Entropy production

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class="vector-toc-numb">2</span> <span>First and second law</span> </div> </a> <ul id="toc-First_and_second_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_irreversible_processes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_of_irreversible_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples of irreversible processes</span> </div> </a> <ul id="toc-Examples_of_irreversible_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Performance_of_heat_engines_and_refrigerators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Performance_of_heat_engines_and_refrigerators"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Performance of heat engines and refrigerators</span> </div> </a> <button aria-controls="toc-Performance_of_heat_engines_and_refrigerators-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Performance of heat engines and refrigerators subsection</span> </button> <ul id="toc-Performance_of_heat_engines_and_refrigerators-sublist" class="vector-toc-list"> <li id="toc-Engines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Engines"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Engines</span> </div> </a> <ul id="toc-Engines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Refrigerators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Refrigerators"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Refrigerators</span> </div> </a> <ul id="toc-Refrigerators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_dissipation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_dissipation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Power dissipation</span> </div> </a> <ul id="toc-Power_dissipation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalence_with_other_formulations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equivalence_with_other_formulations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Equivalence with other formulations</span> </div> </a> <ul id="toc-Equivalence_with_other_formulations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expressions_for_the_entropy_production" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expressions_for_the_entropy_production"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Expressions for the entropy production</span> </div> </a> <button aria-controls="toc-Expressions_for_the_entropy_production-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Expressions for the entropy production subsection</span> </button> <ul id="toc-Expressions_for_the_entropy_production-sublist" class="vector-toc-list"> <li id="toc-Heat_flow" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_flow"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Heat flow</span> </div> </a> <ul id="toc-Heat_flow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_of_mass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flow_of_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Flow of mass</span> 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<ul id="toc-Microscopic_interpretation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_inequalities_and_stability_conditions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_inequalities_and_stability_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Basic inequalities and stability conditions</span> </div> </a> <ul id="toc-Basic_inequalities_and_stability_conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homogeneous_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Homogeneous_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Homogeneous systems</span> </div> </a> <ul id="toc-Homogeneous_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy_production_in_stochastic_processes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Entropy_production_in_stochastic_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Entropy production in stochastic processes</span> </div> </a> <ul id="toc-Entropy_production_in_stochastic_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> 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0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Lead_too_short plainlinks metadata ambox ambox-content ambox-lead_too_short" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article's <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section#Length" title="Wikipedia:Manual of Style/Lead section">lead section</a> <b>may be too short to adequately <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">summarize</a> the key points</b>.<span class="hide-when-compact"> Please consider expanding the lead to <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section#Provide_an_accessible_overview" title="Wikipedia:Manual of Style/Lead section">provide an accessible overview</a> of all important aspects of the article.</span> <span class="date-container"><i>(<span class="date">February 2024</span>)</i></span></div></td></tr></tbody></table> <p><b>Entropy production</b> (or generation) is the amount of entropy which is produced during heat process to evaluate the efficiency of the process. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Clausius.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Clausius.jpg/250px-Clausius.jpg" decoding="async" width="250" height="296" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/40/Clausius.jpg 1.5x" data-file-width="275" data-file-height="326" /></a><figcaption><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a></figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Short_history">Short history</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=1" title="Edit section: Short history"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Entropy" title="Entropy">Entropy</a> is produced in irreversible processes. The importance of avoiding irreversible processes (hence reducing the entropy production) was recognized as early as 1824 by Carnot.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In 1865 <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> expanded his previous work from 1854<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> on the concept of "unkompensierte Verwandlungen" (uncompensated transformations), which, in our modern nomenclature, would be called the entropy production. In the same article in which he introduced the name entropy,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Clausius gives the expression for the entropy production for a cyclical process in a closed system, which he denotes by <i>N</i>, in equation (71) which reads </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=S-S_{0}-\int {\frac {dQ}{T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mi>S</mi> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=S-S_{0}-\int {\frac {dQ}{T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13204120b8c845baf2952f80a1167218baf4fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.939ex; height:5.843ex;" alt="{\displaystyle N=S-S_{0}-\int {\frac {dQ}{T}}.}"></span></dd></dl> <p>Here <i>S</i> is the entropy in the final state and <i>S<sub>0</sub></i> the entropy in the initial state; <i>S<sub>0</sub>-S</i> is the entropy difference for the backwards part of the process. The integral is to be taken from the initial state to the final state, giving the entropy difference for the forwards part of the process. From the context, it is clear that <span class="nowrap"><i>N</i> = 0</span> if the process is reversible and <span class="nowrap"><i>N</i> > 0</span> in case of an irreversible process. </p> <div class="mw-heading mw-heading2"><h2 id="First_and_second_law">First and second law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=2" title="Edit section: First and second law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Thermodynamic_system01b.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Thermodynamic_system01b.jpg/300px-Thermodynamic_system01b.jpg" decoding="async" width="300" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Thermodynamic_system01b.jpg/450px-Thermodynamic_system01b.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Thermodynamic_system01b.jpg/600px-Thermodynamic_system01b.jpg 2x" data-file-width="2356" data-file-height="2020" /></a><figcaption> Fig. 1 General representation of an inhomogeneous system that consists of a number of subsystems. The interaction of the system with the surroundings is through exchange of heat and other forms of energy, flow of matter, and changes of shape. The internal interactions between the various subsystems are of a similar nature and lead to entropy production.