1. Formulas for molecular speeds in an ideal gas

For an ideal gas at temperature $T$:

• Root mean square speed: $v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$

• Average speed: $v_{\text{avg}}=\sqrt{\frac{8RT}{\pi M}}$

Here, $R$ is the gas constant and $M$ is molar mass.

2. Find the required ratio

We need $\frac{v_{\text{rms}}}{v_{\text{avg}}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}}$

Using $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}},$ we get $\frac{v_{\text{rms}}}{v_{\text{avg}}}=\sqrt{\frac{\frac{3RT}{M}}{\frac{8RT}{\pi M}}}$

Now simplify: $\frac{\frac{3RT}{M}}{\frac{8RT}{\pi M}}=\frac{3RT}{M}\cdot \frac{\pi M}{8RT}=\frac{3\pi}{8}$

Therefore, $\frac{v_{\text{rms}}}{v_{\text{avg}}}=\sqrt{\frac{3\pi}{8}}$

3. Match with options

This corresponds to: $\boxed{\sqrt{\frac{3\pi}{8}}}$

So the correct option is A.

4. Note

The ratio is independent of $T$, $R$, and $M$, so the given values are not actually needed in the final calculation.