Carnot & the second law
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You should be comfortable with reversible processes, the definition of a heat reservoir, and the idea of a thermodynamic cycle returning a system to its initial state — see PU-203 / L17.
Why this matters
No engine, however cleverly built, can turn heat into work more efficiently than a reversible one running between the same two temperatures. That ceiling is not an engineering limit — it is a law of nature.
Engineers spent the nineteenth century chasing better steam engines and kept bumping into the same wall. Carnot's insight was to stop asking how to build a better engine and ask instead what the best possible engine could do. The answer turned out to depend only on the two temperatures involved — not on the working substance, not on the mechanism, not on any detail of the design. This single result is where the second law of thermodynamics first shows its face, and where the concept of absolute temperature quietly enters physics. This lecture teaches and interprets the result; we compose it from the derivation established in the vault rather than re-deriving it here.
What Carnot actually claimed
Carnot's theorem has two parts. First: no heat engine operating between two reservoirs can be more efficient than a reversible engine operating between the same two reservoirs. Second: all reversible engines operating between the same two reservoirs have exactly the same efficiency. Both follow from a single move — assume the contrary and you can wire two engines together to build a device that shuttles heat from cold to hot with no other effect, which the second law forbids. The proof is a reductio: the efficiency ceiling exists because violating it violates the Clausius statement of the second law.
The derivation
Carnot efficiency from the second law
Combining the impossibility of net cold-to-hot heat flow with the definition of absolute temperature, the maximum efficiency of any engine between reservoirs at TH and TC is:
Reading the result
Read η = 1 − TC/TH carefully. The efficiency depends only on the ratio of the two absolute temperatures — nothing about the gas, the pistons, or the fuel appears. It reaches 1 only if TC = 0, an unreachable limit, so perfect conversion of heat to work is impossible in principle, not merely in practice. It vanishes when TH = TC: no temperature difference, no available work. And because this is the maximum, every real engine — burdened by friction, finite-time operation, and non-ideal cycles — falls strictly below it. The formula is simultaneously a promise and a verdict.
The Carnot efficiency describes a reversible engine, which by definition runs infinitely slowly and therefore delivers zero power. A real engine optimised for power output, not efficiency, does not run at ηmax; at maximum power its efficiency is closer to the Curzon–Ahlborn value η = 1 − √(TC/TH). Quoting the Carnot bound as the target for a working power plant confuses the thermodynamic ceiling with the engineering optimum — they are different questions. C
Three quick tests
Why does the working substance drop out of the efficiency entirely? What second-law statement is violated if a super-Carnot engine exists? Why does ηmax = 1 require an unreachable reservoir, and what does that say about the third law?
Problem set · PU-203
Chain two Carnot engines through an intermediate reservoir, derive the entropy form of the second law from the cycle, and rank a set of real engines against their Carnot ceilings.
Derivations cited: D-015 (Carnot efficiency from the second law) · Threads: energy, chance