Question 73 of 190

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 :

A$\mathrm{P}_0 \mathrm{~V}_0(2 \ln 2-0.75)$B$\mathrm{P}_0 \mathrm{~V}_0(\ln 2-0.75)$C$\mathrm{P}_0 \mathrm{~V}_0(\ln 2-0.25)$D$P_0 V_0(2 \ln 2-0.25)$

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

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