Halliday, Resnick & Walker — Entropy and the Second Law of Thermodynamics: Problem 61

An inventor has built an engine X and claims that its efficiency \(\epsilon_X\) is greater than the efficiency \(\epsilon\) of an ideal engine operating between the same two temperatures. Suppose you couple engine X to an ideal refrigerator (Fig. 20.16a) 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 \(\epsilon_X > \epsilon\)), the combined unit would act as a perfect refrigerator (Fig. 20.16b), transferring energy as heat from the low-temperature reservoir to the high-temperature reservoir without the need for work.