An ideal gas is initially at `P_1,V_1` is expands isothermally to `P_2,V_2` and then compressed adiabatically to the same volume `V_1` and pressure `P_3.` If W is the net work done by the gas in complete process which of the following is true.

Options

A. `W gt O, P_3 gt P_1`
B. `W lt 0, P_3 gt P_1`
C. `W gt 0 , P_3 lt P_1`
D. `W lt , P_3 gt P_1`

Explanation

To solve the problem, we need to analyze the processes the ideal gas undergoes: isothermal expansion and adiabatic compression. We will determine the net work done (W) by the gas during these processes.

### Step-by-Step Solution:

1. **Identify the Processes:**
- The gas expands isothermally from state (P1, V1) to state (P2, V2).
- Then, it is compressed adiabatically back to the original volume (V1) resulting in a new pressure (P3).

2. **Work Done in Isothermal Expansion:**
- For an isothermal process, the work done (W_isothermal) by the gas can be calculated using the formula:
\[ W_{\text{isothermal}} = nRT \ln\left(\frac{V_2}{V_1}\right) \]
- Here, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature (constant during isothermal expansion).

3. **Work Done in Adiabatic Compression:**
- For adiabatic processes, the work done (W_adiabatic) can be expressed as:
\[ W_{\text{adiabatic}} = \frac{P_2 V_2 - P_3 V_1}{\gamma - 1} \]
- Here, \( \gamma \) is the adiabatic index (ratio of specific heats).

4. **Net Work Done (W):**
- The net work done by the gas during the complete process is the work done during the isothermal expansion minus the work done during the adiabatic compression:
\[ W = W_{\text{isothermal}} - W_{\text{adiabatic}} \]

5. **Determine the Sign of W:**
- Since the work done during the adiabatic compression is generally greater than the work done during the isothermal expansion (as compression requires energy input), we can conclude that:
\[ W < 0 \quad \text{(Net work done is negative)} \]

6. **Final Pressure Comparison:**
- After the processes, we have \( P_3 \) (pressure after adiabatic compression) which is greater than \( P_1 \) and \( P_2 \) (initial pressures). Thus, we can conclude:
\[ P_3 > P_1 \quad \text{and} \quad P_3 > P_2 \]

### Conclusion:
The net work done by the gas in the complete process is negative, and the final pressure \( P_3 \) is greater than both \( P_1 \) and \( P_2 \).