The ratio of the velocity of sound in hydrogen gas to that in helium gas at the same temperature is

Options

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

Explanation

To find the ratio of the velocity of sound in hydrogen gas (H₂) to that in helium gas (He) at the same temperature, we will use the formula for the velocity of sound in a gas, which is given by:

\[ v = \sqrt{\frac{\gamma R T}{M}} \]

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 write:

\[ \frac{v_{H_2}}{v_{He}} = \sqrt{\frac{\gamma_{H_2} R T / M_{H_2}}{\gamma_{He} R T / M_{He}}} \]

Since \( R \) and \( T \) are constant for both gases, they will cancel out:

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

Now we need the values of \( \gamma \) and \( M \) for both gases:

1. For hydrogen (H₂):
- \( \gamma_{H_2} = \frac{7}{5} \)
- \( M_{H_2} = 2 \, \text{g/mol} \)

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

Now substituting these values into the equation:

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

This simplifies to:

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

Calculating the numerator and denominator:

Numerator: \( 7 \cdot 4 \cdot 3 = 84 \)

Denominator: \( 5 \cdot 5 \cdot 2 = 50 \)

So we have:

\[ \frac{v_{H_2}}{v_{He}} = \sqrt{\frac{84}{50}} = \sqrt{\frac{42}{25}} = \frac{\sqrt{42}}{5} \]

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

\[ \frac{v_{H_2}}{v_{He}} = \frac{\sqrt{42}}{5} \]

**Final Answer:**
The ratio of the velocity of sound in hydrogen gas to that in helium gas at the same temperature is \( \frac{\sqrt{42}}{5} \).

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