Given:
- Number of moles of gas (n) = 2
- Initial pressure (P1) = 2 ATM
- Initial temperature (T1) = 27 degree Celsius
- Final volume (V2) = 1/2 V1 (where V1 is the initial volume)
- External pressure (P2) = 4 ATM

Formula:
The work done during an isothermal process can be calculated using the formula:

Work (W) = nRT ln(V2/V1)

Where:
- n is the number of moles of the gas
- R is the ideal gas constant (0.0821 L·atm/(mol·K))
- T is the temperature in Kelvin (T = 273 + ºC)
- ln is the natural logarithm
- V2/V1 is the ratio of the final volume to the initial volume

Calculations:
1. Convert the initial temperature from Celsius to Kelvin:
T1 = 273 + 27 = 300 K

2. Calculate the initial volume (V1):
Since the ideal gas equation is PV = nRT, we can rearrange it to V = nRT/P.
V1 = (2 * 0.0821 * 300) / 2 = 24.63 L

3. Calculate the final volume (V2):
V2 = 1/2 * V1 = 1/2 * 24.63 = 12.31 L

4. Calculate the work done (W):
W = 2 * 0.0821 * 300 * ln(12.31/24.63)
W = 2 * 0.0821 * 300 * (-0.693)
W = -36.12 L·atm

Explanation:
The given problem involves an isothermal process, which means the temperature remains constant throughout the process. In this case, the work done by the gas can be calculated using the formula mentioned above.

To apply the formula, we need to convert the temperature from Celsius to Kelvin. Then, we calculate the initial volume using the ideal gas equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

The final volume is given as half of the initial volume, and the external pressure is provided. Using these values, we substitute them into the formula to calculate the work done.

The negative sign in the calculated work indicates that work is done on the system, meaning energy is transferred from the surroundings to the gas during compression.