In an isothermal expansion of one mole of an ideal gas against vacuum from 5 litre to 50 litre at 300 K, the quantity of heat absorbed by the gas is

Options

A. zero
B. 80 lit. Atm
C. 1380 cal
D. `-1380 cal`

Answer

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

### Step 1: Understand the Conditions of the Problem
- We have an isothermal expansion, which means the temperature remains constant (T = 300 K).
- The gas expands from a volume of 5 liters to 50 liters against a vacuum.

### Step 2: Analyze the Work Done
- In an isothermal expansion against a vacuum, the external pressure is zero.
- The work done (W) by the gas during expansion can be calculated using the formula:
\[ W = -P_{\text{ext}} \Delta V \]
where \( P_{\text{ext}} \) is the external pressure and \( \Delta V \) is the change in volume.
- Since \( P_{\text{ext}} = 0 \) (because it is against vacuum), the work done is:
\[ W = -0 \cdot (50 - 5) = 0 \]

### Step 3: Apply the First Law of Thermodynamics
- The First Law of Thermodynamics states:
\[ \Delta U = Q - W \]
where \( \Delta U \) is the change in internal energy, \( Q \) is the heat absorbed, and \( W \) is the work done.
- For an ideal gas undergoing an isothermal process, the change in internal energy (\( \Delta U \)) is zero because the temperature is constant:
\[ \Delta U = 0 \]

### Step 4: Substitute Values into the First Law Equation
- Since \( \Delta U = 0 \) and \( W = 0 \), we can substitute these values into the First Law equation:
\[ 0 = Q - 0 \]
- This simplifies to:
\[ Q = 0 \]

### Conclusion
- Therefore, the quantity of heat absorbed by the gas during this isothermal expansion against vacuum is:
\[ Q = 0 \]

### Final Answer
The quantity of heat absorbed by the gas is **0 Joules**.

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