The perfect heat engine no real engine can beat — and the ceiling it sets on all of them. Two isotherms and two adiabats slung between a hot reservoir Th and a cold one Tc, and the whole of the second law falls out of one fraction: η = 1 − Tc/Th. Down the center, data flows: the reservoirs go in, the engine computes the ceiling, the proven bound comes out. The blue team builds it; the red team tries to slip past the limit.
source Carnot, Réflexions sur la puissance motrice du feu (1824), Bachelier — scanned original archive.org/details/bub_gb_QX9iIWF3yOMC AMBER (facsimile, no canonical DOI). Rendered, not quoted — ideal gas, γ = 7/5.
The efficiency is not measured; it falls out of the reversible construction. Heat crosses only on the two isotherms, and both cross at a fixed temperature:
1→2 isothermal expansion at Th: absorbs Qh = ThΔS. 2→3 adiabatic expansion: no heat, T drops to Tc. 3→4 isothermal compression at Tc: rejects Qc = TcΔS. 4→1 adiabatic compression back up.
Live for the current reservoirs (ΔS = ln 3 per unit gas):
| stroke | heat | at T |
|---|
η = W/Qh = (Qh−Qc)/Qh = (Th−Tc)/Th = 1 − Tc/Th.
Carnot, 1824, built the perfect engine: reversible, zero entropy generated, η = 1 − Tc/Th set by the reservoirs alone. That number is the ceiling on every heat engine.
the-otto-cycle and the-rankine-cycle both fall short of it, and the-clausius-inequality is the theorem that proves no cycle can exceed it. This sphere is their common bound — each neighbour is measured against the number computed here.
The blue team's live check: re-run the engine on a fixed reservoir pair and confirm it against the honest formula, plus η∈(0,1) and Otto<Carnot. If the red team pushes η above the limit, this badge is where it shows.
A heat engine needs exactly two things it cannot make: a hot reservoir Th to draw from and a cold reservoir Tc to dump into. Between them runs an ideal gas (γ = 7/5, diatomic). Nothing else — no fuel chemistry, no material, no size — enters the ceiling.
Only the ratio Tc/Th matters. A perfect engine would need Tc = 0 (absolute zero) to reach η = 1 — unreachable, so η < 1 always. Feed the two temperatures into the panel below.
Move any slider — every number is computed from 1 − Tc/Th and the reversible heat balance on the spot, never looked up.
What the machine proves: η = 1 − Tc/Th depends on the reservoir temperatures alone, always lies strictly in (0,1), and is the maximum — a same-reservoir Otto cycle comes in below it. The reversible cycle's closed ∫dQ/T = 0, so the T–S diagram is a rectangle whose area (Th−Tc)ΔS equals the net work, and Qc/Qh = Tc/Th.
The blue team's witness (left) confirms this live; the red team (right) tries to make η exceed it.
Finite-time thermodynamics (Curzon–Ahlborn, 1975) shows a real engine run at maximum power peaks near η = 1 − √(Tc/Th) — well below Carnot. The ceiling is real and unbeatable, but it is a limit you approach only by giving up the very thing you built the engine for.
"A Carnot engine is the most efficient engine, full stop." Cut. Most efficient reversible engine between two fixed reservoirs — and it delivers zero power. Real engines optimize power, not this bound.
"Efficiency can, in principle, reach 100%." Cut. Only if Tc = 0 K, which the third law forbids. For any finite Tc > 0, η < 1 — computed live in the panel.
"Carnot's caloric theory was right, so the result stands." Kept, corrected. Carnot used the (wrong) caloric fluid picture; the conclusion survived re-derivation by Clausius & Kelvin on energy conservation. Right answer, retired premise.
The red team's move: claim an engine that beats the limit — swap the formula for η = 1 − Tc/Th + 0.1, an efficiency above Carnot that can exceed 1 and violate the second law. The blue team's witness (window 7) is watching.
Push η above the ceiling and the witness recomputes, disagrees with the honest 1 − Tc/Th, and turns red. Nothing is faked; the over-unity claim is real and it is caught.