= Clausius statement of the second law
{c}
No cyclic device can have as its sole effect the transfer of <heat> from a colder <thermal reservoir> to a hotter <thermal reservoir>. The word “sole” includes the surroundings: the device must return to its starting state, and there must be no net supplied <work>, depletion of a stored resource, or accompanying change elsewhere. The statement excludes a self-running <refrigerator>, while allowing an ordinary <refrigerator> or <heat pump> supplied with <work>.<Thermodynamics and Propulsion/5.1 Concept and Statements of the Second Law (Why do we need a second law?)>[§5.1, statement 2 and Figure 5.2: sole-result restriction]{ref}
For a proposed transfer $Q>0$ from a fixed cold <thermal reservoir> at $T_c$ to a fixed hot <thermal reservoir> at $T_h>T_c>0$, with no other effect, the total <entropy> change would be
$$
\Delta S_{\rm tot}=\frac{Q}{T_h}-\frac{Q}{T_c}<0.
$$
The <first law of thermodynamics> permits this <energy> balance; the <Second law of thermodynamics> forbids the negative total <entropy> change. For $Q=600\,\mathrm J$, $T_c=300\,\mathrm K$ and $T_h=600\,\mathrm K$, it is $-1\,\mathrm{J\,K^{-1}}$. Direct <heat transfer> in the opposite direction has positive total <entropy> change. A powered <heat pump> has a different budget: it supplies <work> and delivers more <heat> to the hot side than it extracts from the cold side.
A short equivalence argument links this restriction to the <Kelvin-Planck statement of the second law>. Couple a hypothetical device that moves $600\,\mathrm J$ from cold to hot with no other effect to the <heat engine> illustrated above, which absorbs $1000\,\mathrm J$, rejects $600\,\mathrm J$, and delivers $400\,\mathrm J$. The cold-side transfers cancel; the combined cyclic device takes $400\,\mathrm J$ from the hot <thermal reservoir> and delivers exactly $400\,\mathrm J$ of <work>, violating the <Kelvin-Planck statement of the second law>. Matching transfers is an ideal composition argument, not a claim that such a device exists.<8.333 Statistical Mechanics I: Statistical Mechanics of Particles/Statistical Mechanics I: Lecture 2>[§I.D, printed pp. 7–8: equivalence by coupling devices]{ref}
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