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 step by step, we will analyze the two processes that the ideal gas undergoes and then sketch the corresponding P-V diagram.

### Step 1: Identify the Initial and Final States
- The initial state (A) has a volume of \( V \).
- The final state (B) has a volume of \( V \) after going through two processes.

### Step 2: Process 1 - Isothermal Expansion
- The gas undergoes an isothermal expansion from volume \( V \) to \( 3V \).
- In an isothermal process, the temperature remains constant. According to the ideal gas law, \( PV = nRT \), where \( n \) is the number of moles and \( R \) is the gas constant.
- Since \( n \) and \( R \) are constants, the product \( PV \) remains constant during this process.
- As the volume increases from \( V \) to \( 3V \), the pressure must decrease. This relationship is inversely proportional, which means the graph will be a hyperbola that slopes downwards.

### Step 3: Process 2 - Isobaric Compression
- The second process is a constant pressure (isobaric) process where the volume decreases from \( 3V \) to \( V \).
- Since the pressure remains constant during this process, the P-V diagram will show a horizontal line at the pressure level corresponding to the pressure at volume \( 3V \).
- The line will extend from \( 3V \) to \( V \), indicating that the volume is decreasing while the pressure remains constant.

### Step 4: Sketch the P-V Diagram
- On the P-V diagram:
- Start at point A (volume \( V \), pressure \( P_A \)).
- Draw a curve from point A to point C (volume \( 3V \), pressure \( P_C \)), indicating the isothermal expansion.
- From point C, draw a horizontal line to point B (volume \( V \), pressure \( P_C \)), indicating the isobaric compression.

### Step 5: Identify the Correct P-V Diagram
- Based on the description of the processes, we can now identify the correct P-V diagram from the options provided. The correct diagram should show:
- A downward-sloping curve from \( V \) to \( 3V \) for the isothermal process.
- A horizontal line from \( 3V \) to \( V \) for the isobaric process.

### Conclusion
The correct P-V diagram representing the two processes is option D, as it accurately reflects the isothermal expansion followed by the isobaric compression.