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Step-by-Step Explanation
Step 1: Identify the Various Thermodynamic Processes
1. Isochoric (constant volume): In this process, there is no change in volume ($\Delta V=0$). Hence, the work done $W=\int P\,dV=0$.
2. Isobaric (constant pressure): The pressure remains constant while volume changes. The work done in compressing the gas from $V_0$ to $\tfrac{V_0}{2}$ would be $W=P\,\Delta V$.
3. Isothermal (constant temperature): For an ideal gas undergoing isothermal compression, $PV = \text{constant}$. The work done is given by
$W = nRT \ln{\dfrac{V_i}{V_f}}$.
4. Adiabatic (no heat exchange): In an adiabatic process, $Q=0$, and the relation $PV^\gamma=\text{constant}$ holds for an ideal gas, where $\gamma = \dfrac{C_p}{C_v}$.
Step 2: Relate Work Done to the Area Under the PāV Curve
From thermodynamics, the work done on (or by) the gas in any process between two fixed volumes on a $P$ā$V$ diagram is obtained by the area under the process curve. Mathematically,
$W = \int_{V_i}^{V_f} P \, dV.$
When compressing an ideal gas from $V_0$ to $\tfrac{V_0}{2}$, the curve that encompasses the largest area under it (from $V_0$ to $\tfrac{V_0}{2}$) corresponds to the maximum work done on the gas.
Step 3: Determine Which Process Has Maximum Work Done
Comparing the shapes of these curves on a $P$ā$V$ diagram for the same volume change:
Isochoric: No area under the curve; $W=0.$
Isobaric: Gives a rectangular area on the $P$ā$V$ plane.
Isothermal: Curve lies between the isobaric and adiabatic curves for the same end volumes.
Adiabatic: Has a steeper slope than the isothermal curve. Thus, for the same volume change, it encloses a larger area (meaning greater work done on the gas during compression).
Step 4: Conclusion
Because the adiabatic curve on a $P$ā$V$ diagram is typically steeper than the isothermal curve (and certainly covers more area than an isobaric curve for the same volume change), the greatest work is done on the gas in the adiabatic process. Hence, adiabatic compression results in the maximum work done on the gas.
