Heat engine & piston.
Follow the gas. Close the cycle.
Where does the useful work come from?
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P–V
Q = ΔU + W
Q into gas is positive.
W by gas is positive.
Increase the expansion ratio. Does the work per cycle change? Does the efficiency change?
Animation shows stage progress, not physical time. No power or flywheel dynamics is inferred.Equations, assumptions & source
A fixed monatomic ideal gas: PV = nRT and U = nCᵥT, with Cᵥ = R/(γ − 1). The cycle has two isotherms and two reversible adiabats, for which PVᵞ and TVᵞ⁻¹ are constant.
Work by the gas is W = ∫P dV. On an isotherm, W = nRT ln(Vf/Vi) and ΔU = 0, so Q = W. On an adiabat, Q = 0, so W = −ΔU. The full cycle returns to the same U, hence Wnet = Qₕ + Q꜀.
The four states are linked: VB = rVA, VD = VA(Th/Tc)¹⧸⁽ᵞ⁻¹⁾ and VC = rVD. They cannot be dragged independently. Engine efficiency is η = 1 − Tc/Th. In reverse, refrigerator COP = Tc/(Th − Tc); net work must be supplied.
Quasistatic, frictionless piston; perfect thermal contact during each isotherm and complete insulation during each adiabat. Reservoir temperatures are fixed. Real heat transfer needs a temperature difference: this reversible cycle is an ideal limit. Finite-speed losses, friction, gas flow and inertia are excluded. Apparatus dimensions, heat chevrons and mechanical linkage are schematic.
Source: original analytical implementation of the ideal-gas Carnot cycle. R = 8.314462618 J mol⁻¹ K⁻¹. Values are model predictions, not measurements. No external assets or live data.