An ideal gas, initially in state $\left( P _{12}, V _1, T _1\right)$ is expanded isobarically to $\left( P _{12}, V _2, T _2\right)$, then adiabatically $\left( P _{34}, V _3, T _3\right)$. It is then contracted isobarically to $\left( P _{34}, V _4, T _4\right)$ and finally adiabatically back to the initial state. The efficiency of this cycle is
KVPY 2021, Advanced
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(A)
$P _{12} V _2^\gamma= P _{34} V_3^\gamma \quad \dots(i)$
$P _{34} V_4^\gamma= P _{12} V_1^\gamma \quad \dots(ii)$
Multiply $(i)$ and $(ii)$
$V _2^\gamma \cdot V _4^\gamma= V _1^\gamma \cdot V _3^\gamma$
$V _2 V _4= V _1 V _3 \Rightarrow\left(\frac{ V _2}{ V _3}=\frac{ V _1}{ V _4}\right)= K _4$
$\eta=\frac{\text { Work done }}{ Q _{\text {suppl. }}}=1-\frac{ Q _{\text {rej }}}{ Q _{\text {supp }}}=1-\frac{ n _{ C / P } \Delta T _{34}}{ n _{ C / P } \Delta T _{12}}$
$=1-\frac{\Delta( PV )_{34}}{\Delta( PV )_{12}}=1-\frac{ P _{34}\left( V _3- V _4\right)}{ P _{12}\left( V _2- V _1\right)}$
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