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 concept of combining gases in two containers. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Variables
Let:
- Volume of each container = \( V \)
- Pressure in the first container = \( P_1 \)
- Temperature in the first container = \( T_1 \)
- Pressure in the second container = \( P_2 \)
- Temperature in the second container = \( T_2 \)

### Step 2: Calculate the Number of Moles in Each Container
Using the ideal gas law \( PV = nRT \), we can express the number of moles in each container.

For the first container:
\[ n_1 = \frac{P_1 V}{R T_1} \]

For the second container:
\[ n_2 = \frac{P_2 V}{R T_2} \]

### Step 3: Total Number of Moles After Joining the Containers
When the two containers are joined, the total volume becomes \( 2V \). The total number of moles after joining the containers is:
\[ n = n_1 + n_2 = \frac{P_1 V}{R T_1} + \frac{P_2 V}{R T_2} \]

### Step 4: Express Total Number of Moles in Terms of Final Pressure and Temperature
After joining, the total number of moles can also be expressed in terms of the final pressure \( P \) and temperature \( T \):
\[ n = \frac{P (2V)}{R T} \]

### Step 5: Set the Two Expressions for Total Moles Equal
Now we set the two expressions for \( n \) equal to each other:
\[ \frac{P_1 V}{R T_1} + \frac{P_2 V}{R T_2} = \frac{P (2V)}{R T} \]

### Step 6: Cancel Common Terms
We can cancel \( V \) and \( R \) from both sides:
\[ \frac{P_1}{T_1} + \frac{P_2}{T_2} = \frac{2P}{T} \]

### Step 7: Rearranging to Find the Ratio \( \frac{P}{T} \)
Rearranging the equation gives:
\[ \frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) \]

### Final Result
Thus, the ratio \( \frac{P}{T} \) is:
\[ \frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) \]