The thermodynamic skeleton of every gasoline engine: two adiabats and two isochores, and an efficiency that rides on how hard it squeezes. Down the center the gas flows — the charge goes in, the four strokes run, the proven efficiency η = 1 − r1−γ comes out. Raise the compression ratio r and η rises — until knock stops you. The blue team builds and defends the cycle; the red team tries to break it.
source N. A. Otto, four-stroke internal-combustion engine (1876); Deutsches Reichspatent DRP 532 (1877) — no stable primary-source link, so AMBER; air-standard idealization (ideal gas, γ=7/5). Rendered, not quoted.
The air-standard cycle is four processes, each an exact law:
1→2 adiabatic compression (Q=0): T·Vγ−1 = const. 2→3 isochoric heat-in (spark): V fixed, so W=0, all Q raises T. 3→4 adiabatic expansion (the power stroke). 4→1 isochoric heat-out (exhaust): W=0.
Live state points for the current r, γ, T₁, Qin (cv=1, one mole, V₂=1):
| state | V | T | P |
|---|
This is the gasoline engine, idealized: Otto's 1876 four-stroke, its η rising with compression through adiabats and isochores.
It is a real cycle, and it sits below its ceiling — the-carnot-cycle bounds every engine between the same two temperatures. Swap the constant-volume burn for a constant-pressure one and you get its compression-ignition cousin, the-diesel-cycle. Each sphere is the next one's premise.
The blue team's live check: recompute η from the formula and from the heat balance, confirm η rises with r, and confirm η≈0.5647 at r=8. If red tampers with the exponent, this badge is where it shows.
Feed the cycle four numbers: the compression ratio r=V₁/V₂ (how far the piston squeezes), the gas's γ=cₚ/cᵥ (7/5 for diatomic air), the intake temperature T₁, and the heat Qin the spark dumps in at top-dead-centre. Everything else — the four temperatures, the two heats, the net work — is forced by the exact laws below.
r is the one knob that sets efficiency: η depends on r and γ alone, not on how much heat you add.
Move any control — η is computed live from 1 − r1−γ and cross-checked against the heat balance Qin−Qout, never looked up. The P-V loop's enclosed area is the net work.
What the machine proves: η = 1 − r1−γ = 1 − 1/rγ−1, rising monotonically with r and with γ, equal to 0.5647 at r=8, γ=7/5 — and always strictly below the Carnot ceiling 1 − Tmin/Tmax between the same extremes. The current run is above; the law is the output.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
And it is bounded twice: below Carnot (thermodynamics) and below the knock limit (chemistry) — squeeze past r≈10–12 on pump gas and the end-gas autoignites, so the formula's "just raise r" is a promise the fuel refuses to keep (AMBER — fuel/geometry dependent).
"Efficiency depends on how much fuel you burn." Cut. η = 1 − r1−γ contains no Qin — burning more raises the temperatures and the work, but leaves the ratio untouched.
"A higher compression ratio always wins." Kept, corrected. η does rise with r forever on paper, but real fuel knocks; the win is real only up to the knock limit.
"Otto invented the four-stroke cycle." Cut. A. Beau de Rochas patented the four-stroke principle in 1862; Otto built the first working engine on it (1876), which is why it bears his name.
The red team's move: flip the sign of the exponent — use η = 1 − rγ−1 instead of r1−γ, so "squeezing harder" appears to hurt. The blue team's witness (window 7) is watching.
Flip the sign and η falls as r climbs — physically backwards. The witness recomputes, sees η no longer matches the heat balance nor 0.5647 at r=8, and turns red. Nothing is faked; the attack is real and it is caught.