# Power required to keep a freezer at constant temperature

Alright, I'm completely stuck on this and so is everyone else I've asked.

A freezer has a temperature $T_\mathrm{c} = -23\,\rm ^\circ C$. The air in the kitchen has a temperature $T_\mathrm{h} = 27\,\rm ^\circ C$. The freezer in not perfectly insulated and heat leaks through the walls of the freezer at a rate of $50\,\rm W$. Find the power of the motor that is needed to maintain the temperature in the freezer.

The answer to this question is $10\,\rm W$.

I don't understand how this is possible: $50\,\rm W$ is literally $\mathrm{J/s}$ in heat. To keep the temperature the same, the amount of heat in the freezer has to be the same. If the motor only removes heat at a rate of $10\,\rm J/s$, then $40\,\rm J$ is being added to the fridge every second, and the temperature changes in response.

Is there something I'm missing?

• Think of energy conservation, and think of the motor in the fridge adding heat to the inside of the refrigerator (cold reservoir) and flowing to the outside (hot reservoir). In other words, the motor takes heat away from the cold and brings it to the hot. Compare this to what you expect the "leaky walls" to do... Jul 4 '14 at 5:24

Note that, in refrigeration, this is the norm: removed heat is (desirably) much bigger than the input power. We call it the beta of the cycle. In your example, $\beta = \frac{50}{10}=5$. It's the ratio of "useful energy transfer" (in this case, removed heat) over "input power used" (in this case, electrical power to the compressor). It means that, for every watt spent, you get 5 watts of useful stuff.