One mole of an ideal gas at `25^(@)C` expands in volume from 1.0 L to 4.0 L at constant temperature. What work (in J) is done if the gas expands against vacuum `(P_("external") = 0)`?

Options

A. `-4.0 xx 10^(2)`
B. `-3.0 xx 10^(2)`
C. `-1.0 xx 10^(2)`
D. Zero

Explanation

To solve the problem of calculating the work done by one mole of an ideal gas expanding from 1.0 L to 4.0 L at constant temperature against a vacuum (where the external pressure is zero), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Work Done**:
The work done by a gas during expansion can be calculated using the formula:
\[ W = -P_{\text{external}} \Delta V \]
where \(W\) is the work done, \(P_{\text{external}}\) is the external pressure, and \(\Delta V\) is the change in volume.

2. **Identify the Given Values**:
- Initial volume (\(V_i\)) = 1.0 L
- Final volume (\(V_f\)) = 4.0 L
- External pressure (\(P_{\text{external}}\)) = 0 (since it expands against a vacuum)

3. **Calculate the Change in Volume**:
The change in volume (\(\Delta V\)) can be calculated as:
\[ \Delta V = V_f - V_i = 4.0 \, \text{L} - 1.0 \, \text{L} = 3.0 \, \text{L} \]

4. **Convert Volume to Appropriate Units**:
Since we need the work done in Joules, we convert the volume from liters to cubic meters (1 L = 0.001 m³):
\[ \Delta V = 3.0 \, \text{L} = 3.0 \times 0.001 \, \text{m}^3 = 0.003 \, \text{m}^3 \]

5. **Substitute Values into the Work Formula**:
Now, substituting the values into the work formula:
\[ W = -P_{\text{external}} \Delta V = -0 \times 0.003 \, \text{m}^3 = 0 \, \text{J} \]

6. **Conclusion**:
Therefore, the work done by the gas during the expansion against a vacuum is:
\[ W = 0 \, \text{J} \]

### Final Answer:
The work done by the gas is **0 J**.

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