</figcaption></figure> <p>The laws of thermodynamics system apply to well-defined systems. Fig. 1 is a general representation of a thermodynamic system. We consider systems which, in general, are inhomogeneous. Heat and mass are transferred across the boundaries (nonadiabatic, open systems), and the boundaries are moving (usually through pistons). In our formulation we assume that heat and mass transfer and volume changes take place only separately at well-defined regions of the system boundary. The expression, given here, are not the most general formulations of the first and second law. E.g. kinetic energy and potential energy terms are missing and exchange of matter by diffusion is excluded. </p><p>The rate of entropy production, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fb101f17394253a0e5bb8816e28386c92391e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.262ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}}"></span>, is a key element of the second law of thermodynamics for open inhomogeneous systems which reads </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=\sum _{k}{\frac {{\dot {Q}}_{k}}{T_{k}}}+\sum _{k}{\dot {S}}_{k}+\sum _{k}{\dot {S}}_{{\text{i}}k}{\text{ with }}{\dot {S}}_{{\text{i}}k}\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> with </mtext> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> <mi>k</mi> </mrow> </msub> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=\sum _{k}{\frac {{\dot {Q}}_{k}}{T_{k}}}+\sum _{k}{\dot {S}}_{k}+\sum _{k}{\dot {S}}_{{\text{i}}k}{\text{ with }}{\dot {S}}_{{\text{i}}k}\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7442dcbb6486a6fef3d94149c6781dfd660199b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.887ex; height:7.176ex;" alt="{\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=\sum _{k}{\frac {{\dot {Q}}_{k}}{T_{k}}}+\sum _{k}{\dot {S}}_{k}+\sum _{k}{\dot {S}}_{{\text{i}}k}{\text{ with }}{\dot {S}}_{{\text{i}}k}\geq 0.}"></span></dd></dl> <p>Here <i>S</i> is the entropy of the system; <i>T</i><sub><i>k</i></sub> is the temperature at which the heat enters the system at heat flow rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94544741116b2dfb2bbf6979d8eca04a5fb79f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.927ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{k}}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{k}={\dot {n}}_{k}S_{{\text{m}}k}={\dot {m}}_{k}s_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{k}={\dot {n}}_{k}S_{{\text{m}}k}={\dot {m}}_{k}s_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9dadd3119bb58cdc5a7b9c45c6830bc828810a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.533ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{k}={\dot {n}}_{k}S_{{\text{m}}k}={\dot {m}}_{k}s_{k}}"></span> represents the entropy flow into the system at position <i>k</i>, due to matter flowing into the system (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {n}}_{k},{\dot {m}}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {n}}_{k},{\dot {m}}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffc3b510c08e85a0683f341a703250bcd6b489e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.647ex; height:2.509ex;" alt="{\displaystyle {\dot {n}}_{k},{\dot {m}}_{k}}"></span> are the molar flow rate and mass flow rate and <i>S</i><sub>m<i>k</i></sub> and <i>s</i><sub><i>k</i></sub> are the molar entropy (i.e. entropy per unit <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a>) and specific entropy (i.e. entropy per unit mass) of the matter, flowing into the system, respectively); <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{{\text{i}}k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{{\text{i}}k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60170a454b12669d0b713dc79cdff463076d6df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.119ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{{\text{i}}k}}"></span> represents the entropy production rates due to internal processes. The subscript 'i' in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{{\text{i}}k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{{\text{i}}k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60170a454b12669d0b713dc79cdff463076d6df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.119ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{{\text{i}}k}}"></span> refers to the fact that the entropy is produced due to irreversible processes. The entropy-production rate of every process in nature is always positive or zero. This is an essential aspect of the second law. </p><p>The Σ's indicate the algebraic sum of the respective contributions if there are more heat flows, matter flows, and internal processes. </p><p>In order to demonstrate the impact of the second law, and the role of entropy production, it has to be combined with the first law which reads </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\mathrm {d} V_{k}}{\mathrm {d} t}}+P,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\mathrm {d} V_{k}}{\mathrm {d} t}}+P,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c648bad8ffee0c003be13063756e918ac12177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.06ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\mathrm {d} V_{k}}{\mathrm {d} t}}+P,}"></span></dd></dl> <p>with <i>U</i> the internal energy of the system; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {H}}_{k}={\dot {n}}_{k}H_{{\text{m}}k}={\dot {m}}_{k}h_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {H}}_{k}={\dot {n}}_{k}H_{{\text{m}}k}={\dot {m}}_{k}h_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dae807290b2886db357870f11f13a7162e18a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.778ex; height:3.009ex;" alt="{\displaystyle {\dot {H}}_{k}={\dot {n}}_{k}H_{{\text{m}}k}={\dot {m}}_{k}h_{k}}"></span> the <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> flows into the system due to the matter that flows into the system (<i>H</i><sub>m<i>k</i></sub> its molar enthalpy, <i>h</i><sub><i>k</i></sub> the specific enthalpy (i.e. enthalpy per unit mass)), and d<i>V</i><sub><i>k</i></sub>/d<i>t</i> are the rates of change of the volume of the system due to a moving boundary at position <i>k</i> while <i>p</i><sub><i>k</i></sub> is the pressure behind that boundary; <i>P</i> represents all other forms of power application (such as electrical). </p><p>The first and second law have been formulated in terms of time derivatives of <i>U</i> and <i>S</i> rather than in terms of total differentials d<i>U</i> and d<i>S</i> where it is tacitly assumed that d<i>t</i> > 0. So, the formulation in terms of time derivatives is more elegant. An even bigger advantage of this formulation is, however, that it emphasizes that <i>heat flow rate</i> and <i>power</i> are the basic thermodynamic properties and that heat and work are derived quantities being the time integrals of the heat flow rate and the power respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_irreversible_processes">Examples of irreversible processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=3" title="Edit section: Examples of irreversible processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Entropy is produced in <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">irreversible processes</a>. Some important irreversible processes are: </p> <ul><li>heat flow through a thermal resistance</li> <li>fluid flow through a flow resistance such as in the <a href="/wiki/Joule_expansion" title="Joule expansion">Joule expansion</a> or the <a href="/wiki/Joule%E2%80%93Thomson_effect" title="Joule–Thomson effect">Joule–Thomson effect</a></li> <li>heat transfer</li> <li>Joule heating</li> <li>friction between solid surfaces</li> <li>fluid viscosity within a system.</li></ul> <p>The expression for the rate of entropy production in the first two cases will be derived in separate sections. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Heat_engine_and_refrigerator01.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Heat_engine_and_refrigerator01.jpg/400px-Heat_engine_and_refrigerator01.jpg" decoding="async" width="400" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Heat_engine_and_refrigerator01.jpg/600px-Heat_engine_and_refrigerator01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Heat_engine_and_refrigerator01.jpg/800px-Heat_engine_and_refrigerator01.jpg 2x" data-file-width="2712" data-file-height="1772" /></a><figcaption>Fig.2 <b>a</b>: Schematic diagram of a heat engine. A heating power <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03275e4e60988b0e4b7468eeef5b3cf025300aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.303ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{H}}}"></span> enters the engine at the high temperature <i>T</i><sub>H</sub>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446088ace716aeb989e48ba28d942a84ad802262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.893ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{a}}}"></span> is released at ambient temperature <i>T</i><sub>a</sub>. A power <i>P</i> is produced and the entropy production rate is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fb101f17394253a0e5bb8816e28386c92391e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.262ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}}"></span>.<br /><b>b</b>: Schematic diagram of a refrigerator. