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$n$ 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
$0$
$nRT \left(\ln 2+\frac{1}{2}\right)$
$nRTIn2$
$nRT \left(\ln 2-\frac{1}{2}\right)$
Solution
$W _{\text {Isothermal }}= nRT \ln \left(\frac{ v _{2}}{ v _{1}}\right)$
$W _{\text {Isobaric }}= P \Delta V = nR \Delta T$
$W _{\text {Isochoric }}=0$
$W _{1}= nRT \ln \left(\frac{2 V }{ V }\right)= nRT \ln 2$
$W _{2}= nR \left(\frac{ T }{2}- T \right)=- nR \frac{ T }{2}$
$W _{3}=0$
$\Rightarrow W _{ net }= W _{1}+ W _{2}+ W _{3}$
$W _{ net }= nRT \left(\ln 2-\frac{1}{2}\right)$
Similar Questions
One mole of a monatomic ideal gas is taken along two cyclic processes $E \rightarrow F \rightarrow G \rightarrow E$ and $E \rightarrow F \rightarrow H \rightarrow$ E as shown in the $PV$ diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic. $Image$
Match the paths in List $I$ with the magnitudes of the work done in List $II$ and select the correct answer using the codes given below the lists.
| List $I$ | List $I$ |
| $P.$ $\quad G \rightarrow E$ | $1.$ $\quad 160 P_0 V_0 \ln 2$ |
| $Q.$ $\quad G \rightarrow H$ | $2.$ $\quad 36 P _0 V _0$ |
| $R.$ $\quad F \rightarrow H$ | $3.$ $\quad 24 P _0 V _0$ |
| $S.$ $\quad F \rightarrow G$ | $4.$ $\quad 31 P_0 V_0$ |
Codes: $ \quad \quad P \quad Q \quad R \quad S $
