C

To find the ratio of the velocity of sound in hydrogen to that in helium, we use the formula for the speed of sound in a gas, which is given by:
$$ v = ext{sqrt}\left( \frac{\gamma R T}{M} \right) $$
where:
- $v$ is the velocity of sound
- $\gamma$ is the adiabatic index
- $R$ is the universal gas constant
- $T$ is the absolute temperature
- $M$ is the molar mass of the gas.
Given that:
For hydrogen: $\gamma = \frac{7}{5}$ and $M_{H_2} = 2 ext{ g/mol}$
For helium: $\gamma = \frac{5}{3}$ and $M_{He} = 4 ext{ g/mol}$
The ratio of the velocities can be calculated as follows:
$$ \frac{v_{H_2}}{v_{He}} = \frac{\text{sqrt}\left( \frac{\gamma_{H_2} R T}{M_{H_2}} \right)}{\text{sqrt}\left( \frac{\gamma_{He} R T}{M_{He}} \right)} $$
This simplifies to:
$$ = \sqrt{\frac{\gamma_{H_2} \cdot M_{He}}{\gamma_{He} \cdot M_{H_2}}} $$
Plugging in the values:
$$ = \sqrt{\frac{(\frac{7}{5}) \cdot 4}{(\frac{5}{3}) \cdot 2}} $$.
Calculating inside the square root:
$$ = \sqrt{\frac{(\frac{28}{5})}{(\frac{10}{3})}} = \sqrt{\frac{28}{5} \cdot \frac{3}{10}} = \sqrt{\frac{84}{50}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}.
Therefore, after evaluating we find that the correct ratio is $$ \sqrt{\frac{21}{5}} $$, matching option C.