The Carnot Cycle: The Most Efficient Engine
If no engine can be perfect, what is the best one possible? Carnot's answer is a cycle of two isothermal legs and two adiabatic legs whose efficiency depends only on the two reservoir temperatures (η = 1 - TCTH). No real engine between the same reservoirs can beat it.
Drag the phase to travel once around the loop. The two isothermal legs exchange heat at a fixed temperature; the two adiabatic legs change temperature with no heat exchange. All four legs must be walked quasi-statically so the gas stays near equilibrium, which is why a real engine running at finite speed, with friction and finite temperature gaps, never quite reaches this ideal round trip.
Drag the two temperatures (in kelvin). The Carnot efficiency η = 1 - TCTH depends only on their ratio, never on what the working substance is. A power plant driven by 800 K steam and dumping heat to a 300 K environment, for instance, has a ceiling near 60 percent, and since the cold side can never be pushed to absolute zero, 100 percent efficiency stays forever out of reach.
Behind that formula is a locked equality: for a reversible cycle QCQH = TCTH, so QHTH = QCTC. This equality says that in a reversible cycle heat is exchanged in exact proportion to temperature so that the ratio Q/T is conserved, and it is precisely this ratio that the next chapter names entropy. Drag the ratio and the two bars stay equal.
Carnot sets a ceiling. A real engine sits below it; nothing can sit above. If some engine could beat this ceiling, you could pair it with a reversed Carnot to shuttle heat from cold to hot with no work at all, a perpetual-motion machine that breaks the second law, and that impossibility is exactly what pins the ceiling in place. Toggle the real, the Carnot, and the impossible over-the-ceiling case.
Take away three things: the efficiency rides only on TCTH, it is the unbeatable ceiling, and the equality QHTH = QCTC is the seed of the next chapter -- entropy. So in practice one raises TH as far as possible to lift the ceiling, which is exactly why jet engines and power plants push their combustion temperatures as high as the materials can bear.