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba465815bf1c62ba9119587e25ec5a6e9d8a69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.098ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{L}}}"></span> is the cooling power at the low temperature <i>T</i><sub>L</sub>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446088ace716aeb989e48ba28d942a84ad802262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.893ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{a}}}"></span> is released at ambient temperature. The power <i>P</i> is supplied and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fb101f17394253a0e5bb8816e28386c92391e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.262ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}}"></span> is the entropy production rate. The arrows define the positive directions of the flows of heat and power in the two cases. They are positive under normal operating conditions.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Performance_of_heat_engines_and_refrigerators">Performance of heat engines and refrigerators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=4" title="Edit section: Performance of heat engines and refrigerators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most heat engines and refrigerators are closed cyclic machines.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In the steady state the internal energy and the entropy of the machines after one cycle are the same as at the start of the cycle. Hence, on average, d<i>U</i>/d<i>t</i> = 0 and d<i>S</i>/d<i>t</i> = 0 since <i>U</i> and <i>S</i> are functions of state. Furthermore, they are closed systems (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {n}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {n}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0954accae5be3956e53cb5fbe41ccd7b74f747b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle {\dot {n}}=0}"></span>) and the volume is fixed (d<i>V</i>/d<i>t</i> = 0). This leads to a significant simplification of the first and second law: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\sum _{k}{\dot {Q}}_{k}+P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\sum _{k}{\dot {Q}}_{k}+P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4789ea32c7c2832b4df337969d729d24e32ab3ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.516ex; height:5.509ex;" alt="{\displaystyle 0=\sum _{k}{\dot {Q}}_{k}+P}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\sum _{k}{\frac {{\dot {Q}}_{k}}{T_{k}}}+{\dot {S}}_{\text{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\sum _{k}{\frac {{\dot {Q}}_{k}}{T_{k}}}+{\dot {S}}_{\text{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57e4ee2e8b108c0c3806e86af7a17be31a7220cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.516ex; height:7.176ex;" alt="{\displaystyle 0=\sum _{k}{\frac {{\dot {Q}}_{k}}{T_{k}}}+{\dot {S}}_{\text{i}}.}"></span></dd></dl> <p>The summation is over the (two) places where heat is added or removed. </p> <div class="mw-heading mw-heading3"><h3 id="Engines">Engines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=5" title="Edit section: Engines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a heat engine (Fig. 2a) the first and second law obtain the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\dot {Q}}_{\text{H}}-{\dot {Q}}_{\text{a}}-P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo>−</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\dot {Q}}_{\text{H}}-{\dot {Q}}_{\text{a}}-P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295d1fd4e7fc757ff7ecb3a7117f5264bc387eb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.883ex; height:3.343ex;" alt="{\displaystyle 0={\dot {Q}}_{\text{H}}-{\dot {Q}}_{\text{a}}-P}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {{\dot {Q}}_{\text{H}}}{T_{\text{H}}}}-{\frac {{\dot {Q}}_{\text{a}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\frac {{\dot {Q}}_{\text{H}}}{T_{\text{H}}}}-{\frac {{\dot {Q}}_{\text{a}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b230ecebf5fba3305d532e6128d443565e38ab4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.719ex; height:6.343ex;" alt="{\displaystyle 0={\frac {{\dot {Q}}_{\text{H}}}{T_{\text{H}}}}-{\frac {{\dot {Q}}_{\text{a}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}.}"></span></dd></dl> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03275e4e60988b0e4b7468eeef5b3cf025300aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.303ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{H}}}"></span> is the heat supplied at the high temperature <i>T</i><sub>H</sub>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446088ace716aeb989e48ba28d942a84ad802262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.893ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{a}}}"></span> is the heat removed at ambient temperature <i>T</i><sub>a</sub>, and <i>P</i> is the power delivered by the engine. Eliminating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446088ace716aeb989e48ba28d942a84ad802262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.893ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{a}}}"></span> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {T_{\text{H}}-T_{\text{a}}}{T_{\text{H}}}}{\dot {Q}}_{\text{H}}-T_{\text{a}}{\dot {S}}_{\text{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {T_{\text{H}}-T_{\text{a}}}{T_{\text{H}}}}{\dot {Q}}_{\text{H}}-T_{\text{a}}{\dot {S}}_{\text{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb97d90392feabc2feb061595ea95d5d9bbb696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.219ex; height:5.509ex;" alt="{\displaystyle P={\frac {T_{\text{H}}-T_{\text{a}}}{T_{\text{H}}}}{\dot {Q}}_{\text{H}}-T_{\text{a}}{\dot {S}}_{\text{i}}.}"></span></dd></dl> <p>The efficiency is defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ={\frac {P}{{\dot {Q}}_{\text{H}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {P}{{\dot {Q}}_{\text{H}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293683524afa46e6bd52ead998d61248cd9f0a1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.054ex; height:6.343ex;" alt="{\displaystyle \eta ={\frac {P}{{\dot {Q}}_{\text{H}}}}.}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d9768d0c8c928255c4b1f09aeac799329edd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.523ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}=0}"></span> the performance of the engine is at its maximum and the efficiency is equal to the Carnot efficiency </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\text{C}}={\frac {T_{\text{H}}-T_{\text{a}}}{T_{\text{H}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{C}}={\frac {T_{\text{H}}-T_{\text{a}}}{T_{\text{H}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e06e5931d34ec037e82d43c6ae3435b39af6c7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.23ex; height:5.509ex;" alt="{\displaystyle \eta _{\text{C}}={\frac {T_{\text{H}}-T_{\text{a}}}{T_{\text{H}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Refrigerators">Refrigerators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=6" title="Edit section: Refrigerators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For refrigerators (Fig. 2b) holds </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\dot {Q}}_{\text{L}}-{\dot {Q}}_{\text{a}}+P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo>+</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\dot {Q}}_{\text{L}}-{\dot {Q}}_{\text{a}}+P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece5bd06df493c3c5a86e546f01f191790929ab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.678ex; height:3.343ex;" alt="{\displaystyle 0={\dot {Q}}_{\text{L}}-{\dot {Q}}_{\text{a}}+P}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {{\dot {Q}}_{\text{L}}}{T_{\text{L}}}}-{\frac {{\dot {Q}}_{\text{a}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\frac {{\dot {Q}}_{\text{L}}}{T_{\text{L}}}}-{\frac {{\dot {Q}}_{\text{a}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f96108cc05865d9af8946bcdd30b76f2c9b68b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.514ex; height:6.343ex;" alt="{\displaystyle 0={\frac {{\dot {Q}}_{\text{L}}}{T_{\text{L}}}}-{\frac {{\dot {Q}}_{\text{a}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}.}"></span></dd></dl> <p>Here <i>P</i> is the power, supplied to produce the cooling power <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba465815bf1c62ba9119587e25ec5a6e9d8a69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.098ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{L}}}"></span> at the low temperature <i>T</i><sub>L</sub>. Eliminating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446088ace716aeb989e48ba28d942a84ad802262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.893ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{a}}}"></span> now gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{L}}={\frac {T_{\text{L}}}{T_{\text{a}}-T_{\text{L}}}}(P-T_{\text{a}}{\dot {S}}_{\text{i}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{L}}={\frac {T_{\text{L}}}{T_{\text{a}}-T_{\text{L}}}}(P-T_{\text{a}}{\dot {S}}_{\text{i}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72dfb92709617fc22b21285b9e2f4ed875c77d05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.618ex; height:5.509ex;" alt="{\displaystyle {\dot {Q}}_{\text{L}}={\frac {T_{\text{L}}}{T_{\text{a}}-T_{\text{L}}}}(P-T_{\text{a}}{\dot {S}}_{\text{i}}).