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(B) The root mean square speed $({v_{rms}})$ of gas molecules is given by the formula: ${v_{rms}} = \sqrt{\frac{3RT}{M}}$,where $R$ is the universal g

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$\frac{1}{4}$

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(B) The root mean square speed $({v_{rms}})$ of gas molecules 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 the temperature $T$ is constant,we have ${v_{rms}} \propto \frac{1}{\sqrt{M}}$.Therefore,the ratio of the ${v_{rms}}$ of ${O_2}$ molecules to that of ${H_2}$ molecules is:$\frac{{v_{rms, O_2}}}{{v_{rms, H_2}}} = \sqrt{\frac{M_{H_2}}{M_{O_2}}}$Given that the molar mass of ${H_2}$ is $M_{H_2} = 2 	ext{ g/mol}$ and the molar mass of ${O_2}$ is $M_{O_2} = 32 	ext{ g/mol}$.Substituting these values:$\frac{{v_{rms, O_2}}}{{v_{rms, H_2}}} = \sqrt{\frac{2}{32}} = \sqrt{\frac{1}{16}} = \frac{1}{4}$.

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