Two gases have the same initial pressure, volume and temperature. They expand to the same final volume, one adiabatically and the other isothermally

Options

A. The final temperature is greater for the isothermal process
B. The final pressure is greater for the isothermal process
C. The work done by the gas is greater for the isothermal process
D. All the above options are incorrect

Explanation

To solve the problem, we need to analyze the behavior of two gases expanding from the same initial state to the same final volume, one through an adiabatic process and the other through an isothermal process. We will compare their final temperatures, pressures, and the work done during the expansion.

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- Both gases start with the same initial pressure \( P_0 \), volume \( V_0 \), and temperature \( T_0 \).
- The final volume for both gases is the same, denoted as \( V_f \).

2. **Isothermal Process**:
- In an isothermal process, the temperature remains constant. Therefore, the final temperature \( T_f \) for the isothermal gas is equal to the initial temperature:
\[ T_f^{\text{isothermal}} = T_0 \]
- The pressure at the final volume can be calculated using the ideal gas law:
\[ P_f^{\text{isothermal}} = \frac{P_0 V_0}{V_f} \]

3. **Adiabatic Process**:
- In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure, volume, and temperature for an adiabatic process is given by:
\[ P V^\gamma = \text{constant} \]
- The final temperature for the adiabatic process can be derived using the formula:
\[ T_f^{\text{adiabatic}} = T_0 \left(\frac{V_0}{V_f}\right)^{\gamma - 1} \]
- The final pressure can also be calculated:
\[ P_f^{\text{adiabatic}} = P_0 \left(\frac{V_0}{V_f}\right)^\gamma \]

4. **Comparing Final Temperatures**:
- Since \( V_f > V_0 \), it follows that:
\[ T_f^{\text{adiabatic}} < T_0 \]
- Therefore, the final temperature for the isothermal process is greater than that for the adiabatic process:
\[ T_f^{\text{isothermal}} > T_f^{\text{adiabatic}} \]

5. **Comparing Final Pressures**:
- For the final pressures, since \( \gamma > 1 \):
\[ P_f^{\text{adiabatic}} < P_f^{\text{isothermal}} \]
- Thus, the final pressure for the isothermal process is greater than that for the adiabatic process.

6. **Work Done by the Gases**:
- The work done by the gas during expansion can be calculated as the area under the PV curve.
- For the isothermal process, the work done is given by:
\[ W_{\text{isothermal}} = nRT_0 \ln\left(\frac{V_f}{V_0}\right) \]
- For the adiabatic process, the work done can be expressed as:
\[ W_{\text{adiabatic}} = \frac{P_0 V_0}{\gamma - 1} \left(1 - \left(\frac{V_0}{V_f}\right)^{\gamma - 1}\right) \]
- Generally, the work done in the isothermal process is greater than in the adiabatic process due to the constant temperature allowing for more energy to be transferred.

7. **Conclusion**:
- From the analysis, we conclude:
- The final temperature is greater for the isothermal process.
- The final pressure is greater for the isothermal process.
- The work done by the gas is greater for the isothermal process.
- Therefore, all statements provided in the question are correct.

### Final Answer:
All the options are correct.