Two samples `1` and `2` are initially kept in the same state. The sample `1` is expanded through an isothermal process where as sample `2` through an adiabatic process upto the same final volume. The final temperature in process `1` and `2` are `T_(1)` and `T_(2)` respectively, then

Options

A. `T_(1) gt T_(2)`
B. `T_(1) = T_(2)`
C. `T_(1) lt T_(2)`
D. The relation between `T_(1)` and `T_(2)` cannot be deduced.

Explanation

To solve the problem, we need to analyze the two processes: isothermal expansion for sample 1 and adiabatic expansion for sample 2.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Both samples (1 and 2) are initially in the same state, meaning they have the same initial temperature \( T_0 \), pressure, and volume.

2. **Isothermal Process for Sample 1**:
- Sample 1 undergoes an isothermal expansion. In an isothermal process, the temperature remains constant throughout the process.
- Therefore, the final temperature \( T_1 \) after the isothermal expansion is equal to the initial temperature:
\[ T_1 = T_0 \]

3. **Adiabatic Process for Sample 2**:
- Sample 2 undergoes an adiabatic expansion. In an adiabatic process, there is no heat exchange with the surroundings.
- The relationship between temperature and volume for an adiabatic process is given by:
\[ T V^{\gamma - 1} = \text{constant} \]
- Since the volume increases during expansion, the temperature must decrease. Therefore, we can express the final temperature \( T_2 \) as:
\[ T_2 < T_0 \]

4. **Comparing Final Temperatures**:
- From the isothermal process, we have \( T_1 = T_0 \).
- From the adiabatic process, we have \( T_2 < T_0 \).
- Thus, we can conclude that:
\[ T_1 > T_2 \]

5. **Final Conclusion**:
- The relationship between the final temperatures after the respective processes is:
\[ T_1 > T_2 \]