Second law of thermodynamics

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statements of the law</span> </div> </a> <button aria-controls="toc-Various_statements_of_the_law-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 Various statements of the law subsection</span> </button> <ul id="toc-Various_statements_of_the_law-sublist" class="vector-toc-list"> <li id="toc-Carnot's_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Carnot's_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Carnot's principle</span> </div> </a> <ul id="toc-Carnot's_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Clausius_statement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clausius_statement"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Clausius statement</span> </div> </a> <ul id="toc-Clausius_statement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kelvin_statements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kelvin_statements"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Kelvin statements</span> </div> </a> <ul id="toc-Kelvin_statements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalence_of_the_Clausius_and_the_Kelvin_statements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence_of_the_Clausius_and_the_Kelvin_statements"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Equivalence of the Clausius and the Kelvin statements</span> </div> </a> <ul id="toc-Equivalence_of_the_Clausius_and_the_Kelvin_statements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planck's_proposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Planck's_proposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Planck's proposition</span> </div> </a> <ul id="toc-Planck's_proposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_between_Kelvin's_statement_and_Planck's_proposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_between_Kelvin's_statement_and_Planck's_proposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Relation between Kelvin's statement and Planck's proposition</span> </div> </a> <ul id="toc-Relation_between_Kelvin's_statement_and_Planck's_proposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planck's_statement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Planck's_statement"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Planck's statement</span> </div> </a> <ul id="toc-Planck's_statement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Principle_of_Carathéodory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Principle_of_Carathéodory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Principle of Carathéodory</span> </div> </a> <ul id="toc-Principle_of_Carathéodory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planck's_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Planck's_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>Planck's principle</span> </div> </a> <ul id="toc-Planck's_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relating_the_second_law_to_the_definition_of_temperature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relating_the_second_law_to_the_definition_of_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.10</span> <span>Relating the second law to the definition of temperature</span> </div> </a> <ul id="toc-Relating_the_second_law_to_the_definition_of_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_law_statements,_such_as_the_Clausius_inequality,_involving_radiative_fluxes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_law_statements,_such_as_the_Clausius_inequality,_involving_radiative_fluxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.11</span> <span>Second law statements, such as the Clausius inequality, involving radiative fluxes</span> </div> </a> <ul id="toc-Second_law_statements,_such_as_the_Clausius_inequality,_involving_radiative_fluxes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_conceptual_statement_of_the_second_law_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_conceptual_statement_of_the_second_law_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.12</span> <span>Generalized conceptual statement of the second law principle</span> </div> </a> <ul id="toc-Generalized_conceptual_statement_of_the_second_law_principle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Corollaries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Corollaries"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Corollaries</span> </div> </a> <button aria-controls="toc-Corollaries-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 Corollaries subsection</span> </button> <ul id="toc-Corollaries-sublist" class="vector-toc-list"> <li id="toc-Perpetual_motion_of_the_second_kind" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perpetual_motion_of_the_second_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Perpetual motion of the second kind</span> </div> </a> <ul id="toc-Perpetual_motion_of_the_second_kind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carnot's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Carnot's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Carnot's theorem</span> </div> </a> <ul id="toc-Carnot's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Clausius_inequality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clausius_inequality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Clausius inequality</span> </div> </a> <ul id="toc-Clausius_inequality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermodynamic_temperature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamic_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Thermodynamic temperature</span> </div> </a> <ul id="toc-Thermodynamic_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Entropy</span> </div> </a> <ul id="toc-Entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Energy,_available_useful_work" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy,_available_useful_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Energy, available useful work</span> </div> </a> <ul id="toc-Energy,_available_useful_work-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Direction_of_spontaneous_processes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Direction_of_spontaneous_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Direction of spontaneous processes</span> </div> </a> <button aria-controls="toc-Direction_of_spontaneous_processes-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 Direction of spontaneous processes subsection</span> </button> <ul id="toc-Direction_of_spontaneous_processes-sublist" class="vector-toc-list"> <li id="toc-Second_law_in_chemical_thermodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_law_in_chemical_thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Second law in chemical thermodynamics</span> </div> </a> <ul id="toc-Second_law_in_chemical_thermodynamics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <button aria-controls="toc-History-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 History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Account_given_by_Clausius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Account_given_by_Clausius"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Account given by Clausius</span> </div> </a> <ul id="toc-Account_given_by_Clausius-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Statistical_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Statistical_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Statistical mechanics</span> </div> </a> <ul id="toc-Statistical_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_from_statistical_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivation_from_statistical_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Derivation from statistical mechanics</span> </div> </a> <button aria-controls="toc-Derivation_from_statistical_mechanics-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 Derivation from statistical mechanics subsection</span> </button> <ul id="toc-Derivation_from_statistical_mechanics-sublist" class="vector-toc-list"> <li id="toc-Derivation_of_the_entropy_change_for_reversible_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_the_entropy_change_for_reversible_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Derivation of the entropy change for reversible processes</span> </div> </a> <ul id="toc-Derivation_of_the_entropy_change_for_reversible_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_for_systems_described_by_the_canonical_ensemble" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_for_systems_described_by_the_canonical_ensemble"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Derivation for systems described by the canonical ensemble</span> </div> </a> <ul id="toc-Derivation_for_systems_described_by_the_canonical_ensemble-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Initial_conditions_at_the_Big_Bang" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Initial_conditions_at_the_Big_Bang"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Initial conditions at the Big Bang</span> </div> </a> <ul id="toc-Initial_conditions_at_the_Big_Bang-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Living_organisms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Living_organisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Living organisms</span> </div> </a> <ul id="toc-Living_organisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gravitational_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Gravitational_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Gravitational systems</span> </div> </a> <ul id="toc-Gravitational_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-equilibrium_states" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Non-equilibrium_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Non-equilibrium states</span> </div> </a> <ul id="toc-Non-equilibrium_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arrow_of_time" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Arrow_of_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Arrow of time</span> </div> </a> <ul id="toc-Arrow_of_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irreversibility" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Irreversibility"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Irreversibility</span> </div> </a> <button aria-controls="toc-Irreversibility-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 Irreversibility subsection</span> </button> <ul id="toc-Irreversibility-sublist" class="vector-toc-list"> <li id="toc-Loschmidt's_paradox" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Loschmidt's_paradox"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>Loschmidt's paradox</span> </div> </a> <ul id="toc-Loschmidt's_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poincaré_recurrence_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poincaré_recurrence_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.2</span> <span>Poincaré recurrence theorem</span> </div> </a> <ul id="toc-Poincaré_recurrence_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maxwell's_demon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maxwell's_demon"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.3</span> <span>Maxwell's demon</span> </div> </a> <ul id="toc-Maxwell's_demon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quotations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Quotations</span> </div> </a> <ul id="toc-Quotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">15.1</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label 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mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%A7%D9%84%D8%AB%D8%A7%D9%86%D9%8A_%D9%84%D9%84%D8%AF%D9%8A%D9%86%D8%A7%D9%85%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D8%AD%D8%B1%D8%A7%D8%B1%D9%8A%D8%A9" title="القانون الثاني للديناميكا الحرارية – Arabic" lang="ar" hreflang="ar" data-title="القانون الثاني للديناميكا الحرارية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Segundu_principiu_de_la_termodin%C3%A1mica" title="Segundu principiu de la termodinámica – Asturian" lang="ast" hreflang="ast" data-title="Segundu principiu de la termodinámica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Termodinamikan%C4%B1n_ikinci_qanunu" title="Termodinamikanın ikinci qanunu – Azerbaijani" lang="az" hreflang="az" data-title="Termodinamikanın ikinci qanunu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%BE%E0%A6%AA%E0%A6%97%E0%A6%A4%E0%A6%BF%E0%A6%AC%E0%A6%BF%E0%A6%A6%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B0_%E0%A6%A6%E0%A7%8D%E0%A6%AC%E0%A6%BF%E0%A6%A4%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="তাপগতিবিদ্যার দ্বিতীয় সূত্র – Bangla" lang="bn" hreflang="bn" data-title="তাপগতিবিদ্যার দ্বিতীয় সূত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%80%D1%83%D0%B3%D1%96_%D0%BF%D0%B0%D1%87%D0%B0%D1%82%D0%B0%D0%BA_%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D0%BA%D1%96" title="Другі пачатак тэрмадынамікі – Belarusian" lang="be" hreflang="be" data-title="Другі пачатак тэрмадынамікі" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%82%D0%BE%D1%80%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BD%D0%B0_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0%D1%82%D0%B0" title="Втори закон на термодинамиката – Bulgarian" lang="bg" hreflang="bg" data-title="Втори закон на термодинамиката" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Drugi_zakon_termodinamike" title="Drugi zakon termodinamike – Bosnian" lang="bs" hreflang="bs" data-title="Drugi zakon termodinamike" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Segon_principi_de_la_termodin%C3%A0mica" title="Segon principi de la termodinàmica – Catalan" lang="ca" hreflang="ca" data-title="Segon principi de la termodinàmica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%C4%83%D0%BD_%D0%B8%D0%BA%D0%BA%C4%95%D0%BC%C4%95%D1%88_%D0%BF%D1%83%C3%A7%D0%BB%D0%B0%D0%BC%C4%83%D1%88%C4%95" title="Термодинамикăн иккĕмĕш пуçламăшĕ – Chuvash" lang="cv" hreflang="cv" data-title="Термодинамикăн иккĕмĕш пуçламăшĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Druh%C3%BD_termodynamick%C3%BD_z%C3%A1kon" title="Druhý termodynamický zákon – Czech" lang="cs" hreflang="cs" data-title="Druhý termodynamický zákon" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Termodynamikkens_2._lov" title="Termodynamikkens 2. lov – Danish" lang="da" hreflang="da" data-title="Termodynamikkens 2. lov" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zweiter_Hauptsatz_der_Thermodynamik" title="Zweiter Hauptsatz der Thermodynamik – German" lang="de" hreflang="de" data-title="Zweiter Hauptsatz der Thermodynamik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Termod%C3%BCnaamika_teine_seadus" title="Termodünaamika teine seadus – Estonian" lang="et" hreflang="et" data-title="Termodünaamika teine seadus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B5%CF%8D%CF%84%CE%B5%CF%81%CE%BF%CF%82_%CE%B8%CE%B5%CF%81%CE%BC%CE%BF%CE%B4%CF%85%CE%BD%CE%B1%CE%BC%CE%B9%CE%BA%CF%8C%CF%82_%CE%BD%CF%8C%CE%BC%CE%BF%CF%82" title="Δεύτερος θερμοδυναμικός νόμος – Greek" lang="el" hreflang="el" data-title="Δεύτερος θερμοδυναμικός νόμος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Segundo_principio_de_la_termodin%C3%A1mica" title="Segundo principio de la termodinámica – Spanish" lang="es" hreflang="es" data-title="Segundo principio de la termodinámica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Dua_le%C4%9Do_de_termodinamiko" title="Dua leĝo de termodinamiko – Esperanto" lang="eo" hreflang="eo" data-title="Dua leĝo de termodinamiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Termodinamikaren_bigarren_legea" title="Termodinamikaren bigarren legea – Basque" lang="eu" hreflang="eu" data-title="Termodinamikaren bigarren legea" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%AF%D9%88%D9%85_%D8%AA%D8%B1%D9%85%D9%88%D8%AF%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9" title="قانون دوم ترمودینامیک – Persian" lang="fa" hreflang="fa" data-title="قانون دوم ترمودینامیک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Deuxi%C3%A8me_principe_de_la_thermodynamique" title="Deuxième principe de la thermodynamique – French" lang="fr" hreflang="fr" data-title="Deuxième principe de la thermodynamique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Segunda_lei_da_termodin%C3%A1mica" title="Segunda lei da termodinámica – Galician" lang="gl" hreflang="gl" data-title="Segunda lei da termodinámica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B4%EC%97%AD%ED%95%99_%EC%A0%9C2%EB%B2%95%EC%B9%99" title="열역학 제2법칙 – Korean" lang="ko" hreflang="ko" data-title="열역학 제2법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8B%D5%A5%D6%80%D5%B4%D5%A1%D5%A4%D5%AB%D5%B6%D5%A1%D5%B4%D5%AB%D5%AF%D5%A1%D5%B5%D5%AB_%D5%A5%D6%80%D5%AF%D6%80%D5%B8%D6%80%D5%A4_%D6%85%D6%80%D5%A5%D5%B6%D6%84" title="Ջերմադինամիկայի երկրորդ օրենք – Armenian" lang="hy" hreflang="hy" data-title="Ջերմադինամիկայի երկրորդ օրենք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80_%E0%A4%95%E0%A4%BE_%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="ऊष्मागतिकी का द्वितीय नियम – Hindi" lang="hi" hreflang="hi" data-title="ऊष्मागतिकी का द्वितीय नियम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Drugi_zakon_termodinamike" title="Drugi zakon termodinamike – Croatian" lang="hr" hreflang="hr" data-title="Drugi zakon termodinamike" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_termodinamika_kedua" title="Hukum termodinamika kedua – Indonesian" lang="id" hreflang="id" data-title="Hukum termodinamika kedua" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Secondo_principio_della_termodinamica" title="Secondo principio della termodinamica – Italian" lang="it" hreflang="it" data-title="Secondo principio della termodinamica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%97%D7%95%D7%A7_%D7%94%D7%A9%D7%A0%D7%99_%D7%A9%D7%9C_%D7%94%D7%AA%D7%A8%D7%9E%D7%95%D7%93%D7%99%D7%A0%D7%9E%D7%99%D7%A7%D7%94" title="החוק השני של התרמודינמיקה – Hebrew" lang="he" hreflang="he" data-title="החוק השני של התרמודינמיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%97%E1%83%94%E1%83%A0%E1%83%9B%E1%83%9D%E1%83%93%E1%83%98%E1%83%9C%E1%83%90%E1%83%9B%E1%83%98%E1%83%99%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%98" title="თერმოდინამიკის მეორე კანონი – Georgian" lang="ka" hreflang="ka" data-title="თერმოდინამიკის მეორე კანონი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0%D0%BD%D1%8B%D2%A3_%D0%B5%D0%BA%D1%96%D0%BD%D1%88%D1%96_%D0%B7%D0%B0%D2%A3%D1%8B" title="Термодинамиканың екінші заңы – Kazakh" lang="kk" hreflang="kk" data-title="Термодинамиканың екінші заңы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Dezy%C3%A8m_lwa_t%C3%A8modinamik" title="Dezyèm lwa tèmodinamik – Haitian Creole" lang="ht" hreflang="ht" data-title="Dezyèm lwa tèmodinamik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Altera_lex_thermodynamica" title="Altera lex thermodynamica – Latin" lang="la" hreflang="la" data-title="Altera lex thermodynamica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Otrais_termodinamikas_likums" title="Otrais termodinamikas likums – Latvian" lang="lv" hreflang="lv" data-title="Otrais termodinamikas likums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/A_termodinamika_m%C3%A1sodik_f%C5%91t%C3%A9tele" title="A termodinamika második főtétele – Hungarian" lang="hu" hreflang="hu" data-title="A termodinamika második főtétele" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hukum_termodinamik_kedua" title="Hukum termodinamik kedua – Malay" lang="ms" hreflang="ms" data-title="Hukum termodinamik kedua" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tweede_wet_van_de_thermodynamica" title="Tweede wet van de thermodynamica – Dutch" lang="nl" hreflang="nl" data-title="Tweede wet van de thermodynamica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E6%B3%95%E5%89%87" title="熱力学第二法則 – Japanese" lang="ja" hreflang="ja" data-title="熱力学第二法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Termodynamikkens_andre_hovedsetning" title="Termodynamikkens andre hovedsetning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Termodynamikkens andre hovedsetning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Segond_principi_de_la_termodinamica" title="Segond principi de la termodinamica – Occitan" lang="oc" hreflang="oc" data-title="Segond principi de la termodinamica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Termodinamikaning_ikkinchi_qonuni" title="Termodinamikaning ikkinchi qonuni – Uzbek" lang="uz" hreflang="uz" data-title="Termodinamikaning ikkinchi qonuni" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A5%E0%A8%B0%E0%A8%AE%E0%A9%8B%E0%A8%A1%E0%A8%BE%E0%A8%87%E0%A8%A8%E0%A8%BE%E0%A8%AE%E0%A8%BF%E0%A8%95%E0%A8%B8_%E0%A8%A6%E0%A8%BE_%E0%A8%A6%E0%A9%82%E0%A8%9C%E0%A8%BE_%E0%A8%A8%E0%A8%BF%E0%A8%AF%E0%A8%AE" title="ਥਰਮੋਡਾਇਨਾਮਿਕਸ ਦਾ ਦੂਜਾ ਨਿਯਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਥਰਮੋਡਾਇਨਾਮਿਕਸ ਦਾ ਦੂਜਾ ਨਿਯਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AA%DA%BE%D8%B1%D9%85%D9%88%DA%88%D8%A7%D8%A6%D9%86%D8%A7%D9%85%DA%A9%D8%B3_%D8%AF%D8%A7_%D8%AF%D9%88%D8%AC%D8%A7_%D9%82%D9%86%D9%88%D9%86" title="تھرموڈائنامکس دا دوجا قنون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="تھرموڈائنامکس دا دوجا قنون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Druga_zasada_termodynamiki" title="Druga zasada termodynamiki – Polish" lang="pl" hreflang="pl" data-title="Druga zasada termodynamiki" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Segunda_lei_da_termodin%C3%A2mica" title="Segunda lei da termodinâmica – Portuguese" lang="pt" hreflang="pt" data-title="Segunda lei da termodinâmica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Principiul_al_doilea_al_termodinamicii" title="Principiul al doilea al termodinamicii – Romanian" lang="ro" hreflang="ro" data-title="Principiul al doilea al termodinamicii" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%82%D0%BE%D1%80%D0%BE%D0%B5_%D0%BD%D0%B0%D1%87%D0%B0%D0%BB%D0%BE_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B8" title="Второе начало термодинамики – Russian" lang="ru" hreflang="ru" data-title="Второе начало термодинамики" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8F%E0%B6%B4%E0%B6%9C%E0%B6%AD%E0%B7%92_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80%E0%B7%9A_%E0%B6%AF%E0%B7%99%E0%B7%80%E0%B6%B1_%E0%B6%B1%E0%B7%92%E0%B6%BA%E0%B6%B8%E0%B6%BA" title="තාපගති විද්‍යාවේ දෙවන නියමය – Sinhala" lang="si" hreflang="si" data-title="තාපගති