1. Recall the formulas from kinetic theory of gases

For an ideal gas at temperature $T$:

• Most probable speed: $v_p = \sqrt{\frac{2RT}{M}}$

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

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

2. Find the relation between $v_{\mathrm{rms}}$ and $v_p$

Take the ratio:

$\frac{v_{\mathrm{rms}}}{v_p} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{2RT}{M}}} = \sqrt{\frac{3}{2}}$

Therefore,

$v_{\mathrm{rms}} = \sqrt{\frac{3}{2}}\, v_p$

3. Match with the options

• A: $v_{\mathrm{rms}} = \sqrt{\frac{2}{3}} v_p$ ❌
• B: $v_{\mathrm{rms}} = \sqrt{\frac{3}{2}} v_p$ ✅
• C: $v_{\mathrm{rms}} = v_p$ ❌
• D: $v_{\mathrm{rms}} = \sqrt{\frac{1}{3}} v_p$ ❌

4. Conclusion

The correct relation is:

$\boxed{v_{\mathrm{rms}} = \sqrt{\frac{3}{2}}\, v_p}$

So, the correct option is B.