Two different ideal diatomic gases `A` and `B` are initially in the same state. `A` and `B` are then expanded to same final volume through adiabatic and isothermal process respectively. If `P_(A), P_(B)` and `T_(A), T_(B)` represents the final pressure and temperature of `A` and `B` respectively then.

Options

A. `P_(A) lt P_(B)` and `T_(A) lt T_(B)`
B. `P_(A) gt P_(B)` and `T_(A) gt T_(B)`
C. `P_(A) gt P_(B)` and `T_(A) lt T_(B)`
D. `P_(A) lt P_(B)` and `T_(A) gt T_(B)`

Explanation

To solve the problem, we need to analyze the behavior of two different ideal diatomic gases, A and B, during their respective processes: adiabatic expansion for gas A and isothermal expansion for gas B.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
Both gases A and B are initially in the same state, meaning they have the same initial pressure (P₀), volume (V₀), and temperature (T₀).

2. **Identify the Processes**:
- Gas A undergoes an **adiabatic expansion**.
- Gas B undergoes an **isothermal expansion**.

3. **Final Volume**:
Both gases are expanded to the same final volume (V_f).

4. **Analyze the Adiabatic Process (Gas A)**:
For an ideal gas undergoing adiabatic expansion, the relationship between pressure, volume, and temperature is given by:
\[ P_f V_f^{\gamma} = P_0 V_0^{\gamma} \]
where \(\gamma\) (gamma) is the heat capacity ratio (C_p/C_v) for diatomic gases, which is approximately 1.4. Since the volume increases, the final pressure (P_A) will be lower than the initial pressure (P_0).

5. **Analyze the Isothermal Process (Gas B)**:
For an ideal gas undergoing isothermal expansion, the pressure and volume relationship is given by:
\[ P_f V_f = P_0 V_0 \]
Since the temperature remains constant during an isothermal process, the final pressure (P_B) will also be lower than the initial pressure, but the decrease will be less pronounced than in the adiabatic case.

6. **Comparison of Final Pressures**:
Since gas B is expanded isothermally and gas A is expanded adiabatically, the pressure of gas B (P_B) will be greater than the pressure of gas A (P_A) after expansion:
\[ P_B > P_A \]

7. **Determine Final Temperatures**:
- For gas A (adiabatic), the temperature decreases as it expands, leading to a final temperature (T_A) that is lower than the initial temperature (T_0).
- For gas B (isothermal), the temperature remains constant, so the final temperature (T_B) is equal to the initial temperature (T_0).

8. **Final Comparison**:
- Since \(P_B > P_A\) and \(T_B = T_0\) while \(T_A < T_0\), we conclude:
\[ P_B > P_A \quad \text{and} \quad T_B > T_A \]

### Conclusion:
The final result is that the pressure of gas B is greater than that of gas A, and the temperature of gas B is greater than that of gas A.