The ratio of pressure of the same gas in two containers is `(n_(1) T_(1))/(n_(2) T_(2))` where `n_(1)` & `n_(2)` are the number of moles and `T_(1)` & `T_(2)` are respective temperatures. If the containers are now joined find the ratio of final pressure to the temperature :

Options

A. `(P_(1) T_(2) + P_(2) T_(1))/(2 T_(1) T_(2))`
B. `(P_(1) T_(1) + P_(2) T_(1))/(T_(1) T_(2))`
C. `(P_(1) T_(1) + P_(2) T_(2))/(2 T_(1) T_(2))`
D. none of these

Explanation

To solve the problem, we need to find the ratio of the final pressure to the temperature after the two containers of gas are joined. We will use the ideal gas law and the relationship between pressure, volume, temperature, and the number of moles.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- We have two containers with gases at different pressures, temperatures, and number of moles.
- The ratio of the pressures of the gases in the two containers is given by:
\[ \frac{P_1}{P_2} = \frac{n_1 T_1}{n_2 T_2} \]

2. **Using the Ideal Gas Law**:
- The ideal gas law states that:
\[ PV = nRT \]
- For both containers, we can express the pressures as:
\[ P_1 = \frac{n_1 R T_1}{V} \quad \text{and} \quad P_2 = \frac{n_2 R T_2}{V} \]

3. **Combining the Containers**:
- When the two containers are joined, the total volume becomes \(2V\).
- The total number of moles of gas after mixing is \(n_1 + n_2\).

4. **Applying the Ideal Gas Law to the Combined System**:
- The pressure \(P\) in the combined volume can be expressed as:
\[ P \cdot (2V) = (n_1 + n_2) R T \]
- Rearranging gives:
\[ P = \frac{(n_1 + n_2) R T}{2V} \]

5. **Finding the Ratio of Final Pressure to Temperature**:
- We need to find the ratio \( \frac{P}{T} \):
\[ \frac{P}{T} = \frac{(n_1 + n_2) R}{2V} \]

6. **Substituting for \(P_1\) and \(P_2\)**:
- From the earlier expressions for \(P_1\) and \(P_2\):
\[ P_1 = \frac{n_1 R T_1}{V} \quad \text{and} \quad P_2 = \frac{n_2 R T_2}{V} \]
- We can express \(n_1\) and \(n_2\) in terms of \(P_1\) and \(P_2\):
\[ n_1 = \frac{P_1 V}{RT_1} \quad \text{and} \quad n_2 = \frac{P_2 V}{RT_2} \]

7. **Substituting Back into the Ratio**:
- Substitute \(n_1\) and \(n_2\) into the ratio:
\[ \frac{P}{T} = \frac{\left(\frac{P_1 V}{RT_1} + \frac{P_2 V}{RT_2}\right) R}{2V} \]
- Simplifying gives:
\[ \frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) \]

8. **Final Result**:
- Thus, the final ratio of pressure to temperature after the containers are joined is:
\[ \frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) \]