විද්‍යාවේ දෙවන නියමය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Second law of thermodynamics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Druh%C3%A1_termodynamick%C3%A1_veta" title="Druhá termodynamická veta – Slovak" lang="sk" hreflang="sk" data-title="Druhá termodynamická veta" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Drugi_zakon_termodinamike" title="Drugi zakon termodinamike – Slovenian" lang="sl" hreflang="sl" data-title="Drugi zakon termodinamike" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D1%80%D1%83%D0%B3%D0%B8_%D0%BF%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B5" title="Други принцип термодинамике – Serbian" lang="sr" hreflang="sr" data-title="Други принцип термодинамике" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Drugi_zakon_termodinamike" title="Drugi zakon termodinamike – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Drugi zakon termodinamike" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Termodynamiikan_toinen_p%C3%A4%C3%A4s%C3%A4%C3%A4nt%C3%B6" title="Termodynamiikan toinen pääsääntö – Finnish" lang="fi" hreflang="fi" data-title="Termodynamiikan toinen pääsääntö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Termodynamikens_andra_huvudsats" title="Termodynamikens andra huvudsats – Swedish" lang="sv" hreflang="sv" data-title="Termodynamikens andra huvudsats" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA_%E0%AE%87%E0%AE%AF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%87%E0%AE%B0%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%BE%E0%AE%AE%E0%AF%8D_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="வெப்ப இயக்கவியலின் இரண்டாம் விதி – Tamil" lang="ta" hreflang="ta" data-title="வெப்ப இயக்கவியலின் இரண்டாம் விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B8%82%E0%B9%89%E0%B8%AD%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%AA%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AD%E0%B8%B8%E0%B8%93%E0%B8%AB%E0%B8%9E%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="กฎข้อที่สองของอุณหพลศาสตร์ – Thai" lang="th" hreflang="th" data-title="กฎข้อที่สองของอุณหพลศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Termodinami%C4%9Fin_ikinci_yasas%C4%B1" title="Termodinamiğin ikinci yasası – Turkish" lang="tr" hreflang="tr" data-title="Termodinamiğin ikinci yasası" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%80%D1%83%D0%B3%D0%B8%D0%B9_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D1%96%D0%BA%D0%B8" title="Другий закон термодинаміки – Ukrainian" lang="uk" hreflang="uk" data-title="Другий закон термодинаміки" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%B1%D8%AD%D8%B1%DA%A9%DB%8C%D8%A7%D8%AA_%DA%A9%D8%A7_%D8%AF%D9%88%D8%B3%D8%B1%D8%A7_%D9%82%D8%A7%D9%86%D9%88%D9%86" title="حرحرکیات کا دوسرا قانون – Urdu" lang="ur" hreflang="ur" data-title="حرحرکیات کا دوسرا قانون" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_lu%E1%BA%ADt_th%E1%BB%A9_hai_c%E1%BB%A7a_nhi%E1%BB%87t_%C4%91%E1%BB%99ng_l%E1%BB%B1c_h%E1%BB%8Dc" title="Định luật thứ hai của nhiệt động lực học – Vietnamese" lang="vi" hreflang="vi" data-title="Định luật thứ hai của nhiệt động lực học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%B8%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B" title="熱力學第二定律 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="熱力學第二定律" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B" title="热力学第二定律 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mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><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 c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><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 \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><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 \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><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 \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><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 \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><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 U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><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(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><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 A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><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(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Other</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>second law of thermodynamics</b> is a <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">physical law</a> based on <a href="/wiki/Universal_(metaphysics)" title="Universal (metaphysics)">universal</a> <a href="/wiki/Empirical" class="mw-redirect" title="Empirical">empirical</a> <a href="/wiki/Observation" title="Observation">observation</a> concerning <a href="/wiki/Heat" title="Heat">heat</a> and <a href="/wiki/Energy_transformation" title="Energy transformation">energy interconversions</a>. A simple statement of the law is that heat always flows spontaneously from hotter to colder regions of matter (or 'downhill' in terms of the temperature gradient). Another statement is: "Not all heat can be converted into <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> in a <a href="/wiki/Cyclic_process" class="mw-redirect" title="Cyclic process">cyclic process</a>."<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><sup id="cite_ref-Rao_2-0" class="reference"><a href="#cite_note-Rao-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Young&Freedman11th_3-0" class="reference"><a href="#cite_note-Young&Freedman11th-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The second law of thermodynamics establishes the concept of <a href="/wiki/Entropy" title="Entropy">entropy</a> as a physical property of a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>. It predicts whether processes are forbidden despite obeying the requirement of <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> as expressed in the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> and provides necessary criteria for <a href="/wiki/Spontaneous_process" title="Spontaneous process">spontaneous processes</a>. For example, the first law allows the process of a cup falling off a table and breaking on the floor, as well as allowing the reverse process of the cup fragments coming back together and 'jumping' back onto the table, while the second law allows the former and denies the latter. The second law may be formulated by the observation that the entropy of <a href="/wiki/Isolated_system" title="Isolated system">isolated systems</a> left to spontaneous evolution cannot decrease, as they always tend toward a state of <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> where the entropy is highest at the given internal energy.<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> An increase in the combined entropy of system and surroundings accounts for the <a href="/wiki/Irreversibility" class="mw-redirect" title="Irreversibility">irreversibility</a> of natural processes, often referred to in the concept of the <a href="/wiki/Arrow_of_time" title="Arrow of time">arrow of time</a>.<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><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><p>Historically, the second law was an <a href="/wiki/Empirical_evidence" title="Empirical evidence">empirical finding</a> that was accepted as an <a href="/wiki/Axiom" title="Axiom">axiom</a> of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamic theory</a>. <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a> provides a microscopic explanation of the law in terms of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> of the states of large assemblies of <a href="/wiki/Atom" title="Atom">atoms</a> or <a href="/wiki/Molecule" title="Molecule">molecules</a>. The second law has been expressed in many ways. Its first formulation, which preceded the proper definition of entropy and was based on <a href="/wiki/Caloric_theory" title="Caloric theory">caloric theory</a>, is <a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a>, formulated by the French scientist <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Sadi Carnot</a>, who in 1824 showed that the efficiency of conversion of heat to work in a heat engine has an upper limit.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The first rigorous definition of the second law based on the concept of entropy came from German scientist <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> in the 1850s and included his statement that heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. </p><p>The second law of thermodynamics allows the definition of the concept of <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a>, but this has been formally delegated to the <a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">zeroth law of thermodynamics</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Heat_flow_hot_to_cold.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Heat_flow_hot_to_cold.png/170px-Heat_flow_hot_to_cold.png" decoding="async" width="170" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Heat_flow_hot_to_cold.png/255px-Heat_flow_hot_to_cold.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Heat_flow_hot_to_cold.png/340px-Heat_flow_hot_to_cold.png 2x" data-file-width="400" data-file-height="582" /></a><figcaption>Heat flowing from hot water to cold water</figcaption></figure> <p>The <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> provides the definition of the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> of a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>, and expresses its change for a <a href="/wiki/Closed_system" title="Closed system">closed system</a> in terms of <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> and <a href="/wiki/Heat" title="Heat">heat</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> It can be linked to the law of <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Conceptually, the first law describes the fundamental principle that systems do not consume or 'use up' energy, that energy is neither created nor destroyed, but is simply converted from one form to another. </p><p>The second law is concerned with the direction of natural processes.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It asserts that a natural process runs only in one sense, and is not reversible. That is, the state of a natural system itself can be reversed, but not without increasing the entropy of the system's surroundings, that is, both the state of the system plus the state of its surroundings cannot be together, fully reversed, without implying the destruction of entropy. </p><p>For example, when a path for conduction or <a href="/wiki/Radiation" title="Radiation">radiation</a> is made available, heat always flows spontaneously from a hotter to a colder body. Such <a href="/wiki/Phenomenon" title="Phenomenon">phenomena</a> are accounted for in terms of <a href="/wiki/Entropy" title="Entropy">entropy change</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> A heat pump can reverse this heat flow, but the reversal process and the original process, both cause entropy production, thereby increasing the entropy of the system's surroundings. If an isolated system containing distinct subsystems is held initially in internal thermodynamic equilibrium by internal partitioning by impermeable walls between the subsystems, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>, and its total entropy, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, increases. </p><p>In a <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible</a> or <a href="/wiki/Quasistatic_process" title="Quasistatic process">quasi-static</a>, idealized process of transfer of energy as heat to a <a href="/wiki/Closed_system" title="Closed system">closed</a> thermodynamic system of interest, (which allows the entry or exit of energy – but not transfer of matter), from an auxiliary thermodynamic system, an infinitesimal increment (<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} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72ba425d7d7a0f229457dea3c0be4a47ea303cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.792ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} S}"></span>) in the entropy of the system of interest is defined to result from an infinitesimal <a href="/wiki/Transfer_of_heat" class="mw-redirect" title="Transfer of heat">transfer of heat</a> (<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 \delta Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf715eece146c816847a8c5d56eae97798453d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle \delta Q}"></span>) to the system of interest, divided by the common <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a> <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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f9a82bdcb95d511ed6965b0534a3eda98ceb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.446ex; height:2.843ex;" alt="{\displaystyle (T)}"></span> of the system of interest and the auxiliary thermodynamic system:<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; idealized, reversible process)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(closed system; idealized, reversible process)</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; idealized, reversible process)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4422a72c7a3c220fba255df2cdc2466a34b1d62e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.304ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; idealized, reversible process)}}.}"></span></dd></dl> <p>Different notations are used for an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> amount of heat <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 (\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea20ade7c72d2d4722a03cbbfc3f54a03a7c2b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.858ex; height:2.843ex;" alt="{\displaystyle (\delta )}"></span> and infinitesimal change of entropy <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} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {d} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b16fdc1501515bc9d24841a920087022a08c9d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.102ex; height:2.843ex;" alt="{\displaystyle (\mathrm {d} )}"></span> because entropy is a <a href="/wiki/Function_of_state" class="mw-redirect" title="Function of state">function of state</a>, while heat, like work, is not. </p><p>For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the <a href="/wiki/Clausius_Theorem" class="mw-redirect" title="Clausius Theorem">inequality</a><sup id="cite_ref-MortimerBook_15-0" class="reference"><a href="#cite_note-MortimerBook-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FermiBook_16-0" class="reference"><a href="#cite_note-FermiBook-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </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} S>{\frac {\delta Q}{T_{\text{surr}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; actually possible, irreversible process).}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>surr</mtext> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(closed system; actually possible, irreversible process).</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S>{\frac {\delta Q}{T_{\text{surr}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; actually possible, irreversible process).}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae63708f5ca5b7fdd07bcafe5d5d320053eb0b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:74.194ex; height:5.676ex;" alt="{\displaystyle \mathrm {d} S>{\frac {\delta Q}{T_{\text{surr}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; actually possible, irreversible process).}}}"></span></dd></dl> <p>This is because a general process for this case (no mass exchange between the system and its surroundings) may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature (<span class="texhtml"><i>T</i></span>) and the temperature of the surroundings (<span class="texhtml"><i>T</i><sub>surr</sub></span>).<sup id="cite_ref-:0_17-0" class="reference"><a href="#cite_note-:0-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Munster_45_18-0" class="reference"><a href="#cite_note-Munster_45-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The equality still applies for pure heat flow (only heat flow, no change in chemical composition and mass), </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} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(actually possible quasistatic irreversible process without composition change).}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(actually possible quasistatic irreversible process without composition change).</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(actually possible quasistatic irreversible process without composition change).}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be2f9b652feaf2f5fc9882a803732f834f00bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:97.316ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(actually possible quasistatic irreversible process without composition change).}}}"></span></dd></dl> <p>which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<sup id="cite_ref-Oxtoby8th_19-0" class="reference"><a href="#cite_note-Oxtoby8th-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-MortimerBook_15-1" class="reference"><a href="#cite_note-MortimerBook-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">zeroth law of thermodynamics</a> in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<sup id="cite_ref-dugdale_20-0" class="reference"><a href="#cite_note-dugdale-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (August 2018)">clarification needed</span></a></i>]</sup> a distinguished temperature scale, which defines an absolute, <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a>, independent of the properties of any particular reference thermometric body.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Various_statements_of_the_law">Various statements of the law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=2" title="Edit section: Various statements of the law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The second law of thermodynamics may be expressed in many specific ways,<sup id="cite_ref-MIT_23-0" class="reference"><a href="#cite_note-MIT-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> the most prominent classical statements<sup id="cite_ref-FOOTNOTELiebYngvason1999_24-0" class="reference"><a href="#cite_note-FOOTNOTELiebYngvason1999-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> being the statement by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> (1854), the statement by <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">Lord Kelvin</a> (1851), and the statement in <a href="/wiki/Axiomatic" class="mw-redirect" title="Axiomatic">axiomatic</a> thermodynamics by <a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Constantin Carathéodory</a> (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<sup id="cite_ref-FOOTNOTERao2004213_25-0" class="reference"><a href="#cite_note-FOOTNOTERao2004213-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Carnot's_principle"><span id="Carnot.27s_principle"></span>Carnot's principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=3" title="Edit section: Carnot's principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The historical origin<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> of the second law of thermodynamics was in <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Sadi Carnot</a>'s theoretical analysis of the flow of heat in steam engines (1824). The centerpiece of that analysis, now known as a <a href="/wiki/Carnot_engine" class="mw-redirect" title="Carnot engine">Carnot engine</a>, is an ideal <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a> fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>. It represents the theoretical maximum efficiency of a heat engine operating between any two given thermal or heat reservoirs at different temperatures. Carnot's principle was recognized by Carnot at a time when the <a href="/wiki/Caloric_theory" title="Caloric theory">caloric theory</a> represented the dominant understanding of the nature of heat, before the recognition of the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, Carnot's analysis is physically equivalent to the second law of thermodynamics, and remains valid today. Some samples from his book are: </p> <dl><dd><dl><dd>...<i>wherever there exists a difference of temperature, motive power can be produced.</i><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></dd> <dd>The production of motive power is then due in <a href="/wiki/Steam_engines" class="mw-redirect" title="Steam engines">steam engines</a> not to an actual consumption of caloric, but <i>to its transportation from a warm body to a cold body ...</i><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></dd> <dd><i>The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.</i><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p>In modern terms, Carnot's principle may be stated more precisely: </p> <dl><dd>The efficiency of a quasi-static or reversible <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot cycle</a> depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Clausius_statement">Clausius statement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=4" title="Edit section: Clausius statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The German scientist <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.<sup id="cite_ref-FOOTNOTEClausius1850_36-0" class="reference"><a href="#cite_note-FOOTNOTEClausius1850-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> His formulation of the second law, which was published in German in 1854, is known as the <i>Clausius statement</i>: </p> <blockquote><p>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.