Heat engines and entropy
Heat never flows uphill for free: every engine pays a tax to the cold reservoir, and entropy is the tax collector.
Heat engines
An engine takes heat Qh from a hot reservoir, converts some to work W, and dumps the rest Qc into a cold reservoir. Its efficiency is e = W/Qh = 1 - Qc/Qh. Carnot's theorem caps this at e = 1 - Tc/Th using absolute temperatures — no engine can beat a reversible one running between the same two reservoirs.
e_{max} = 1 - \frac{T_c}{T_h}Refrigerators and entropy
A refrigerator runs the flow backwards: work input moves heat from the cold box into the warm room. Entropy change is delta-S = Q/T for heat Q entering at temperature T, and the second law says the universe's total entropy never decreases. That single inequality is why perfectly efficient engines and perpetual motion machines are impossible.
\Delta S_{universe} \ge 0Worked example
An engine runs between a boiler at 500 K and the environment at 300 K. The Carnot bound is e = 1 - 300/500 = 0.40, so at best 40 percent of the absorbed heat becomes work. If the engine takes in 1000 J per cycle, the maximum work is 400 J and at least 600 J must be exhausted as waste heat.
e = 1 - \frac{300}{500} = 0.40Flashcard
Carnot efficiency limit
Check your understanding
An inventor claims an engine whose hot reservoir is at 227 C and whose cold reservoir is at 27 C. What is the maximum possible efficiency?