A student plotted the following four graphs representing the variation of velocity of sound in a gas with the pressure P. Whic one is correct?

Options

A.
B.
C.
D.

Explanation

To determine the correct graph representing the variation of the velocity of sound in a gas with pressure, we can follow these steps:

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

### Step 2: Analyze how pressure and density change
When the pressure \( P \) of a gas increases, the density \( \rho \) also increases. However, the ratio \( \frac{P}{\rho} \) remains constant for a given gas under adiabatic conditions. This means that as pressure increases, density increases in such a way that the ratio \( \frac{P}{\rho} \) does not change.

### Step 3: Determine the implication for the speed of sound
Since the ratio \( \frac{P}{\rho} \) is constant, we can conclude that the speed of sound \( v \) does not depend on the pressure \( P \). Therefore, the speed of sound remains constant regardless of changes in pressure.

### Step 4: Identify the correct graph
Given that the speed of sound is constant with respect to pressure, the correct graph would be a horizontal line (constant value) indicating that the velocity of sound does not change with pressure.

From the options provided, the graph that represents this constant relationship is:

**Option D**: A horizontal line indicating constant velocity.

### Final Conclusion
The correct answer is **Option D**, which shows that the velocity of sound in a gas is independent of pressure.

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