Heat Engines And Efficiency

The thermal efficiency of a heat engine measures the fraction of absorbed heat converted into useful work, defined as $\eta =\frac{W}{Q_H}=1-\frac{Q_C}{Q_H}$. The net work per cycle is $W=Q_H-Q_C=800-600=200\text{ J}$. Dividing by the input heat gives $\eta =200/800=0.25$, or 25 percent. The value 75 percent mistakenly reports the rejected fraction $Q_C/Q_H=0.75$ rather than the useful fraction. The value 33 percent comes from dividing work by the rejected heat, $200/600$, which is not the definition of efficiency. The value 20 percent results from dividing the net work by the sum of the heats instead of the input heat. The efficiency can never reach 100 percent for a cyclic engine because the Second Law forbids converting all absorbed heat into work without rejecting some to a colder reservoir. Notice also that only the difference $Q_H-Q_C$ appears as work, so reducing the rejected heat is the only way to raise efficiency for a fixed intake. As a plausibility check, real engines always reject most of their input heat, so an efficiency well below fifty percent is expected, and 25 percent is consistent with rejecting 600 of the 800 joules supplied.