}"></span></dd></dl> <p>The <a href="/wiki/Coefficient_of_performance" title="Coefficient of performance">coefficient of performance</a> of refrigerators is defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi ={\frac {{\dot {Q}}_{\text{L}}}{P}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mi>P</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi ={\frac {{\dot {Q}}_{\text{L}}}{P}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbce05fd7772ab95705b8d112a9aef856df5697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.709ex; height:6.009ex;" alt="{\displaystyle \xi ={\frac {{\dot {Q}}_{\text{L}}}{P}}.}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d9768d0c8c928255c4b1f09aeac799329edd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.523ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}=0}"></span> the performance of the cooler is at its maximum. The COP is then given by the Carnot coefficient of performance </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{\text{C}}={\frac {T_{\text{L}}}{T_{\text{a}}-T_{\text{L}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{\text{C}}={\frac {T_{\text{L}}}{T_{\text{a}}-T_{\text{L}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e80c3d3c66bc6b4f7fc30945a7854cb52b8e65f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.888ex; height:5.509ex;" alt="{\displaystyle \xi _{\text{C}}={\frac {T_{\text{L}}}{T_{\text{a}}-T_{\text{L}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Power_dissipation">Power dissipation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=7" title="Edit section: Power dissipation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In both cases we find a contribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{a}}{\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{a}}{\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78570fffa7b6b34bf540fb635d9e75dda2b4a23d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.674ex; height:3.176ex;" alt="{\displaystyle T_{\text{a}}{\dot {S}}_{\text{i}}}"></span> which reduces the system performance. This product of ambient temperature and the (average) entropy production rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{diss}}=T_{\text{a}}{\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>diss</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{diss}}=T_{\text{a}}{\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d953867441df5a6e55431170b46861d126340d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.164ex; height:3.176ex;" alt="{\displaystyle P_{\text{diss}}=T_{\text{a}}{\dot {S}}_{\text{i}}}"></span> is called the dissipated power. </p> <div class="mw-heading mw-heading2"><h2 id="Equivalence_with_other_formulations">Equivalence with other formulations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=8" title="Edit section: Equivalence with other formulations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is interesting to investigate how the above mathematical formulation of the second law relates with other well-known formulations of the second law. </p><p>We first look at a heat engine, assuming that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{a}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{a}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f45e664d4f79a85f05df4b0c9bbb014f5fb3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.153ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{a}}=0}"></span>. In other words: the heat flow rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03275e4e60988b0e4b7468eeef5b3cf025300aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.303ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{H}}}"></span> is completely converted into power. In this case the second law would reduce to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {{\dot {Q}}_{\text{H}}}{T_{\text{H}}}}+{\dot {S}}_{\text{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\frac {{\dot {Q}}_{\text{H}}}{T_{\text{H}}}}+{\dot {S}}_{\text{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a8afc4129aa90863f339d611b66d621dfce24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.15ex; height:6.343ex;" alt="{\displaystyle 0={\frac {{\dot {Q}}_{\text{H}}}{T_{\text{H}}}}+{\dot {S}}_{\text{i}}.}"></span></dd></dl> <p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{H}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{H}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae827d037040cde9d828e647f92193fa8a3a129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.564ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{H}}\geq 0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{H}}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{H}}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2c5b9c7db46f4837a5ff26f89ed50148671d8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.083ex; height:2.509ex;" alt="{\displaystyle T_{\text{H}}>0}"></span> this would result in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9d318b33a455eb70b127f45411a59c0d0c9c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.523ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}\leq 0}"></span> which violates the condition that the entropy production is always positive. Hence: <i>No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.</i> This is the Kelvin statement of the second law. </p><p>Now look at the case of the refrigerator and assume that the input power is zero. In other words: heat is transported from a low temperature to a high temperature without doing work on the system. The first law with <span class="nowrap"><i>P</i> = 0</span> would give </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{L}}={\dot {Q}}_{\text{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{L}}={\dot {Q}}_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a585c123ebeac0258fbd7212622267f5d58611bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.089ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{L}}={\dot {Q}}_{\text{a}}}"></span></dd></dl> <p>and the second law then yields </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {{\dot {Q}}_{\text{L}}}{T_{\text{L}}}}-{\frac {{\dot {Q}}_{\text{L}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\frac {{\dot {Q}}_{\text{L}}}{T_{\text{L}}}}-{\frac {{\dot {Q}}_{\text{L}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abd578ad109d60cdb35d39703c816b150d87f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.072ex; height:6.343ex;" alt="{\displaystyle 0={\frac {{\dot {Q}}_{\text{L}}}{T_{\text{L}}}}-{\frac {{\dot {Q}}_{\text{L}}}{T_{\text{a}}}}+{\dot {S}}_{\text{i}}}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}={\dot {Q}}_{\text{L}}\left({\frac {1}{T_{\text{a}}}}-{\frac {1}{T_{\text{L}}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}={\dot {Q}}_{\text{L}}\left({\frac {1}{T_{\text{a}}}}-{\frac {1}{T_{\text{L}}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d80ce02d86bfa0ba9702b6578f9e63fdab08e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.842ex; height:6.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}={\dot {Q}}_{\text{L}}\left({\frac {1}{T_{\text{a}}}}-{\frac {1}{T_{\text{L}}}}\right).}"></span></dd></dl> <p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}_{\text{L}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}_{\text{L}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9297d739ca2a94c714200703f16ed58a8699b66e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.359ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}_{\text{L}}\geq 0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{a}}>T_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mo>></mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{a}}>T_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c5dbb2ba0baba436d9e68d18152325878af887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.127ex; height:2.509ex;" alt="{\displaystyle T_{\text{a}}>T_{\text{L}}}"></span> this would result in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9d318b33a455eb70b127f45411a59c0d0c9c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.523ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}\leq 0}"></span> which again violates the condition that the entropy production is always positive. Hence: <i>No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.</i> This is the Clausius statement of the second law. </p> <div class="mw-heading mw-heading2"><h2 id="Expressions_for_the_entropy_production">Expressions for the entropy production</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=9" title="Edit section: Expressions for the entropy production"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Heat_flow">Heat flow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=10" title="Edit section: Heat flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In case of a heat flow rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ef647ca15bb236a9473fcbe17f16fe87c95ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:3.176ex;" alt="{\displaystyle {\dot {Q}}}"></span> from <i>T</i><sub>1</sub> to <i>T</i><sub>2</sub> (with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1}\geq T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}\geq T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14cf99c5fe68c9b16f505bc6ab3528ab2cbda05c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle T_{1}\geq T_{2}}"></span>) the rate of entropy production is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}={\dot {Q}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}={\dot {Q}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56840a86f3ae8715d02b20e1848e7a106d71a429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.378ex; height:6.