THM-10

Carnot efficiency

η = 1 − Tc/Th for a reversible engine between two reservoirs.

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First lawCarnot

Governing equation

\eta=1-T_c/T_h

$T_h \in \left[300, 1200\right]$

where

T_h: Hot reservoir (K)
T_c: Cold reservoir (K)
\eta: Efficiency (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\eta=1-T_c/T_h

$T_h \in \left[300, 1200\right]$

$T_h = 300$ $\eta = 0$

Along the curve

②

Domain of the variable — blue

T_h_{\min} = 300,\quad T_h_{\max} = 1200

$T_h \in \left[300, 1200\right]$

$T_h = 300$ $T_h = 300$

Along the curve

③

Response curve — green

\eta=1-T_c/T_h

$T_h \in \left[300, 1200\right]$

$T_h = 300$ $\eta = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\eta}{\mathrm{d}T_h}

$T_h \in \left[300, 1200\right]$

$T_h = 305.7$ $\eta^{\prime} = 0.003$

Along the curve

Lecture brief

Historical brief

Carnot (1824), Clausius entropy, the first law and later van der Waals and Gibbs potentials turned heat into a state science. The sheets compute work, efficiency and vapour pressure. This sheet (THM-10 — Carnot efficiency) is the form associated with Carnot. Working symbols: $T_h$ , $T_c$ $\rightarrow$ $\eta$ . No engine between the same two temperatures can beat Carnot. Equality for reversible cycles.

Purpose

Purpose: compute

\eta

from

T_h

T_c

in Thermodynamics via

\eta=1-T_c/T_h

η = 1 − Tc/Th for a reversible engine between two reservoirs. Use it when a real thermodynamics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

T_h = 600.000\,\mathrm{K}

T_c = 300.000\,\mathrm{K}

, the governing relation

\eta=1-T_c/T_h

yields

\eta = 0.500\,\mathrm{—}

. Two baths, a cycle lozenge, an efficiency bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Hot reservoir T_h600 KCold reservoir T_c300 K

Outputs

Efficiency \eta0.500 —

Reading speed

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Narration of this film

Two baths, a cycle lozenge, an efficiency bar.

No engine between the same two temperatures can beat Carnot. Equality for reversible cycles.

Reading speed

Watch on YouTube