Ex. 2

When a fridge stands in a room at $20^{\rm o}$C, the motor has to extract 500W of heat from the cabinet, at $4^{\rm o}$C, to compensate for less than perfect insulation. How much power must be supplied to the motor if its efficiency is 80% of the maximium achievable?

The maximum efficiency comes from using a Carnot (reversible) fridge, for which $Q_H/Q_C=T_H/T_C$. Here $Q_C$ is known, and so

$$W=Q_H-Q_C=Q_C\left({T_H\over T_C}-1\right)$$

(In this question $W$, $Q_H$ and $Q_C$ will refer to energy transfer per second, measured in Watts.) With $T_H=293$K and $T_C=277$K, $Q_C=500$W gives $W_{\rm max}=29$W. However the real fridge works at 80% of the maximum efficiency, so $W=(29/0.8)=36$W