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}={\dot {Q}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).}"></span></dd></dl> <p>If the heat flow is in a bar with length <i>L</i>, cross-sectional area <i>A</i>, and thermal conductivity <i>κ</i>, and the temperature difference is small </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}=\kappa {\frac {A}{L}}(T_{1}-T_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}=\kappa {\frac {A}{L}}(T_{1}-T_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd699ab0794f59334d53d4993bd83bbb6a15dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.328ex; height:5.343ex;" alt="{\displaystyle {\dot {Q}}=\kappa {\frac {A}{L}}(T_{1}-T_{2})}"></span></dd></dl> <p>the entropy production rate is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}=\kappa {\frac {A}{L}}{\frac {(T_{1}-T_{2})^{2}}{T_{1}T_{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>L</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}=\kappa {\frac {A}{L}}{\frac {(T_{1}-T_{2})^{2}}{T_{1}T_{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6eab55b152ecda876f99ba9f0495ab421d2256d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.289ex; height:6.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}=\kappa {\frac {A}{L}}{\frac {(T_{1}-T_{2})^{2}}{T_{1}T_{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Flow_of_mass">Flow of mass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=11" title="Edit section: Flow of mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In case of a volume flow rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49ef9fa9f410331b94e4578bab90e9edda5c919b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.676ex;" alt="{\displaystyle {\dot {V}}}"></span> from a pressure <i>p</i><sub>1</sub> to <i>p</i><sub>2</sub> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}=-\int _{p_{1}}^{p_{2}}{\frac {\dot {V}}{T}}\mathrm {d} p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}=-\int _{p_{1}}^{p_{2}}{\frac {\dot {V}}{T}}\mathrm {d} p.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/554a467b223843699d419363289491248f250910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.03ex; height:6.676ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}=-\int _{p_{1}}^{p_{2}}{\frac {\dot {V}}{T}}\mathrm {d} p.}"></span></dd></dl> <p>For small pressure drops and defining the flow conductance <i>C</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {V}}=C(p_{1}-p_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {V}}=C(p_{1}-p_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0261cf064826208df6292af2bd8351fdf6d4fe98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.749ex; height:3.176ex;" alt="{\displaystyle {\dot {V}}=C(p_{1}-p_{2})}"></span> we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}=C{\frac {(p_{1}-p_{2})^{2}}{T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}=C{\frac {(p_{1}-p_{2})^{2}}{T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2b6f1c44c59b37c1f69e1f12a2b42e694a0bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.761ex; height:5.843ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}=C{\frac {(p_{1}-p_{2})^{2}}{T}}.}"></span></dd></dl> <p>The dependences of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fb101f17394253a0e5bb8816e28386c92391e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.262ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}}"></span> on <span class="nowrap"><i>T</i><sub>1</sub> − <i>T</i><sub>2</sub></span> and on <span class="nowrap"><i>p</i><sub>1</sub> − <i>p</i><sub>2</sub></span> are quadratic. </p><p>This is typical for expressions of the entropy production rates in general. They guarantee that the entropy production is positive. </p> <div class="mw-heading mw-heading3"><h3 id="Entropy_of_mixing">Entropy of mixing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=12" title="Edit section: Entropy of mixing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this Section we will calculate the <a href="/wiki/Entropy_of_mixing" title="Entropy of mixing">entropy of mixing</a> when two ideal gases diffuse into each other. Consider a volume <i>V</i><sub>t</sub> divided in two volumes <i>V</i><sub>a</sub> and <i>V</i><sub>b</sub> so that <span class="nowrap"><i>V</i><sub>t</sub> = <i>V</i><sub>a</sub> + <i>V</i><sub>b</sub></span>. The volume <i>V</i><sub>a</sub> contains <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> <i>n</i><sub>a</sub> of an ideal gas a and <i>V</i><sub>b</sub> contains amount of substance <i>n</i><sub>b</sub> of gas b. The total amount of substance is <span class="nowrap"><i>n</i><sub>t</sub> = <i>n</i><sub>a</sub> + <i>n</i><sub>b</sub></span>. The temperature and pressure in the two volumes is the same. The entropy at the start is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{t1}}=S_{\text{a1}}+S_{\text{b1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t1</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a1</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b1</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{t1}}=S_{\text{a1}}+S_{\text{b1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db0c87f4fd4d2bd78ec5447fbc1a1dffb4dd5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.399ex; height:2.509ex;" alt="{\displaystyle S_{\text{t1}}=S_{\text{a1}}+S_{\text{b1}}.}"></span></dd></dl> <p>When the division between the two gases is removed the two gases expand, comparable to a Joule–Thomson expansion. In the final state the temperature is the same as initially but the two gases now both take the volume <i>V</i><sub>t</sub>. The relation of the entropy of an amount of substance <i>n</i> of an ideal gas is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=nC_{\text{V}}\ln {\frac {T}{T_{0}}}+nR\ln {\frac {V}{V_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>n</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>V</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mi>n</mi> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=nC_{\text{V}}\ln {\frac {T}{T_{0}}}+nR\ln {\frac {V}{V_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d698a94bce2e6039960755567bb014be1a9d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.039ex; height:5.676ex;" alt="{\displaystyle S=nC_{\text{V}}\ln {\frac {T}{T_{0}}}+nR\ln {\frac {V}{V_{0}}}}"></span></dd></dl> <p>where <i>C</i><sub>V</sub> is the molar heat capacity at constant volume and <i>R</i> is the molar gas constant. The system is an adiabatic closed system, so the entropy increase during the mixing of the two gases is equal to the entropy production. It is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\Delta }=S_{\text{t2}}-S_{\text{t1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t2</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t1</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\Delta }=S_{\text{t2}}-S_{\text{t1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2d294cd65ff470a183667c9f80b243354d8f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.849ex; height:2.509ex;" alt="{\displaystyle S_{\Delta }=S_{\text{t2}}-S_{\text{t1}}.}"></span></dd></dl> <p>As the initial and final temperature are the same, the temperature terms cancel, leaving only the volume terms. The result is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\Delta }=n_{\text{a}}R\ln {\frac {V_{\text{t}}}{V_{\text{a}}}}+n_{\text{b}}R\ln {\frac {V_{\text{t}}}{V_{\text{b}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>a</mtext> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\Delta }=n_{\text{a}}R\ln {\frac {V_{\text{t}}}{V_{\text{a}}}}+n_{\text{b}}R\ln {\frac {V_{\text{t}}}{V_{\text{b}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c0f460a5b8a3fcc5d37fd2cb2bd27bec9297e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.14ex; height:5.676ex;" alt="{\displaystyle S_{\Delta }=n_{\text{a}}R\ln {\frac {V_{\text{t}}}{V_{\text{a}}}}+n_{\text{b}}R\ln {\frac {V_{\text{t}}}{V_{\text{b}}}}.}"></span></dd></dl> <p>Introducing the concentration <i>x</i> = <i>n</i><sub>a</sub>/<i>n</i><sub>t</sub> = <i>V</i><sub>a</sub>/<i>V</i><sub>t</sub> we arrive at the well-known expression </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\Delta }=-n_{\text{t}}R[x\ln x+(1-x)\ln(1-x)].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\Delta }=-n_{\text{t}}R[x\ln x+(1-x)\ln(1-x)].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3a4288d9f96e1474dd1bb6403427576cdc2145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.727ex; height:2.843ex;" alt="{\displaystyle S_{\Delta }=-n_{\text{t}}R[x\ln x+(1-x)\ln(1-x)].