Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 4.16 – 4.17.
A heat engine has a maximum efficiency of
where is the temperature of the hot reservoir from which heat is extracted and is the temperature of the cold reservoir to which heat is expelled after work is extracted. From the second law of thermodynamics, the heat and temperature ratios satisfy (at the optimum efficiency):
Thus the work done by an optimal heat engine is
For an optimal refrigerator, the work required to extract from the cold reservoir and expel to the hot reservoir is exactly the same as 3. Therefore, if we use the work from a Carnot engine (which operates at the maximum efficiency) to power a Carnot refrigerator, the refrigerator will exactly cancel the effect of the heat engine, since it removes exactly the same amount of heat from the cold reservoir that the Carnot engine deposited there, and expels exactly the same amount of heat back into the hot reservoir that the Carnot engine extracted in the first place.
If, however, we could make an engine that was more efficient than a Carnot engine, that engine could produce the same amount of work by extracting a smaller amount of heat and expelling a smaller amount of waste heat , provided that as before. In this case, the heat engine still provides the amount of work required to power the refrigerator, but it produces less waste heat than the refrigerator extracts. The net effect of combining the two is to produce a system that requires no input of work, but still extracts a net amount of heat from the cold reservoir. That is, we’ve produced a refrigerator that requires no input work.
Conversely, if we started with a refrigerator that had a coefficient of performance COP that is greater than the Carnot value of
this means that it requires an amount of work to extract the heat from the cold reservoir. Hooking this up to a Carnot engine that extracts from the hot reservoir, produces work and expels heat to the cold reservoir means that the compound system produces a net excess of work while producing no waste heat, since the refrigerator extracts exactly the same amount of heat that the Carnot engine expels.
Both of these systems satisfy the first law of thermodynamics (conservation of energy) but violate the second law (entropy cannot decrease), which is why we can’t make them in real life.