The velocity of sound at 15°C and 380 mm pressure is 340 m/s. If the temperature remains constant and pressure is doubled, the velocity of sound will be

Options

A. 340m/s
B. 170m/s
C. 680 m/s
D. None of these

Explanation

To solve the problem of how the velocity of sound changes with pressure while keeping the temperature constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The velocity of sound in a medium is given by the formula:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]
where \( v \) is the velocity of sound, \( \gamma \) is the adiabatic index (a constant for a given gas), \( P \) is the pressure, and \( \rho \) is the density of the medium.

2. **Identify Constants**: In this problem, we are told that the temperature remains constant. This means that while pressure is changing, the density of the gas will also change in such a way that the ratio \( \frac{P}{\rho} \) remains constant.

3. **Apply Ideal Gas Law**: According to the ideal gas law, the relationship between pressure, volume, and temperature can be expressed as:
\[ PV = nRT \]
Rearranging this gives:
\[ P = \frac{nRT}{V} \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

4. **Density Relation**: We can express density \( \rho \) as:
\[ \rho = \frac{m}{V} \]
where \( m \) is the mass of the gas.

5. **Constant Ratio**: Since temperature \( T \), gas constant \( R \), and molar mass \( M \) are constants, the ratio \( \frac{P}{\rho} \) remains constant when the temperature does not change. This means that if pressure \( P \) is doubled, the density \( \rho \) will also change in such a way that the ratio \( \frac{P}{\rho} \) remains constant.

6. **Conclusion on Velocity**: Since \( v \) is dependent on the ratio \( \frac{P}{\rho} \), and this ratio does not change when pressure is doubled (because density also increases), the velocity of sound \( v \) remains unchanged.

7. **Final Answer**: Therefore, the velocity of sound remains the same at 340 m/s.