Graph between speed v of sound in air and pressure P at a constant temperature is given by

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Answer

To solve the question regarding the graph between the speed \( v \) of sound in air and pressure \( P \) at a constant temperature, we can follow these steps:

### Step 1: Understand the Formula for Speed of Sound
The speed of sound in a medium is given by the formula:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]
where:
- \( v \) is the speed of sound,
- \( \gamma \) is the adiabatic index (constant for a given gas),
- \( P \) is the pressure,
- \( \rho \) is the density of the medium.

### Step 2: Relate Density and Pressure
For an ideal gas, the equation of state is given by:
\[ PV = nRT \]
Rearranging this, we can express pressure in terms of density:
\[ P = \frac{nRT}{V} \]
Since \( n = \frac{m}{M} \) (where \( m \) is mass and \( M \) is molar mass), we can express \( P \) as:
\[ P = \frac{mRT}{MV} \]
From the definition of density \( \rho = \frac{m}{V} \), we can substitute \( \rho \) into the equation:
\[ P = \rho \cdot \frac{RT}{M} \]

### Step 3: Establish a Relationship at Constant Temperature
At constant temperature, \( T \) and \( R \) are constants, which means:
\[ \frac{P}{\rho} = \text{constant} \]
This implies that if the pressure \( P \) increases, the density \( \rho \) must decrease to maintain the ratio constant.

### Step 4: Substitute Back into the Speed of Sound Formula
Substituting \( \frac{P}{\rho} \) back into the speed of sound formula:
\[ v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\gamma \cdot \text{constant}} \]
Since both \( \gamma \) and the ratio \( \frac{P}{\rho} \) are constants, we can conclude that the speed of sound \( v \) remains constant as pressure changes.

### Conclusion
Thus, the graph between the speed \( v \) of sound in air and pressure \( P \) at a constant temperature is a horizontal line, indicating that the speed of sound does not change with pressure at constant temperature.

### Final Answer
The correct answer is that the speed of sound remains constant as pressure changes at constant temperature.

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