An ideal gas,initially in state (P_{12}, V_1, | vedclass

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(A) For the adiabatic processes $BC$ and $DA$:$P_{12} V_2^\gamma = P_{34} V_3^\gamma \quad \dots(i)$$P_{34} V_4^\gamma = P_{12} V_1^\gamma \quad \dots

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$1-\frac{T_4}{T_1}$

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(A) For the adiabatic processes $BC$ and $DA$:$P_{12} V_2^\gamma = P_{34} V_3^\gamma \quad \dots(i)$$P_{34} V_4^\gamma = P_{12} V_1^\gamma \quad \dots(ii)$Multiplying $(i)$ and $(ii)$:$P_{12} P_{34} (V_2 V_4)^\gamma = P_{12} P_{34} (V_1 V_3)^\gamma$$V_2 V_4 = V_1 V_3 \Rightarrow \frac{V_2}{V_3} = \frac{V_1}{V_4}$Using the ideal gas law $PV = nRT$ for isobaric processes:$T_1 = \frac{P_{12} V_1}{nR}, T_2 = \frac{P_{12} V_2}{nR}, T_3 = \frac{P_{34} V_3}{nR}, T_4 = \frac{P_{34} V_4}{nR}$Efficiency $\eta = 1 - \frac{Q_{rejected}}{Q_{supplied}} = 1 - \frac{n C_p (T_3 - T_4)}{n C_p (T_2 - T_1)} = 1 - \frac{T_3 - T_4}{T_2 - T_1}$Substituting $T$ values:$\eta = 1 - \frac{\frac{P_{34}}{nR}(V_3 - V_4)}{\frac{P_{12}}{nR}(V_2 - V_1)} = 1 - \frac{P_{34}}{P_{12}} \left( \frac{V_3 - V_4}{V_2 - V_1} \right)$Since $\frac{V_2}{V_3} = \frac{V_1}{V_4} = k$,then $V_2 = k V_3$ and $V_1 = k V_4$.$\eta = 1 - \frac{P_{34}}{P_{12}} \left( \frac{V_3 - V_4}{k(V_3 - V_4)} \right) = 1 - \frac{P_{34}}{P_{12} k} = 1 - \frac{P_{34}}{P_{12}} \cdot \frac{V_3}{V_2} = 1 - \frac{T_3}{T_2} = 1 - \frac{T_4}{T_1}$

Column $I$	Column $II$
$(A)$ An insulated container has two chambers separated by a valve. Chamber $I$ contains an ideal gas and Chamber $II$ has a vacuum. The valve is opened.	$(p)$ The temperature of the gas decreases
$(B)$ An ideal monoatomic gas expands to twice its original volume such that its pressure $P \propto V^{-2}$	$(q)$ The temperature of the gas increases or remains constant
$(C)$ An ideal monoatomic gas expands to twice its original volume such that its pressure $P \propto V^{-4/3}$	$(r)$ The gas loses heat
$(D)$ An ideal monoatomic gas expands such that its pressure $P$ and volume $V$ follow the behavior shown in the graph	$(s)$ The gas gains heat

An ideal gas,initially in state $(P_{12}, V_1, T_1)$ is expanded isobarically to $(P_{12}, V_2, T_2)$,then adiabatically to $(P_{34}, V_3, T_3)$. It is then contracted isobarically to $(P_{34}, V_4, T_4)$ and finally adiabatically back to the initial state. The efficiency of this cycle is

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