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The Second law of thermodynamics states that total entropy of an isolated composite system cannot decrease. For a cyclic heat engine exchanging heat only with reservoirs at $T_h>T_c>0$, it gives $-Q_h/T_h+Q_c/T_c\ge 0$. Since $W=Q_h-Q_c$,The bound is attained by a reversible heat engine: no friction, no finite-temperature-difference heat exchange and no other entropy production. A Carnot cycle provides an implementation with two reversible isotherms and two reversible adiabats. All temperatures in these ratios are absolute.

Let $n_{\mathrm{mol}}$ be the fixed gas amount, $r=V_2/V_1>1$, $\Delta T=T_h-T_c$, and let $C_V$ denote the heat capacity of the whole sample. First take the unregenerated Stirling cycle, so that heat rejected during isochoric cooling is not stored and returned during isochoric heating. The reversible quasistatic gas work iswhile the heat inputs are $Q_{12}=n_{\mathrm{mol}}R{T}_h\ln r$ and $Q_{41}=C_V\Delta T$. HenceThe strict inequality follows because $C_V\Delta T>0$ adds heat input without additional work. Quasistatic motion alone does not make heat exchange with a single hot or cold reservoir reversible while the gas temperature changes on an isochore. Indeed, using those two reservoirs gives total external entropy production $C_V\Delta T(1/T_c-1/T_h)>0$ per cycle.

The strict comparison requires the no-regeneration assumption. A perfect reversible regenerator stores $C_V\Delta T$ during $2\to 3$ and returns it during $4\to 1$; only the isothermal heat transfers then enter the external heat budget, giving $\eta =1-T_c/T_h$ exactly. For example, perfect regeneration with $T_h=600\mathrm{K}$, $T_c=300\mathrm{K}$ and $r=2$ gives $\eta =1/2={\eta }_{\max }$. Thus the four gas paths by themselves do not enforce an efficiency strictly below the Carnot bound.

For the gas itself, the four entropy changes areExpansion increases the available positional states; compression reverses that change. Isochoric cooling reduces thermal populations and isochoric heating restores them. The sum is zero because entropy is a state function. This does not imply zero total entropy production in the gas plus external reservoirs.

Taking $S_1$ as reference, the isotherms are horizontal on a $T$–$S$ diagram. The isochore through state 1 obeys $S-S_1=C_V\ln (T/T_h)$, and the isochore through state 2 is shifted right by $n_{\mathrm{mol}}R\ln r$. For the internally reversible quasistatic gas paths, $\delta Q=TdS$, soThe enclosed area is the net work per engine cycle, with a clockwise orientation. It is not the total heat input.

For the refrigerator, the stated rates give the cooling coefficient of performance $\mathcal{C}={\dot{Q}}_c/P=1.9/0.1=19$. But the absolute reservoir temperatures are $T_c=277.15\mathrm{K}$ and $T_h=293.15\mathrm{K}$, so Carnot's theorem requiresAt $0.1\mathrm{kW}$ input the largest allowed cooling rate is only $1.7322\mathrm{kW}$. Equivalently the stated $2.0\mathrm{kW}$ heat rejection gives the reservoir entropy ratea direct violation of the Second law of thermodynamics for this cyclic two-reservoir device.

The unregenerated neon model makes the algebraic obstruction explicit. With $C_V/(n_{\mathrm{mol}}R)=3/2$, reversing the cycle givesSetting this to 19 produces $\ln r=-0.89385$, or $r=0.4091$, contradicting the required expansion ratio $r>1$. Perfect regeneration can raise performance only to the Carnot bound and does not fix the problem. There is no physically admissible $V_2/V_1$ for the stated data.

An ideal cyclic heat engine producing no entropy overall. Between two equilibrium reservoirs its efficiency reaches the Carnot bound; finite reservoir temperatures can be treated as varying through successive reversible cycles.

A Stirling engine repeatedly heats and cools a confined gas, whose expansion and compression drive a moving mechanical load. This makes it a particularly accessible example of a heat engine: both the repeating motion and the two regions exchanging heat can be observed.

In the published Demonstration experiments with a Stirling engine by Christopher G Deacon and others, an electrical heater warms one end of a cylinder and flowing water cools the other at $12{}^{\circ }\mathrm{C}$. A moving displacer shuttles gas through copper wool that temporarily stores and returns internal energy. A laser spot traces a closed pressure-volume path on a screen: mechanical motion controls one coordinate, pressure the other. Calibrating both scales converts the enclosed area to work per thermodynamic cycle; multiplying by rotation frequency gives gas-cycle power. At $97\mathrm{W}$ heater input, the paper reports $4.4\mathrm{W}$ gas-cycle power, while a separate Prony brake measures $3.4\mathrm{W}$ shaft output. The cold-side temperature is controlled and two higher heater settings are compared. The gas has a nonuniform temperature, and heater energy also escapes to the room: neither output/input ratio measures $W/Q_h$ for an ideal two-thermal reservoir engine. These observations demonstrate work production and losses, not a direct measurement of the bound in Carnot's theorem. The inspected apparatus, readouts and results are on pp. 180–182, Figures 1–3 and Table 1.

A thermodynamic cycle returns a working substance to its initial state. Its net internal-energy change is zero, so the net work output equals net heat input.

Thermodynamics studies energy transfers and the constraints on macroscopic changes. Its vocabulary connects internal energy, entropy, temperature, pressure and volume to heat and work. The first law of thermodynamics balances energy; the Second law of thermodynamics decides which changes can occur spontaneously and limits how much work a heat engine can deliver. These principles let us bound performance before choosing a material or a detailed machine design. Applications include power generation, propulsion and refrigerators. ^[ref]

Start by choosing a system boundary. A closed system can exchange energy but no matter; an isolated system exchanges neither. A state function, such as internal energy or entropy, depends on the equilibrium state. A process specifies how that state changes. Thus a body's temperature is a property it has, whereas heat records energy crossing its boundary during a process. A body contains internal energy, rather than a stored quantity of heat.

Two insulated bodies initially at different positive temperatures approach thermal equilibrium when brought into thermal contact. Their combined energy is conserved, but their combined entropy increases. They do not spontaneously separate again into a hotter body and a colder one. This familiar asymmetry makes thermodynamics interesting: conserving energy leaves many imaginable changes available, and entropy supplies a further constraint. The same distinction underlies both entropy maximization under thermal contact and the thermal efficiency limit in Carnot's theorem.

A physicist whose thermodynamics distinguished conservation of energy from restrictions on cyclic heat-to-work conversion.

A Scottish engineer and physicist who contributed to thermodynamics and classical mechanics. Institutional historical profile: Imperial College historical plaque booklet: William John Macquorn Rankine.