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

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Question

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

Vedclass · Answer

(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-$II$
$(a)$ Heat engine	$(i)$ $\text{COP} = \frac{Q_2}{Q_1 - Q_2}$
$(b)$ Heat pump	$(ii)$ $\eta = \frac{Q_1 - Q_2}{Q_1}$
	$(iii)$ $\eta = \frac{T_1 - T_2}{T_1}$

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

Similar Questions

An ideal gas expands isothermally from volume $V_1$ to volume $V_2$. It is then compressed to the original volume $V_1$ adiabatically. If $p_1$ and $p_2$ represent the initial pressure and final pressure respectively, and $W$ represents the net work done by the gas during the entire process, then:

$A$ real gas within a closed chamber at $27^{\circ} C$ undergoes the cyclic process as shown in the figure. The gas obeys the $PV^3 = RT$ equation for the path $A$ to $B$. The net work done in the complete cycle is (assuming $R = 8 \, J/mol \cdot K$): (in $ \, J$)

Match the devices in Column-$I$ with their efficiency/coefficient of performance in Column-$II$:

Column-$I$ Column-$II$

$(a)$ Heat engine $(i)$ $\text{COP} = \frac{Q_2}{Q_1 - Q_2}$

$(b)$ Heat pump $(ii)$ $\eta = \frac{Q_1 - Q_2}{Q_1}$

$(iii)$ $\eta = \frac{T_1 - T_2}{T_1}$

$A$ bubble containing $8$ moles of Helium is submerged at a certain depth in water. When the temperature of the water increases by $30 \ ^\circ C$,how much heat in $J$ is added during the expansion of the Helium bubble?

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