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Match of the following.
$\begin{array}{ll}
\hline \text{ Column I } & \text{ Column II } \\
\hline \begin{array}{ll}
\text{A. Ratio of } \frac{\Delta Q}{\Delta U} \text{ in an isobaric} \\
\text{process}
\end{array} & 1. \frac{T_1}{\left(T_1-T_2\right)} \\
\hline \begin{array}{ll}
\text{B. Ratio of } \frac{\Delta Q}{\Delta W} \text{ in an isobaric} \\
\text{process}
\end{array} & 2. \frac{T_2}{\left(T_1-T_2\right)} \\
\hline \begin{array}{l}
\text{C. Coefficient of performance} \\
\text{of a refrigerator}
\end{array} & 3. \frac{\gamma}{\gamma-1} \\
\hline \begin{array}{l}
\text{D. Coefficient of performance} \\
\text{of a heat pump}
\end{array} & 4. \gamma \\
\hline
\end{array}$
Codes
$\begin{array}{llll}A & B & C & D\end{array}$
Options:
$\begin{array}{ll}
\hline \text{ Column I } & \text{ Column II } \\
\hline \begin{array}{ll}
\text{A. Ratio of } \frac{\Delta Q}{\Delta U} \text{ in an isobaric} \\
\text{process}
\end{array} & 1. \frac{T_1}{\left(T_1-T_2\right)} \\
\hline \begin{array}{ll}
\text{B. Ratio of } \frac{\Delta Q}{\Delta W} \text{ in an isobaric} \\
\text{process}
\end{array} & 2. \frac{T_2}{\left(T_1-T_2\right)} \\
\hline \begin{array}{l}
\text{C. Coefficient of performance} \\
\text{of a refrigerator}
\end{array} & 3. \frac{\gamma}{\gamma-1} \\
\hline \begin{array}{l}
\text{D. Coefficient of performance} \\
\text{of a heat pump}
\end{array} & 4. \gamma \\
\hline
\end{array}$
Codes
$\begin{array}{llll}A & B & C & D\end{array}$
Solution:
2998 Upvotes
Verified Answer
The correct answer is:
$\begin{array}{llll}4 & 3 & 2 & 1\end{array}$
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 $\mathrm{D} \rightarrow 1$.
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 $\mathrm{D} \rightarrow 1$.
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