To solve the problem, we need to analyze the statements given regarding the expansion of an ideal gas under different conditions. Let's break down the statements one by one.
### Step-by-Step Solution:
1. **Understanding the Process**:
- An ideal gas is expanded from state (P1, V1, T1) to (P2, V2, T2) under different conditions. We need to evaluate the work done and changes in internal energy for various processes.
2. **Statement 1**: "The work done on the gas is maximum when it is compressed irreversibly from (P2, V2) to (P1, V1) against constant pressure P1."
- **Analysis**: When compressing the gas irreversibly against a constant external pressure, the work done is indeed maximized because the process is not constrained by equilibrium conditions. The area under the pressure-volume graph for irreversible processes is larger than that for reversible processes.
- **Conclusion**: This statement is **correct**.
3. **Statement 2**: "The change in internal energy of the gas is zero if it is expanded reversibly with T1 = T2."
- **Analysis**: In an isothermal process (where temperature remains constant), the change in internal energy (ΔU) for an ideal gas is zero because internal energy is a function of temperature alone. Thus, if T1 = T2, ΔU = 0.
- **Conclusion**: This statement is **correct**.
4. **Statement 3**: "If it is expanded reversibly under adiabatic conditions where T1 ≠ T2, then the change in internal energy is positive."
- **Analysis**: In a reversible adiabatic expansion, the gas does work on the surroundings, and since no heat is exchanged (Q = 0), the internal energy decreases (ΔU < 0). Therefore, the statement that ΔU is positive is incorrect.
- **Conclusion**: This statement is **incorrect**.
5. **Statement 4**: "If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic."
- **Analysis**: In free expansion, the gas expands into a vacuum without doing work and without heat exchange. Since there is no change in temperature (ΔT = 0), it is isothermal, and because no heat is exchanged, it is also adiabatic.
- **Conclusion**: This statement is **correct**.
6. **Statement 5**: "The work done by the gas is less when it is expanded reversibly from V1 to V2 under adiabatic conditions as compared to when expanded reversibly under isothermal conditions."
- **Analysis**: The work done in an isothermal expansion is greater than that in an adiabatic expansion because in isothermal expansion, the gas absorbs heat, allowing it to do more work. The area under the curve for isothermal expansion is larger than that for adiabatic expansion.
- **Conclusion**: This statement is **correct**.
### Final Conclusion:
The correct statements are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Incorrect
- Statement 4: Correct
- Statement 5: Correct
Thus, the correct options are **1, 2, 4, and 5**.
