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

An inventor has built an engine X and claims that its

efficiency εx is greater than the efficiency ε of an ideal engine

operating between the same two temperatures. Suppose you couple

engine X to an ideal refrigerator (Fig. 2a) 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 (Fig.

2b), transferring energy as heat from the low-temperature reservoir to

the high-temperature reservoir without the need for work.