Rms Speed

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Hydrogen and oxygen samples are kept at the same temperature, and given that oxygen is sixteen times as massive per molecule as hydrogen, how do their rms speeds compare?

RankGuru · Accepted Answer

The comparison hinges on how rms speed depends on molar mass, a relationship given in NCERT Class 11, Chapter 13 (Kinetic Theory) as $v_{rms} = \sqrt{\frac{3RT}{M}}$. At a common temperature the factor $3RT$ is identical for both gases, so the rms speed varies as $\frac{1}{\sqrt{M}}$, meaning lighter molecules move faster. The ratio of speeds is $\frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{M_{O_2}}{M_{H_2}}} = \sqrt{16} = 4$, so hydrogen molecules travel four times faster than oxygen molecules. The option of sixteen times ignores the square root and uses the raw mass ratio. The option of identical speeds forgets that speed depends on mass even when temperature is shared. The option that oxygen is faster inverts the physical relationship, since heavier molecules are actually slower. This inverse dependence on the square root of mass is a direct consequence of every molecule sharing the same average translational kinetic energy at a common temperature. A sanity check supports the result: because kinetic energy is equal at equal temperature, the heavier oxygen must move more slowly to hold the same energy, and a mass ratio of sixteen gives a speed ratio of exactly its square root, four.

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