In an isothermal expansion of one mole of an ideal gas against vacuum from 10 litre to 100 litre at `27^(@)C`, the quantity of heat absorbed by the gas is

Answer

To solve the problem of the quantity of heat absorbed by one mole of an ideal gas during an isothermal expansion against vacuum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Process**:
The process described is an isothermal expansion of an ideal gas against vacuum. This means that the gas expands without doing any work against an external pressure (since the external pressure is zero).

2. **Identify Key Parameters**:
- Number of moles (n) = 1 mole
- Initial volume (V1) = 10 liters
- Final volume (V2) = 100 liters
- Temperature (T) = 27°C = 300 K (convert to Kelvin)

3. **Recognize the Nature of the Expansion**:
Since the expansion is against vacuum, it is termed as a "free expansion." In free expansion, no external pressure is applied, and thus no work is done by the gas.

4. **Calculate Work Done (W)**:
The work done by the gas during expansion is given by the formula:
\[ W = P_{\text{external}} \Delta V \]
Here, \(P_{\text{external}} = 0\) (since it is against vacuum), and therefore:
\[ W = 0 \times (V2 - V1) = 0 \]

5. **Apply the First Law of Thermodynamics**:
The first law of thermodynamics states:
\[ \Delta U = Q + W \]
Where:
- \(\Delta U\) = change in internal energy
- \(Q\) = heat absorbed by the system
- \(W\) = work done by the system

For an ideal gas undergoing an isothermal process, the change in internal energy (\(\Delta U\)) is zero because the temperature is constant. Thus:
\[ \Delta U = 0 \]

6. **Substituting Values**:
Since \(\Delta U = 0\) and \(W = 0\), we can substitute these values into the first law equation:
\[ 0 = Q + 0 \]
Therefore:
\[ Q = 0 \]

7. **Conclusion**:
The quantity of heat absorbed by the gas during this isothermal expansion against vacuum is **0**.