Halliday, Resnick & Walker — Entropy and the Second Law of Thermodynamics: Problem 28

In the first stage of a two-stage Carnot engine, energy is absorbed as heat \(Q_1\) at temperature \(T_1\), work \(W_1\) is done, and energy is expelled as heat \(Q_2\) at a lower temperature \(T_2\). The second stage absorbs that energy as heat \(Q_2\), does work \(W_2\), and expels energy as heat \(Q_3\) at a still lower temperature \(T_3\). Prove that the efficiency of the engine is \((T_1 - T_3)/T_1\).