An ideal gas exists in a state with pressure \(P_0\) volume \(\mathrm{V}_0\).It is isothermally expanded to 4 times of its initial volume \(\left(\mathrm{V}_0\right)\), then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is :

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Correct Answer

(A) \(\mathrm{P}_0 \mathrm{~V}_0(2 \ln 2-0.75)\)

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Detailed explanation

\(\begin{aligned} & \omega_1=P_0 v_0 \ln 4 \\ & \omega_2=\frac{P_0}{4}\left(-3 v_0\right)=-\frac{3 P_0 v_0}{4}\end{aligned}\)
\(\begin{aligned} & \omega_3=0 \\ & \mathrm{Q}_{\mathrm{T}}=\Delta \mathrm{U}_{\text {cyclic }}+\omega\end{aligned}\)
\(\mathrm{Q}_{\mathrm{T}}=\omega \quad\left(\Delta \mathrm{U}_{\text {cyclic }}=0\right)\)
\(\begin{aligned} & \mathrm{Q}_{\mathrm{T}}=\mathrm{P}_0 \mathrm{v}_0\left(\ln 4-\frac{3}{4}\right) \\ & =\mathrm{P}_0 \mathrm{v}_0(2 \ln 2-0.75)\end{aligned}\)

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List-I	List-II
A. Spring constant	I. \(ML ^2 T^{-2} K^{-1}\)
B. Thermal conductivity	II. \(ML ^0 T^{-2}\)
C. Boltzmann constant	III. \(ML ^2 T^{-3} A^{-2}\)
D. Inductive reactance	IV. \(MLT ^{-3} K^{-1}\)