= Carnot's theorem
{c}
{title2=$\eta\le1-T_c/T_h$}
{wiki=Carnot's_theorem_(thermodynamics)}
= Carnot theorem
{c}
{synonym}
No cyclic <heat engine> exchanging <heat> only with two fixed <thermal reservoirs> at positive absolute <temperatures> $T_h>T_c$ can exceed the <thermal efficiency> of an engine whose cycle is a <reversible thermodynamic process> between them. All such reversible engines have the same <thermal efficiency>, regardless of their working substance. Let $Q_h>0$ be <heat> absorbed from the hot <thermal reservoir> and $Q_c>0$ the magnitude rejected to the cold one. Then
$$
W=Q_h-Q_c,\qquad
\eta=\frac W{Q_h}\le\eta_C=1-\frac{T_c}{T_h}.
$$
The <work> convention is the one in the <first law of thermodynamics>: $W>0$ is net output by the engine. Use absolute <thermodynamic temperatures>, such as values in <kelvin>; ratios of readings in <degree Celsius> are invalid because their zero is shifted.
For $T_h=600\,\mathrm K$, $T_c=300\,\mathrm K$ and $Q_h=1000\,\mathrm J$, the maximum <thermal efficiency> is $50\%$. A reversible engine delivers $W=500\,\mathrm J$ and rejects $Q_c=500\,\mathrm J$. The <thermal reservoirs> have <entropy> changes $-1000/600$ and $+500/300\,\mathrm{J\,K^{-1}}$, which cancel. The engine's own <entropy> change is zero over its <thermodynamic cycle>.
\Image[/carnot-engine-efficiency.png]
{title=Carnot engine energy transfers and the reversible cycle between 600 K and 300 K}
A short derivation now becomes transparent: the <Second law of thermodynamics> gives $-Q_h/T_h+Q_c/T_c\ge0$, and the <first law of thermodynamics> gives $W=Q_h-Q_c$. Together they yield the bound and, more precisely,
$$
W=Q_h\left(1-\frac{T_c}{T_h}\right)-T_c\Delta S_{\rm tot}.
$$
At fixed $Q_h,T_h,T_c$, every extra unit of total <entropy production> reduces output by $T_c$ units of <energy>. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node44.html[MIT, Maximum heat-engine work]{ref}
Fixed <thermal reservoirs> are essential. Finite bodies cool and warm, so their initial <temperature> ratio cannot be used throughout an entire extraction process. As a computed contrast, the bodies with equal constant <heat capacity> in the <entropy> example would reach $T_f=\sqrt{600\times300}=424.26\,\mathrm K$ if a reversible engine extracted <work> until their <temperatures> matched. Zero total <entropy> change gives this geometric mean; the <first law of thermodynamics> then gives $W_{\max}=C(600+300-2T_f)=5147.19\,\mathrm J$. Direct contact reaches $450\,\mathrm K$ and extracts no <work>.
Carnot's 1824 book asked whether the useful output of machines driven by <heat> was fundamentally limited and whether changing the working substance could improve that limit. Its reversible comparisons are the historical setting of <Carnot's theorem>. Carnot used the then-current caloric picture, treating <heat> as conserved in its passage through an engine; the modern balance $Q_h-Q_c=W$ should not be read back into that text.<Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance by Sadi Carnot>[pp. 6–7 and 20–23]{ref}
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