A gas is compressed isothermally to half its initial volume. The same gas is compressed separately through an adiabatic process until its volume is again reduced to half. Then:

Solution & Explanation

\textbf{Answer:} (b) Compressing the gas through an adiabatic process will require more work to be done \textbf{Solution:} On a $P$-$V$ diagram, an adiabatic curve ($P{V}^{\gamma }=$ const) is steeper than an isothermal curve ($PV=$ const), because $\gamma >1$. During compression to half the volume, the adiabatic curve rises above the isothermal curve, since in an adiabatic compression no heat escapes and the temperature (hence pressure) increases more rapidly. The magnitude of work done in compression equals the area under the $P$-$V$ curve. As the adiabatic curve lies above the isothermal curve over the same volume range, the area under it is larger: $$|W_{\text{adiabatic}}|>|W_{\text{isothermal}}|.$$ Therefore compressing the gas adiabatically requires more work. This matches option (b).