}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Joule_expansion">Joule expansion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=13" title="Edit section: Joule expansion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Joule_expansion" title="Joule expansion">Joule expansion</a> is similar to the mixing described above. It takes place in an adiabatic system consisting of a gas and two rigid vessels a and b of equal volume, connected by a valve. Initially, the valve is closed. Vessel a contains the gas while the other vessel b is empty. When the valve is opened, the gas flows from vessel a into b until the pressures in the two vessels are equal. The volume, taken by the gas, is doubled while the internal energy of the system is constant (adiabatic and no work done). Assuming that the gas is ideal, the molar internal energy is given by <span class="nowrap"><i>U</i><sub>m</sub> = <i>C</i><sub>V</sub><i>T</i></span>. As <i>C</i><sub>V</sub> is constant, constant <i>U</i> means constant <i>T</i>. The molar entropy of an ideal gas, as function of the molar volume <i>V</i><sub>m</sub> and <i>T</i>, is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{m}}=C_{\text{V}}\ln {\frac {T}{T_{0}}}+R\ln {\frac {V_{\text{m}}}{V_{0}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>V</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{m}}=C_{\text{V}}\ln {\frac {T}{T_{0}}}+R\ln {\frac {V_{\text{m}}}{V_{0}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fcfe6d3c22d615093215b71c94faef9fd75aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.97ex; height:5.676ex;" alt="{\displaystyle S_{\text{m}}=C_{\text{V}}\ln {\frac {T}{T_{0}}}+R\ln {\frac {V_{\text{m}}}{V_{0}}}.}"></span></dd></dl> <p>The system consisting of the two vessels and the gas is closed and adiabatic, so the entropy production during the process is equal to the increase of the entropy of the gas. So, doubling the volume with <i>T</i> constant gives that the molar entropy produced is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\text{mi}}=R\ln 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>mi</mtext> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\text{mi}}=R\ln 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ebb8925795dd41eba84a013d1c95d38588550b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.869ex; height:2.509ex;" alt="{\displaystyle S_{\text{mi}}=R\ln 2.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Microscopic_interpretation">Microscopic interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=14" title="Edit section: Microscopic interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Joule expansion provides an opportunity to explain the entropy production in statistical mechanical (i.e., microscopic) terms. At the expansion, the volume that the gas can occupy is doubled. This means that, for every molecule there are now two possibilities: it can be placed in container a or b. If the gas has amount of substance <i>n</i>, the number of molecules is equal to <i>n</i>⋅<i>N</i><sub>A</sub>, where <i>N</i><sub>A</sub> is the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro constant</a>. The number of microscopic possibilities increases by a factor of 2 per molecule due to the doubling of volume, so in total the factor is 2<sup><i>n</i>⋅<i>N</i><sub>A</sub></sup>. Using the well-known Boltzmann expression for the <a href="/wiki/Entropy" title="Entropy">entropy</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=k\ln \Omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k\ln \Omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92012f4b5e04d8e0cfd0eabb75a7e98b2d197b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.847ex; height:2.509ex;" alt="{\displaystyle S=k\ln \Omega ,}"></span></dd></dl> <p>where <i>k</i> is the Boltzmann constant and Ω is the number of microscopic possibilities to realize the macroscopic state. This gives change in molar entropy of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{{\text{m}}\Delta }=S_{\Delta }/n=k\ln(2^{n\cdot N_{\text{A}}})/n=kN_{\text{A}}\ln 2=R\ln 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>⋅</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>=</mo> <mi>k</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mo>=</mo> <mi>R</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{{\text{m}}\Delta }=S_{\Delta }/n=k\ln(2^{n\cdot N_{\text{A}}})/n=kN_{\text{A}}\ln 2=R\ln 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed4067c84360cc20be3974b82223f479d2fe5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.304ex; height:3.176ex;" alt="{\displaystyle S_{{\text{m}}\Delta }=S_{\Delta }/n=k\ln(2^{n\cdot N_{\text{A}}})/n=kN_{\text{A}}\ln 2=R\ln 2.}"></span></dd></dl> <p>So, in an irreversible process, the number of microscopic possibilities to realize the macroscopic state is increased by a certain factor. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_inequalities_and_stability_conditions">Basic inequalities and stability conditions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=15" title="Edit section: Basic inequalities and stability conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this section we derive the basic inequalities and stability conditions for closed systems. For closed systems the first law reduces to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}-p{\frac {\mathrm {d} V}{\mathrm {d} t}}+P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}-p{\frac {\mathrm {d} V}{\mathrm {d} t}}+P.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a3643727c47e7c3009bb051214ff60dde10dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.006ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}-p{\frac {\mathrm {d} V}{\mathrm {d} t}}+P.}"></span></dd></dl> <p>The second law we write as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}-{\frac {\dot {Q}}{T}}\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}-{\frac {\dot {Q}}{T}}\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ffa26df253d4d99688af1113ab0c3f87148ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.051ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}-{\frac {\dot {Q}}{T}}\geq 0.}"></span></dd></dl> <p>For <i>adiabatic systems</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f0b3424bd099214a72fbd84cf05902cc8e6dfad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:3.176ex;" alt="{\displaystyle {\dot {Q}}=0}"></span> so <span class="nowrap">d<i>S</i>/d<i>t</i> ≥ 0</span>. In other words: the entropy of adiabatic systems cannot decrease. In equilibrium the entropy is at its maximum. Isolated systems are a special case of adiabatic systems, so this statement is also valid for isolated systems. </p><p>Now consider systems with <i>constant temperature and volume</i>. In most cases <i>T</i> is the temperature of the surroundings with which the system is in good thermal contact. Since <i>V</i> is constant the first law gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}=\mathrm {d} U/\mathrm {d} t-P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>−</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}=\mathrm {d} U/\mathrm {d} t-P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc3e34c28d35e00a9fa698fa689e5d5fd3877dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.892ex; height:3.343ex;" alt="{\displaystyle {\dot {Q}}=\mathrm {d} U/\mathrm {d} t-P}"></span>. Substitution in the second law, and using that <i>T</i> is constant, gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} (TS)}{\mathrm {d} t}}-{\frac {\mathrm {d} U}{\mathrm {d} t}}+P\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>P</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} (TS)}{\mathrm {d} t}}-{\frac {\mathrm {d} U}{\mathrm {d} t}}+P\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d622b9145c39736a692b11719be4fd048b1c18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.319ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} (TS)}{\mathrm {d} t}}-{\frac {\mathrm {d} U}{\mathrm {d} t}}+P\geq 0.}"></span></dd></dl> <p>With the Helmholtz free energy, defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=U-TS,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=U-TS,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7e2112b4ea54d4ffd8ea31af77cd224c34a6ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.245ex; height:2.509ex;" alt="{\displaystyle F=U-TS,}"></span></dd></dl> <p>we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} F}{\mathrm {d} t}}-P\leq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>F</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>P</mi> <mo>≤</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} F}{\mathrm {d} t}}-P\leq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3bdb1e6cf3ae361bf3eb8b71975d3180f439eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.363ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} F}{\mathrm {d} t}}-P\leq 0.