<sup id="cite_ref-FOOTNOTEClausius185486_37-0" class="reference"><a href="#cite_note-FOOTNOTEClausius185486-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other. </p><p>Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of <a href="/wiki/Refrigeration" title="Refrigeration">refrigeration</a>, for example. In a refrigerator, heat is transferred from cold to hot, but only when forced by an external agent, the refrigeration system. </p> <div class="mw-heading mw-heading3"><h3 id="Kelvin_statements">Kelvin statements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=5" title="Edit section: Kelvin statements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">Lord Kelvin</a> expressed the second law in several wordings. </p> <dl><dd><dl><dd>It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.</dd> <dd>It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<sup id="cite_ref-FOOTNOTEThomson1851_38-0" class="reference"><a href="#cite_note-FOOTNOTEThomson1851-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Equivalence_of_the_Clausius_and_the_Kelvin_statements">Equivalence of the Clausius and the Kelvin statements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=6" title="Edit section: Equivalence of the Clausius and the Kelvin statements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Deriving_Kelvin_Statement_from_Clausius_Statement.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Deriving_Kelvin_Statement_from_Clausius_Statement.svg/220px-Deriving_Kelvin_Statement_from_Clausius_Statement.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Deriving_Kelvin_Statement_from_Clausius_Statement.svg/330px-Deriving_Kelvin_Statement_from_Clausius_Statement.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Deriving_Kelvin_Statement_from_Clausius_Statement.svg/440px-Deriving_Kelvin_Statement_from_Clausius_Statement.svg.png 2x" data-file-width="662" data-file-height="589" /></a><figcaption>Derive Kelvin Statement from Clausius Statement</figcaption></figure> <p>Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work (the drained heat is fully converted to work) in a cyclic fashion without any other result. Now pair it with a reversed <a href="/wiki/Carnot_engine" class="mw-redirect" title="Carnot engine">Carnot engine</a> as shown by the right figure. The <a href="/wiki/Heat_engine#Efficiency" title="Heat engine">efficiency</a> of a normal heat engine is <i>η</i> and so the efficiency of the reversed heat engine is 1/<i>η</i>. The net and sole effect of the combined pair of engines is to transfer heat <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="{\textstyle \Delta Q=Q\left({\frac {1}{\eta }}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> <mo>=</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>η</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Delta Q=Q\left({\frac {1}{\eta }}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a360321a75c2b741008172ac76a3d4836ddd7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.539ex; height:4.843ex;" alt="{\textstyle \Delta Q=Q\left({\frac {1}{\eta }}-1\right)}"></span> from the cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>, as for the total system's energy to remain the same; <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="{\textstyle {\text{Input}}+{\text{Output}}=0\implies (Q+Q_{c})-{\frac {Q}{\eta }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Input</mtext> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Output</mtext> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>η</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\text{Input}}+{\text{Output}}=0\implies (Q+Q_{c})-{\frac {Q}{\eta }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08875c9431acfc6eb4c7580bdacf002f34374de5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.112ex; height:4.176ex;" alt="{\textstyle {\text{Input}}+{\text{Output}}=0\implies (Q+Q_{c})-{\frac {Q}{\eta }}=0}"></span>, so therefore <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="{\textstyle Q_{c}=Q\left({\frac {1}{\eta }}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>η</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Q_{c}=Q\left({\frac {1}{\eta }}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03702804374f47cb9ddb39fffcd69daa65a64366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.548ex; height:4.843ex;" alt="{\textstyle Q_{c}=Q\left({\frac {1}{\eta }}-1\right)}"></span>, where (1) the sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) <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 {Q}{\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>η</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Q}{\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdda20674365c00854ab1eb7607f880ca2969af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.674ex; height:5.843ex;" alt="{\displaystyle {\frac {Q}{\eta }}}"></span> is obtained by <a href="/wiki/Heat_engine#Efficiency" title="Heat engine">the definition of efficiency</a> of the engine when the engine operation is not reversed. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent. </p> <div class="mw-heading mw-heading3"><h3 id="Planck's_proposition"><span id="Planck.27s_proposition"></span>Planck's proposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=7" title="Edit section: Planck's proposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law. </p> <dl><dd><dl><dd>It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the production of work and cooling of a heat reservoir.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relation_between_Kelvin's_statement_and_Planck's_proposition"><span id="Relation_between_Kelvin.27s_statement_and_Planck.27s_proposition"></span>Relation between Kelvin's statement and Planck's proposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=8" title="Edit section: Relation between Kelvin's statement and Planck's proposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is almost customary in textbooks to speak of the "Kelvin–Planck statement" of the law, as for example in the text by <a href="/wiki/Dirk_ter_Haar" title="Dirk ter Haar">ter Haar</a> and <a href="/wiki/Harald_Wergeland" title="Harald Wergeland">Wergeland</a>.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> This version, also known as the <b>heat engine statement</b>, of the second law states that </p> <dl><dd><dl><dd>It is impossible to devise a <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">cyclically</a> operating device, the sole effect of which is to absorb energy in the form of heat from a single <a href="/wiki/Heat_reservoir" class="mw-redirect" title="Heat reservoir">thermal reservoir</a> and to deliver an equivalent amount of <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>.<sup id="cite_ref-Rao_2-1" class="reference"><a href="#cite_note-Rao-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Planck's_statement"><span id="Planck.27s_statement"></span>Planck's statement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=9" title="Edit section: Planck's statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> stated the second law as follows. </p> <dl><dd><dl><dd>Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<sup id="cite_ref-Planck_100_42-0" class="reference"><a href="#cite_note-Planck_100-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Planck_463_43-0" class="reference"><a href="#cite_note-Planck_463-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Roberts_&_Miller_382_44-0" class="reference"><a href="#cite_note-Roberts_&_Miller_382-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p>Rather like Planck's statement is that of <a href="/wiki/George_Uhlenbeck" title="George Uhlenbeck">George Uhlenbeck</a> and G. W. Ford for <i>irreversible phenomena</i>. </p> <dl><dd><dl><dd>... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Principle_of_Carathéodory"><span id="Principle_of_Carath.C3.A9odory"></span>Principle of Carathéodory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=10" title="Edit section: Principle of Carathéodory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Constantin Carathéodory</a> formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <blockquote><p>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>With this formulation, he described the concept of <a href="/wiki/Adiabatic_accessibility" title="Adiabatic accessibility">adiabatic accessibility</a> for the first time and provided the foundation for a new subfield of classical thermodynamics, often called <a href="/wiki/Ruppeiner_geometry" title="Ruppeiner geometry">geometrical thermodynamics</a>. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic <a href="/wiki/Process_function" title="Process function">process function</a>, in other words, <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 \delta Q=TdS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mi>T</mi> <mi>d</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q=TdS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8142742fd8644554e5a89912f03dcac94097b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.676ex;" alt="{\displaystyle \delta Q=TdS}"></span>.<sup id="cite_ref-Sychev1991_48-0" class="reference"><a href="#cite_note-Sychev1991-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<sup id="cite_ref-Munster_45_18-1" class="reference"><a href="#cite_note-Munster_45-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTELiebYngvason199949_49-0" class="reference"><a href="#cite_note-FOOTNOTELiebYngvason199949-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Planck_1926_50-0" class="reference"><a href="#cite_note-Planck_1926-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (February 2014)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Planck's_principle"><span id="Planck.27s_principle"></span>Planck's principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=11" title="Edit section: Planck's principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1926, Max Planck wrote an important paper on the basics of thermodynamics.<sup id="cite_ref-Planck_1926_50-1" class="reference"><a href="#cite_note-Planck_1926-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> He indicated the principle </p> <dl><dd><dl><dd>The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<sup id="cite_ref-Munster_45_18-2" class="reference"><a href="#cite_note-Munster_45-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTELiebYngvason199949_49-1" class="reference"><a href="#cite_note-FOOTNOTELiebYngvason199949-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p>This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> Planck wrote: "The production of heat by friction is irreversible."<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> It is relevant that for a system at constant volume and <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole numbers</a>, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy. </p><p>A statement that in a sense is complementary to Planck's principle is made by Claus Borgnakke and Richard E. Sonntag. They do not offer it as a full statement of the second law: </p> <dl><dd><dl><dd>... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p>Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy. </p> <div class="mw-heading mw-heading3"><h3 id="Relating_the_second_law_to_the_definition_of_temperature">Relating the second law to the definition of temperature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=12" title="Edit section: Relating the second law to the definition of temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The second law has been shown to be equivalent to the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> <span class="texhtml"><i>U</i></span> defined as a <a href="/wiki/Convex_function" title="Convex function">convex function</a> of the other extensive properties of the system.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> That is, when a system is described by stating its <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> <span class="texhtml"><i>U</i></span>, an extensive variable, as a function of its <a href="/wiki/Entropy" title="Entropy">entropy</a> <span class="texhtml"><i>S</i></span>, volume <span class="texhtml"><i>V</i></span>, and mol number <span class="texhtml"><i>N</i></span>, i.e. <span class="texhtml"><i>U</i> = <i>U</i> (<i>S</i>, <i>V</i>, <i>N</i></span>), then the temperature is equal to the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of the internal energy with respect to the entropy<sup id="cite_ref-Callen_146–148_59-0" class="reference"><a href="#cite_note-Callen_146–148-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> (essentially equivalent to the first <span class="texhtml"><i>TdS</i></span> equation for <span class="texhtml"><i>V</i></span> and <span class="texhtml"><i>N</i></span> held constant): </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=\left({\frac {\partial U}{\partial S}}\right)_{V,N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mo>,</mo> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V,N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/818d012b1eb2878592bb811562c29772d322ca9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.505ex; height:6.509ex;" alt="{\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V,N}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Second_law_statements,_such_as_the_Clausius_inequality,_involving_radiative_fluxes"><span id="Second_law_statements.2C_such_as_the_Clausius_inequality.2C_involving_radiative_fluxes"></span>Second law statements, such as the Clausius inequality, involving radiative fluxes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=13" title="Edit section: Second law statements, such as the Clausius inequality, involving radiative fluxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Clausius inequality, as well as some other statements of the second law, must be re-stated to have general applicability for all forms of heat transfer, i.e. scenarios involving radiative fluxes. For example, the integrand (đQ/T) of the Clausius expression applies to heat conduction and convection, and the case of ideal infinitesimal blackbody radiation (BR) transfer, but does not apply to most radiative transfer scenarios and in some cases has no physical meaning whatsoever. Consequently, the Clausius inequality was re-stated<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> so that it is applicable to cycles with processes involving any form of heat transfer. The entropy transfer with radiative fluxes (<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="\delta S_{\text{NetRad}}"> <semantics> <mrow> <mi>δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NetRad</mtext> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">\delta S_{\text{NetRad}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0fc3cd3f47c2dbfa232ae4655bfb54617fcce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.254ex; height:2.676ex;" alt="\delta S_{\text{NetRad}}"></span>) is taken separately from that due to heat transfer by conduction and convection (<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="\delta Q_{CC}"> <semantics> <mrow> <mi>δ</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>C</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">\delta Q_{CC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cdb90175f1dd1778f623a3fe2ceb8a4b6c13a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.617ex; height:2.676ex;" alt="\delta Q_{CC}"></span>), where the temperature is evaluated at the system boundary where the heat transfer occurs. The modified Clausius inequality, for all heat transfer scenarios, can then be expressed as,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\text{cycle}}({\frac {\delta Q_{CC}}{T_{b}}}+\delta S_{\text{NetRad}})\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>cycle</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>C</mi> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NetRad</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\text{cycle}}({\frac {\delta Q_{CC}}{T_{b}}}+\delta S_{\text{NetRad}})\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e19b8c1382167d1f084d641ac129ca318c0526" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.658ex; height:6.176ex;" alt="{\displaystyle \int _{\text{cycle}}({\frac {\delta Q_{CC}}{T_{b}}}+\delta S_{\text{NetRad}})\leq 0}"></span> </p><p>In a nutshell, the Clausius inequality is saying that when a cycle is completed, the change in the state property S will be zero, so the entropy that was produced during the cycle must have transferred out of the system by heat transfer. The <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="\delta "> <semantics> <mi>δ</mi> <annotation encoding="application/x-tex">\delta</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3086070a07409fa7e760fea6e7c932b2a590b1ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="\delta "></span> (or đ) indicates a path dependent integration. </p><p>Due to the inherent emission of radiation from all matter, most entropy flux calculations involve incident, reflected and emitted radiative fluxes. The energy and entropy of unpolarized blackbody thermal radiation, is calculated using the spectral energy and entropy radiance expressions derived by Max Planck<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> using equilibrium statistical mechanics,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\nu }={\frac {2h}{c^{2}}}{\frac {\nu ^{3}}{\exp \left({\frac {h\nu }{kT}}\right)-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>h</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\nu }={\frac {2h}{c^{2}}}{\frac {\nu ^{3}}{\exp \left({\frac {h\nu }{kT}}\right)-1}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e50f5b8c7cd3b2886ce1f4557da23a2f04860e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:24.176ex; height:8.343ex;" alt="{\displaystyle K_{\nu }={\frac {2h}{c^{2}}}{\frac {\nu ^{3}}{\exp \left({\frac {h\nu }{kT}}\right)-1}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\nu }={\frac {2k\nu ^{2}}{c^{2}}}((1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln(1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})-({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\nu }={\frac {2k\nu ^{2}}{c^{2}}}((1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln(1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})-({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f22d938aa5bb3492e1902cb00404344e0205c32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:59.747ex; height:6.009ex;" alt="{\displaystyle L_{\nu }={\frac {2k\nu ^{2}}{c^{2}}}((1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln(1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})-({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}}))}"></span> where <i>c</i> is the speed of light, <i>k</i> is the Boltzmann constant, <i>h</i> is the Planck constant, <i>ν</i> is frequency, and the quantities <i>K</i><sub>v</sub> and <i>L</i><sub>v</sub> are the energy and entropy fluxes per unit frequency, area, and solid angle. In deriving this blackbody spectral entropy radiance, with the goal of deriving the blackbody energy formula, Planck postulated that the energy of a photon was quantized (partly to simplify the mathematics), thereby starting quantum theory. </p><p>A non-equilibrium statistical mechanics approach has also been used to obtain the same result as Planck, indicating it has wider significance and represents a non-equilibrium entropy.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> A plot of <i>K</i><sub>v</sub> versus frequency (v) for various values of temperature (<i>T)</i> gives a family of blackbody radiation energy spectra, and likewise for the entropy spectra. For non-blackbody radiation (NBR) emission fluxes, the spectral entropy radiance <i>L</i><sub>v</sub> is found by substituting <i>K</i><sub>v</sub> spectral energy radiance data into the <i>L</i><sub>v</sub> expression (noting that emitted and reflected entropy fluxes are, in general, not independent). For the emission of NBR, including graybody radiation (GR), the resultant emitted entropy flux, or radiance <i>L</i>, has a higher ratio of entropy-to-energy (<i>L/K</i>), than that of BR. That is, the entropy flux of NBR emission is farther removed from the conduction and convection <i>q</i>/<i>T</i> result, than that for BR emission.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> This observation is consistent with Max Planck's blackbody radiation energy and entropy formulas and is consistent with the fact that blackbody radiation emission represents the maximum emission of entropy for all materials with the same temperature, as well as the maximum entropy emission for all radiation with the same energy radiance. </p> <div class="mw-heading mw-heading3"><h3 id="Generalized_conceptual_statement_of_the_second_law_principle">Generalized conceptual statement of the second law principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=14" title="Edit section: Generalized conceptual statement of the second law principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Second law analysis is valuable in scientific and engineering analysis in that it provides a number of benefits over energy analysis alone, including the basis for determining energy quality (exergy content<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup>), understanding fundamental physical phenomena, and improving performance evaluation and optimization. As a result, a conceptual statement of the principle is very useful in engineering analysis. Thermodynamic systems can be categorized by the four combinations of either entropy (S) up or down, and uniformity (Y) – between system and its environment – up or down. This 'special' category of processes, category IV, is characterized by movement in the direction of low disorder and low uniformity, counteracting the second law tendency towards uniformity and disorder.