Comparison Of Molecular Speeds

Because root-mean-square speed varies as $v_{rms}=\sqrt{3RT/M}$, at a fixed temperature it depends only on molar mass, giving $\frac{v_{H_2}}{v_{O_2}}=\sqrt{\frac{M_{O_2}}{M_{H_2}}}$. The temperature cancels because both gases share the same average thermal energy per molecule, a direct consequence of being in thermal equilibrium at the same temperature. Inserting $M_{H_2}=2$ g/mol and $M_{O_2}=32$ g/mol gives $\sqrt{32/2}=\sqrt{16}=4$. Thus hydrogen molecules move four times faster on average than oxygen molecules. This large difference explains why light gases such as hydrogen and helium diffuse and effuse much more rapidly and even escape planetary atmospheres more easily over geological time. The value 16 results from forgetting the square root and using the mass ratio directly. The value 2 comes from misreading hydrogen's molar mass as 8. The value 8 arises from an incorrect molar mass of 0.5 for hydrogen. This problem applies the NCERT principle that at equal temperatures all gases possess equal average translational kinetic energy, so lighter molecules must move proportionally faster to carry the same energy. As a plausibility check, the kinetic energies $\frac{1}{2}M{v}_{rms}^2$ of the two gases are indeed equal when the speed ratio is 4 and the mass ratio is 16, confirming consistency with equipartition.