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(B) For an ideal gas, the equation of state is $pV = \mu RT$, which implies $p = (\frac{\mu R}{V})T$. In a $p-T$ diagram, lines passing through the or

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$20$

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(B) For an ideal gas, the equation of state is $pV = \mu RT$, which implies $p = (\frac{\mu R}{V})T$. In a $p-T$ diagram, lines passing through the origin represent isochoric processes (constant volume) because $p \propto T$ implies $V = 	ext{constant}$. Processes $AB$ and $CD$ are straight lines passing through the origin, so they are isochoric processes. Work done in an isochoric process is zero. Thus, $W_{AB} = 0$ and $W_{CD} = 0$. Processes $BC$ and $DA$ are isobaric processes (constant pressure). Work done in an isobaric process is $W = p\Delta V = \mu R \Delta T$. For process $BC$ (at pressure $p_2$): $W_{BC} = \mu R(T_C - T_B) = 3 	imes R 	imes (2400 - 800) = 3R 	imes 1600 = 4800R$. For process $DA$ (at pressure $p_1$): $W_{DA} = \mu R(T_A - T_D) = 3 	imes R 	imes (400 - 1200) = 3R 	imes (-800) = -2400R$. The total work done in the cycle is $W = W_{AB} + W_{BC} + W_{CD} + W_{DA} = 0 + 4800R + 0 - 2400R = 2400R$. Substituting $R = 8.314 \, J/mol \cdot K$: $W = 2400 	imes 8.314 = 19953.6 \, J \approx 20 \, kJ$.

Column $I$	Column $II$
$A$. Ratio of $\frac{\Delta Q}{\Delta U}$ in an isobaric process	$1$. $\frac{T_1}{T_1-T_2}$
$B$. Ratio of $\frac{\Delta Q}{\Delta W}$ in an isobaric process	$2$. $\frac{T_2}{T_1-T_2}$
$C$. Coefficient of performance of a refrigerator	$3$. $\frac{\gamma}{\gamma-1}$
$D$. Coefficient of performance of a heat pump	$4$. $\gamma$

$3$ moles of an ideal monoatomic gas performs an $ABCDA$ cyclic process as shown in the figure below. The gas temperatures are $T_A=400 \, K$, $T_B=800 \, K$, $T_C=2400 \, K$, and $T_D=1200 \, K$. The work done by the gas is (approximately) $(R=8.314 \, J/mol \cdot K)$. (in $ \, kJ$)

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