<sup id="cite_ref-Wright_12–18_67-0" class="reference"><a href="#cite_note-Wright_12–18-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Exergysun1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Exergysun1.png/220px-Exergysun1.png" decoding="async" width="220" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Exergysun1.png/330px-Exergysun1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Exergysun1.png/440px-Exergysun1.png 2x" data-file-width="1246" data-file-height="580" /></a><figcaption>Four categories of processes given entropy up or down and uniformity up or down</figcaption></figure> <p>The second law can be conceptually stated<sup id="cite_ref-Wright_12–18_67-1" class="reference"><a href="#cite_note-Wright_12–18-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> as follows: Matter and energy have the tendency to reach a state of uniformity or internal and external equilibrium, a state of maximum disorder (entropy). Real non-equilibrium processes always produce entropy, causing increased disorder in the universe, while idealized reversible processes produce no entropy and no process is known to exist that destroys entropy. The tendency of a system to approach uniformity may be counteracted, and the system may become more ordered or complex, by the combination of two things, a work or exergy source and some form of instruction or intelligence. Where 'exergy' is the thermal, mechanical, electric or chemical work potential of an energy source or flow, and 'instruction or intelligence', although subjective, is in the context of the set of category IV processes. </p><p>Consider a category IV example of robotic manufacturing and assembly of vehicles in a factory. The robotic machinery requires electrical work input and instructions, but when completed, the manufactured products have less uniformity with their surroundings, or more complexity (higher order) relative to the raw materials they were made from. Thus, system entropy or disorder decreases while the tendency towards uniformity between the system and its environment is counteracted. In this example, the instructions, as well as the source of work may be internal or external to the system, and they may or may not cross the system boundary. To illustrate, the instructions may be pre-coded and the electrical work may be stored in an energy storage system on-site. Alternatively, the control of the machinery may be by remote operation over a communications network, while the electric work is supplied to the factory from the local electric grid. In addition, humans may directly play, in whole or in part, the role that the robotic machinery plays in manufacturing. In this case, instructions may be involved, but intelligence is either directly responsible, or indirectly responsible, for the direction or application of work in such a way as to counteract the tendency towards disorder and uniformity. </p><p>There are also situations where the entropy spontaneously decreases by means of energy and entropy transfer. When thermodynamic constraints are not present, spontaneously energy or mass, as well as accompanying entropy, may be transferred out of a system in a progress to reach external equilibrium or uniformity in intensive properties of the system with its surroundings. This occurs spontaneously because the energy or mass transferred from the system to its surroundings results in a higher entropy in the surroundings, that is, it results in higher overall entropy of the system plus its surroundings. Note that this transfer of entropy requires dis-equilibrium in properties, such as a temperature difference. One example of this is the cooling crystallization of water that can occur when the system's surroundings are below freezing temperatures. Unconstrained heat transfer can spontaneously occur, leading to water molecules freezing into a crystallized structure of reduced disorder (sticking together in a certain order due to molecular attraction). The entropy of the system decreases, but the system approaches uniformity with its surroundings (category III). </p><p>On the other hand, consider the refrigeration of water in a warm environment. Due to refrigeration, as heat is extracted from the water, the temperature and entropy of the water decreases, as the system moves further away from uniformity with its warm surroundings or environment (category IV). The main point, take-away, is that refrigeration not only requires a source of work, it requires designed equipment, as well as pre-coded or direct operational intelligence or instructions to achieve the desired refrigeration effect. </p> <div class="mw-heading mw-heading2"><h2 id="Corollaries">Corollaries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=15" title="Edit section: Corollaries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Perpetual_motion_of_the_second_kind">Perpetual motion of the second kind</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=16" title="Edit section: Perpetual motion of the second kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Perpetual_motion" title="Perpetual motion">Perpetual motion</a></div> <p>Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines. </p> <div class="mw-heading mw-heading3"><h3 id="Carnot's_theorem"><span id="Carnot.27s_theorem"></span>Carnot's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=17" title="Edit section: Carnot's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a> (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states: </p> <ul><li>All irreversible heat engines between two heat reservoirs are less efficient than a <a href="/wiki/Carnot_engine" class="mw-redirect" title="Carnot engine">Carnot engine</a> operating between the same reservoirs.</li> <li>All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.</li></ul> <p>In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as <a href="/wiki/Thermodynamic_reversibility" class="mw-redirect" title="Thermodynamic reversibility">thermodynamic reversibility</a>. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realize the <a href="/wiki/Carnot_cycle" title="Carnot cycle">Carnot cycle</a>'s reversibility and was condemned to be less efficient. </p><p>Though formulated in terms of caloric (see the obsolete <a href="/wiki/Caloric_theory" title="Caloric theory">caloric theory</a>), rather than <a href="/wiki/Entropy" title="Entropy">entropy</a>, this was an early insight into the second law. </p> <div class="mw-heading mw-heading3"><h3 id="Clausius_inequality">Clausius inequality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=18" title="Edit section: Clausius inequality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a> (1854) states that in a cyclic process </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 \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>surr</mtext> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17daae51489d5c2e13b9e241d27f51c811b2f6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.766ex; height:5.843ex;" alt="{\displaystyle \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0.}"></span></dd></dl> <p>The equality holds in the reversible case<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> and the strict inequality holds in the irreversible case, with <i>T</i><sub>surr</sub> as the temperature of the heat bath (surroundings) here. The reversible case is used to introduce the state function <a href="/wiki/Entropy" title="Entropy">entropy</a>. This is because in cyclic processes the variation of a state function is zero from state functionality. </p> <div class="mw-heading mw-heading3"><h3 id="Thermodynamic_temperature">Thermodynamic temperature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=19" title="Edit section: Thermodynamic temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Thermodynamic temperature</a></div> <p>For an arbitrary heat engine, the efficiency is: </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {|W_{n}|}{q_{\text{H}}}}={\frac {q_{H}+q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|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> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {|W_{n}|}{q_{\text{H}}}}={\frac {q_{H}+q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e53d36d564969fef4e9194d3e2a1dd5198d91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.502ex; height:6.509ex;" alt="{\displaystyle \eta ={\frac {|W_{n}|}{q_{\text{H}}}}={\frac {q_{H}+q_{\text{C}}}{q_{\text{H}}}}=1-{\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}}"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table> <p>where <i>W</i><sub><i>n</i></sub> is the net work done by the engine per cycle, <i>q</i><sub>H</sub> > 0 is the heat added to the engine from a hot reservoir, and <i>q</i><sub>C</sub> = −|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>q</i><sub><i>C</i></sub></span>| < 0<sup id="cite_ref-PlanckBook_69-0" class="reference"><a href="#cite_note-PlanckBook-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> is waste <a href="/wiki/Heat" title="Heat">heat given off</a> to a cold reservoir from the engine. Thus the efficiency depends only on the ratio |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>q</i><sub>C</sub></span>| / |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>q</i><sub>H</sub></span>|. </p><p><a href="/wiki/Carnot_theorem_(thermodynamics)" class="mw-redirect" title="Carnot theorem (thermodynamics)">Carnot's theorem</a> states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures <i>T</i><sub>H</sub> and <i>T</i><sub>C</sub> must have the same efficiency, that is to say, the efficiency is a function of temperatures only: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {|q_{\text{C}}|}{|q_{\text{H}}|}}=f(T_{\text{H}},T_{\text{C}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <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>C</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=f(T_{\text{H}},T_{\text{C}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d554b985f18f933b7e398ae4e638199bfd05fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.098ex; height:6.509ex;" alt="{\displaystyle {\frac {|q_{\text{C}}|}{|q_{\text{H}}|}}=f(T_{\text{H}},T_{\text{C}}).}"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table> <p>In addition, a reversible heat engine operating between temperatures <i>T</i><sub>1</sub> and <i>T</i><sub>3</sub> must have the same efficiency as one consisting of two cycles, one between <i>T</i><sub>1</sub> and another (intermediate) temperature <i>T</i><sub>2</sub>, and the second between <i>T</i><sub>2</sub> and <i>T</i><sub>3</sub>, where <i>T</i><sub>1</sub> > <i>T</i><sub>2</sub> > <i>T</i><sub>3</sub>. This is because, if a part of the two cycle engine is hidden such that it is recognized as an engine between the reservoirs at the temperatures <i>T</i><sub>1</sub> and <i>T</i><sub>3</sub>, then the efficiency of this engine must be same to the other engine at the same reservoirs. If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as the below. </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 _{1}=1-{\frac {|q_{3}|}{|q_{1}|}}=1-f(T_{1},T_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <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>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{1}=1-{\frac {|q_{3}|}{|q_{1}|}}=1-f(T_{1},T_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef691688f84f677c564d434b44b223364508f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.579ex; height:6.509ex;" alt="{\displaystyle \eta _{1}=1-{\frac {|q_{3}|}{|q_{1}|}}=1-f(T_{1},T_{3})}"></span>,</dd> <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 _{2}=1-{\frac {|q_{2}|}{|q_{1}|}}=1-f(T_{1},T_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <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 \eta _{2}=1-{\frac {|q_{2}|}{|q_{1}|}}=1-f(T_{1},T_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9055592a7fe9375eafd0f8d49bbfb93a9627b0e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.579ex; height:6.509ex;" alt="{\displaystyle \eta _{2}=1-{\frac {|q_{2}|}{|q_{1}|}}=1-f(T_{1},T_{2})}"></span>,</dd> <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 _{3}=1-{\frac {|q_{3}|}{|q_{2}|}}=1-f(T_{2},T_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{3}=1-{\frac {|q_{3}|}{|q_{2}|}}=1-f(T_{2},T_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08fe873bb80a7441a2ddeffd1594884816e41545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.579ex; height:6.509ex;" alt="{\displaystyle \eta _{3}=1-{\frac {|q_{3}|}{|q_{2}|}}=1-f(T_{2},T_{3})}"></span>.</dd></dl> <p>Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at <i>T</i><sub>2</sub>. We also have used the fact that the heat <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> passes through the intermediate thermal reservoir at <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> without losing its energy. (I.e., <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> is not lost during its passage through the reservoir at <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>.) This fact can be proved by the following. </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 {\begin{aligned}&{{\eta }_{2}}=1-{\frac {|{{q}_{2}}|}{|{{q}_{1}}|}}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\&{{\eta }_{3}}=1-{\frac {|{{q}_{3}}|}{|{{q}_{2}}^{*}|}}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\&|{{w}_{2}}|+|{{w}_{3}}|=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\&{{\eta }_{1}}=1-{\frac {|{{q}_{3}}|}{|{{q}_{1}}|}}={\frac {(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}}={\frac {(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}}.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{{\eta }_{2}}=1-{\frac {|{{q}_{2}}|}{|{{q}_{1}}|}}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\&{{\eta }_{3}}=1-{\frac {|{{q}_{3}}|}{|{{q}_{2}}^{*}|}}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\&|{{w}_{2}}|+|{{w}_{3}}|=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\&{{\eta }_{1}}=1-{\frac {|{{q}_{3}}|}{|{{q}_{1}}|}}={\frac {(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}}={\frac {(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}}.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6694df25e96a6fb3c0901682dc159bb7a0ff10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:62.207ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}&{{\eta }_{2}}=1-{\frac {|{{q}_{2}}|}{|{{q}_{1}}|}}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\&{{\eta }_{3}}=1-{\frac {|{{q}_{3}}|}{|{{q}_{2}}^{*}|}}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\&|{{w}_{2}}|+|{{w}_{3}}|=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\&{{\eta }_{1}}=1-{\frac {|{{q}_{3}}|}{|{{q}_{1}}|}}={\frac {(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}}={\frac {(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}}.\\\end{aligned}}}"></span></dd></dl> <p>In order to have the consistency in the last equation, the heat <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> flown from the engine 2 to the intermediate reservoir must be equal to the heat <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_{2}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97230842c1738b56bf51d10956cb0aee9e1ce62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.134ex; height:2.843ex;" alt="{\displaystyle q_{2}^{*}}"></span> flown out from the reservoir to the engine 3. </p><p>Then </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(T_{1},T_{3})={\frac {|q_{3}|}{|q_{1}|}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <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>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <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> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{3})={\frac {|q_{3}|}{|q_{1}|}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3629b1ae7b290d7bc2aaf2cfb64a1eb53d701ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.606ex; height:6.509ex;" alt="{\displaystyle f(T_{1},T_{3})={\frac {|q_{3}|}{|q_{1}|}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}"></span></dd></dl> <p>Now consider the case where <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> is a fixed reference temperature: the temperature of the <a href="/wiki/Triple_point" title="Triple point">triple point</a> of water as 273.16 K; <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}=\mathrm {273.16~K} }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mn>273.16</mn> <mtext> </mtext> <mi mathvariant="normal">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}=\mathrm {273.16~K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190548cc892fecb01a8bc653bd2991ad7d4e0ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.358ex; height:2.509ex;" alt="{\displaystyle T_{1}=\mathrm {273.16~K} }"></span>. Then for any <i>T</i><sub>2</sub> and <i>T</i><sub>3</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 f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16{\text{ K}}\cdot f(T_{1},T_{3})}{273.16{\text{ K}}\cdot f(T_{1},T_{2})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <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>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>f</mi> <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> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>273.16</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> K</mtext> </mrow> <mo>⋅</mo> <mi>f</mi> <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>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>273.16</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> K</mtext> </mrow> <mo>⋅</mo> <mi>f</mi> <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> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16{\text{ K}}\cdot f(T_{1},T_{3})}{273.16{\text{ K}}\cdot f(T_{1},T_{2})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f5fd94175a7a5a9fb7215542051165299986f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.879ex; height:6.509ex;" alt="{\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16{\text{ K}}\cdot f(T_{1},T_{3})}{273.16{\text{ K}}\cdot f(T_{1},T_{2})}}.}"></span></dd></dl> <p>Therefore, if thermodynamic temperature <i>T</i>* 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 T^{*}=273.16{\text{ K}}\cdot f(T_{1},T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>273.16</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> K</mtext> </mrow> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}=273.16{\text{ K}}\cdot f(T_{1},T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2885e8d3418c6385a99c5d5b0614a2dedc04c0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.57ex; height:2.843ex;" alt="{\displaystyle T^{*}=273.16{\text{ K}}\cdot f(T_{1},T)}"></span></dd></dl> <p>then the function <i>f</i>, viewed as a function of thermodynamic temperatures, is simply </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(T_{2},T_{3})=f(T_{2}^{*},T_{3}^{*})={\frac {T_{3}^{*}}{T_{2}^{*}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{2},T_{3})=f(T_{2}^{*},T_{3}^{*})={\frac {T_{3}^{*}}{T_{2}^{*}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa02d7d0c882e760651f096eddf51dd60b50f642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.07ex; height:6.509ex;" alt="{\displaystyle f(T_{2},T_{3})=f(T_{2}^{*},T_{3}^{*})={\frac {T_{3}^{*}}{T_{2}^{*}}},}"></span></dd></dl> <p>and the reference temperature <i>T</i><sub>1</sub>* = 273.16 K × <i>f</i>(<i>T</i><sub>1</sub>,<i>T</i><sub>1</sub>) = 273.16 K. (Any reference temperature and any positive numerical value could be used – the choice here corresponds to the <a href="/wiki/Kelvin" title="Kelvin">Kelvin</a> scale.) </p> <div class="mw-heading mw-heading3"><h3 id="Entropy">Entropy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=20" title="Edit section: Entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Entropy_(classical_thermodynamics)" title="Entropy (classical thermodynamics)">Entropy (classical thermodynamics)</a></div> <p>According to the <a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius equality</a>, for a <i>reversible process</i> </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 \oint {\frac {\delta Q}{T}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint {\frac {\delta Q}{T}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c592a64d03a3decd4011e6c1454e62a40a6c9f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.565ex; height:5.843ex;" alt="{\displaystyle \oint {\frac {\delta Q}{T}}=0}"></span></dd></dl> <p>That means the line integral <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 \int _{L}{\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{L}{\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2dd542ebc69408366001d232890632960f7197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.754ex; height:5.843ex;" alt="{\displaystyle \int _{L}{\frac {\delta Q}{T}}}"></span> is path independent for reversible processes. </p><p>So we can define a state function <i>S</i> called entropy, which for a reversible process or for pure heat transfer satisfies </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 dS={\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS={\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a490ba698a8c5716f119d0932af9a11ed483b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.537ex; height:5.343ex;" alt="{\displaystyle dS={\frac {\delta Q}{T}}}"></span></dd></dl> <p>With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the <a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">third law of thermodynamics</a>, which states that <i>S</i> = 0 at <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a> for perfect crystals. </p><p>For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy. </p><p>Now reverse the reversible process and combine it with the said irreversible process. Applying the <a href="/wiki/Clausius_inequality" class="mw-redirect" title="Clausius inequality">Clausius inequality</a> on this loop, with <i>T</i><sub>surr</sub> as the temperature of the surroundings, </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 -\Delta S+\int {\frac {\delta Q}{T_{\text{surr}}}}=\oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>+</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>surr</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>surr</mtext> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\Delta S+\int {\frac {\delta Q}{T_{\text{surr}}}}=\oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/556bdefc4ff8c38dc8c9db40e45d7b03a9c53e38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.159ex; height:5.843ex;" alt="{\displaystyle -\Delta S+\int {\frac {\delta Q}{T_{\text{surr}}}}=\oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0}"></span></dd></dl> <p>Thus, </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 \Delta S\geq \int {\frac {\delta Q}{T_{\text{surr}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>≥</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>surr</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S\geq \int {\frac {\delta Q}{T_{\text{surr}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80a8a4cac34747ec74534ab04869a96374fcd16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.391ex; height:5.843ex;" alt="{\displaystyle \Delta S\geq \int {\frac {\delta Q}{T_{\text{surr}}}}}"></span></dd></dl> <p>where the equality holds if the transformation is reversible. If the