1. Use the formula for RMS speed

For a gas, the root mean square speed is

$v_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}}$

where:

• $R$ is the gas constant,
• $T$ is the absolute temperature,
• $M$ is the molar mass.

At the same temperature, we have

$v_{\mathrm{rms}} \propto \frac{1}{\sqrt{M}}$

2. Compare hydrogen and oxygen

For hydrogen gas $\mathrm{H_2}$:

$M_{\mathrm{H_2}} = 2$

For oxygen gas $\mathrm{O_2}$:

$M_{\mathrm{O_2}} = 32$

Therefore,

$\frac{v_{\mathrm{O_2}}}{v_{\mathrm{H_2}}} = \sqrt{\frac{M_{\mathrm{H_2}}}{M_{\mathrm{O_2}}}} = \sqrt{\frac{2}{32}} = \sqrt{\frac{1}{16}} = \frac{1}{4}$

3. Substitute the given speed

Given:

$v_{\mathrm{H_2}} = 2\ \text{km/s}$

So,

$v_{\mathrm{O_2}} = 2 \times \frac{1}{4} = 0.5\ \text{km/s}$

4. Match with the options

$0.5\ \text{km/s}$

So the correct option is:

D: 0.5