A sound wave passing volume `1 m^(3)` is divided into two equal compartments by a partition. One of these compartments contains an ideal gas at `300 K`. The other compartment is vacuum. 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. `8.97 xx 10^(-4) K`
B. `8.97 xx 10^(-6) K`
C. `8.97 xx 10^(-8) K`
D. none of these

Explanation

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

### Step 1: Understand the System
We have a volume of 1 m³ divided into two compartments. One compartment contains an ideal gas at a temperature of 300 K, and the other compartment is a vacuum. The system is thermally isolated from its surroundings.

**Hint**: Visualize the setup to understand the initial conditions of the gas and vacuum.

### Step 2: Identify the Process
When the partition is removed, the gas expands to fill the entire volume. This process is known as free expansion, where the gas expands into a vacuum without doing any work on the surroundings.

**Hint**: Recall the definition of free expansion and how it differs from other types of gas expansion.

### Step 3: Apply the First Law of Thermodynamics
According to the first law of thermodynamics, the change in internal energy (ΔU) is related to heat added to the system (ΔQ) and work done by the system (ΔW) by the equation:
\[ \Delta Q = \Delta U + \Delta W \]

Since the system is thermally isolated, there is no heat exchange with the surroundings:
\[ \Delta Q = 0 \]

**Hint**: Remember that in an isolated system, energy cannot enter or leave the system.

### Step 4: Analyze Work Done
In free expansion, the gas does not perform any work because it expands into a vacuum (no external pressure to work against):
\[ \Delta W = 0 \]

**Hint**: Consider the implications of expanding into a vacuum on the work done.

### Step 5: Relate Internal Energy Change to Temperature Change
From the first law, we have:
\[ 0 = \Delta U + 0 \]
This implies:
\[ \Delta U = 0 \]

For an ideal gas, the change in internal energy is given by:
\[ \Delta U = nC_V \Delta T \]
where \( n \) is the number of moles, \( C_V \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.

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

Given that \( n \) and \( C_V \) cannot be zero, we conclude:
\[ \Delta T = 0 \]

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

### Step 6: Conclude the Final Temperature
Since the change in temperature is zero, the final temperature of the gas after expansion remains the same as the initial temperature:
\[ T_{\text{final}} = T_{\text{initial}} = 300 \, \text{K} \]

**Hint**: Understand that the temperature of an ideal gas does not change in free expansion.

### Final Answer
The temperature of the gas after it expands to occupy the whole volume is **300 K**.