Match of the following. Column I Column II array ll A. Ratio of (Δ...

We know that, at constant volume, Heat supplied, $\Delta Q_1=\Delta U=C_V \Delta T$ At constant pressure, heat supplied, $\Delta Q=C_p \Delta T=\Delta U+\Delta W$ Ratio of heat capacity at constant pressure to constant volume is $\gamma=\frac{C_p}{C_V}$ Now, $\frac{\Delta Q}{\Delta U}=\frac{C_p \Delta T}{C_V \Delta T}=\frac{C_p}{C_V}=\gamma$ Now, $\frac{\Delta Q}{\Delta W}=\frac{\Delta Q}{\Delta Q-\Delta U} =\frac{C_p \Delta T}{C_p \Delta T-C_V \Delta T}$$\begin{aligned} & =\frac{C_p}{C_p-C_V}=\frac{C_p}{C_V\left(\frac{C_p}{C_V}-1\right)} \\ & =\frac{\gamma}{\gamma-1} \end{aligned}$$$For refrigerator, coefficient of performance,$$\begin{aligned} \beta & =\frac{Q_2}{W}=\frac{\text{ Heat absorbed from food stuff }}{\text{ Electrical work done }} \\ \therefore \quad \beta & =\frac{Q_2}{Q_1-Q_2} \end{aligned}$$Coefficient of heat pump$=\frac{\text{ Heat released }}{\text{ Electrical work }}$i.e.$\quad \alpha=\frac{Q_1}{W}=\frac{Q_1}{Q_1-Q_2}$We know that,$\frac{Q_1}{Q_2}=\frac{T_1}{T_2}$So, we get,$\beta=\frac{Q_2}{Q_1-Q_2}=\frac{\frac{Q_2}{Q_1}}{1-\frac{Q_2}{Q_1}} =\frac{\frac{T_2}{T_1}}{1-\frac{T_2}{T_1}}=\frac{T_2}{T_1-T_2}$and$\alpha=\frac{Q_1}{Q_1-Q_2} =\frac{\frac{Q_1}{Q_2}}{\frac{Q_1}{Q_2}-1}=\frac{\frac{T_1}{T_2}}{\frac{T_1}{T_2}-1} =\frac{T_1}{T_1-T_2}$Hence, we get the best match as given by$\mathrm{A} \rightarrow 4, \mathrm{ B} \rightarrow 3, \mathrm{C} \rightarrow 2$and$\mathrm{D} \rightarrow 1$.