One mole of an ideal gas goes from an initial state A to final state B via two processs : It first undergoes isothermal expansion from volume `V` to `3 V` and then its volume is reduced from `3 V` to `V` at constant pressure. The correct `P-V` diagram representing the two process in (figure)

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the two processes that the ideal gas undergoes and how they are represented on a P-V diagram.

### Step-by-Step Solution:

1. **Identify the Initial and Final States**:
- The initial state (A) is at volume \( V \) and some pressure \( P_A \).
- The final state (B) is at volume \( V \) after undergoing two processes.

2. **Process 1: Isothermal Expansion**:
- The gas undergoes isothermal expansion from volume \( V \) to \( 3V \).
- During an isothermal process, the temperature remains constant, and we can use the ideal gas law \( PV = nRT \). Since \( n \) and \( R \) are constants, \( P \) is inversely proportional to \( V \).
- As the volume increases from \( V \) to \( 3V \), the pressure decreases from \( P_A \) to \( P_B \) (where \( P_B \) is the pressure at volume \( 3V \)).
- This process will be represented as a hyperbolic curve on the P-V diagram, moving from point A to point C.

3. **Process 2: Isobaric Compression**:
- After reaching volume \( 3V \), the gas is compressed back to volume \( V \) at constant pressure.
- Since the pressure remains constant during this process, the line on the P-V diagram will be horizontal.
- This process will move from point C to point B.

4. **Combine the Two Processes**:
- The complete path from A to B on the P-V diagram consists of:
- A downward curve from A to C (isothermal expansion).
- A horizontal line from C to B (isobaric compression).

5. **Identify the Correct P-V Diagram**:
- Among the given options, we need to identify the one that shows a downward curve followed by a horizontal line.
- The correct option will be the one that represents the isothermal expansion followed by the isobaric compression.

### Conclusion:
The correct P-V diagram representing the two processes is option D, which shows the isothermal expansion from \( V \) to \( 3V \) followed by isobaric compression from \( 3V \) back to \( V \).