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 undergoing isothermal expansion 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 process is isothermal, meaning the temperature remains constant throughout the expansion.

2. **Identify the Key Formula**: For an ideal gas, the change in internal energy (ΔE) is given by the formula:
\[ \Delta E = n C_v \Delta T \]
where:
- \( n \) = number of moles
- \( C_v \) = molar heat capacity at constant volume
- \( \Delta T \) = change in temperature

3. **Determine the Change in Temperature**: Since the process is isothermal, the temperature does not change. Therefore:
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 250 K - 250 K = 0 K \]

4. **Calculate the Change in Internal Energy**: Substituting the values into the formula:
\[ \Delta E = n C_v \Delta T = 1 \text{ mol} \times C_v \times 0 = 0 \]
Since \( \Delta T = 0 \), the change in internal energy (ΔE) is:
\[ \Delta E = 0 \]

5. **Conclusion**: The change in internal energy for the isothermal expansion of one mole of an ideal gas from 5 liters to 10 liters at 250 K is:
\[ \Delta E = 0 \text{ cal} \]