process is an <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic process</a>, then <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 \delta Q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90687641a4ec52e9db4cf73695d57f8b49899f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.148ex; height:2.676ex;" alt="{\displaystyle \delta Q=0}"></span>, so <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 \Delta S\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa331aee4875f49fa0f9bc35b807212533254270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.696ex; height:2.343ex;" alt="{\displaystyle \Delta S\geq 0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Energy,_available_useful_work"><span id="Energy.2C_available_useful_work"></span>Energy, available useful work</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=21" title="Edit section: Energy, available useful work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Exergy" title="Exergy">Exergy</a></div> <p>An important and revealing idealized special case is to consider applying the second law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an <i>unlimited</i> heat reservoir at temperature <i>T</i><sub>R</sub> and pressure <i>P</i><sub>R</sub>  – so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain <i>T</i><sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain <i>P</i><sub>R</sub>. </p><p>Whatever changes to <i>dS</i> and <i>dS</i><sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, the entropy <i>S</i><sub>tot</sub> of the isolated total system must not decrease according to the second law of thermodynamics: </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 dS_{\mathrm {tot} }=dS+dS_{\text{R}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>d</mi> <mi>S</mi> <mo>+</mo> <mi>d</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS_{\mathrm {tot} }=dS+dS_{\text{R}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ee0cb66a20406514aa651942169ce5da79de16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.972ex; height:2.509ex;" alt="{\displaystyle dS_{\mathrm {tot} }=dS+dS_{\text{R}}\geq 0}"></span></dd></dl> <p>According to the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>, the change <i>dU</i> in the internal energy of the sub-system is the sum of the heat <i>δq</i> added to the sub-system, <i>minus</i> any work <i>δw</i> done <i>by</i> the sub-system, <i>plus</i> any net chemical energy entering the sub-system <i>d</i> Σ<i>μ<sub>iR</sub>N<sub>i</sub></i>, so that: </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 dU=\delta q-\delta w+d\left(\sum \mu _{iR}N_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <mi>δ</mi> <mi>q</mi> <mo>−</mo> <mi>δ</mi> <mi>w</mi> <mo>+</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mo>∑</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>R</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=\delta q-\delta w+d\left(\sum \mu _{iR}N_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f590f84d5e98aa9aa7845e92f6d554846acf5192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.844ex; height:4.843ex;" alt="{\displaystyle dU=\delta q-\delta w+d\left(\sum \mu _{iR}N_{i}\right)}"></span></dd></dl> <p>where <i>μ</i><sub><i>iR</i></sub> are the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potentials</a> of chemical species in the external surroundings. </p><p>Now the heat leaving the reservoir and entering the sub-system 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 \delta q=T_{\text{R}}(-dS_{\text{R}})\leq T_{\text{R}}dS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>q</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mi>d</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta q=T_{\text{R}}(-dS_{\text{R}})\leq T_{\text{R}}dS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e1e8bbcbeda8267224a93444f2313b2c280074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.329ex; height:2.843ex;" alt="{\displaystyle \delta q=T_{\text{R}}(-dS_{\text{R}})\leq T_{\text{R}}dS}"></span></dd></dl> <p>where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the second law inequality from above. </p><p>It therefore follows that any net work <i>δw</i> done by the sub-system must obey </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 \delta w\leq -dU+T_{\text{R}}dS+\sum \mu _{iR}dN_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>w</mi> <mo>≤</mo> <mo>−</mo> <mi>d</mi> <mi>U</mi> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mi>d</mi> <mi>S</mi> <mo>+</mo> <mo>∑</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>R</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta w\leq -dU+T_{\text{R}}dS+\sum \mu _{iR}dN_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8f7943e892e4c9ea8e98ca9921ff904c995db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:32.886ex; height:3.843ex;" alt="{\displaystyle \delta w\leq -dU+T_{\text{R}}dS+\sum \mu _{iR}dN_{i}}"></span></dd></dl> <p>It is useful to separate the work <i>δw</i> done by the subsystem into the <i>useful</i> work <i>δw<sub>u</sub></i> that can be done <i>by</i> the sub-system, over and beyond the work <i>p<sub>R</sub> dV</i> done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done: </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 \delta w_{u}\leq -d\left(U-T_{\text{R}}S+p_{\text{R}}V-\sum \mu _{iR}N_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>≤</mo> <mo>−</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mi>S</mi> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mi>V</mi> <mo>−</mo> <mo>∑</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>R</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta w_{u}\leq -d\left(U-T_{\text{R}}S+p_{\text{R}}V-\sum \mu _{iR}N_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c93c927d9430f9e7172367c34c6e7b6d08b5c58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.028ex; height:4.843ex;" alt="{\displaystyle \delta w_{u}\leq -d\left(U-T_{\text{R}}S+p_{\text{R}}V-\sum \mu _{iR}N_{i}\right)}"></span></dd></dl> <p>It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the <i>availability</i> or <i><a href="/wiki/Exergy" title="Exergy">exergy</a></i> <i>E</i> of the subsystem, </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=U-T_{\text{R}}S+p_{\text{R}}V-\sum \mu _{iR}N_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>U</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mi>S</mi> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mi>V</mi> <mo>−</mo> <mo>∑</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>R</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=U-T_{\text{R}}S+p_{\text{R}}V-\sum \mu _{iR}N_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb948b0c889a01e8aa04e170ffce992a1641b2ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.732ex; height:3.843ex;" alt="{\displaystyle E=U-T_{\text{R}}S+p_{\text{R}}V-\sum \mu _{iR}N_{i}}"></span></dd></dl> <p>The second law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact, </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 dE+\delta w_{u}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dE+\delta w_{u}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263c3052f2f3f8c38885c3c83a2d9188b932976a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.978ex; height:2.676ex;" alt="{\displaystyle dE+\delta w_{u}\leq 0}"></span></dd></dl> <p>i.e. the change in the subsystem's exergy plus the useful work done <i>by</i> the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done <i>on</i> the system) must be less than or equal to zero. </p><p>In sum, if a proper <i>infinite-reservoir-like</i> reference state is chosen as the system surroundings in the real world, then the second law predicts a decrease in <i>E</i> for an irreversible process and no change for a reversible process. </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 dS_{\text{tot}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>tot</mtext> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS_{\text{tot}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667b194847be239ef7a0b33078b4018496476625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.235ex; height:2.509ex;" alt="{\displaystyle dS_{\text{tot}}\geq 0}"></span> is equivalent to <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 dE+\delta w_{u}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dE+\delta w_{u}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263c3052f2f3f8c38885c3c83a2d9188b932976a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.978ex; height:2.676ex;" alt="{\displaystyle dE+\delta w_{u}\leq 0}"></span></dd></dl> <p>This expression together with the associated reference state permits a <a href="/wiki/Design_engineer" title="Design engineer">design engineer</a> working at the macroscopic scale (above the <a href="/wiki/Thermodynamic_limit" title="Thermodynamic limit">thermodynamic limit</a>) to utilize the second law without directly measuring or considering entropy change in a total isolated system (see also <i><a href="/wiki/Process_engineer" class="mw-redirect" title="Process engineer">Process engineer</a></i>). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (see <i><a href="/wiki/Exergy_efficiency" title="Exergy efficiency">Exergy efficiency</a></i>). </p><p>This approach to the second law is widely utilized in <a href="/wiki/Engineering" title="Engineering">engineering</a> practice, <a href="/wiki/Environmental_accounting" title="Environmental accounting">environmental accounting</a>, <a href="/wiki/Systems_ecology" title="Systems ecology">systems ecology</a>, and other disciplines. </p> <div class="mw-heading mw-heading2"><h2 id="Direction_of_spontaneous_processes">Direction of spontaneous processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=22" title="Edit section: Direction of spontaneous processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The second law determines whether a proposed physical or chemical process is forbidden or may occur spontaneously. For <a href="/wiki/Isolated_system" title="Isolated system">isolated systems</a>, no energy is provided by the surroundings and the second law requires that the entropy of the system alone must increase: Δ<i>S</i> > 0. Examples of spontaneous physical processes in isolated systems include the following: </p> <ul><li>1) <a href="/wiki/Heat_transfer" title="Heat transfer">Heat can be transferred</a> from a region of higher temperature to a lower temperature (but not the reverse).</li> <li>2) Mechanical energy can be converted to thermal energy (but not the reverse).</li> <li>3) A solute can move from a region of higher concentration to a region of lower concentration (but not the reverse).</li></ul> <p>However, for some non-isolated systems which can exchange energy with their surroundings, the surroundings exchange enough heat with the system, or do sufficient work on the system, so that the processes occur in the opposite direction. This is possible provided the total entropy change of the system plus the surroundings is positive as required by the second law: Δ<i>S</i><sub>tot</sub> = Δ<i>S</i> + Δ<i>S</i><sub>R</sub> > 0. For the three examples given above: </p> <ul><li>1) Heat can be transferred from a region of lower temperature to a higher temperature in a <a href="/wiki/Refrigerator" title="Refrigerator">refrigerator</a> or in a <a href="/wiki/Heat_pump" title="Heat pump">heat pump</a>. These machines must provide sufficient work to the system.</li> <li>2) Thermal energy can be converted to mechanical work in a <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a>, if sufficient heat is also expelled to the surroundings.</li> <li>3) A solute can move from a region of lower concentration to a region of higher concentration in the biochemical process of <a href="/wiki/Active_transport" title="Active transport">active transport</a>, if sufficient work is provided by a concentration gradient of a chemical such as <a href="/wiki/Adenosine_triphosphate" title="Adenosine triphosphate">ATP</a> or by an <a href="/wiki/Electrochemical_gradient" title="Electrochemical gradient">electrochemical gradient</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Second_law_in_chemical_thermodynamics">Second law in chemical thermodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=23" title="Edit section: Second law in chemical thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Spontaneous_process" title="Spontaneous process">spontaneous chemical process</a> in a closed system at constant temperature and pressure without non-<i>PV</i> work, the Clausius inequality Δ<i>S</i> > <i>Q/T</i><sub>surr</sub> transforms into a condition for the change in <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</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 \Delta G<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>G</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41ad1c1ad25e7df4a1e249f0955f799538de5208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.024ex; height:2.176ex;" alt="{\displaystyle \Delta G<0}"></span></dd></dl> <p>or d<i>G</i> < 0. For a similar process at constant temperature and volume, the change in <a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a> must be negative, <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 \Delta A<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>A</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta A<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ac4e13b480ec981508056714912f9aebcdb931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.94ex; height:2.176ex;" alt="{\displaystyle \Delta A<0}"></span>. Thus, a negative value of the change in free energy (<i>G</i> or <i>A</i>) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<sup id="cite_ref-Oxtoby8th_19-1" class="reference"><a href="#cite_note-Oxtoby8th-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-MortimerBook_15-2" class="reference"><a href="#cite_note-MortimerBook-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The chemical equilibrium condition at constant <i>T</i> and <i>p</i> without electrical work is d<i>G</i> = 0. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=24" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_entropy" title="History of entropy">History of entropy</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sadi_Carnot.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Sadi_Carnot.jpeg/170px-Sadi_Carnot.jpeg" decoding="async" width="170" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Sadi_Carnot.jpeg/255px-Sadi_Carnot.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Sadi_Carnot.jpeg/340px-Sadi_Carnot.jpeg 2x" data-file-width="956" data-file-height="1286" /></a><figcaption>Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the <a href="/wiki/%C3%89cole_Polytechnique" class="mw-redirect" title="École Polytechnique">École Polytechnique</a></figcaption></figure> <p>The first theory of the conversion of heat into mechanical work is due to <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Nicolas Léonard Sadi Carnot</a> in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its surroundings. </p><p>Recognizing the significance of <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Prescott Joule</a>'s work on the conservation of energy, <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> was the first to formulate the second law during 1850, in this form: heat does not flow <i>spontaneously</i> from cold to hot bodies. While common knowledge now, this was contrary to the <a href="/wiki/Caloric_theory" title="Caloric theory">caloric theory</a> of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865). </p><p>Established during the 19th century, the <a href="/wiki/Kelvin-Planck_statement" class="mw-redirect" title="Kelvin-Planck statement">Kelvin-Planck statement of the second law</a> says, "It is impossible for any device that operates on a <a href="/wiki/Cyclic_process" class="mw-redirect" title="Cyclic process">cycle</a> to receive heat from a single <a href="/wiki/Heat_reservoir" class="mw-redirect" title="Heat reservoir">reservoir</a> and produce a net amount of work." This statement was shown to be equivalent to the statement of Clausius. </p><p>The <a href="/wiki/Ergodic_hypothesis" title="Ergodic hypothesis">ergodic hypothesis</a> is also important for the <a href="/wiki/Boltzmann" class="mw-redirect" title="Boltzmann">Boltzmann</a> approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same. </p><p>There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a <a href="/wiki/Macroscopic_bodies" class="mw-redirect" title="Macroscopic bodies">macroscopic system</a>. This doctrine is obsolescent.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lambert_72-0" class="reference"><a href="#cite_note-Lambert-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Account_given_by_Clausius">Account given by Clausius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=25" title="Edit section: Account given by Clausius"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Clausius-1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Clausius-1.jpg/170px-Clausius-1.jpg" decoding="async" width="170" height="265" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Clausius-1.jpg/255px-Clausius-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Clausius-1.jpg/340px-Clausius-1.jpg 2x" data-file-width="578" data-file-height="900" /></a><figcaption>Rudolf Clausius</figcaption></figure> <p>In 1865, the German physicist <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> stated what he called the "second fundamental theorem in the <a href="/wiki/Mechanical_theory_of_heat" class="mw-redirect" title="Mechanical theory of heat">mechanical theory of heat</a>" in the following form:<sup id="cite_ref-FOOTNOTEClausius1867_73-0" class="reference"><a href="#cite_note-FOOTNOTEClausius1867-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> </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 \int {\frac {\delta Q}{T}}=-N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {\delta Q}{T}}=-N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788c11865876c1029258d9f6c21a942a0909650f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.274ex; height:5.843ex;" alt="{\displaystyle \int {\frac {\delta Q}{T}}=-N}"></span></dd></dl> <p>where <i>Q</i> is heat, <i>T</i> is temperature and <i>N</i> is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes: </p> <blockquote><p>The entropy of the universe tends to a maximum.</p></blockquote> <p>This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. <a href="/wiki/Universe" title="Universe">universe</a>, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description. </p><p>In terms of time variation, the mathematical statement of the second law for an <a href="/wiki/Isolated_system" title="Isolated system">isolated system</a> undergoing an arbitrary transformation 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 {\frac {dS}{dt}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dS}{dt}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c43dbbd79e66bed761e987211afc239aa28a4c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.812ex; height:5.509ex;" alt="{\displaystyle {\frac {dS}{dt}}\geq 0}"></span></dd></dl> <p>where </p> <dl><dd><i>S</i> is the entropy of the system and</dd> <dd><i>t</i> is <a href="/wiki/Time" title="Time">time</a>.</dd></dl> <p>The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems 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 {\frac {dS}{dt}}={\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </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 {\frac {dS}{dt}}={\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb29f3ca9895dedba07fa26fa94cde318dc85a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.912ex; height:5.509ex;" alt="{\displaystyle {\frac {dS}{dt}}={\dot {S}}_{\text{i}}}"></span> 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 {\dot {S}}_{\text{i}}\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>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}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c178cbfb1ca94089d12f36d625731fd1a00e227d" 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}}\geq 0}"></span></dd></dl> <p>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 {\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> the sum of the rate of <a href="/wiki/Entropy_production" title="Entropy production">entropy production</a> by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <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}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2e4686f46533e2d4a091bfde3783858ddb1d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{\text{a}}}"></span> it gives the so-called dissipated energy <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>. </p><p>The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) 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 {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mi>d</mi> <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> <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 {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7fd070768bac72ee61142bb627daac7fa647f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.427ex; height:6.009ex;" alt="{\displaystyle {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}_{\text{i}}}"></span> 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 {\dot {S}}_{\text{i}}\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>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}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c178cbfb1ca94089d12f36d625731fd1a00e227d" 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}}\geq 0}"></span></dd></dl> <p>Here, </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}}}"> <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> is the heat flow into the system</dd> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is the temperature at the point where the heat enters the system.