At `25^(@)C`, a 0.01 mole sample of a gas is compressed from 4.0 L to 1.0 L at constant temperature. What is the work done for this process if the external pressure is 4.0 bar?

Options

A. `1.6 xx 10^(3) J`
B. `8.0 xx 10^(2) J`
C. `4.0 xx 10^(2) J`
D. `1.2 xx 10^(3) J`

Explanation

To find the work done during the compression of a gas at constant temperature, we can use the formula for work done in a thermodynamic process:

\[ W = -P_{\text{ext}} \Delta V \]

Where:
- \( W \) is the work done on the gas,
- \( P_{\text{ext}} \) is the external pressure,
- \( \Delta V \) is the change in volume.

### Step 1: Identify the given values
- External pressure, \( P_{\text{ext}} = 4.0 \, \text{bar} \)
- Initial volume, \( V_i = 4.0 \, \text{L} \)
- Final volume, \( V_f = 1.0 \, \text{L} \)

### Step 2: Calculate the change in volume
\[ \Delta V = V_f - V_i \]
\[ \Delta V = 1.0 \, \text{L} - 4.0 \, \text{L} \]
\[ \Delta V = -3.0 \, \text{L} \]

### Step 3: Substitute the values into the work formula
Now substituting the values into the work formula:
\[ W = -P_{\text{ext}} \Delta V \]
\[ W = - (4.0 \, \text{bar}) \times (-3.0 \, \text{L}) \]

### Step 4: Calculate the work done
\[ W = 4.0 \, \text{bar} \times 3.0 \, \text{L} \]
\[ W = 12.0 \, \text{bar} \cdot \text{L} \]

### Step 5: Convert bar·L to Joules
To convert the work from bar·L to Joules, we use the conversion factor:
\[ 1 \, \text{bar} \cdot \text{L} = 100 \, \text{J} \]
Thus,
\[ W = 12.0 \, \text{bar} \cdot \text{L} \times 100 \, \text{J/bar·L} \]
\[ W = 1200 \, \text{J} \]

### Final Answer
The work done on the gas during the compression is:
\[ W = 1200 \, \text{J} \]