}"></span></dd></dl> <p>If <i>P</i> = 0 this is the mathematical formulation of the general property that the free energy of systems with fixed temperature and volume tends to a minimum. The expression can be integrated from the initial state i to the final state f resulting in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\text{S}}\leq F_{\text{i}}-F_{\text{f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo>≤</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{S}}\leq F_{\text{i}}-F_{\text{f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229bc80ef2a9ca4853f9243c995751ad8cbca890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.802ex; height:2.509ex;" alt="{\displaystyle W_{\text{S}}\leq F_{\text{i}}-F_{\text{f}}}"></span></dd></dl> <p>where <i>W</i><sub>S</sub> is the work done <i>by</i> the system. If the process inside the system is completely reversible the equality sign holds. Hence the maximum work, that can be extracted from the system, is equal to the free energy of the initial state minus the free energy of the final state. </p><p>Finally we consider systems with <i>constant temperature and pressure</i> and take <span class="nowrap"><i>P</i> = 0</span>. As <i>p</i> is constant the first laws gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}-{\frac {\mathrm {d} (pV)}{\mathrm {d} t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}-{\frac {\mathrm {d} (pV)}{\mathrm {d} t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d90efdad08e1116228ea04a9330b61e908d4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.23ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}-{\frac {\mathrm {d} (pV)}{\mathrm {d} t}}.}"></span></dd></dl> <p>Combining with the second law, and using that <i>T</i> is constant, gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} (TS)}{\mathrm {d} t}}-{\frac {\mathrm {d} U}{\mathrm {d} t}}-{\frac {\mathrm {d} (pV)}{\mathrm {d} t}}\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} (TS)}{\mathrm {d} t}}-{\frac {\mathrm {d} U}{\mathrm {d} t}}-{\frac {\mathrm {d} (pV)}{\mathrm {d} t}}\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc11b5dea1777f9f3c5c53b8b7e1e34ff7eea6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.468ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} (TS)}{\mathrm {d} t}}-{\frac {\mathrm {d} U}{\mathrm {d} t}}-{\frac {\mathrm {d} (pV)}{\mathrm {d} t}}\geq 0.}"></span></dd></dl> <p>With the Gibbs free energy, defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=U+pV-TS,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=U+pV-TS,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0d711dab2c4db6f44f16fa49933a1c2f7c5962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.128ex; height:2.509ex;" alt="{\displaystyle G=U+pV-TS,}"></span></dd></dl> <p>we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} G}{\mathrm {d} t}}\leq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>G</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>≤</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} G}{\mathrm {d} t}}\leq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed162ce860011e61b24ee18f109c1bd13c75a7d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.863ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} G}{\mathrm {d} t}}\leq 0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Homogeneous_systems">Homogeneous systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=16" title="Edit section: Homogeneous systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In homogeneous systems the temperature and pressure are well-defined and all internal processes are reversible. Hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {S}}_{\text{i}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}_{\text{i}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d9768d0c8c928255c4b1f09aeac799329edd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.523ex; height:3.176ex;" alt="{\displaystyle {\dot {S}}_{\text{i}}=0}"></span>. As a result, the second law, multiplied by <i>T</i>, reduces to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T{\frac {\mathrm {d} S}{\mathrm {d} t}}={\dot {Q}}+{\dot {n}}TS_{\text{m}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>T</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T{\frac {\mathrm {d} S}{\mathrm {d} t}}={\dot {Q}}+{\dot {n}}TS_{\text{m}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a472e873a4cf81020087ca4a1d944a35f2b35f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.745ex; height:5.509ex;" alt="{\displaystyle T{\frac {\mathrm {d} S}{\mathrm {d} t}}={\dot {Q}}+{\dot {n}}TS_{\text{m}}.}"></span></dd></dl> <p>With <i>P</i> = 0 the first law becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}+{\dot {n}}H_{\text{m}}-p{\frac {\mathrm {d} V}{\mathrm {d} t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}+{\dot {n}}H_{\text{m}}-p{\frac {\mathrm {d} V}{\mathrm {d} t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1033d6884a29f58b9ef8a639f2f2505bf59992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.188ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\dot {Q}}+{\dot {n}}H_{\text{m}}-p{\frac {\mathrm {d} V}{\mathrm {d} t}}.}"></span></dd></dl> <p>Eliminating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ef647ca15bb236a9473fcbe17f16fe87c95ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:3.176ex;" alt="{\displaystyle {\dot {Q}}}"></span> and multiplying with d<i>t</i> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+(H_{\text{m}}-TS_{\text{m}})\mathrm {d} n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>−</mo> <mi>T</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+(H_{\text{m}}-TS_{\text{m}})\mathrm {d} n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe14005317a043c0215d64d60b4680386a58abde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.71ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+(H_{\text{m}}-TS_{\text{m}})\mathrm {d} n.}"></span></dd></dl> <p>Since </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{m}}-TS_{\text{m}}=G_{\text{m}}=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>−</mo> <mi>T</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{m}}-TS_{\text{m}}=G_{\text{m}}=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a0fa805ab14850ec59e8b478626203c4546bc7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.062ex; height:2.676ex;" alt="{\displaystyle H_{\text{m}}-TS_{\text{m}}=G_{\text{m}}=\mu }"></span></dd></dl> <p>with <i>G</i><sub>m</sub> the molar <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> and <i>μ</i> the molar <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> we obtain the well-known result </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+\mu \mathrm {d} n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>+</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+\mu \mathrm {d} n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ba58a3e51dd80b9e4cc8b07a413b2c890796de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.267ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+\mu \mathrm {d} n.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Entropy_production_in_stochastic_processes">Entropy production in stochastic processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=17" title="Edit section: Entropy production in stochastic processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since physical processes can be described by stochastic processes, such as Markov chains and diffusion processes, entropy production can be defined mathematically in such processes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>For a continuous-time Markov chain with instantaneous probability distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bac02321854cf8683e6e53e9a1da8b498f663b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.707ex; height:2.843ex;" alt="{\displaystyle p_{i}(t)}"></span> and transition rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b08ec83005828a8789b639e4944b11905e9b18b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.343ex;" alt="{\displaystyle q_{ij}}"></span>, the instantaneous entropy production rate is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{p}(t)={\frac {1}{2}}\sum _{i,j}[p_{i}(t)q_{ij}-p_{j}(t)q_{ji}]\log {\frac {p_{i}(t)q_{ij}}{p_{j}(t)q_{ji}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{p}(t)={\frac {1}{2}}\sum _{i,j}[p_{i}(t)q_{ij}-p_{j}(t)q_{ji}]\log {\frac {p_{i}(t)q_{ij}}{p_{j}(t)q_{ji}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0ef3f06192b588d3109969a982faf1dee3a966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.61ex; height:7.176ex;" alt="{\displaystyle e_{p}(t)={\frac {1}{2}}\sum _{i,j}[p_{i}(t)q_{ij}-p_{j}(t)q_{ji}]\log {\frac {p_{i}(t)q_{ij}}{p_{j}(t)q_{ji}}}.}"></span></dd></dl> <p>The long-time behavior of entropy production is kept after a proper lifting of the process. This approach provides a dynamic explanation for the Kelvin statement and the Clausius statement of the second law of thermodynamics.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First law of thermodynamics</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second law of thermodynamics</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversible process</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium thermodynamics</a></li> <li><a href="/wiki/High_entropy_alloys" class="mw-redirect" title="High entropy alloys">High entropy alloys</a></li> <li><a href="/wiki/General_equation_of_heat_transfer" title="General equation of heat transfer">General equation of heat transfer</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">S. Carnot <i>Reflexions sur la puissance motrice du feu</i> Bachelier, Paris, 1824</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFClausius1854" class="citation journal cs1">Clausius, R. (1854). