One mole of an ideal gas at 250 K is expanded isothermally from an initial volume of 5 litre to 10 litres. The `Delta E` for this process is `(R = 2 cal. Mol^(-1)K^(-1))`

Options

A. `1381.1 cal`
B. Zero
C. `163.7 cal`
D. `9 L atm`

Answer

To solve the problem of finding the change in internal energy (ΔE) for one mole of an ideal gas expanding isothermally from an initial volume of 5 liters to 10 liters at a constant temperature of 250 K, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Process**: The gas is expanding isothermally, which means the temperature remains constant throughout the process.

2. **Recall the Relationship for Internal Energy**: For an ideal gas, the change in internal energy (ΔE) is related to the change in temperature (ΔT) by the equation:
\[ ΔE = nC_VΔT \]
where:
- \( n \) = number of moles of gas
- \( C_V \) = molar heat capacity at constant volume
- \( ΔT \) = change in temperature

3. **Identify the Given Values**:
- Number of moles, \( n = 1 \) mole
- Temperature, \( T = 250 \) K (constant)
- Initial volume, \( V_1 = 5 \) L
- Final volume, \( V_2 = 10 \) L
- \( R = 2 \, \text{cal/mol K} \)

4. **Determine ΔT**: Since the process is isothermal, the change in temperature (ΔT) is:
\[ ΔT = T_{final} - T_{initial} = 250 \, K - 250 \, K = 0 \, K \]

5. **Calculate ΔE**: Substitute the values into the equation for ΔE:
\[ ΔE = nC_VΔT = 1 \cdot C_V \cdot 0 = 0 \]
Therefore, the change in internal energy (ΔE) is:
\[ ΔE = 0 \, \text{cal} \]

### Final Answer:
The change in internal energy (ΔE) for the isothermal expansion of the gas is **0 cal**.

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