A container of volume `1m^3` is divided into two equal parts by a partition. One part has an ideal gas at 300K and the other part is vacuum. The whole system is thermally isolated from the surroundings. When the partition is removed, the gas expands to occupy the whole volume. Its temperature will now be .......

Options

A. `300 K`
B. `250 K`
C. `200 K`
D. `10 K`

Explanation

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

### Step 1: Understand the System
We have a container of volume \(1 \, m^3\) divided into two equal parts. One part contains an ideal gas at a temperature of \(300 \, K\), and the other part is a vacuum. The system is thermally isolated from the surroundings.

**Hint:** Identify the components of the system and the conditions provided (ideal gas, vacuum, thermal isolation).

### Step 2: Identify the Process
When the partition is removed, the gas expands into the vacuum. This type of expansion is known as "free expansion," where the gas expands without doing any work on the surroundings and without heat exchange.

**Hint:** Recognize that free expansion means no external work is done and no heat is exchanged.

### 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,
- \(W\) is the work done by the system.

In free expansion:
- \(Q = 0\) (no heat exchange),
- \(W = 0\) (no work done).

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

**Hint:** Understand that in free expansion, the internal energy of the gas does not change.

### Step 4: Relate Internal Energy to Temperature
For an ideal gas, the change in internal energy (\(\Delta U\)) is related to the change in temperature (\(\Delta T\)) by the equation:
\[ \Delta U = nC_V\Delta T \]
Where:
- \(n\) is the number of moles,
- \(C_V\) is the molar heat capacity at constant volume.

Since \(\Delta U = 0\), it follows that:
\[ nC_V\Delta T = 0 \]
This implies that:
\[ \Delta T = 0 \]

**Hint:** Remember that for an ideal gas, internal energy is a function of temperature.

### Step 5: Conclude the Final Temperature
Since \(\Delta T = 0\), the final temperature (\(T_f\)) is equal to the initial temperature (\(T_i\)):
\[ T_f = T_i = 300 \, K \]

**Hint:** The final temperature remains the same as the initial temperature in free expansion.

### Final Answer
The final temperature of the gas after the partition is removed will be \(300 \, K\).