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k15178k/f499.image">"Ueber eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheoriein"</a>. <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik und Chemie</a></i>. <b>93</b> (12): 481–506. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1854AnP...169..481C">1854AnP...169..481C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.18541691202">10.1002/andp.18541691202</a><span class="reference-accessdate">. Retrieved <span class="nowrap">25 June</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik+und+Chemie&rft.atitle=Ueber+eine+ver%C3%A4nderte+Form+des+zweiten+Hauptsatzes+der+mechanischen+W%C3%A4rmetheoriein&rft.volume=93&rft.issue=12&rft.pages=481-506&rft.date=1854&rft_id=info%3Adoi%2F10.1002%2Fandp.18541691202&rft_id=info%3Abibcode%2F1854AnP...169..481C&rft.aulast=Clausius&rft.aufirst=R.&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k15178k%2Ff499.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+production" class="Z3988"></span>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1856" class="citation journal cs1">Clausius, R. (August 1856). <a rel="nofollow" class="external text" href="https://archive.org/stream/londonedinburghd12lond#page/80/mode/2up">"On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat"</a>. <i><a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Phil. Mag.</a></i> 4. <b>12</b> (77): 81–98. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786445608642141">10.1080/14786445608642141</a><span class="reference-accessdate">. Retrieved <span class="nowrap">25 June</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phil.+Mag.&rft.atitle=On+a+Modified+Form+of+the+Second+Fundamental+Theorem+in+the+Mechanical+Theory+of+Heat&rft.volume=12&rft.issue=77&rft.pages=81-98&rft.date=1856-08&rft_id=info%3Adoi%2F10.1080%2F14786445608642141&rft.aulast=Clausius&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Flondonedinburghd12lond%23page%2F80%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+production" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">R. Clausius <i>Über verschiedene für die Anwendung bequeme Formen der Hauptgleigungen der mechanische Wärmetheorie</i> in Abhandlungen über die Anwendung bequeme Formen der Haubtgleichungen der mechanischen Wärmetheorie Ann.Phys. [2] 125, 390 (1865). This paper is translated and can be found in: The second law of thermodynamics, Edited by J. Kestin, Dowden, Hutchinson, & Ross, Inc., Stroudsburg, Pennsylvania, pp. 162–193.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">A.T.A.M. de Waele, Basic operation of cryocoolers and related thermal machines, Review article, Journal of Low Temperature Physics, Vol.164, pp. 179–236, (2011), DOI: 10.1007/s10909-011-0373-x.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiangQianQian2004" class="citation book cs1">Jiang, Da-Quan; Qian, Min; Qian, Min-Ping (2004). <i>Mathematical theory of nonequilibrium steady states: on the frontier of probability and dynamical systems</i>. Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-40957-1" title="Special:BookSources/978-3-540-40957-1"><bdi>978-3-540-40957-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+theory+of+nonequilibrium+steady+states%3A+on+the+frontier+of+probability+and+dynamical+systems&rft.place=Berlin&rft.pub=Springer&rft.date=2004&rft.isbn=978-3-540-40957-1&rft.aulast=Jiang&rft.aufirst=Da-Quan&rft.au=Qian%2C+Min&rft.au=Qian%2C+Min-Ping&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+production" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangQian2020" class="citation journal cs1">Wang, Yue; Qian, Hong (2020). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007%2Fs10955-020-02556-6">"Mathematical Representation of Clausius' and Kelvin's Statements of the Second Law and Irreversibility"</a>. <i>Journal of Statistical Physics</i>. <b>179</b> (3): 808–837. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1805.09530">1805.09530</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020JSP...179..808W">2020JSP...179..808W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10955-020-02556-6">10.1007/s10955-020-02556-6</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:254745126">254745126</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Physics&rft.atitle=Mathematical+Representation+of+Clausius%27+and+Kelvin%27s+Statements+of+the+Second+Law+and+Irreversibility&rft.volume=179&rft.issue=3&rft.pages=808-837&rft.date=2020&rft_id=info%3Aarxiv%2F1805.09530&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A254745126%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10955-020-02556-6&rft_id=info%3Abibcode%2F2020JSP...179..808W&rft.aulast=Wang&rft.aufirst=Yue&rft.au=Qian%2C+Hong&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%252Fs10955-020-02556-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+production" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_production&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrooks1999" class="citation journal cs1">Crooks, G. (1999). "Entropy production fluctuation theorem and the non-equilibrium work relation for free energy differences". <i>Physical Review E</i> (Free PDF). <b>60</b> (3): 2721–2726. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9901352">cond-mat/9901352</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999PhRvE..60.2721C">1999PhRvE..60.2721C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.60.2721">10.1103/PhysRevE.60.2721</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11970075">11970075</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1813818">1813818</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+E&rft.atitle=Entropy+production+fluctuation+theorem+and+the+non-equilibrium+work+relation+for+free+energy+differences&rft.volume=60&rft.issue=3&rft.pages=2721-2726&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1813818%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1999PhRvE..60.2721C&rft_id=info%3Aarxiv%2Fcond-mat%2F9901352&rft_id=info%3Apmid%2F11970075&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.60.2721&rft.aulast=Crooks&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+production" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeifert2005" class="citation journal cs1">Seifert, Udo (2005). "Entropy Production along a Stochastic Trajectory and an Integral Fluctuation Theorem". <i>Physical Review Letters</i> (Free PDF). <b>95</b> (4): 040602. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0503686">cond-mat/0503686</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005PhRvL..95d0602S">2005PhRvL..95d0602S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.95.040602">10.1103/PhysRevLett.95.040602</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16090792">16090792</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:31706268">31706268</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Entropy+Production+along+a+Stochastic+Trajectory+and+an+Integral+Fluctuation+Theorem&rft.volume=95&rft.issue=4&rft.pages=040602&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A31706268%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2005PhRvL..95d0602S&rft_id=info%3Aarxiv%2Fcond-mat%2F0503686&rft_id=info%3Apmid%2F16090792&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.95.040602&rft.aulast=Seifert&rft.aufirst=Udo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+production" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Entropy_production&oldid=1246167368">https://en.wikipedia.org/w/index.php?title=Entropy_production&oldid=1246167368</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Cooling_technology" title="Category:Cooling technology">Cooling technology</a></li><li><a href="/wiki/Category:Cryogenics" title="Category:Cryogenics">Cryogenics</a></li><li><a href="/wiki/Category:Heat_pumps" title="Category:Heat pumps">Heat pumps</a></li><li><a href="/wiki/Category:Thermodynamic_entropy" title="Category:Thermodynamic entropy">Thermodynamic entropy</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_introduction_cleanup_from_February_2024" title="Category:Wikipedia introduction cleanup from February 2024">Wikipedia introduction cleanup from February 2024</a></li><li><a href="/wiki/Category:All_pages_needing_cleanup" title="Category:All pages needing cleanup">All pages needing cleanup</a></li><li><a href="/wiki/Category:Articles_covered_by_WikiProject_Wikify_from_February_2024" title="Category:Articles covered by WikiProject Wikify from February 2024">Articles covered by WikiProject Wikify from February 2024</a></li><li><a href="/wiki/Category:All_articles_covered_by_WikiProject_Wikify" title="Category:All articles covered by WikiProject Wikify">All articles covered by WikiProject Wikify</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 17 September 2024, at 09:49<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Entropy production - Wikipedia