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 involved in the transformation of the ideal gas and how they are represented in a P-V graph.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The gas starts at pressure \( P_1 \) and volume \( V_1 \).

2. **Isothermal Compression**:
- The gas is compressed isothermally. In an isothermal process, the temperature remains constant, and the relationship between pressure and volume is given by Boyle's Law: \( P_1 V_1 = P_2 V_2 \).
- On a P-V graph, this process will be represented by a hyperbolic curve that slopes downwards to the right, indicating that as volume decreases, pressure increases.

3. **Adiabatic Expansion**:
- After isothermal compression, the gas undergoes adiabatic expansion until it returns to pressure \( P_1 \). During an adiabatic process, no heat is exchanged with the surroundings, and the relationship between pressure and volume is given by \( PV^{\gamma} = \text{constant} \), where \( \gamma \) is the heat capacity ratio.
- The P-V graph for this process will also slope downwards but steeper than the isothermal curve, indicating that pressure decreases more rapidly with volume increase.

4. **Return to Original Volume**:
- Finally, the gas is allowed to expand to its original volume \( V_1 \) at constant pressure \( P_1 \). This is an isobaric process, represented by a horizontal line on the P-V graph, as the pressure remains constant while the volume increases.

5. **Combining the Processes**:
- The complete cycle on the P-V graph will start from point \( (P_1, V_1) \), move downwards along the isothermal curve to a new point \( (P_2, V_2) \), then steeply downwards along the adiabatic curve until it reaches back to pressure \( P_1 \) at a new volume \( V_3 \), and finally move horizontally to the right back to the original volume \( V_1 \) at pressure \( P_1 \).

6. **Identifying the Correct P-V Graph**:
- The correct P-V graph will show:
- A downward curve for the isothermal compression.
- A steeper downward curve for the adiabatic expansion.
- A horizontal line returning to the original volume at constant pressure.

### Conclusion:
The correct P-V graph will show these three distinct processes clearly, with the isothermal curve having a gentler slope than the adiabatic curve, followed by a horizontal line indicating isobaric expansion.