Two identical rooms in a perfectly insulated house are connected by an open doorway. The temperature in the two rooms are maintained at different values. The room which contains more air molecules is :

Options

A. the one with higher temperature
B. the one with lower temperature
C. the one with higher pressure
D. neither since both have same volume

Explanation

To solve the problem, we need to analyze the conditions of the two identical rooms connected by an open doorway, focusing on the relationship between temperature, pressure, volume, and the number of air molecules.

### Step-by-Step Solution:

1. **Understand the Setup**: We have two identical rooms that are perfectly insulated and connected by an open doorway. This means that the volume of both rooms is the same.

2. **Identify Given Conditions**: The problem states that the temperatures in the two rooms are maintained at different values. Let's denote:
- Room A: Higher temperature (T_A)
- Room B: Lower temperature (T_B)

3. **Apply the Ideal Gas Law**: The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \(P\) = pressure
- \(V\) = volume
- \(n\) = number of moles (which is proportional to the number of air molecules)
- \(R\) = ideal gas constant
- \(T\) = temperature

4. **Analyze Pressure and Volume**: Since both rooms are identical and connected, we can assume:
- The pressure (P) in both rooms is the same.
- The volume (V) in both rooms is the same.

5. **Rearranging the Ideal Gas Law**: Since \(P\) and \(V\) are constant for both rooms, we can express the ideal gas law as:
\[ n = \frac{PV}{RT} \]
This shows that the number of moles (n) is inversely proportional to the temperature (T) when pressure and volume are constant.

6. **Comparing the Two Rooms**:
- For Room A (higher temperature \(T_A\)):
\[ n_A = \frac{PV}{RT_A} \]
- For Room B (lower temperature \(T_B\)):
\[ n_B = \frac{PV}{RT_B} \]

7. **Conclusion**: Since \(T_A > T_B\), it follows that:
\[ n_A < n_B \]
This means that Room B, which has the lower temperature, contains more air molecules than Room A.

8. **Final Answer**: Therefore, the room that contains more air molecules is the one with the lower temperature (Room B).