= Second law of thermodynamics
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For an <isolated system>, the second law requires $\Delta S\ge0$. More generally, include all affected surroundings to obtain
$$
\Delta S_{\rm tot}=\Delta S_{\rm system}+\Delta S_{\rm surroundings}\ge0.
$$
Equality describes the reversible limit; irreversible macroscopic processes produce <entropy>. A <closed system> can lose <entropy> by exporting <heat>, so nondecrease does not apply to that system alone. The <Clausius statement of the second law> forbids a cyclic device whose sole effect is transferring <heat> from cold to hot. The <Kelvin-Planck statement of the second law> forbids a cyclic device whose sole effect is converting <heat> from one <thermal reservoir> completely into <work>. "Sole effect" matters: a <refrigerator> moves <heat> from cold to hot by consuming <work>. <Thermodynamics and Propulsion/5.1 Concept and Statements of the Second Law (Why do we need a second law?)>{ref}
For direct transfer of $Q=1000\,\mathrm J$ from a <thermal reservoir> at $600\,\mathrm K$ to one at $300\,\mathrm K$, with no <work> extracted,
$$
\Delta S_{\rm tot}=-\frac{1000}{600}+\frac{1000}{300}
=\frac53\,\mathrm{J\,K^{-1}}>0.
$$
The reverse transfer alone would have negative total <entropy> change. Both directions conserve <energy>, but only the hot-to-cold direction is spontaneous. The finite-body example under <entropy> replaces these constant <temperatures> by changing ones.
<Friction>, <free expansion> and <heat transfer> across a finite <temperature> difference all introduce irreversibility. Moving arbitrarily slowly is insufficient if <friction> remains. A <reversible thermodynamic process> must permit restoration of both the system and surroundings. This gives engineers a useful ideal benchmark: irreversibility spends an opportunity to obtain <work>, even though <energy> itself is conserved. https://www.ocw.mit.edu/ans7870/16/16.unified/thermoF03/chapter_7.htm[MIT, Irreversibility and lost opportunities for work]{ref}
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