A sample of gas follow process represented by `PV^(2)` = constant . Bulk modulus for this process is `B`, then which of the following graph is correct ?

Options

A.
B.
C.
D.

Answer

To solve the problem step by step, we will analyze the given process \( PV^2 = \text{constant} \) and derive the relationship for the bulk modulus \( B \).

### Step-by-Step Solution:

1. **Understanding the Process**:
The process is given by the equation \( PV^2 = \text{constant} \). This implies that as the pressure \( P \) and volume \( V \) change, their relationship is defined by this equation.

2. **Rearranging the Equation**:
We can express pressure \( P \) in terms of volume \( V \):
\[ P = k V^{-2} \]
where \( k \) is a constant.

3. **Differentiating the Equation**:
To find the relationship between changes in pressure and volume, we differentiate \( P \):
\[ \frac{dP}{dV} = -2k V^{-3} \]

4. **Using the Definition of Bulk Modulus**:
The bulk modulus \( B \) is defined as:
\[ B = -V \frac{dP}{dV} \]
Substituting the expression for \( \frac{dP}{dV} \):
\[ B = -V \left(-2k V^{-3}\right) = 2k V^{-2} \]

5. **Relating Bulk Modulus to Pressure**:
From our earlier expression for \( P \):
\[ P = k V^{-2} \implies k = P V^2 \]
Substituting this back into the expression for \( B \):
\[ B = 2P \]
This shows that the bulk modulus \( B \) is directly proportional to the pressure \( P \).

6. **Analyzing the Relationship**:
Since \( B \) is proportional to \( P \), we can conclude that if we plot \( B \) against \( P \), the graph will be a straight line passing through the origin.

7. **Considering Temperature and Volume**:
Using the ideal gas equation \( PV = nRT \), we can express \( B \) in terms of temperature \( T \) and volume \( V \):
\[ B = 2P \implies B \propto T \quad \text{(since } P \propto T \text{ for a fixed volume)} \]
Additionally, since \( B \) is inversely proportional to \( V \), the relationship can be expressed as:
\[ B \propto \frac{1}{V} \]
This indicates that if we plot \( B \) against \( T \) and \( V \), we will observe a hyperbolic relationship.

### Conclusion:
The correct graph that represents the relationship between the bulk modulus \( B \) and pressure \( P \) is a straight line, while the relationship between \( B \) and temperature \( T \) and volume \( V \) will be hyperbolic.