One mole of a monoatomic ideal gas $\left(c_{ V }=\frac{3}{2} R \right)$ undergoes a cycle where it first goes isochorically from the state $\left(\frac{3}{2} P _0, V _0\right)$ to $\left( P _0, V _0\right)$, and then is isobarically contracted to the volume $\frac{1}{2} V _0$. It is then taken back to the initial state by a path which is a quarter ellipse on the $P - V$ diagram. The efficiency of this cycle is
KVPY 2021, Advanced
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(C)
$n =\frac{ w _{\text {net }}}{ Q _{\text {sup plied }}}$
$w _{\text {net }}=\frac{\pi}{4} \times \frac{ V _0}{2} \times \frac{ P _0}{2}=\frac{\pi P _0 V _0}{16}$
Heat is supplied to the gas only during process $CA$.
$\text { so } Q _{ CA }= w _{ CA }+\Delta U _{ CA }$
$Q _{ CA }=\left(\frac{\pi P _0 V _0}{16}+\frac{ P _0 V _0}{2}\right)+\frac{3}{2} \times P _0 V _0$
$Q _{\text {supplied }}=\frac{\pi P _0 V _0}{16}+2 P _0 V _0$
$n =\frac{\frac{\pi P _0 V _0}{16}}{\frac{\pi P _0 V _0}{16}+2 P _0 V _0}=\frac{\pi}{\pi+32}$
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