Question

(II) A restaurant refrigerator has a coefficient of performance of 4.6. If the temperature in the kitchen outside the refrigerator is 32°C, what is the lowest temperature that could be obtained inside the refrigerator if it were ideal?

Short answer

The lowest temperature at ideal condition is \(251\;{\rm{K}}\).

Step-by-step solution

Understanding the coefficient of performance of a refrigerator

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

The expression for the coefficient of performance is given as:

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

Given data

The coefficient of performance is \(COP = 4.6\).

The temperature outside the refrigerator is \({T_2} = {\rm{32}}\circ {\rm{C}}\).

Evaluation of the lowest temperature inside the refrigerator

The expression for COP is given by:

\({\rm{COP}} = \frac{{{T_1}}}{{{T_2} - {T_1}}}\)

Substitute the values in the above expression.

\(\begin{array}{c}4.6 = \left( {\frac{{{T_1}}}{{\left( {{\rm{32}}\circ {\rm{C}} + 273} \right)\;{\rm{K}} - {T_1}}}} \right)\\{T_1} = 251\;{\rm{K}}\end{array}\)

Thus, the lowest temperature at ideal condition is \(251\;{\rm{K}}\).