</dd></dl> <p>The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms. </p><p>For open systems (also allowing exchange of matter): </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 {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}+{\dot {S}}_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mi>d</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </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 {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}+{\dot {S}}_{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b9febd48a49541bd3b1ace200e6d9a5a3bb183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.84ex; height:6.009ex;" alt="{\displaystyle {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}+{\dot {S}}_{\text{i}}}"></span> 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 {\dot {S}}_{\text{i}}\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>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}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c178cbfb1ca94089d12f36d625731fd1a00e227d" 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}}\geq 0}"></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 {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/151eef166d15fbc6d73b76167f158c45b7670427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.573ex; height:2.843ex;" alt="{\displaystyle {\dot {S}}}"></span> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions. </p> <div class="mw-heading mw-heading2"><h2 id="Statistical_mechanics">Statistical mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=26" title="Edit section: Statistical mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 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-Unreferenced plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Second_law_of_thermodynamics" title="Special:EditPage/Second law of thermodynamics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Second+law+of+thermodynamics%22">"Second law of thermodynamics"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Second+law+of+thermodynamics%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Second+law+of+thermodynamics%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Second+law+of+thermodynamics%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Second+law+of+thermodynamics%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Second+law+of+thermodynamics%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2025</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a> gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a <a href="/wiki/Microstate_(statistical_mechanics)" title="Microstate (statistical mechanics)">microstate</a> of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>N</i></span></span> where <i>N</i> is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation_from_statistical_mechanics">Derivation from statistical mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=27" title="Edit section: Derivation from statistical mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/H-theorem" title="H-theorem">H-theorem</a></div> <p>The first mechanical argument of the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">Kinetic theory of gases</a> that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> in 1860;<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> with his <a href="/wiki/H-theorem" title="H-theorem">H-theorem</a> of 1872 also argued that due to collisions gases should over time tend toward the <a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann distribution</a>. </p><p>Due to <a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a>, derivations of the second law have to make an assumption regarding the past, namely that the system is <a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">uncorrelated</a> at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary condition</a>, and thus the second law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a>), though <a href="/wiki/Boltzmann_brain" title="Boltzmann brain">other scenarios</a> have also been suggested.<sup id="cite_ref-Hawking_AOT_75-0" class="reference"><a href="#cite_note-Hawking_AOT-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lebowitz_77-0" class="reference"><a href="#cite_note-Lebowitz-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> </p><p>Given these assumptions, in statistical mechanics, the second law is not a postulate, rather it is a consequence of the <a href="/wiki/Statistical_mechanics#Fundamental_postulate" title="Statistical mechanics">fundamental postulate</a>, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2012)">citation needed</span></a></i>]</sup> The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> 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=k_{\mathrm {B} }\ln \left[\Omega \left(E\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k_{\mathrm {B} }\ln \left[\Omega \left(E\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98aaf7e3918c01dcd3623db44cf34dd698594eda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.475ex; height:2.843ex;" alt="{\displaystyle S=k_{\mathrm {B} }\ln \left[\Omega \left(E\right)\right]}"></span></dd></dl> <p>where <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 \Omega \left(E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \left(E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab95b1220d150818ca655e3491155e0468ce2923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.65ex; height:2.843ex;" alt="{\displaystyle \Omega \left(E\right)}"></span> is the number of quantum states in a small interval between <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></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 E+\delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E+\delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083aab9a05dd601cd79d43fdb9006c40e9baa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.44ex; height:2.509ex;" alt="{\displaystyle E+\delta E}"></span>. 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 \delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bd788b183deb3082d2f76b8e4101d950292bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.824ex; height:2.343ex;" alt="{\displaystyle \delta E}"></span> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice 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 \delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bd788b183deb3082d2f76b8e4101d950292bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.824ex; height:2.343ex;" alt="{\displaystyle \delta E}"></span>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <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 \delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bd788b183deb3082d2f76b8e4101d950292bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.824ex; height:2.343ex;" alt="{\displaystyle \delta E}"></span>. </p><p>Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is maximized at the given energy of the isolated system<sup id="cite_ref-Young&FreedmanIS_78-0" class="reference"><a href="#cite_note-Young&FreedmanIS-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> as that is the most probable situation in equilibrium. </p><p>If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value). Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability 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 1/\Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c69a7440bc1d3fa40206529c31c03122d2ad15e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.003ex; height:2.843ex;" alt="{\displaystyle 1/\Omega }"></span>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's <a href="/wiki/H-theorem" title="H-theorem">H-theorem</a>, however, proves that the quantity <span class="texhtml"><i>H</i></span> increases monotonically as a function of time during the intermediate out of equilibrium state. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_the_entropy_change_for_reversible_processes">Derivation of the entropy change for reversible processes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=28" title="Edit section: Derivation of the entropy change for reversible processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The second part of the second law states that the entropy change of a system undergoing a reversible process 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 dS={\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS={\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a490ba698a8c5716f119d0932af9a11ed483b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.537ex; height:5.343ex;" alt="{\displaystyle dS={\frac {\delta Q}{T}}}"></span></dd></dl> <p>where the temperature is 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 {\frac {1}{k_{\mathrm {B} }T}}\equiv \beta \equiv {\frac {d\ln \left[\Omega \left(E\right)\right]}{dE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>≡</mo> <mi>β</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>E</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{k_{\mathrm {B} }T}}\equiv \beta \equiv {\frac {d\ln \left[\Omega \left(E\right)\right]}{dE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e720432da60639a2f9cdcefb4ac56845da4f36b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.931ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{k_{\mathrm {B} }T}}\equiv \beta \equiv {\frac {d\ln \left[\Omega \left(E\right)\right]}{dE}}}"></span></dd></dl> <p>See <i><a href="/wiki/Microcanonical_ensemble" title="Microcanonical ensemble">Microcanonical ensemble</a></i> for the justification for this definition. Suppose that the system has some external parameter, <i>x</i>, that can be changed. In general, the energy eigenstates of the system will depend on <i>x</i>. According to the <a href="/wiki/Adiabatic_theorem" title="Adiabatic theorem">adiabatic theorem</a> of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in. </p><p>The generalized force, <i>X</i>, corresponding to the external variable <i>x</i> is defined such 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 Xdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Xdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6331a545837b58e5cc6261e85c7318af63f42a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.526ex; height:2.176ex;" alt="{\displaystyle Xdx}"></span> is the work performed by the system if <i>x</i> is increased by an amount <i>dx</i>. For example, if <i>x</i> is the volume, then <i>X</i> is the pressure. The generalized force for a system known to be in energy eigenstate <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_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293328c22d5276d48118b81fdd39033c93c100e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.689ex; height:2.509ex;" alt="{\displaystyle E_{r}}"></span> 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 X=-{\frac {dE_{r}}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=-{\frac {dE_{r}}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef59b97cf23d098fd2c99ac242f58d488c090efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.628ex; height:5.509ex;" alt="{\displaystyle X=-{\frac {dE_{r}}{dx}}}"></span></dd></dl> <p>Since the system can be in any energy eigenstate within an interval 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 \delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bd788b183deb3082d2f76b8e4101d950292bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.824ex; height:2.343ex;" alt="{\displaystyle \delta E}"></span>, we define the generalized force for the system as the expectation value of the above 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 X=-\left\langle {\frac {dE_{r}}{dx}}\right\rangle \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=-\left\langle {\frac {dE_{r}}{dx}}\right\rangle \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d3cc55dd6b04f150999033a1d7e431f79b8e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.888ex; height:6.176ex;" alt="{\displaystyle X=-\left\langle {\frac {dE_{r}}{dx}}\right\rangle \,}"></span></dd></dl> <p>To evaluate the average, we partition the <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 \Omega \left(E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \left(E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab95b1220d150818ca655e3491155e0468ce2923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.65ex; height:2.843ex;" alt="{\displaystyle \Omega \left(E\right)}"></span> energy eigenstates by counting how many of them have a value for <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 {dE_{r}}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dE_{r}}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b091eafec4a232dda6869608928ab35c3274c67a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.741ex; height:5.509ex;" alt="{\displaystyle {\frac {dE_{r}}{dx}}}"></span> within a range between <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></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 Y+\delta Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>+</mo> <mi>δ</mi> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y+\delta Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533bb09ebf7bf9f9046510b9dba2f3cfca16a190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.436ex; height:2.509ex;" alt="{\displaystyle Y+\delta Y}"></span>. Calling this number <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 \Omega _{Y}\left(E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{Y}\left(E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63384aa0c99e2f757304c001f61b31435af36deb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.136ex; height:2.843ex;" alt="{\displaystyle \Omega _{Y}\left(E\right)}"></span>, we have: </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 \Omega \left(E\right)=\sum _{Y}\Omega _{Y}\left(E\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </munder> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \left(E\right)=\sum _{Y}\Omega _{Y}\left(E\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b559ecb07a74f58f2c8d0728f9d87f4364333ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.014ex; height:5.509ex;" alt="{\displaystyle \Omega \left(E\right)=\sum _{Y}\Omega _{Y}\left(E\right)\,}"></span></dd></dl> <p>The average defining the generalized force can now be written: </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 X=-{\frac {1}{\Omega \left(E\right)}}\sum _{Y}Y\Omega _{Y}\left(E\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </munder> <mi>Y</mi> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=-{\frac {1}{\Omega \left(E\right)}}\sum _{Y}Y\Omega _{Y}\left(E\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6840998a6a7c46f755361034d5d3db29760be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.799ex; height:6.343ex;" alt="{\displaystyle X=-{\frac {1}{\Omega \left(E\right)}}\sum _{Y}Y\Omega _{Y}\left(E\right)\,}"></span></dd></dl> <p>We can relate this to the derivative of the entropy with respect to <i>x</i> at constant energy <i>E</i> as follows. Suppose we change <i>x</i> to <i>x</i> + <i>dx</i>. Then <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 \Omega \left(E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \left(E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab95b1220d150818ca655e3491155e0468ce2923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.65ex; height:2.843ex;" alt="{\displaystyle \Omega \left(E\right)}"></span> will change because the energy eigenstates depend on <i>x</i>, causing energy eigenstates to move into or out of the range between <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></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 E+\delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E+\delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083aab9a05dd601cd79d43fdb9006c40e9baa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.44ex; height:2.509ex;" alt="{\displaystyle E+\delta E}"></span>. Let's focus again on the energy eigenstates for which <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="{\textstyle {\frac {dE_{r}}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {dE_{r}}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91adf0d73e7f6ee3c340b4dad742eae9d199cd9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.675ex; height:4.009ex;" alt="{\textstyle {\frac {dE_{r}}{dx}}}"></span> lies within the range between <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></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 Y+\delta Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>+</mo> <mi>δ</mi> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y+\delta Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533bb09ebf7bf9f9046510b9dba2f3cfca16a190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.436ex; height:2.509ex;" alt="{\displaystyle Y+\delta Y}"></span>. Since these energy eigenstates increase in energy by <i>Y dx</i>, all such energy eigenstates that are in the interval ranging from <i>E</i> – <i>Y</i> <i>dx</i> to <i>E</i> move from below <i>E</i> to above <i>E</i>. There are </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_{Y}\left(E\right)={\frac {\Omega _{Y}\left(E\right)}{\delta E}}Ydx\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>δ</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mi>Y</mi> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{Y}\left(E\right)={\frac {\Omega _{Y}\left(E\right)}{\delta E}}Ydx\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd141b148408e0a9cb381b84e174722bf6d7e16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.101ex; height:5.843ex;" alt="{\displaystyle N_{Y}\left(E\right)={\frac {\Omega _{Y}\left(E\right)}{\delta E}}Ydx\,}"></span></dd></dl> <p>such energy eigenstates. 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 Ydx\leq \delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mi>d</mi> <mi>x</mi> <mo>≤</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ydx\leq \delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28059c70f58a987bcbac3eb0ec308db900552d14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.242ex; height:2.509ex;" alt="{\displaystyle Ydx\leq \delta E}"></span>, all these energy eigenstates will move into the range between <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></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 E+\delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E+\delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083aab9a05dd601cd79d43fdb9006c40e9baa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.44ex; height:2.509ex;" alt="{\displaystyle E+\delta E}"></span> and contribute to an increase 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. The number of energy eigenstates that move from below <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+\delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E+\delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083aab9a05dd601cd79d43fdb9006c40e9baa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.44ex; height:2.509ex;" alt="{\displaystyle E+\delta E}"></span> to above <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+\delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E+\delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083aab9a05dd601cd79d43fdb9006c40e9baa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.44ex; height:2.509ex;" alt="{\displaystyle E+\delta E}"></span> is given 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 N_{Y}\left(E+\delta E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{Y}\left(E+\delta E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1bf15d06ee12d0a0dc237cb445a674c7dae203f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.989ex; height:2.843ex;" alt="{\displaystyle N_{Y}\left(E+\delta E\right)}"></span>. The difference </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_{Y}\left(E\right)-N_{Y}\left(E+\delta E\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{Y}\left(E\right)-N_{Y}\left(E+\delta E\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece2196401057c889f12660052ba16ae1f544a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.541ex; height:2.843ex;" alt="{\displaystyle N_{Y}\left(E\right)-N_{Y}\left(E+\delta E\right)\,}"></span></dd></dl> <p>is thus the net contribution to the increase 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. If <i>Y dx</i> is larger than <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 \delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bd788b183deb3082d2f76b8e4101d950292bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.824ex; height:2.343ex;" alt="{\displaystyle \delta E}"></span> there will be the energy eigenstates that move from below <i>E</i> to above <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+\delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E+\delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4083aab9a05dd601cd79d43fdb9006c40e9baa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.44ex; height:2.509ex;" alt="{\displaystyle E+\delta E}"></span>. They are counted in both <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_{Y}\left(E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{Y}\left(E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da68e32b7c99e9664a70179f5daf90d40a8849b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.324ex; height:2.843ex;" alt="{\displaystyle N_{Y}\left(E\right)}"></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 N_{Y}\left(E+\delta E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>E</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{Y}\left(E+\delta E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1bf15d06ee12d0a0dc237cb445a674c7dae203f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.989ex; height:2.843ex;" alt="{\displaystyle N_{Y}\left(E+\delta E\right)}"></span>, therefore the above expression is also valid in that case. </p><p>Expressing the above expression as a derivative with respect to <i>E</i> and summing over <i>Y</i> yields the 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 \left({\frac {\partial \Omega }{\partial x}}\right)_{E}=-\sum _{Y}Y\left({\frac {\partial \Omega _{Y}}{\partial E}}\right)_{x}=\left({\frac {\partial \left(\Omega X\right)}{\partial E}}\right)_{x}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </munder> <mi>Y</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Ω</mi> <mi>X</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \Omega }{\partial x}}\right)_{E}=-\sum _{Y}Y\left({\frac {\partial \Omega _{Y}}{\partial E}}\right)_{x}=\left({\frac {\partial \left(\Omega X\right)}{\partial E}}\right)_{x}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8c9ac74ca81b8d1ca2e4ee52425f092d6a46e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.55ex; height:6.843ex;" alt="{\displaystyle \left({\frac {\partial \Omega }{\partial x}}\right)_{E}=-\sum _{Y}Y\left({\frac {\partial \Omega _{Y}}{\partial E}}\right)_{x}=\left({\frac {\partial \left(\Omega X\right)}{\partial E}}\right)_{x}\,}"></span></dd></dl> <p>The