An ideal gas expands isothermally from volume `V_(1)` to `V_(2)` and is then compressed to original volume `V_(1)` adiabatically. Initialy pressure is `P_(1)` and final pressure is `P_(3)`. The total work done is `W`. Then

Options

A. `P_(3) gt P_(1), W gt 0`
B. `P_(3) lt P_(1), W lt 0`
C. `P_(3) gt P_(1), W lt 0`
D. `P_(3) = P_(1), W = 0`

Explanation

To solve the problem step by step, we will analyze the isothermal expansion and the adiabatic compression of an ideal gas, and then determine the relationships between the pressures and the work done.

### Step 1: Understand the Processes
1. The gas undergoes an **isothermal expansion** from volume \( V_1 \) to \( V_2 \) at a constant temperature.
2. After reaching volume \( V_2 \), the gas is then **compressed adiabatically** back to its original volume \( V_1 \).

### Step 2: Analyze the Isothermal Expansion
- During the isothermal expansion, the pressure decreases as the volume increases.
- The initial state is at \( (V_1, P_1) \) and the final state after expansion is at \( (V_2, P_2) \).
- The work done during this process (\( W_1 \)) is positive because the gas is doing work on the surroundings.

### Step 3: Analyze the Adiabatic Compression
- In the adiabatic compression from \( V_2 \) back to \( V_1 \), the pressure increases from \( P_2 \) to \( P_3 \).
- The work done during this process (\( W_2 \)) is negative because work is done on the gas by the surroundings.

### Step 4: Determine Pressure Relationships
- Since the gas is compressed adiabatically, the final pressure \( P_3 \) after the compression will be greater than the initial pressure \( P_1 \) before the expansion. This is due to the nature of adiabatic processes where the pressure increases when the volume decreases.
- Therefore, we conclude that \( P_3 > P_1 \).

### Step 5: Determine Work Done
- The work done during the isothermal expansion \( W_1 \) is positive, while the work done during the adiabatic compression \( W_2 \) is negative.
- Since the magnitude of the work done during the adiabatic compression is greater than that of the isothermal expansion, the total work done \( W \) will be negative.
- Hence, we conclude that \( W < 0 \).

### Final Conclusion
- From the analysis, we have:
- \( P_3 > P_1 \)
- \( W < 0 \)

Thus, the correct answer is that \( P_3 \) is greater than \( P_1 \) and the total work done \( W \) is less than zero.