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(D) The root mean square velocity $(v_{rms})$ of a gas molecule is given by the formula $v_{rms} = \sqrt{\frac{3RT}{M}}$,where $R$ is the universal ga

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$1:4$

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(D) The root mean square velocity $(v_{rms})$ of a gas molecule is given by the formula $v_{rms} = \sqrt{\frac{3RT}{M}}$,where $R$ is the universal gas constant,$T$ is the absolute temperature,and $M$ is the molar mass of the gas.Since $R$ and $T$ are constant for both gases,we have $v_{rms} \propto \frac{1}{\sqrt{M}}$.Therefore,the ratio of the root mean square velocities of oxygen $(O_2)$ and hydrogen $(H_2)$ is given by $\frac{(v_{rms})_{O_2}}{(v_{rms})_{H_2}} = \sqrt{\frac{M_{H_2}}{M_{O_2}}}$.The molar mass of $H_2$ is $M_{H_2} = 2 \ g/mol$ and the molar mass of $O_2$ is $M_{O_2} = 32 \ g/mol$.Substituting these values,we get $\frac{(v_{rms})_{O_2}}{(v_{rms})_{H_2}} = \sqrt{\frac{2}{32}} = \sqrt{\frac{1}{16}} = \frac{1}{4}$.Thus,the ratio is $1:4$.

At a given temperature,the root mean square velocities of oxygen and hydrogen molecules are in the ratio:

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