logarithmic derivative 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> with respect to <i>x</i> is thus 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 \left({\frac {\partial \ln \left(\Omega \right)}{\partial x}}\right)_{E}=\beta X+\left({\frac {\partial X}{\partial E}}\right)_{x}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi mathvariant="normal">Ω</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mi>β</mi> <mi>X</mi> <mo>+</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>X</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \ln \left(\Omega \right)}{\partial x}}\right)_{E}=\beta X+\left({\frac {\partial X}{\partial E}}\right)_{x}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d74fe017cd8b342f52aae4516f33fce4a40b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.243ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial \ln \left(\Omega \right)}{\partial x}}\right)_{E}=\beta X+\left({\frac {\partial X}{\partial E}}\right)_{x}\,}"></span></dd></dl> <p>The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanish in the thermodynamic limit. We have thus found that: </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 \left({\frac {\partial S}{\partial x}}\right)_{E}={\frac {X}{T}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S}{\partial x}}\right)_{E}={\frac {X}{T}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5d8e8236574a8deec8ff1224a67980ea05c6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.864ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial S}{\partial x}}\right)_{E}={\frac {X}{T}}\,}"></span></dd></dl> <p>Combining this with </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 \left({\frac {\partial S}{\partial E}}\right)_{x}={\frac {1}{T}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S}{\partial E}}\right)_{x}={\frac {1}{T}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aac0be1da45a3d87c39638189d95dba75722a58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.481ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial S}{\partial E}}\right)_{x}={\frac {1}{T}}\,}"></span></dd></dl> <p>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 dS=\left({\frac {\partial S}{\partial E}}\right)_{x}dE+\left({\frac {\partial S}{\partial x}}\right)_{E}dx={\frac {dE}{T}}+{\frac {X}{T}}dx={\frac {\delta Q}{T}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>d</mi> <mi>E</mi> <mo>+</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mi>T</mi> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS=\left({\frac {\partial S}{\partial E}}\right)_{x}dE+\left({\frac {\partial S}{\partial x}}\right)_{E}dx={\frac {dE}{T}}+{\frac {X}{T}}dx={\frac {\delta Q}{T}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54ecda4ab4262df6df846cbd733552938471358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.614ex; height:6.176ex;" alt="{\displaystyle dS=\left({\frac {\partial S}{\partial E}}\right)_{x}dE+\left({\frac {\partial S}{\partial x}}\right)_{E}dx={\frac {dE}{T}}+{\frac {X}{T}}dx={\frac {\delta Q}{T}}\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Derivation_for_systems_described_by_the_canonical_ensemble">Derivation for systems described by the canonical ensemble</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=29" title="Edit section: Derivation for systems described by the canonical ensemble"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a system is in thermal contact with a heat bath at some temperature <i>T</i> then, in equilibrium, the probability distribution over the energy eigenvalues are given by the <a href="/wiki/Canonical_ensemble" title="Canonical ensemble">canonical ensemble</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 P_{j}={\frac {\exp \left(-{\frac {E_{j}}{k_{\mathrm {B} }T}}\right)}{Z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mi>Z</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{j}={\frac {\exp \left(-{\frac {E_{j}}{k_{\mathrm {B} }T}}\right)}{Z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d1e1dde851f284a10c78a41c019b292ebfeac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.431ex; height:7.509ex;" alt="{\displaystyle P_{j}={\frac {\exp \left(-{\frac {E_{j}}{k_{\mathrm {B} }T}}\right)}{Z}}}"></span></dd></dl> <p>Here <i>Z</i> is a factor that normalizes the sum of all the probabilities to 1, this function is known as the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a>. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy: </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_{\mathrm {B} }\sum _{j}P_{j}\ln \left(P_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=-k_{\mathrm {B} }\sum _{j}P_{j}\ln \left(P_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c68df1156cc8bf520b6c8b858a89237815ac5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.082ex; height:5.843ex;" alt="{\displaystyle S=-k_{\mathrm {B} }\sum _{j}P_{j}\ln \left(P_{j}\right)}"></span></dd></dl> <p>that </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 dS=-k_{\mathrm {B} }\sum _{j}\ln \left(P_{j}\right)dP_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>d</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS=-k_{\mathrm {B} }\sum _{j}\ln \left(P_{j}\right)dP_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24fc3371f5d4208e2d5e811374a8699a8acf0121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.126ex; height:5.843ex;" alt="{\displaystyle dS=-k_{\mathrm {B} }\sum _{j}\ln \left(P_{j}\right)dP_{j}}"></span></dd></dl> <p>Inserting the formula for <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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5da6c3564a2129f714ef11acd8ba649d18e604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.402ex; height:2.843ex;" alt="{\displaystyle P_{j}}"></span> for the canonical ensemble in here 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 dS={\frac {1}{T}}\sum _{j}E_{j}dP_{j}={\frac {1}{T}}\sum _{j}d\left(E_{j}P_{j}\right)-{\frac {1}{T}}\sum _{j}P_{j}dE_{j}={\frac {dE+\delta W}{T}}={\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> <mo>+</mo> <mi>δ</mi> <mi>W</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS={\frac {1}{T}}\sum _{j}E_{j}dP_{j}={\frac {1}{T}}\sum _{j}d\left(E_{j}P_{j}\right)-{\frac {1}{T}}\sum _{j}P_{j}dE_{j}={\frac {dE+\delta W}{T}}={\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3bfab183d9d6a5bb4728a8f407d270a6452d28d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.554ex; height:6.843ex;" alt="{\displaystyle dS={\frac {1}{T}}\sum _{j}E_{j}dP_{j}={\frac {1}{T}}\sum _{j}d\left(E_{j}P_{j}\right)-{\frac {1}{T}}\sum _{j}P_{j}dE_{j}={\frac {dE+\delta W}{T}}={\frac {\delta Q}{T}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Initial_conditions_at_the_Big_Bang">Initial conditions at the Big Bang</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=30" title="Edit section: Initial conditions at the Big Bang"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Past_hypothesis" title="Past hypothesis">Past hypothesis</a></div> <p>As elaborated above, it is thought that the second law of thermodynamics is a result of the very low-entropy initial conditions at the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a>. From a statistical point of view, these were very special conditions. On the other hand, they were quite simple, as the universe - or at least the part thereof from which the <a href="/wiki/Observable_universe" title="Observable universe">observable universe</a> developed - seems to have been extremely uniform.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p><p>This may seem somewhat paradoxical, since in many physical systems uniform conditions (e.g. mixed rather than separated gases) have high entropy. The paradox is solved once realizing that gravitational systems have <a href="/wiki/Heat_capacity#Negative_heat_capacity" title="Heat capacity">negative heat capacity</a>, so that when gravity is important, uniform conditions (e.g. gas of uniform density) in fact have lower entropy compared to non-uniform ones (e.g. black holes in empty space).<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> Yet another approach is that the universe had high (or even maximal) entropy given its size, but as the universe grew it rapidly came out of thermodynamic equilibrium, its entropy only slightly increased compared to the increase in maximal possible entropy, and thus it has arrived at a very low entropy when compared to the much larger possible maximum given its later size.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> </p><p>As for the reason why initial conditions were such, one suggestion is that <a href="/wiki/Cosmological_inflation" class="mw-redirect" title="Cosmological inflation">cosmological inflation</a> was enough to wipe off non-smoothness, while another is that the universe was <a href="/wiki/Hartle%E2%80%93Hawking_state" title="Hartle–Hawking state">created spontaneously</a> where the mechanism of creation implies low-entropy initial conditions.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Living_organisms">Living organisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=31" title="Edit section: Living organisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. The thermodynamics of living organisms has been considered by many authors, including <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> (in his book <i><a href="/wiki/What_is_Life%3F" class="mw-redirect" title="What is Life?">What is Life?</a></i>) and <a href="/wiki/L%C3%A9on_Brillouin" title="Léon Brillouin">Léon Brillouin</a>.<sup id="cite_ref-Brillouin_2013_p._83-0" class="reference"><a href="#cite_note-Brillouin_2013_p.-83"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> </p><p>To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of <a href="/wiki/Metabolism" title="Metabolism">metabolism</a>, give out breakdown products and heat. Plants <a href="/wiki/Photosynthesis" title="Photosynthesis">take in radiative energy</a> from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. </p><p>Furthermore, the ability of living organisms to grow and increase in complexity, as well as to form correlations with their environment in the form of adaption and memory, is not opposed to the second law – rather, it is akin to general results following from it: Under some definitions, an increase in entropy also results in an increase in complexity,<sup id="cite_ref-Ladyman_Lambert_Wiesner_pp._33–67_84-0" class="reference"><a href="#cite_note-Ladyman_Lambert_Wiesner_pp._33–67-84"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> and for a finite system interacting with finite reservoirs, an increase in entropy is equivalent to an increase in correlations between the system and the reservoirs.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> </p><p>Living organisms may be considered as open systems, because matter passes into and out from them. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are a subject of <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">ongoing research</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Gravitational_systems">Gravitational systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=32" title="Edit section: Gravitational systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Commonly, systems for which gravity is not important have a positive <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a>, meaning that their temperature rises with their internal energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature decreases while the sink temperature is increased; hence temperature differences tend to diminish over time. </p><p>This is not always the case for systems in which the gravitational force is important: systems that are bound by their own gravity, such as stars, can have negative heat capacities. As they contract, both their total energy and their entropy decrease<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> but <a href="/wiki/Kelvin-Helmholtz_mechanism" class="mw-redirect" title="Kelvin-Helmholtz mechanism">their internal temperature may increase</a>. This can be significant for <a href="/wiki/Protostars" class="mw-redirect" title="Protostars">protostars</a> and even gas giant planets such as <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>. When the entropy of the <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a> emitted by the bodies is included, however, the total entropy of the system can be shown to increase even as the entropy of the planet or star decreases.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-equilibrium_states">Non-equilibrium states</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=33" title="Edit section: Non-equilibrium states"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium thermodynamics</a></div> <p>The theory of classical or <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">equilibrium thermodynamics</a> is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<sup id="cite_ref-Grandy_151_88-0" class="reference"><a href="#cite_note-Grandy_151-88"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> </p><p>For purposes of physical analysis, it is often enough convenient to make an assumption of <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable <a href="/wiki/Thermal_fluctuations" title="Thermal fluctuations">fluctuations</a>. There is an exception, the case of <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical states</a>, which exhibit to the naked eye the phenomenon of <a href="/wiki/Critical_opalescence" title="Critical opalescence">critical opalescence</a>. For laboratory studies of critical states, exceptionally long observation times are needed. </p><p>In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system. </p><p>It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> </p><p>There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation,<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> but strictly speaking it is still an approximation, not theoretically ideal. </p><p>For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law. </p><p>The physics of macroscopically observable fluctuations is beyond the scope of this article. </p> <div class="mw-heading mw-heading2"><h2 id="Arrow_of_time">Arrow of time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=34" title="Edit section: Arrow of time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a> and <a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy (arrow of time)</a></div> <p>The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with symmetries observed in the fundamental laws of physics (particularly <a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a>) since the second law applies statistically on time-asymmetric boundary conditions.<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> The second law has been related to the difference between moving forwards and backwards in time, or to the principle that cause precedes effect (<a href="/wiki/Arrow_of_time#Causal_arrow_of_time" title="Arrow of time">the causal arrow of time</a>, or <a href="/wiki/Causality" title="Causality">causality</a>).<sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Irreversibility">Irreversibility</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=35" title="Edit section: Irreversibility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Irreversibility in <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">thermodynamic processes</a> is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this. </p> <div class="mw-heading mw-heading3"><h3 id="Loschmidt's_paradox"><span id="Loschmidt.27s_paradox"></span>Loschmidt's paradox</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=36" title="Edit section: Loschmidt's paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></div> <p><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a>, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system. </p><p>In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropy – or for that matter, all other irreversible statements – so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it. This reaffirms Albert Einstein's postulates that cornerstone Special and General Relativity - that the flow of time is irreversible, however it is relative. Cause must precede effect, but only within the constraints as defined explicitly within <a href="/wiki/General_Relativity" class="mw-redirect" title="General Relativity">General Relativity</a> (or <a href="/wiki/Special_Relativity" class="mw-redirect" title="Special Relativity">Special Relativity</a>, depending on the local spacetime conditions). Good examples of this are the <a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder Paradox</a>, <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a> and <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a> exhibited by objects approaching the velocity of light or within proximity of a super-dense region of mass/energy - e.g. black holes, neutron stars, magnetars and quasars. </p> <div class="mw-heading mw-heading3"><h3 id="Poincaré_recurrence_theorem"><span id="Poincar.C3.A9_recurrence_theorem"></span>Poincaré recurrence theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=37" title="Edit section: Poincaré recurrence theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a></div> <p>The <a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a> considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal. </p> <div class="mw-heading mw-heading3"><h3 id="Maxwell's_demon"><span id="Maxwell.27s_demon"></span>Maxwell's demon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=38" title="Edit section: Maxwell's demon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:James-clerk-maxwell3.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/James-clerk-maxwell3.jpg/170px-James-clerk-maxwell3.jpg" decoding="async" width="170" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/James-clerk-maxwell3.jpg/255px-James-clerk-maxwell3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/James-clerk-maxwell3.jpg/340px-James-clerk-maxwell3.jpg 2x" data-file-width="898" data-file-height="1192" /></a><figcaption>James Clerk Maxwell</figcaption></figure> <p><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> imagined one container divided into two parts, <i>A</i> and <i>B</i>. Both parts are filled with the same <a href="/wiki/Gas" title="Gas">gas</a> at equal temperatures and placed next to each other, separated by a wall. Observing the <a href="/wiki/Molecule" title="Molecule">molecules</a> on both sides, an imaginary <a href="/wiki/Demon" title="Demon">demon</a> guards a microscopic trapdoor in the wall. When a faster-than-average molecule from <i>A</i> flies towards the trapdoor, the demon opens it, and the molecule will fly from <i>A</i> to <i>B</i>. The average <a href="/wiki/Speed" title="Speed">speed</a> of the molecules in <i>B</i> will have increased while in <i>A</i> they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in <i>A</i> and increases in <i>B</i>, contrary to the second law of thermodynamics.<sup id="cite_ref-:1_98-0" class="reference"><a href="#cite_note-:1-98"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> </p><p>One response to this question was suggested in 1929 by <a href="/wiki/Le%C3%B3_Szil%C3%A1rd" class="mw-redirect" title="Leó Szilárd">Leó Szilárd</a> and later by <a href="/wiki/L%C3%A9on_Brillouin" title="Léon Brillouin">Léon Brillouin</a>. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<sup id="cite_ref-:2_99-0" class="reference"><a href="#cite_note-:2-99"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> Likewise, Brillouin demonstrated that the decrease in entropy caused by the demon would be less than the entropy produced by choosing molecules based on their speed.<sup id="cite_ref-:1_98-1" class="reference"><a href="#cite_note-:1-98"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> </p><p>Maxwell's 'demon' repeatedly alters the permeability of the wall between <i>A</i> and <i>B</i>. It is therefore performing <a href="/wiki/Thermodynamic_operation" title="Thermodynamic operation">thermodynamic operations</a> on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<sup id="cite_ref-:2_99-1" class="reference"><a href="#cite_note-:2-99"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Quotations">Quotations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=39" title="Edit section: Quotations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Second_law_of_thermodynamics" class="extiw" title="q:Special:Search/Second law of thermodynamics">Second law of thermodynamics</a></b></i>.</div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>The law that entropy always increases holds, I think, the supreme position among the <a href="/wiki/Laws_of_science" class="mw-redirect" title="Laws of science">laws of Nature</a>. If someone points out to you that your pet theory of the <a href="/wiki/Universe" title="Universe">universe</a> is in disagreement with <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.</p><div class="templatequotecite">— <cite>Sir <a href="/wiki/Arthur_Stanley_Eddington" class="mw-redirect" title="Arthur Stanley Eddington">Arthur Stanley Eddington</a>, <i>The Nature of the Physical World</i> (1927)</cite></div></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>There have been nearly as many formulations of the second law as there have been discussions of it.</p><div class="templatequotecite">— <cite>Philosopher / Physicist <a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">P.W. Bridgman</a>, (1941)</cite></div></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.</p><div class="templatequotecite">— <cite><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a>, Muncaster, R. G. (1980). <i>Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics</i>, Academic Press, New York, <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-701350-4" title="Special:BookSources/0-12-701350-4">0-12-701350-4</a>, p. 17.