If the root mean square speed of hydrogen molecules at a temperature $T$ is $v$, then the root mean square speed of oxygen molecules at the same temperature is

Solution

1. The root mean square speed is given by the formula $v_{rms}=\sqrt{\frac{3RT}{M}}$, which implies $v_{rms}\propto \frac{1}{\sqrt{M}}$.
2. The molar mass of hydrogen ($H_2$) is $M_{H_2}=2\text{ g/mol}$ and the molar mass of oxygen ($O_2$) is $M_{O_2}=32\text{ g/mol}$.
3. Setting up the ratio, we get $$\begin{array}{rl}\frac{v_{O_2}}{v_{H_2}}& =\sqrt{\frac{M_{H_2}}{M_{O_2}}}\\ & =\sqrt{\frac{2}{32}}\\ & =\sqrt{\frac{1}{16}}\\ & =\frac{1}{4}\end{array}$$.
4. Since $v_{H_2}=v$, it follows that $v_{O_2}=\frac{v}{4}$.
5. Hence the answer is (A).