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

Q: 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

c (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 $CA$. $\text { so } Q _{ CA }= w _{ CA }+\Delta U _{ CA }$ $Q _{ CA }=\left(\frac{\pi P _0 V _0}{16}+\frac{ P _0 V _0}{2}\right)+\frac{3}{2} \times P _0 V _0$ $Q _{\text {supplied }}=\frac{\pi P _0 V _0}{16}+2 P _0 V _0$ $n =\frac{\frac{\pi P _0 V _0}{16}}{\frac{\pi P _0 V _0}{16}+2 P _0 V _0}=\frac{\pi}{\pi+32}$

Question

A$\frac{1}{\pi}$
B$\frac{\pi}{16+\pi}$
C$\frac{\pi}{32+\pi}$
D$\frac{2 \pi}{32+\pi}$

KVPY 2021, Advanced

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