A certain mass of an ideal gas is at pressure `P_(1)` and volume `V_(1)`. If is compressed isothermally and then allowed to expand adiabatically untill its pressure returns to `P_(1)`. The gas is then allowed to expand its original volume. Which of the following `P -V` graphs are these process correctly shown?

Options

A.
B.
C.
D.

Explanation

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

1. **Isothermal Compression**: The gas is compressed at a constant temperature.
2. **Adiabatic Expansion**: The gas expands without heat exchange with the surroundings until it returns to its initial pressure \( P_1 \).
3. **Expansion to Original Volume**: The gas expands to its original volume \( V_1 \).

### Step-by-Step Solution:

**Step 1: Initial State**
- The gas starts at pressure \( P_1 \) and volume \( V_1 \). This point is our starting point on the graph.

**Step 2: Isothermal Compression**
- During isothermal compression, the gas's volume decreases while the pressure increases, but the temperature remains constant. This process is represented by a downward curve on the P-V graph.
- The curve will start from point \( (V_1, P_1) \) and move to a new point \( (V_2, P_2) \) where \( V_2 < V_1 \) and \( P_2 > P_1 \).

**Step 3: Adiabatic Expansion**
- After the isothermal compression, the gas undergoes adiabatic expansion. In this process, the gas expands and does work on the surroundings without any heat exchange, resulting in a decrease in pressure.
- This process will be represented by a steeper curve moving from point \( (V_2, P_2) \) to a new point \( (V_3, P_1) \) where the pressure returns to \( P_1 \) but the volume \( V_3 > V_2 \).

**Step 4: Expansion to Original Volume**
- Finally, the gas expands to its original volume \( V_1 \) at constant pressure \( P_1 \). This is represented by a horizontal line moving from \( (V_3, P_1) \) to \( (V_1, P_1) \).

**Step 5: Complete P-V Graph**
- The complete P-V graph will show:
- A downward curve for isothermal compression from \( (V_1, P_1) \) to \( (V_2, P_2) \).
- A steeper downward curve for adiabatic expansion from \( (V_2, P_2) \) to \( (V_3, P_1) \).
- A horizontal line from \( (V_3, P_1) \) to \( (V_1, P_1) \).

### Conclusion
The correct P-V graph that represents these processes is option A.