An ideal gas at initial pressur e`P_(1)` and volume `V_(1)` undergoes reversible expansion to the same volume `V_(f)` either isothermally or adiabatically . Consider the following statements `:`
1. `|P_(f)(` adiabatic `) lt P_(f)(` isothermal `)|" "` 2. `|W(` adiabatic `) |lt|W(` isothermal `)|`
3. `|T_(f)(` adiabatic `) lt T_(f)(` isothermal `)|" "` 4. `|q(` adiabatic `)|lt|q(` isothermal `)|`
where the symbols have their usual meaning.
How many statements of the above are correct?

Options

A. Only one
B. Only two
C. Only three
D. All

Explanation

To solve the problem, we need to analyze the four statements regarding the behavior of an ideal gas undergoing reversible expansion either isothermally or adiabatically.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- In an **isothermal expansion**, the temperature remains constant throughout the process.
- In an **adiabatic expansion**, there is no heat exchange with the surroundings, and the temperature changes.

2. **Analyzing Statement 1**:
- **Statement**: |P_f (adiabatic) < P_f (isothermal)|
- **Analysis**: For a given final volume (V_f), the pressure in the adiabatic process will be lower than in the isothermal process because the gas expands without heat input, leading to a drop in temperature and thus pressure.
- **Conclusion**: This statement is **True**.

3. **Analyzing Statement 2**:
- **Statement**: |W (adiabatic)| < |W (isothermal)|
- **Analysis**: The work done (W) during isothermal expansion is greater than that during adiabatic expansion for the same change in volume because the area under the isothermal curve (work done) is larger than that under the adiabatic curve.
- **Conclusion**: This statement is **True**.

4. **Analyzing Statement 3**:
- **Statement**: |T_f (adiabatic) < T_f (isothermal)|
- **Analysis**: At the final volume V_f, the temperature in the isothermal process remains constant and is higher than the temperature in the adiabatic process, which decreases due to the expansion.
- **Conclusion**: This statement is **True**.

5. **Analyzing Statement 4**:
- **Statement**: |q (adiabatic)| < |q (isothermal)|
- **Analysis**: In an adiabatic process, there is no heat exchange (q = 0), while in an isothermal process, heat is absorbed from the surroundings. Therefore, the heat in the isothermal process is greater than in the adiabatic process.
- **Conclusion**: This statement is **True**.

### Final Conclusion:
All four statements are correct. Therefore, the answer to the question is that **all four statements are correct**.