Thought Toys · Cycles & change · Exhibit 68
Give yourself an engine with no friction, no leaks, no wasted motion — parts machined to the atom. Run it between a flame and the open air and it still can't turn much more than half the heat into work. The rest has to be dumped. Not because the engineering isn't good enough: because of what heat is.
The cycle, and where the heat actually goes work you get out heat you must dump the ceiling
—
—
On the left is one trip round the cycle. A gas is let expand while touching something hot, so it draws heat in and pushes on a piston. It expands a little further while touching nothing, which cools it. Then it's squeezed back while touching something cold, which is where heat gets dumped. Then squeezed the rest of the way, which warms it back up to where it started. The loop closes, and the amber area inside it is the work a reversible engine gets out — the most this pair of temperatures allows anyone. Your own engine's claim is the column on the right. (The axes rescale to keep the loop in frame, so compare shapes on the left and numbers on the right — not areas across two different settings.)
On the right is the accounting. The whole column is the heat you drew from the hot side. The amber block on top is the part you turned into work; the red block underneath is the part you had to hand over to the cold side. Every engine has that red block. It is not leakage, and no amount of precision machining shrinks it — it is the fee for moving heat downhill at all.
The dashed line is the ceiling, and it moves when you move either temperature. Push the two temperatures apart and it rises; bring them together and it falls toward nothing. What it never depends on is your engine. Not the gas inside it, not the design, not the century it was built in. Only the two numbers.
Now drag your claim above the dashed line. Nothing on screen breaks — but look at the entropy figure. It goes negative, and that is the interesting part. An engine that beats the ceiling wouldn't just be very hard to build; it would require the total disorder of the universe to decrease, which is the one thing the second law says never happens on its own. Below the line you're merely inefficient, which is allowed, and which is where every real engine sits. Above it you're not an engineer with a hard problem. You're proposing a different universe.
Two settings are worth reaching deliberately. Push the hot side down until it meets the cold side: the ceiling hits exactly zero. The engine is still swallowing thousands of joules of heat and delivering nothing. An ocean's worth of warmth, all at one temperature, is worth precisely no work at all — which is why the heat in the sea around a ship can't move the ship. And a single degree of difference is already worth something: it isn't the heat that's valuable, it's the gap.
improve/verify/68-carnot.js): the efficiency is never
substituted from the formula, it is derived — Simpson's rule integrates the real ∫P dV
along all four legs and the ratio of the integrated areas lands on 1 − Tc/Th to within
6×10⁻¹³ across 5 hot temperatures × 4 cold ones × 3 compression ratios × 2 gases, with the first law
(W = Qhot − Qcold) and the entropy balance confirmed from the same integrated numbers.
Four negative controls: any efficiency above the ceiling drives the total entropy strictly negative — exactly
zero at the ceiling, and already negative one part in a billion above it; three different gases trace cycles whose
expansion volumes differ by a factor of 5.2 yet land on the identical efficiency; an engine driven by a
finite temperature gap falls strictly short at every gap and reaches the ceiling only as the gap — and so the
power — goes to zero; and at Tc = Th the engine absorbs 4,567 J and delivers 10⁻¹¹ J.
Real engines land well below the ceiling for exactly the third reason: an engine that
actually reaches the Carnot limit has to run infinitely slowly.
Also in Cycles & change: Squeeze a reaction and it pushes back →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 68.