In an isothermal expansion of an ideal gas against vacuum, the work is involved is :

Options

A. zero
B. maximum
C. minimum
D. none of these

Answer

To solve the question regarding the work done in an isothermal expansion of an ideal gas against a vacuum, we can follow these steps:

### Step 1: Understand the Conditions
In an isothermal process, the temperature of the system remains constant. For an ideal gas, this means that the internal energy (U) is also constant because the internal energy is a function of temperature.

**Hint:** Remember that for an ideal gas, internal energy depends solely on temperature.

### Step 2: Identify the External Pressure
During an expansion against a vacuum, the external pressure (P_ext) is effectively zero. This is crucial because work done (W) in thermodynamics is calculated using the formula:
\[ W = -P_{ext} \Delta V \]
where \( \Delta V \) is the change in volume.

**Hint:** Consider how the external pressure affects the work done during expansion.

### Step 3: Calculate Work Done
Since the external pressure is zero (P_ext = 0), we can substitute this into the work formula:
\[ W = -0 \times \Delta V = 0 \]
Thus, the work done during the isothermal expansion against a vacuum is zero.

**Hint:** Think about what happens to the work equation when the external pressure is zero.

### Step 4: Apply the First Law of Thermodynamics
The first law of thermodynamics states:
\[ \Delta U = Q + W \]
Since we established that \( \Delta U = 0 \) (no change in internal energy) and \( W = 0 \) (no work done), we can conclude:
\[ 0 = Q + 0 \]
This means that the heat transfer (Q) is also zero.

**Hint:** Relate the first law of thermodynamics to the conditions of the system.

### Conclusion
Therefore, the work done in an isothermal expansion of an ideal gas against a vacuum is zero.

**Final Answer:** The work done is 0.