Which of the following process is used to do maximum work done on the ideal gas if the gas is compressed to half of its initial volume?

The correct option is Adiabatic.

To understand why the maximum work is done on an ideal gas during an adiabatic compression to half its initial volume, we can analyze the behavior of the pressure and volume during different thermodynamic processes.

Work done during compression (which is work done on the gas) is given by the area under the curve on a Pressure-Volume (P-V) diagram. Since the gas is compressed from an initial volume $V_1$ to a final volume $V_2=\frac{V_1}{2}$, the magnitude of work done on the gas is:
$W=-{\int }_{V_1}^{V_2}P dV$

Let's examine how the pressure $P$ changes with volume $V$ for the given processes starting from the same initial state $(P_1,V_1)$:
1. Isochoric (Constant Volume): Since the volume must change to be compressed, an isochoric process cannot compress the gas to half its volume. Technically, the work done in a purely isochoric process is zero because $dV=0$.
2. Isobaric (Constant Pressure): The pressure remains constant at $P=P_1$ throughout the compression. The work done is $W_{\text{isobaric}}=P_1(V_1-V_2)$.
3. Isothermal (Constant Temperature): The relation is $PV=\text{constant}$, or $P\propto V^{-1}$. As volume decreases, pressure increases.
4. Adiabatic (No heat exchange): The relation is $P{V}^{\gamma }=\text{constant}$, where $\gamma$ is the adiabatic index (ratio of specific heats, $\gamma =\frac{C_p}{C_v}>1$). Here, $P\propto V^{-\gamma }$.

Comparing the slopes of the curves on a P-V indicator diagram during compression:
The slope of the adiabatic curve is steeper than that of the isothermal curve because $\gamma >1$. Specifically:
${\left(\frac{dP}{dV}\right)}_{\text{adiabatic}}=-\gamma \frac{P}{V}$
${\left(\frac{dP}{dV}\right)}_{\text{isothermal}}=-\frac{P}{V}$

Since $\gamma >1$, as the volume decreases from $V_1$ to $V_2$, the pressure in the adiabatic process rises much faster than in the isothermal process, which in turn rises faster than in the isobaric process (where pressure remains constant). Therefore, at any intermediate compressed volume, we have:
$P_{\text{adiabatic}}>P_{\text{isothermal}}>P_{\text{isobaric}}$

Since the pressure is highest at every stage of the compression for the adiabatic path, the area under the P-V curve (representing the work done on the gas) is the largest. Thus, the maximum work is done on the ideal gas during the Adiabatic process.