Velocity of sound in a gaseous medium is `330 m s^(-1)`. If the pressure is increased by 4 times without change in temperature, the velocity of sound in the gas is

Options

A. `330 m s^(-1)`
B. `660 m s^(-1)`
C. `156 m s^(-1)`
D. `990 m s^(-1)`

Explanation

To find the new velocity of sound in the gas after the pressure is increased by 4 times without changing the temperature, we can follow these steps:

### Step-by-Step Solution:

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

2. **Initial Conditions**:
We know that the initial velocity of sound is \( v_1 = 330 \, m/s \) and the initial pressure is \( P \).

3. **Change in Pressure**:
The pressure is increased by 4 times, so the new pressure \( P' \) is:
\[ P' = 4P \]

4. **Effect on Density**:
When the pressure increases, the density \( \rho \) of the gas also changes. According to the ideal gas law, if the temperature remains constant, the relationship between pressure and density can be expressed as:
\[ P = \rho RT \]
Therefore, if pressure increases, density will also increase. However, the ratio \( \frac{P}{\rho} \) remains constant because both \( P \) and \( \rho \) increase proportionally.

5. **New Velocity Calculation**:
The new velocity of sound \( v' \) can be expressed as:
\[ v' = \sqrt{\frac{\gamma P'}{\rho'}} \]
Substituting \( P' = 4P \) and recognizing that \( \frac{P'}{\rho'} = \frac{P}{\rho} \):
\[ v' = \sqrt{\frac{\gamma (4P)}{\rho'}} = \sqrt{4 \cdot \frac{\gamma P}{\rho'}} = 2 \sqrt{\frac{\gamma P}{\rho'}} = 2v \]

6. **Conclusion**:
Since \( \frac{P}{\rho} \) remains constant, the velocity of sound does not change with the increase in pressure. Therefore, the new velocity of sound remains:
\[ v' = v = 330 \, m/s \]

### Final Answer:
The velocity of sound in the gas after the pressure is increased by 4 times without changing the temperature is **330 m/s**.