The work done on a gas during compression is given by the area under the pressure-volume ($P-V$) curve ($w = -\int P_{ex} dV$) .

1. Isochoric Process: The volume is constant ($dV = 0$), so the work done is zero .
2. Isobaric Process: The pressure is constant. The work done is the area of the rectangle under the horizontal line.
3. Isothermal vs. Adiabatic: For an ideal gas, the slope of the adiabatic curve ($PV^\gamma = \text{constant}$) is steeper than that of the isothermal curve ($PV = \text{constant}$) because $\gamma > 1$. During compression from an initial volume $V$ to $V/2$, the final pressure attained in an adiabatic process is higher than in an isothermal process (since temperature increases in adiabatic compression but remains constant in isothermal).

• Consequently, the P-V curve for adiabatic compression lies above the isothermal curve, enclosing a larger area.

Therefore, the maximum work is done in the adiabatic compression.