An engine operates between reservoirs at 600 K and 300 K. Its manufacturer claims that each cycle absorbs 1200 J, produces 720 J of work and rejects 480 J. Check first-law consistency, calculate claimed efficiency and compare it with the Carnot limit. Calculate the total entropy change of the two reservoirs per claimed cycle and use it to diagnose the claim. State one way a real engine could legitimately produce 720 J of work per cycle.

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Energy is conserved, but claimed efficiency 60% exceeds the 50% Carnot limit. Reservoir entropy change is −1200/600 + 480/300 = −0.400 J K⁻¹, violating the second law. More heat input or a hotter source/lower sink is required. First-law accounting gives Q_H = W + Q_C: 1200 = 720 + 480 J, so the numerical energy balance is internally consistent. Claimed efficiency is W/Q_H = 720/1200 = 0.600. The maximum reversible efficiency is 1−T_C/T_H = 1−300/600 = 0.500. The hot reservoir entropy change is −Q_H/T_H = −1200/600 = −2.00 J K⁻¹. The cold reservoir gains +480/300 = +1.60 J K⁻¹. Total reservoir entropy change is −0.400 J K⁻¹. A cyclic engine returns to its own state, so it contributes no net entropy; the negative total violates the second law and confirms the claim is impossible. At the same temperatures, even a reversible engine needs Q_H ≥ W/η_C = 720/0.5 = 1440 J. Alternatively, a hotter source or colder sink could raise the Carnot limit, subject to real irreversibility lowering actual efficiency.

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