Solution:

The relation between the root mean square (r.m.s.) velocity \( v_{\text{rms}} \) and the most probable velocity \( v_{\text{mp}} \) of a gas is derived from their respective formulas:

1. **Most probable velocity** \( v_{\text{mp}} \):
\[
v_{\text{mp}} = \sqrt{\frac{2 k_B T}{m}}
\]

2. **Root mean square velocity** \( v_{\text{rms}} \):
\[
v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}
\]

Dividing \( v_{\text{rms}} \) by \( v_{\text{mp}} \):

\[
\frac{v_{\text{rms}}}{v_{\text{mp}}} = \sqrt{\frac{3}{2}}
\]

Thus:

\[
v_{\text{rms}} = \sqrt{\frac{3}{2}} \, v_{\text{mp}}
\]