= Carnot cycle
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{title2=$W=(T_h-T_c)\Delta S$}
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A Carnot cycle realizes the reversible bound in <Carnot's theorem>. Its four steps for an <ideal gas> working substance are:
* Reversible <isothermal expansion> at $T_h$, absorbing $Q_h$ and increasing <entropy> by $\Delta S$.
* Reversible <adiabatic expansion>, doing <work> while the <temperature> falls from $T_h$ to $T_c$ and <entropy> stays fixed.
* Reversible <isothermal compression> at $T_c$, rejecting $Q_c$ and decreasing <entropy> by $\Delta S$.
* Reversible <adiabatic compression>, requiring <work> to restore $T_h$ at fixed <entropy>.
The <temperature> versus <entropy> plot in the <Carnot's theorem> figure is therefore a rectangle traversed clockwise. Because $\delta Q_{\rm rev}=T\,dS$, its width is $\Delta S=Q_h/T_h$. For the illustrated engine, $\Delta S=5/3\,\mathrm{J\,K^{-1}}$: the upper step absorbs $600\times5/3=1000\,\mathrm J$, the lower step rejects $300\times5/3=500\,\mathrm J$, and the enclosed area is the $500\,\mathrm J$ net <work>. Geometry displays the <thermal efficiency> without solving the gas motion. The two vertical steps are <isentropic processes>, not constant-<temperature> steps. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node24.html[MIT, Four steps of a Carnot cycle]{ref} https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node46.html[MIT, Temperature-entropy diagrams]{ref}
Reversible <heat transfer> is a limit of vanishing <temperature> differences; ordinary finite-rate transfer introduces <entropy production>. The <Carnot cycle> is a performance benchmark rather than a guarantee of useful <power>. Running it backward gives the ideal operation of a <refrigerator> or <heat pump>, with <work> supplied to move <heat> toward the hotter <thermal reservoir>. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node25.html[MIT, Reversed cycles and refrigeration]{ref}
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