An inventor has built an engine X and claims that its efficiency x is greater than the efficiency e of an ideal engine operating between

An inventor has built an engine X and claims that its efficiency εx is greater than the efficiency e of an ideal engine operating between the same two temperatures. Suppose you couple engine X to an ideal refrigerator (Figure a) and adjust the cycle of engine X so that the work per cycle it provides equals the work per cycle required by the ideal refrigerator. Treat this combination as a single unit and show that if the inventor's claim were true (if εX > ε) the combined unit would act as a perfect refrigerator (Figure b), transferring energy as heat from the low-temperature reservoir to the high-temperature reservoir without the need for work.

TH Ideal T refrigerator t o Qu Engine Perfect refrigerator Q'c Qc Te (a) (b)