In a mechanical refrigerator, the low temperature coils are at a temperature of $- 23^{0} C$ and the compressed gas in the condenser has a temperature of $27^{0} C$ . The theoretical coefficient of performance is _____.

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Solution

Before Opening :
$\Rightarrow P_{A} V = n_{A} R T_{A} \Rightarrow n_{A} = \frac{P_{A} V}{R T_{A}} \to (1)$

$\Rightarrow P_{B} V = n_{B} R T_{B} \Rightarrow n_{B} = \frac{P_{B} V}{R T_{B}} \to (2)$
After opening :
As $n = n_{A} + n_{B}$
Putting from $(1) (2)$

$∴ \frac{2 P V}{R T} = \frac{P_{A} V}{R T_{A}} + \frac{P_{B} V}{R T_{B}}$

$\Rightarrow \frac{P}{V} = \frac{1}{2} [\frac{P_{A}}{T_{A}} + \frac{P_{B}}{T_{B}}] \Rightarrow$ co efficient of performance
Hence, the answer is $= \frac{1}{2} [\frac{P_{A}}{T_{A}} + \frac{P_{B}}{T_{B}}] .$

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