Two adiabatic containers have volumes `V_(1)` and `V_(2)` respectively. The first container has monoatomic gas at pressure `p_(1)` and temperature `T_(1)`. The second container has another monoatomic gas at pressure `p_(2)` and temperature `T_(2)`. When the two containers are connected by a narrow tube, the final temperature and pressure of the gases in the containers are P and T respectively. Then

Options

A. `T=(p_(1)V_(1)T_(2)+p_(2)V_(2)T_(1))/(p_(1)V_(1)+p_(2)V_(2))`
B. `T=(p_(1)V_(1)T_(2)+p_(2)V_(2)T_(1))/(p_(1)V_(2)+p_(2)V_(1))`
C. `p=(p_(1)V_(2)+p_(2)V_(1))/(V_(1)+V_(2))`
D. `p=(p_(1)V_(1)+p_(2)V_(2))/(V_(1)+V_(2))`

Answer

To solve the problem, we need to analyze the behavior of the two monoatomic gases in the adiabatic containers when they are connected. We will use the ideal gas law and the principles of thermodynamics to derive the final pressure and temperature.

### Step-by-Step Solution:

1. **Identify Variables**:
- Let the first container have volume \( V_1 \), pressure \( P_1 \), and temperature \( T_1 \).
- Let the second container have volume \( V_2 \), pressure \( P_2 \), and temperature \( T_2 \).
- Let the final pressure and temperature after connecting the containers be \( P \) and \( T \) respectively.

2. **Calculate Moles of Gas**:
- For the first container, using the ideal gas law:
\[ n_1 = \frac{P_1 V_1}{RT_1} \]
- For the second container:
\[ n_2 = \frac{P_2 V_2}{RT_2} \]

3. **Total Moles After Connection**:
- The total number of moles when the containers are connected is:
\[ n_{total} = n_1 + n_2 = \frac{P_1 V_1}{RT_1} + \frac{P_2 V_2}{RT_2} \]

4. **Using the Ideal Gas Law for Final State**:
- The total volume after connecting the containers is \( V_1 + V_2 \).
- The final state can also be described by the ideal gas law:
\[ P(V_1 + V_2) = n_{total}RT \]
- Substituting \( n_{total} \):
\[ P(V_1 + V_2) = \left(\frac{P_1 V_1}{RT_1} + \frac{P_2 V_2}{RT_2}\right)RT \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ P = \frac{P_1 V_1}{T_1} + \frac{P_2 V_2}{T_2} \cdot \frac{1}{V_1 + V_2} \]

6. **Final Result**:
- Thus, the final pressure \( P \) can be expressed as:
\[ P = \frac{P_1 V_1 + P_2 V_2}{V_1 + V_2} \]

### Conclusion:
The final pressure \( P \) of the gases in the containers after they are connected is given by:
\[ P = \frac{P_1 V_1 + P_2 V_2}{V_1 + V_2} \]