= Kelvin-Planck statement of the second law
{c}
No cyclic device can have as its sole effect the extraction of <heat> from one <thermal reservoir> and its complete conversion into net <work>. A putative device absorbing $Q_h>0$ and returning to its starting state with $W=Q_h$ and no other transfer would conserve <energy>, yet violate the <Second law of thermodynamics>. Rejecting <heat> to colder surroundings is an essential part of the ordinary two-<thermal reservoir> <heat engine> budget.<Thermodynamics and Propulsion/5.1 Concept and Statements of the Second Law (Why do we need a second law?)>[§5.1, statement 1 and Figure 5.1: complete conversion as the sole result]{ref}
The cyclic and sole-effect qualifications matter. A single <isothermal expansion> of an <ideal gas> can absorb <heat> and deliver equal <work> while leaving its <volume (thermodynamics)> changed; restoring the initial state must also be counted. Likewise, a device discharging stored <internal energy> or consuming fuel has another effect. Converting supplied electrical <work> completely to <heat> through <Joule heating> reverses the proposed conversion and is allowed. The statement also allows cyclic devices that exchange <heat> with one <thermal reservoir> while consuming net <work>; it forbids positive net <work> output with that single <heat> source as the only external change.
Conversely, a hypothetical complete-conversion device could take $100\,\mathrm J$ from the hot <thermal reservoir> and supply $100\,\mathrm J$ of <work> to an ideal <refrigerator> that extracts $100\,\mathrm J$ from the cold <thermal reservoir> and delivers $200\,\mathrm J$ to the hot one. At $600\,\mathrm K$ and $300\,\mathrm K$, that <refrigerator> is itself admissible: its reservoir <entropy> changes cancel. The composite would consume no external <work> and move a net $100\,\mathrm J$ from cold to hot, violating the <Clausius statement of the second law>. Together with the construction there, this proves the equivalence of the two impossibility statements.<8.333 Statistical Mechanics I: Statistical Mechanics of Particles/Statistical Mechanics I: Lecture 2>[§I.D, printed p. 8: engine driving a refrigerator]{ref}
For the two fixed positive-<temperature> <thermal reservoir> hypotheses, <Carnot's theorem> gives the sharper restriction $\eta\le1-T_c/T_h<1$. A <reversible thermodynamic process> reaches that bound in the ideal <Carnot cycle>; an ordinary device's smaller <thermal efficiency> reflects its <entropy production> and measurement boundary. An inefficient observed engine is compatible with the <Second law of thermodynamics>, rather than evidence that the bound has been experimentally determined.
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