5 mole of an ideal gas of volume is expanded against vaccum to make its volume 2 times, then work done by the gas is:

Options

A. `-RT(V_2-V_1)`
B. `-RTln(V_2/V_1)`
C. 0
D. `C_v(T_2-T_1)`

Answer

To solve the problem of calculating the work done by an ideal gas when it expands against a vacuum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Expansion Against Vacuum**:
- When a gas expands against a vacuum, it does not have to do any work against an external pressure because the external pressure is zero. This means that the gas can expand freely without any resistance.

2. **Identify the Work Done Formula**:
- The work done (W) by a gas during expansion can be calculated using the formula:
\[ W = P_{\text{external}} \times (V_2 - V_1) \]
- Here, \(P_{\text{external}}\) is the external pressure, \(V_1\) is the initial volume, and \(V_2\) is the final volume.

3. **Set the External Pressure**:
- Since the gas is expanding against a vacuum, the external pressure \(P_{\text{external}} = 0\) atm.

4. **Calculate the Work Done**:
- Substitute \(P_{\text{external}} = 0\) into the work done formula:
\[ W = 0 \times (V_2 - V_1) = 0 \]
- Therefore, the work done by the gas during this expansion is:
\[ W = 0 \, \text{J} \]

5. **Conclusion**:
- The work done by the gas when it expands against a vacuum is zero. This is characteristic of free expansion, where no work is done.

### Final Answer:
The work done by the gas is \(0 \, \text{J}\).

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