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 up to the same final volume.Let `p_(1)` and `p_(2)` be the final pressure of the samples and 2 respectively in the previous question then.

Options

A. `p_(1)gtp_(2)`
B. `p_(1)=p_(2)`
C. `p_(1)ltp_(2)`
D. The relation between `p_(1)` and `p_(2)` cannot be deduced

Answer

To solve the problem, we need to analyze the behavior of two gas samples undergoing different types of expansion: isothermal for sample 1 and adiabatic for sample 2. We will derive the final pressures \( P_1 \) and \( P_2 \) for both samples and compare them.

### Step-by-Step Solution:

1. **Define Initial Conditions**:
- Let the initial pressure and volume for both samples be \( P_0 \) and \( V_0 \) respectively.

2. **Isothermal Process for Sample 1**:
- In an isothermal process, the temperature remains constant. According to the ideal gas law, we have:
\[ P_1 V_f = P_0 V_0 \]
- Rearranging this equation to find the final pressure \( P_1 \):
\[ P_1 = \frac{P_0 V_0}{V_f} \]

3. **Adiabatic Process for Sample 2**:
- In an adiabatic process, the relationship between pressure and volume is given by:
\[ P_2 V_f^\gamma = P_0 V_0^\gamma \]
- Rearranging this equation to find the final pressure \( P_2 \):
\[ P_2 = \frac{P_0 V_0^\gamma}{V_f^\gamma} \]

4. **Comparison of Final Pressures**:
- Now we have:
\[ P_1 = \frac{P_0 V_0}{V_f} \]
\[ P_2 = \frac{P_0 V_0^\gamma}{V_f^\gamma} \]
- To compare \( P_1 \) and \( P_2 \), we can analyze the ratio:
\[ \frac{P_1}{P_2} = \frac{\frac{P_0 V_0}{V_f}}{\frac{P_0 V_0^\gamma}{V_f^\gamma}} = \frac{V_f^\gamma}{V_f} \cdot \frac{V_0}{V_0^\gamma} = \frac{V_f^{\gamma - 1}}{V_0^{\gamma - 1}} \]
- Since \( \gamma > 1 \) and \( V_f > V_0 \), we know that \( V_0 < V_f \), which implies that \( \frac{V_0}{V_f} < 1 \). Thus, \( P_1 > P_2 \).

5. **Conclusion**:
- Therefore, we conclude that:
\[ P_1 > P_2 \]
- The correct option is (a) \( P_1 \) is greater than \( P_2 \).