Chapter 20, Problem 57P

(III) A general theorem states that the amount of energy that becomes unavailable to do useful work in any process is equal to T_L ∆S, where T_L is the lowest temperature available and ∆S is the total change in entropy during the process. Show that this is valid in the specific cases of (a) a falling rock that comes to rest when it hits the ground; (b) the free adiabatic expansion of an ideal gas; and (c) the conduction of heat, Q, from a high-temperature (T_H) reservoir to a low-temperature (T_L) reservoir. [Hint: In part (c) compare to a Carnot engine.]