The ratio of the root-mean-square (rms) speed and the most probable speed for molecules in an ideal gas is:

From the kinetic theory of gases, the statistical speeds of ideal gas molecules are given by the formulas:
* Root-mean-square speed:

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

* Most probable speed:

$$v_{\text{mp}}=\sqrt{\frac{2RT}{M}}$$

Taking the ratio of $v_{\text{rms}}$ to $v_{\text{mp}}$:

$$\frac{v_{\text{rms}}}{v_{\text{mp}}}=\frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{2RT}{M}}}=\frac{\sqrt{3}}{\sqrt{2}}$$

Thus, the ratio is $\sqrt{3}:\sqrt{2}$.