A real gas within a closed chamber at $27^{\circ} \mathrm{C}$ undergoes the cyclic process as shown in figure. The gas obeys $P V^3=\mathrm{RT}$ equation for the path $A$ to $B$. The net work done in the complete cycle is (assuming $R=8 \mathrm{~J} / \mathrm{molK}$ ):
JEE MAIN 2024, Diffcult
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$\mathrm{W}_{\mathrm{AB}} =\int \mathrm{PdV} \quad \text { (Assuming } \mathrm{T} \text { to be constant) }$
$=\int \frac{\mathrm{RTdV}}{\mathrm{V}^3}$
$=\mathrm{RT} \int_2^4 \mathrm{~V}^{-3} \mathrm{dV}$
$=8 \times 300 \times\left(-\frac{1}{2}\left[\frac{1}{4^2}-\frac{1}{2^2}\right]\right)$
$=225 \mathrm{~J}$
$\mathrm{~W}_{\mathrm{BC}} =\mathrm{P} \int_4^2 \mathrm{dV}=10(2-4)=-20 \mathrm{~J}$
$\mathrm{~W}_{\mathrm{CA}} =0$
$\therefore \mathrm{W}_{\text {cyck }} =205 \mathrm{~J}$
Note: Data is inconsistent in process $A B$.
So needs to be challenged.
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