Equation 12.15 shows that the greater the temperature difference between the reservoirs of a heat engine, the greater the engine’s Carnot efficiency. Suppose you had the choice of raising the temperature of the high-temperature reservoir by a certain number of kelvins or lowering the temperature of the low-temperature reservoir by the same number of kelvins.
(a) To produce the largest increase in efficiency, you should choose (1) to raise the high-temperature reservoir, (2) to lower the low-temperature reservoir, (3) both 1 and 2 produce the same change in efficiency, so it does not matter which you choose. Explain.
(b) Prove your answer to part (a) mathematically.