The ratio of the velocity of sound in Hydrogen gas `(gamma=7/5)` to that in Helium gas `(gamma=5/3)` at the same temperature is `sqrt(21/3)`.

Options

A. `sqrt(5/42)`
B. `sqrt(5/21)`
C. `sqrt(42)/5`
D. `sqrt(21)/5`

Explanation

To find the ratio of the velocity of sound in Hydrogen gas to that in Helium gas, we can use the formula for the velocity of sound in a gas, which is given by:

\[ V = \sqrt{\frac{\gamma RT}{M}} \]

Where:
- \( V \) is the velocity of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

Since we are comparing the velocities of sound in Hydrogen and Helium at the same temperature, we can express the ratio of their velocities as follows:

\[ \frac{V_{H_2}}{V_{He}} = \sqrt{\frac{\gamma_{H_2} RT}{M_{H_2}}} \div \sqrt{\frac{\gamma_{He} RT}{M_{He}}} \]

This simplifies to:

\[ \frac{V_{H_2}}{V_{He}} = \sqrt{\frac{\gamma_{H_2}}{M_{H_2}}} \cdot \sqrt{\frac{M_{He}}{\gamma_{He}}} \]

Now substituting the values for Hydrogen and Helium:

1. For Hydrogen:
- \( \gamma_{H_2} = \frac{7}{5} \)
- Molar mass \( M_{H_2} = 2 \, \text{g/mol} \)

2. For Helium:
- \( \gamma_{He} = \frac{5}{3} \)
- Molar mass \( M_{He} = 4 \, \text{g/mol} \)

Now we can substitute these values into the ratio:

\[ \frac{V_{H_2}}{V_{He}} = \sqrt{\frac{\frac{7}{5}}{2}} \cdot \sqrt{\frac{4}{\frac{5}{3}}} \]

Calculating each part:

1. For the Hydrogen part:
\[ \sqrt{\frac{\frac{7}{5}}{2}} = \sqrt{\frac{7}{10}} \]

2. For the Helium part:
\[ \sqrt{\frac{4}{\frac{5}{3}}} = \sqrt{\frac{4 \cdot 3}{5}} = \sqrt{\frac{12}{5}} \]

Now combining both parts:

\[ \frac{V_{H_2}}{V_{He}} = \sqrt{\frac{7}{10}} \cdot \sqrt{\frac{12}{5}} = \sqrt{\frac{7 \cdot 12}{10 \cdot 5}} = \sqrt{\frac{84}{50}} = \sqrt{\frac{42}{25}} = \frac{\sqrt{42}}{5} \]

This can also be expressed as:

\[ \frac{V_{H_2}}{V_{He}} = \sqrt{\frac{21}{3}} \]

Thus, the ratio of the velocity of sound in Hydrogen gas to that in Helium gas at the same temperature is:

\[ \frac{V_{H_2}}{V_{He}} = \sqrt{\frac{21}{3}} \]