A gas is compressed to half of its initial volume isothermally. The same gas is compressed again until the volume reduces to half through an adiabatic process. Then

The correct option is "Work done is more during the adiabatic process".

To understand why more work is done during the adiabatic compression compared to the isothermal compression, let us analyze the two processes on a Pressure-Volume (P-V) diagram.

For an isothermal process, the temperature remains constant. The relation between pressure and volume is given by Boyle's law:
$$PV=\text{constant}$$

For an adiabatic process, there is no heat exchange with the surroundings. The relation between pressure and volume is given by Poisson's equation:
$$P{V}^{\gamma }=\text{constant}$$
where ^γ (gamma) is the ratio of specific heats (C_p/C_v). Since ^γ is always greater than 1 for any gas (^γ > 1), the slope of the adiabatic curve on a P-V diagram is steeper than the slope of the isothermal curve.

The slope of an isothermal curve is:
$$\frac{dP}{dV}=-\frac{P}{V}$$

The slope of an adiabatic curve is:
$$\frac{dP}{dV}=-\gamma \frac{P}{V}$$

Since both compression processes start from the same initial state (P₁, V₁) and reduce the volume to half (V₂ = V₁/2):
During compression, the volume decreases. Because the adiabatic curve is steeper than the isothermal curve, the pressure rises much more rapidly during the adiabatic compression than during the isothermal compression.
Consequently, at the final volume V₂, the final pressure reached in the adiabatic process is higher than the final pressure reached in the isothermal process.

The magnitude of work done on the gas during compression is represented by the area under the P-V curve from the initial volume V₁ to the final volume V₂.
Since the adiabatic curve lies entirely above the isothermal curve during compression starting from the same initial state, the area under the adiabatic P-V curve is greater than the area under the isothermal P-V curve.

Therefore, the work done during the adiabatic compression is greater than the work done during the isothermal compression.