Show that the coefficients of performance of refrigerators and heat pumps are related by\({\bf{CO}}{{\bf{P}}_{{\bf{ref}}}} = {\bf{CO}}{{\bf{P}}_{{\bf{hp}}}} - {\bf{1}}\).

Start with the definitions of the\({\bf{COP}}\)s and the conservation of energy relationship between\({{\bf{Q}}_{\bf{h}}}\),\({{\bf{Q}}_{\bf{c}}}\), and\({\bf{W}}\).

The conservation of energy is a principle that is used for conserving a particular amount of energy in an object. In this principle, the addition of the initial potential and the kinetic energy is equal to the addition of the final potential and the kinetic energy.

The Coefficient of Performance (COP) is described as the ratio of useful cooling or heating to the required work energy.

The equation of the COP of refrigerator is expressed as:

\(CO{P_{ref}} = \frac{{{Q_c}}}{W}\)

Here,\(CO{P_{ref}}\)is the coefficient of performance of the refrigerator,\({Q_c}\)is the heat removed and\(W\)is the work done.

The equation of the COP of refrigerator can also be expressed as:

\(\begin{array}{c}CO{P_{ref}} = \frac{{{Q_h}}}{W} - 1\\\frac{{{Q_h}}}{W} = CO{P_{ref}} - 1\end{array}\) …(i)

Here,\({Q_h}\)is the heat added to the engine.

The equation of the COP of heat pump is expressed as:

\(CO{P_{hp}} = \frac{{{Q_h}}}{W}\) …(ii)

Here,\(CO{P_{hp}}\)\(CO{P_{ref}}\)is the coefficient of performance of the heat pump,\({Q_h}\)is the heat added and\(W\)is the work done.

Equaling the equation (i) and (ii)

\(CO{P_{hp}} = CO{P_{ref}} - 1\)

Thus, the coefficient of performance of refrigerators and heat pump are related by\(CO{P_{hp}} = CO{P_{ref}} - 1\).