A thermally insulated container is divided into two parts by a screen. In one part the pressure and temperature are `P` and `T` for an ideal gas filled. In the second part it is vacuum. If now a small hole is created in the screen, then the temperature of the gas will

Answer

To solve the problem step-by-step, we can follow these logical steps:

### Step 1: Understand the System
We have a thermally insulated container divided into two parts. One side contains an ideal gas at pressure \( P \) and temperature \( T \), while the other side is a vacuum.

**Hint:** Identify the key elements of the system: the ideal gas, vacuum, and the thermally insulated nature of the container.

### Step 2: Analyze the Process
When a small hole is created in the screen, the gas will rush into the vacuum. This process is known as free expansion, which occurs without any external pressure acting on the gas.

**Hint:** Recall the characteristics of free expansion: it is spontaneous and occurs without doing work on the surroundings.

### Step 3: Apply 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:
- The container is thermally insulated, so \( Q = 0 \) (no heat transfer).
- Since the gas expands into a vacuum, no work is done, so \( W = 0 \).

Thus, we have:
\[ \Delta U = 0 - 0 = 0 \]

**Hint:** Remember that in a thermally insulated system with no work done, the internal energy remains constant.

### Step 4: Relate Internal Energy to Temperature
For an ideal gas, the internal energy \( U \) is a function of temperature \( T \) only. Therefore, if the internal energy does not change (\( \Delta U = 0 \)), it implies that the temperature must also remain constant.

**Hint:** Recall that for an ideal gas, internal energy is solely dependent on temperature, not on pressure or volume.

### Step 5: Conclude the Result
Since the internal energy of the gas remains constant during the free expansion, we conclude that the temperature of the gas after it expands into the vacuum will also remain the same as it was initially.

**Final Answer:** The temperature of the gas will remain \( T \).

---