A container of volume `1m^(3)` is divided into two equal compartments, one of which contains an ideal gas at 300K. The oterh compartment is vaccum. The whole system is thermally isolated from its surroundings. The partition is removed and the gas expands to occupy the whole volume of the container. Its temperature now would be

Options

A. 300K
B. 250K
C. 200K
D. 100K

Explanation

To solve the problem step by step, let's analyze the situation:

### Step 1: Understand the System
We have a container of volume \(1 \, m^3\) divided into two equal compartments. One compartment contains an ideal gas at a temperature of \(300 \, K\) and the other compartment is a vacuum.

### Step 2: Remove the Partition
When the partition is removed, the gas expands to fill the entire volume of the container. Since one side is a vacuum, the gas will undergo free expansion.

### Step 3: Analyze the Thermodynamic Process
The system is thermally isolated, which means there is no heat exchange with the surroundings. According to the first law of thermodynamics:
\[ \Delta U = Q - W \]
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.

### Step 4: Determine Heat Transfer and Work Done
Since the system is thermally isolated:
\[ Q = 0 \]
Also, during free expansion into a vacuum, the work done \(W\) is zero:
\[ W = 0 \]

### Step 5: Calculate Change in Internal Energy
Substituting these values into the first law:
\[ \Delta U = 0 - 0 = 0 \]
This means the change in internal energy is zero.

### Step 6: Relate Change in Internal Energy to Temperature
For an ideal gas, the change in internal energy is given by:
\[ \Delta U = n C_v \Delta T \]
Where:
- \(n\) is the number of moles,
- \(C_v\) is the specific heat at constant volume,
- \(\Delta T\) is the change in temperature.

Since \(\Delta U = 0\), we have:
\[ n C_v \Delta T = 0 \]
This implies that either \(n = 0\), \(C_v = 0\), or \(\Delta T = 0\). Since the number of moles and specific heat cannot be zero for an ideal gas, we conclude:
\[ \Delta T = 0 \]

### Step 7: Final Temperature Calculation
Since \(\Delta T = T_{final} - T_{initial} = 0\), we have:
\[ T_{final} = T_{initial} \]
Given that the initial temperature \(T_{initial} = 300 \, K\), we find:
\[ T_{final} = 300 \, K \]

### Conclusion
The final temperature of the gas after it expands into the vacuum is \(300 \, K\).

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