Question 130 of 190

$n$ mole of a perfect gas undergoes a cyclic process $ABCA$ (see figure) consisting of the following processes.
$A\rightarrow B:$ Isothermal expansion at temperature $T$ so that the volume is doubled from ${V}_{1}$ to ${V}_{2}=2{V}_{1}$ and pressure changes from ${P}_{1}$ to ${P}_{2}$
$B\rightarrow C:$ Isobaric compression at pressure ${P}_{2}$ to initial volume ${V}_{1}.$
$C\rightarrow A:$ Isochoric change leading to change of pressure from ${P}_{2}$ to ${P}_{1}$
Total work done in the complete cycle $ABCA$ is:

A$nRT\left(\ln 2-\dfrac{1}{2}\right)$B$nRT\ln 2$C$nRT\left(\ln 2+\dfrac{1}{2}\right)$D$0$

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Answer: (A) $nRT\left(\ln 2-\dfrac{1}{2}\right)$

${W}_{isothermal}=nRT\ln \left(\dfrac{{V}_{2}}{{V}_{1}}\right)$\
${W}_{isobaric}=P\Delta V=nR\Delta T$
${W}_{isochoric}=0$

${W}_{1}=nRT\ln \left(\dfrac{2V}{V}\right)=nRT\ln 2$
${W}_{2}=nR\left(\dfrac{T}{2}-T\right)=-nR\dfrac{T}{2}$
${W}_{3}=0$
$\Rightarrow {W}_{net}={W}_{1}+{W}_{2}+{W}_{3}$
${W}_{net}=nRT\left(\ln 2-\dfrac{1}{2}\right)$