Step 1: Write the two speeds.
The speed of sound in a gas and the rms speed of its molecules are: \[ v_s = \sqrt{\frac{\gamma RT}{M}}, \qquad v_{rms} = \sqrt{\frac{3RT}{M}} \] We need their ratio.
Step 2: Find gamma for a diatomic gas.
A diatomic gas molecule has $5$ ways to store energy (3 of moving, 2 of rotating). The ratio of specific heats is: \[ \gamma = 1 + \frac{2}{f} = 1 + \frac{2}{5} = \frac{7}{5} \]
Step 3: Divide the two speeds.
\[ \frac{v_s}{v_{rms}} = \frac{\sqrt{\gamma RT/M}}{\sqrt{3RT/M}} \]
Step 4: Cancel the common terms.
The $R$, $T$ and $M$ all cancel, leaving: \[ \frac{v_s}{v_{rms}} = \sqrt{\frac{\gamma}{3}} \]
Step 5: Put in gamma.
\[ \frac{v_s}{v_{rms}} = \sqrt{\frac{7/5}{3}} = \sqrt{\frac{7}{15}} \]
Step 6: State the answer.
So the required ratio is: \[ \boxed{\sqrt{\frac{7}{15}}} \]