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Step-by-Step Solution
Step 1: Identify the Processes
There are two consecutive processes for one mole of an ideal gas:
An isothermal expansion from volume $V$ to $3V$.
A constant-pressure compression from volume $3V$ back to $V$.
Step 2: Analyze the Isothermal Expansion (Process 1)
For an isothermal process at temperature $T$, the product $PV$ remains constant, i.e., $P \propto \frac{1}{V}$.
Starting at volume $V$ and expanding to $3V$, the pressure drops accordingly so the curve on the $P$ā$V$ diagram is a rectangular hyperbola.
Step 3: Analyze the Constant-Pressure Compression (Process 2)
After the isothermal expansion ends at the volume $3V$, the gas is then compressed back to volume $V$ at a fixed pressure. On a $P$ā$V$ diagram, a constant-pressure process is represented by a horizontal line. Thus, the volume decreases from $3V$ to $V$ while $P$ remains the same.
Step 4: Combine the Processes on the PāV Diagram
The correct diagram must first show an isothermal curve decreasing from $(V, P)$ to $(3V, P_{\text{lower}})$, followed by a horizontal line from $(3V, P_{\text{constant}})$ back to $(V, P_{\text{constant}})$. This matches the shape shown in the correct answer figure:
Final Answer
The PāV diagram in Option 4 is the correct representation of the described processes.
