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_1T_1+P_2T_2)/((T_1+T_2))`
C. `(P_1T_2+P_2T_1)/((T_1+T_2))`
D. `P_1/(2T_1)+P_2/(2T_2)`

Answer

To solve the problem, we will use the ideal gas law and the concept of conservation of moles when two containers are combined. Let's break down the solution step by step.

### Step 1: Write the Ideal Gas Law for Each Container
For the first container, we can express the number of moles (μ₁) using the ideal gas law:
\[ \mu_1 = \frac{P_1 V}{RT_1} \]
Where:
- \(P_1\) = Pressure in the first container
- \(V\) = Volume of the container (equal for both)
- \(R\) = Universal gas constant
- \(T_1\) = Absolute temperature in the first container

For the second container, the number of moles (μ₂) is given by:
\[ \mu_2 = \frac{P_2 V}{RT_2} \]

### Step 2: Total Moles After Joining the Containers
When the two containers are joined, the total number of moles of gas remains the same:
\[ \mu = \mu_1 + \mu_2 \]
Substituting the expressions from Step 1:
\[ \mu = \frac{P_1 V}{RT_1} + \frac{P_2 V}{RT_2} \]

### Step 3: Express the Common Pressure and Temperature
After joining, let the common pressure and temperature be \(P\) and \(T\), respectively. The total number of moles can also be expressed as:
\[ \mu = \frac{P (2V)}{RT} \]
Since the volume of both containers is equal, the total volume is \(2V\).

### Step 4: Set the Two Expressions for Moles Equal
Equating the two expressions for the number of moles:
\[ \frac{P (2V)}{RT} = \frac{P_1 V}{RT_1} + \frac{P_2 V}{RT_2} \]
We can cancel \(V\) and \(R\) from both sides:
\[ \frac{2P}{T} = \frac{P_1}{T_1} + \frac{P_2}{T_2} \]

### Step 5: Solve for the Ratio \(\frac{P}{T}\)
Rearranging the equation gives us:
\[ \frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) \]

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

### Conclusion
The correct answer is:
\[ \frac{P}{T} = \frac{P_1}{2T_1} + \frac{P_2}{2T_2} \]