if the velocity of sound in helium at room temperature is 330 m/s, then the velocity of sound in hydrogen is

Options

A. 330 m/s
B. 427.7 m/s
C. 1500 m/s
D. 5900 m/s

Answer

To find the velocity of sound in hydrogen given the velocity of sound in helium, we can use the relationship between the velocities of sound in different gases, which is based on their specific heat ratios (gamma) and molar masses.

### Step-by-Step Solution:

1. **Understanding the Formula**:
The velocity of sound in a gas can be expressed as:
\[ V = \sqrt{\frac{\gamma \cdot P}{\rho}} \]
where \( V \) is the velocity of sound, \( \gamma \) is the specific heat ratio, \( P \) is the pressure, and \( \rho \) is the density of the gas.

2. **Using the Ideal Gas Law**:
From the ideal gas law, we know that:
\[ \frac{P}{\rho} = \frac{RT}{M} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

3. **Substituting into the Velocity Formula**:
Substituting the ideal gas law into the velocity formula gives us:
\[ V = \sqrt{\frac{\gamma \cdot RT}{M}} \]

4. **Comparing Velocities of Two Gases**:
For two gases (helium and hydrogen), we can write:
\[ \frac{V_{H2}}{V_{He}} = \sqrt{\frac{\gamma_{H2}}{M_{H2}} \cdot \frac{M_{He}}{\gamma_{He}}} \]
where \( V_{H2} \) and \( V_{He} \) are the velocities of sound in hydrogen and helium, respectively.

5. **Substituting Known Values**:
We know:
- \( V_{He} = 330 \, \text{m/s} \)
- \( \gamma_{H2} = \frac{7}{5} \)
- \( M_{H2} = 2 \, \text{g/mol} \)
- \( \gamma_{He} = \frac{5}{3} \)
- \( M_{He} = 4 \, \text{g/mol} \)

Plugging these values into the equation:
\[ \frac{V_{H2}}{330} = \sqrt{\frac{\frac{7}{5}}{2} \cdot \frac{4}{\frac{5}{3}}} \]

6. **Calculating the Right Side**:
Simplifying the right side:
\[ \frac{V_{H2}}{330} = \sqrt{\frac{7 \cdot 4 \cdot 3}{5 \cdot 2 \cdot 5}} = \sqrt{\frac{84}{50}} = \sqrt{1.68} \]

7. **Finding \( V_{H2} \)**:
Now, calculate \( V_{H2} \):
\[ V_{H2} = 330 \cdot \sqrt{1.68} \]
Approximating \( \sqrt{1.68} \approx 1.296 \):
\[ V_{H2} \approx 330 \cdot 1.296 \approx 427.7 \, \text{m/s} \]

8. **Final Answer**:
Thus, the velocity of sound in hydrogen is approximately:
\[ V_{H2} \approx 427.7 \, \text{m/s} \]

### Conclusion:
The correct answer is option 2: 427.7 m/s.