Two containers of equal volume contain the same gas at pressure `P_(1)` and `P_(2)` and absolute temperature `T_(1)` and `T_(2)`, respectively. On joining the vessels, the gas reaches a common pressure `P` and common temperature `T`. The ratio `P//T` is equal to

Options

A. `(P_(1))/(T_(1)) + (P_(2))/(T_(2))`
B. `(P_(1)T_(1) + P_(2) T_(2))/((T_(1) + T_(2))^(2))`
C. `(P_(1)T_(2) + P_(2) T_(2))/((T_(1) + T_(1))^(2))`
D. `(P_(1))/(2T_(1)) + (P_(2))/(2T_(2))`

Explanation

To solve the problem, we will use the ideal gas law and the principles of thermodynamics. Let's break down the steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**: The ideal gas law states that for a given amount of gas, the product of pressure (P), volume (V), and temperature (T) is a constant. Mathematically, it is expressed as:
\[ PV = nRT \]
where \( n \) is the number of moles of gas and \( R \) is the universal gas constant.

2. **Calculate Moles in Each Container**:
- For the first container:
\[ P_1 V = n_1 R T_1 \implies n_1 = \frac{P_1 V}{R T_1} \]
- For the second container:
\[ P_2 V = n_2 R T_2 \implies n_2 = \frac{P_2 V}{R T_2} \]

3. **Combine the Containers**: When the two containers are joined, the total volume becomes \( 2V \) and the total number of moles of gas becomes \( n_1 + n_2 \).

4. **Total Moles Calculation**:
\[ n_1 + n_2 = \frac{P_1 V}{R T_1} + \frac{P_2 V}{R T_2} \]

5. **Expressing Total Moles**:
\[ n_1 + n_2 = V \left( \frac{P_1}{R T_1} + \frac{P_2}{R T_2} \right) \]

6. **Using Ideal Gas Law for Combined System**: The pressure \( P \) and temperature \( T \) of the combined gas can be expressed as:
\[ P (2V) = (n_1 + n_2) R T \]

7. **Substituting for \( n_1 + n_2 \)**:
\[ P (2V) = \left( V \left( \frac{P_1}{R T_1} + \frac{P_2}{R T_2} \right) \right) R T \]

8. **Simplifying the Equation**:
\[ 2P = \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) T \]

9. **Finding the Ratio \( \frac{P}{T} \)**:
\[ \frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) \]

10. **Final Result**: Therefore, the ratio \( \frac{P}{T} \) is given by:
\[ \frac{P}{T} = \frac{P_1}{2T_1} + \frac{P_2}{2T_2} \]