One mole of an ideal gas at `27^@C` expanded isothermally from an initial volume of `1 ` litre to `10` litre. The `DeltaU` for this process is : `(R=2 cal K^(-1) mol^(-1))`

Explanation

To solve the problem, we need to determine the change in internal energy (ΔU) for one mole of an ideal gas that expands isothermally from an initial volume of 1 liter to a final volume of 10 liters at a temperature of 27°C.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Number of moles (n) = 1 mole
- Initial volume (V1) = 1 liter
- Final volume (V2) = 10 liters
- Temperature (T) = 27°C = 300 K (since T in Kelvin = T in Celsius + 273)
- R = 2 cal K^(-1) mol^(-1)

2. **Understand the Process:**
- The process is isothermal, which means the temperature remains constant throughout the expansion.
- For an ideal gas undergoing an isothermal process, the change in internal energy (ΔU) is determined by the formula:
\[ \Delta U = n C_v \Delta T \]
- Where \( C_v \) is the molar heat capacity at constant volume and \( \Delta T \) is the change in temperature.

3. **Calculate Change in Temperature (ΔT):**
- Since the process is isothermal, the temperature does not change:
\[ \Delta T = T_{final} - T_{initial} = 300 K - 300 K = 0 K \]

4. **Substitute Values into the ΔU Formula:**
- Now substituting the values into the ΔU formula:
\[ \Delta U = n C_v \Delta T = 1 \text{ mole} \times C_v \times 0 K \]
- Since \( \Delta T = 0 \), we have:
\[ \Delta U = 0 \]

5. **Conclusion:**
- Therefore, the change in internal energy (ΔU) for this isothermal expansion of the gas is:
\[ \Delta U = 0 \text{ cal} \]

### Final Answer:
\[ \Delta U = 0 \text{ cal} \]