Question

In mole 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 workdone in the complete cycle ABCA is

• A. 0
• B. $nRT\left(\ln 2+\frac{1}{2}\right)$
• C. $nRTln2$
• D. $nRT\left(\ln 2-\frac{1}{2}\right)$

Answer 1

Answer: (D)

$W _{\text {Isothermal }}=n R Tln\left(\frac{ v _{2}}{ v _{1}}\right)$
$W _{\text {Isobaric }}= P \Delta V = nR \Delta T$
$W _{\text {Isochoric }}=0$

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