Carnot engine · the most work heat can do

Heat flows from hot to cold — what is the MOST work an engine can wring from that flow, and what does the limit depend on?

How the lab tests it

Run a reversible Carnot cycle of an ideal gas (γ=5/3) between reservoirs T_h and T_c: isothermal expansion at T_h (drinks Q_h) → adiabatic expansion → isothermal compression at T_c (sheds Q_c) → adiabatic compression. Numerically traverse the loop, integrating the net work W=∮P dV (the enclosed P–V area) and the hot-isotherm heat Q_h by trapezoid, and recover the efficiency η=W/Q_h — never plugging into the formula. Repeat across five (T_h,T_c) pairs, sweep the expansion ratio V₂/V₁∈[1.5,12], and run an Otto cycle between the same temperature extremes as a control. ?world=carnot.

What it checks

the Carnot efficiency η = 1 − T_c/T_h — recovered from ∮P dV to machine precision (600 K/300 K ⇒ exactly 50%) and, decisively, INDEPENDENT of the working substance, the amount of gas, and the expansion ratio: η is the SAME to 1e-15 across V₂/V₁ from 1.5 to 12, so a perfect engine's quality is fixed by the two reservoir temperatures alone. And no engine beats it (Carnot's theorem = the second law): an Otto cycle between the same extremes (r_v=8, 300–1500 K) reaches only η=1−1/r_v^(γ−1)=75% against the Carnot ceiling 1−300/1500=80%. The gap is why no power plant reaches Carnot and why raising T_h, not lowering T_c, is the lever — a real coal plant at ~810 K/300 K caps near 63%. The lab's first thermodynamics world.