Question: (I) The low temperature of a freezer cooling coil is\({\bf{ - 8^\circ C}}\)and the discharge temperature is\({\bf{33^\circ C}}\). What is the maximum theoretical coefficient of performance?

The ratio of the heat removed from the low-temperature area inside the refrigerator to the work done by the refrigerator to remove the heat is termed the coefficient of performance of the refrigerator (COP).

\({\rm{COP}} = \frac{{{Q_{\rm{L}}}}}{W}\)

In the case of the ideal refrigerator, the expression for COP is

\({\rm{COP}} = \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}} - {T_{\rm{L}}}}}\).

Here, \({T_{\rm{L}}}\) is the temperature of the low-temperature region, i.e., inside the refrigerator and \({T_{\rm{H}}}\) is the temperature of the high-temperature region, i.e., the room in which the refrigerator is placed.

The lower temperature of the freezer cooling coil is \({T_{\rm{L}}} = - 8^\circ {\rm{C}} = 265\;{\rm{K}}\).

The discharge temperature is \({T_{\rm{H}}} = 33^\circ {\rm{C}} = 306\;{\rm{K}}\).

The coefficient of performance of an ideal refrigerator is maximum and is given by

\({\rm{COP}} = \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}} - {T_{\rm{L}}}}}\).

Substitute the values in the above expression.

\(\begin{aligned}{c}{\rm{COP}} &= \frac{{265\;{\rm{K}}}}{{\left( {306 - 265} \right)\;{\rm{K}}}}\\ &= 6.463\\ \approx 6.5\end{aligned}\)

Thus, the maximum theoretical coefficient of performance is 6.5.