An ideal gas is at pressure P and temperature T in a box, which is kept in vacuum with in a large container. The wall of the box is punctured. What happens as the gas occupies entire container?
(a)It's temperature falls
(b)Its temperature rises
(c)Its temperature remains the same
(d)Unpredictable

Options

A. It's temperature falls
B. Its temperature rises
C. Its temperature remains the same
D. Unpredicatable

Explanation

To solve the problem, we need to analyze what happens when an ideal gas expands into a vacuum after the wall of its container is punctured.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have an ideal gas in a small box at pressure \( P \) and temperature \( T \).
- The box is inside a larger container that is in a vacuum.

2. **Puncturing the Wall**:
- When the wall of the box is punctured, the gas will expand into the vacuum of the larger container.

3. **Nature of Expansion**:
- The expansion of the gas into a vacuum is known as "free expansion."
- In free expansion, the gas does not do any work on the surroundings because there is no external pressure acting against it (the external pressure is zero).

4. **Applying the First Law of Thermodynamics**:
- The First Law of Thermodynamics states:
\[ \Delta U = Q - W \]
where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system.
- In this case:
- Since the gas expands into a vacuum, the work done \( W = 0 \) (because \( P_{\text{external}} = 0 \)).
- There is no heat exchange with the surroundings since the container is in a vacuum, so \( Q = 0 \).

5. **Change in Internal Energy**:
- Therefore, we have:
\[ \Delta U = 0 - 0 = 0 \]
- This implies that the internal energy of the gas does not change.

6. **Temperature and Internal Energy**:
- For an ideal gas, the internal energy is directly related to its temperature. Since the internal energy does not change, the temperature of the gas also remains constant.

7. **Conclusion**:
- As the gas occupies the entire container, its temperature remains the same.

### Final Answer:
(c) Its temperature remains the same.