room temperature lie in the range of a few hundred metres per second and are comparable to the speed of sound in the gas, so 484 m/s is physically reasonable for oxygen.

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