</cite></div></blockquote> <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=Second_law_of_thermodynamics&action=edit&section=40" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth law of 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/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third law of thermodynamics</a></li> <li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem inequality</a></li> <li><a href="/wiki/Fluctuation_theorem" title="Fluctuation theorem">Fluctuation theorem</a></li> <li><a href="/wiki/Heat_death_of_the_universe" title="Heat death of the universe">Heat death of the universe</a></li> <li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">History of thermodynamics</a></li> <li><a href="/wiki/Jarzynski_equality" title="Jarzynski equality">Jarzynski equality</a></li> <li><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws of thermodynamics</a></li> <li><a href="/wiki/Maximum_entropy_thermodynamics" title="Maximum entropy thermodynamics">Maximum entropy thermodynamics</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li> <li><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the Motive Power of Fire</a></li> <li><a href="/wiki/Relativistic_heat_conduction" title="Relativistic heat conduction">Relativistic heat conduction</a></li> <li><a href="/wiki/Thermal_diode" title="Thermal diode">Thermal diode</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li></ul> </div> <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=Second_law_of_thermodynamics&action=edit&section=41" 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 mw-references-columns"><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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReichl1980" class="citation book cs1"><a href="/wiki/Linda_Reichl" title="Linda Reichl">Reichl, Linda</a> (1980). <i>A Modern Course in Statistical Physics</i>. Edward Arnold. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7131-2789-9" title="Special:BookSources/0-7131-2789-9"><bdi>0-7131-2789-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Modern+Course+in+Statistical+Physics&rft.pages=9&rft.pub=Edward+Arnold&rft.date=1980&rft.isbn=0-7131-2789-9&rft.aulast=Reichl&rft.aufirst=Linda&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-Rao-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rao_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rao_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRao1997" class="citation book cs1">Rao, Y. V. C. 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Pearson. p. 764.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000">"5.2 Axiomatic Statements of the Laws of Thermodynamics"</a>. <i>www.web.mit.edu</i>. <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.web.mit.edu&rft.atitle=5.2+Axiomatic+Statements+of+the+Laws+of+Thermodynamics&rft_id=http%3A%2F%2Fweb.mit.edu%2F16.unified%2Fwww%2FFALL%2Fthermodynamics%2Fnotes%2Fnode38.html%23SECTION05224000000000000000&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></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="CITEREFSanborn_Scott2003" class="citation journal cs1">Sanborn Scott, David (2003). "The arrow of time". <i>International Journal of Hydrogen Energy</i>. <b>28</b> (2): <span class="nowrap">147–</span>149. <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/2003IJHE...28..147S">2003IJHE...28..147S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0360-3199%2802%2900019-8">10.1016/S0360-3199(02)00019-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Hydrogen+Energy&rft.atitle=The+arrow+of+time&rft.volume=28&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E147-%3C%2Fspan%3E149&rft.date=2003&rft_id=info%3Adoi%2F10.1016%2FS0360-3199%2802%2900019-8&rft_id=info%3Abibcode%2F2003IJHE...28..147S&rft.aulast=Sanborn+Scott&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" 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="CITEREFCarroll2010" class="citation book cs1">Carroll, Sean (2010). <i><a href="/wiki/From_Eternity_to_Here:_The_Quest_for_the_Ultimate_Theory_of_Time" class="mw-redirect" title="From Eternity to Here: The Quest for the Ultimate Theory of Time">From Eternity to Here: The Quest for the Ultimate Theory of Time</a></i>. Dutton. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-525-95133-9" title="Special:BookSources/978-0-525-95133-9"><bdi>978-0-525-95133-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Eternity+to+Here%3A+The+Quest+for+the+Ultimate+Theory+of+Time&rft.pub=Dutton&rft.date=2010&rft.isbn=978-0-525-95133-9&rft.aulast=Carroll&rft.aufirst=Sean&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaffeTaylor2018" class="citation book cs1">Jaffe, R.L.; Taylor, W. (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=drZDDwAAQBAJ"><i>The Physics of Energy</i></a>. Cambridge UK: Cambridge University Press. p. 150, n259, 772, 743. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-01665-1" title="Special:BookSources/978-1-107-01665-1"><bdi>978-1-107-01665-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Physics+of+Energy&rft.place=Cambridge+UK&rft.pages=150%2C+n259%2C+772%2C+743&rft.pub=Cambridge+University+Press&rft.date=2018&rft.isbn=978-1-107-01665-1&rft.aulast=Jaffe&rft.aufirst=R.L.&rft.au=Taylor%2C+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdrZDDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_L._Chandler2011" class="citation web cs1">David L. Chandler (2011-05-19). <a rel="nofollow" class="external text" href="http://news.mit.edu/2010/explained-carnot-0519">"Explained: The Carnot Limit"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Explained%3A+The+Carnot+Limit&rft.date=2011-05-19&rft.au=David+L.+Chandler&rft_id=http%3A%2F%2Fnews.mit.edu%2F2010%2Fexplained-carnot-0519&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1897/1903), pp. 40–41.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Munster A. (1970), pp. 8–9, 50–51.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFMandl1988">Mandl 1988</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1897/1903), pp. 79–107.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Bailyn, M. (1994), Section 71, pp. 113–154.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Bailyn, M. (1994), p. 120.</span> </li> <li id="cite_note-MortimerBook-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-MortimerBook_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MortimerBook_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-MortimerBook_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortimer2008" class="citation book cs1">Mortimer, R.G. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5CXWAQAACAAJ"><i>Physical Chemistry</i></a>. Elsevier Science. p. 120. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-370617-1" title="Special:BookSources/978-0-12-370617-1"><bdi>978-0-12-370617-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry&rft.pages=120&rft.pub=Elsevier+Science&rft.date=2008&rft.isbn=978-0-12-370617-1&rft.aulast=Mortimer&rft.aufirst=R.G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5CXWAQAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-FermiBook-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FermiBook_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFermi2012" class="citation book cs1">Fermi, E. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xCjDAgAAQBAJ"><i>Thermodynamics</i></a>. Dover Books on Physics. Dover Publications. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13485-7" title="Special:BookSources/978-0-486-13485-7"><bdi>978-0-486-13485-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics&rft.series=Dover+Books+on+Physics&rft.pages=48&rft.pub=Dover+Publications&rft.date=2012&rft.isbn=978-0-486-13485-7&rft.aulast=Fermi&rft.aufirst=E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxCjDAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-:0-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_17-0">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983), p. 75.</span> </li> <li id="cite_note-Munster_45-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-Munster_45_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Munster_45_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Munster_45_18-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Münster, A. (1970), p. 45.</span> </li> <li id="cite_note-Oxtoby8th-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Oxtoby8th_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Oxtoby8th_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Oxtoby, D. W; Gillis, H.P., <a href="/wiki/Laurie_Butler" title="Laurie Butler">Butler, L. J.</a> (2015).<i>Principles of Modern Chemistry</i>, Brooks Cole. p. 617. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1305079113" title="Special:BookSources/978-1305079113">978-1305079113</a></span> </li> <li id="cite_note-dugdale-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-dugdale_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._S._Dugdale1996" class="citation book cs1">J. S. Dugdale (1996). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/entropyitsphysic00dugd"><i>Entropy and its Physical Meaning</i></a></span>. Taylor & Francis. p. <a rel="nofollow" class="external text" href="https://archive.org/details/entropyitsphysic00dugd/page/n23">13</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7484-0569-5" title="Special:BookSources/978-0-7484-0569-5"><bdi>978-0-7484-0569-5</bdi></a>. <q>This law is the basis of temperature.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entropy+and+its+Physical+Meaning&rft.pages=13&rft.pub=Taylor+%26+Francis&rft.date=1996&rft.isbn=978-0-7484-0569-5&rft.au=J.+S.+Dugdale&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fentropyitsphysic00dugd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="/wiki/Mark_Zemansky" title="Mark Zemansky">Zemansky, M.W.</a> (1968), pp. 207–209.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Quinn, T.J. (1983), p. 8.</span> </li> <li id="cite_note-MIT-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-MIT_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html">"Concept and Statements of the Second Law"</a>. web.mit.edu<span class="reference-accessdate">. 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(1824/1986), p. 51.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Carnot, S. (1824/1986), p. 46.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Carnot, S. (1824/1986), p. 68.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a> (1980), Chapter 5.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983), pp. 56–58.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Münster, A. (1970), p. 11.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Kondepudi, D., <a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Prigogine, I.</a> (1998), pp.67–75.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Eu, B.C. (2002), pp. 32–35.</span> </li> <li id="cite_note-FOOTNOTEClausius1850-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEClausius1850_36-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFClausius1850">Clausius (1850)</a>.</span> </li> <li id="cite_note-FOOTNOTEClausius185486-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEClausius185486_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFClausius1854">Clausius (1854)</a>, p. 86.</span> </li> <li id="cite_note-FOOTNOTEThomson1851-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEThomson1851_38-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFThomson1851">Thomson (1851)</a>.</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1897/1903), p. 86.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><a href="/wiki/Dirk_ter_Haar" title="Dirk ter Haar">ter Haar, D.</a>, <a href="/wiki/Harald_Wergeland" title="Harald Wergeland">Wergeland, H.</a> (1966), p. 17.</span> </li> <li id="cite_note-Planck_100-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-Planck_100_42-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1897/1903), p. 100.</span> </li> <li id="cite_note-Planck_463-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-Planck_463_43-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1926), p. 463, translation by Uffink, J. (2003), p. 131.</span> </li> <li id="cite_note-Roberts_&_Miller_382-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-Roberts_&_Miller_382_44-0">^</a></b></span> <span class="reference-text">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a href="/wiki/George_Uhlenbeck" title="George Uhlenbeck">Uhlenbeck, G.E.</a>, Ford, G.W. (1963), p. 16.</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory, C.</a> (1909).</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text">Buchdahl, H.A. (1966), p. 68.</span> </li> <li id="cite_note-Sychev1991-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sychev1991_48-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSychev1991" class="citation book cs1">Sychev, V. V. (1991). <i>The Differential Equations of Thermodynamics</i>. 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University of California Riverside<span class="reference-accessdate">. Retrieved <span class="nowrap">7 June</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=UC+Riverside+Department+of+Mathematics&rft.atitle=Can+Gravity+Decrease+Entropy%3F&rft.date=2000-08-07&rft.aulast=Baez&rft.aufirst=John&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fentropy2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-Grandy_151-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-Grandy_151_88-0">^</a></b></span> <span class="reference-text">Grandy, W.T. (Jr) (2008), p. 151.</span> </li> <li id="cite_note-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-89">^</a></b></span> <span class="reference-text">Callen, H.B. 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J. (1983). <i>Equilibrium thermodynamics</i> (1st ed. 1968, 3rd ed.). 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(2006). <i>Atkins' Physical Chemistry</i>, eighth edition, W.H. Freeman, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-8759-4" title="Special:BookSources/978-0-7167-8759-4">978-0-7167-8759-4</a>.</li> <li>Attard, P. (2012). <i>Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications</i>, Oxford University Press, Oxford UK, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-966276-0" title="Special:BookSources/978-0-19-966276-0">978-0-19-966276-0</a>.</li> <li>Baierlein, R. (1999). <i>Thermal Physics</i>, Cambridge University Press, Cambridge UK, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-59082-5" title="Special:BookSources/0-521-59082-5">0-521-59082-5</a>.</li> <li>Bailyn, M. (1994). <i>A Survey of Thermodynamics</i>, American Institute of Physics, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88318-797-3" title="Special:BookSources/0-88318-797-3">0-88318-797-3</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlundellBlundell2010" class="citation book cs1"><a href="/wiki/Stephen_Blundell" title="Stephen Blundell">Blundell, Stephen J.</a>; <a href="/wiki/Katherine_Blundell" title="Katherine Blundell">Blundell, Katherine M.</a> (2010). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/1235139"><i>Concepts in thermal physics</i></a> (2nd ed.). Oxford: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facprof%3Aoso%2F9780199562091.001.0001">10.1093/acprof:oso/9780199562091.001.0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199562107" title="Special:BookSources/9780199562107"><bdi>9780199562107</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/607907330">607907330</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concepts+in+thermal+physics&rft.place=Oxford&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=2010&rft_id=info%3Aoclcnum%2F607907330&rft_id=info%3Adoi%2F10.1093%2Facprof%3Aoso%2F9780199562091.001.0001&rft.isbn=9780199562107&rft.aulast=Blundell&rft.aufirst=Stephen+J.&rft.au=Blundell%2C+Katherine+M.&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F1235139&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann, L.</a> (1896/1964). <i>Lectures on Gas Theory</i>, translated by S.G. Brush, University of California Press, Berkeley.</li> <li>Borgnakke, C., Sonntag., R.E. (2009). <i>Fundamentals of Thermodynamics</i>, seventh edition, Wiley, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-04192-5" title="Special:BookSources/978-0-470-04192-5">978-0-470-04192-5</a>.</li> <li>Buchdahl, H.A. (1966). <i>The Concepts of Classical Thermodynamics</i>, Cambridge University Press, Cambridge UK.</li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman, P.W.</a> (1943). <i>The Nature of Thermodynamics</i>, Harvard University Press, Cambridge MA.</li> <li><a href="/wiki/Herbert_Callen" title="Herbert Callen">Callen, H.B.</a> (1960/1985). <i>Thermodynamics and an Introduction to Thermostatistics</i>, (1st edition 1960) 2nd edition 1985, Wiley, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-86256-8" title="Special:BookSources/0-471-86256-8">0-471-86256-8</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._Carathéodory1909" class="citation journal cs1"><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">C. Carathéodory</a> (1909). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304213645/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1">"Untersuchungen über die Grundlagen der Thermodynamik"</a>. <i>Mathematische Annalen</i>. <b>67</b> (3): <span class="nowrap">355–</span>386. <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%2Fbf01450409">10.1007/bf01450409</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:118230148">118230148</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1">the original</a> on 2016-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-02-18</span></span>. <q>Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Untersuchungen+%C3%BCber+die+Grundlagen+der+Thermodynamik&rft.volume=67&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E355-%3C%2Fspan%3E386&rft.date=1909&rft_id=info%3Adoi%2F10.1007%2Fbf01450409&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118230148%23id-name%3DS2CID&rft.au=C.+Carath%C3%A9odory&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DPPN235181684_0067%26DMDID%3DDMDLOG_0033%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span>. A translation may be found <a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf">here</a>. Also a mostly reliable <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations">translation is to be found</a> at Kestin, J. (1976). <i>The Second Law of Thermodynamics</i>, Dowden, Hutchinson & Ross, Stroudsburg PA.</li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot, S.</a> (1824/1986). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/812944517"><i>Reflections on the motive power of fire</i></a>, Manchester University Press, Manchester UK, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7190-1741-6" title="Special:BookSources/0-7190-1741-6">0-7190-1741-6</a>. <a rel="nofollow" class="external text" href="https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up">Also here.</a></li> <li><a href="/wiki/Sydney_Chapman_(mathematician)" title="Sydney Chapman (mathematician)">Chapman, S.</a>, <a href="/wiki/Thomas_George_Cowling" class="mw-redirect" title="Thomas George Cowling">Cowling, T.G.</a> (1939/1970). <i>The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases</i>, third edition 1970, Cambridge University Press, London.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1850" class="citation journal cs1"><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1850). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image">"Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen"</a>. <i>Annalen der Physik</i>. <b>79</b> (4): <span class="nowrap">368–</span>397, <span class="nowrap">500–</span>524. <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/1850AnP...155..500C">1850AnP...155..500C</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.18501550403">10.1002/andp.18501550403</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuc1.%24b242250">2027/uc1.$b242250</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">26 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&rft.atitle=Ueber+Die+Bewegende+Kraft+Der+W%C3%A4rme+Und+Die+Gesetze%2C+Welche+Sich+Daraus+F%C3%BCr+Die+W%C3%A4rmelehre+Selbst+Ableiten+Lassen&rft.volume=79&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E368-%3C%2Fspan%3E397%2C+%3Cspan+class%3D%22nowrap%22%3E500-%3C%2Fspan%3E524&rft.date=1850&rft_id=info%3Ahdl%2F2027%2Fuc1.%24b242250&rft_id=info%3Adoi%2F10.1002%2Fandp.18501550403&rft_id=info%3Abibcode%2F1850AnP...155..500C&rft.aulast=Clausius&rft.aufirst=R.&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k15164w%2Ff518.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span> Translated into English: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1851" class="citation journal cs1">Clausius, R. (July 1851). <a rel="nofollow" class="external text" href="https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up">"On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom"</a>. <i>London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science</i>. 4th. <b>2</b> (VIII): <span class="nowrap">1–</span>21, <span class="nowrap">102–</span>119. <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%2F14786445108646819">10.1080/14786445108646819</a><span class="reference-accessdate">. Retrieved <span class="nowrap">26 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=London%2C+Edinburgh%2C+and+Dublin+Philosophical+Magazine+and+Journal+of+Science&rft.atitle=On+the+Moving+Force+of+Heat%2C+and+the+Laws+regarding+the+Nature+of+Heat+itself+which+are+deducible+therefrom&rft.volume=2&rft.issue=VIII&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E21%2C+%3Cspan+class%3D%22nowrap%22%3E102-%3C%2Fspan%3E119&rft.date=1851-07&rft_id=info%3Adoi%2F10.1080%2F14786445108646819&rft.aulast=Clausius&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Flondonedinburghd02lond%23page%2F1%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1854" class="citation journal cs1 cs1-prop-long-vol"><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1854). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140324093633/http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf">"Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie"</a> <span class="cs1-format">(PDF)</span>. <i>Annalen der Physik</i>. xciii (12): <span class="nowrap">481–</span>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>. Archived from <a rel="nofollow" class="external text" href="http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 24 March 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">24 March</span> 2014</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&rft.atitle=%C3%9Cber+eine+ver%C3%A4nderte+Form+des+zweiten+Hauptsatzes+der+mechanischen+W%C3%A4rmetheorie&rft.volume=xciii&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E481-%3C%2Fspan%3E506&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%2Fzfbb.thulb.uni-jena.de%2Fservlets%2FMCRFileNodeServlet%2Fjportal_derivate_00140956%2F18541691202_ftp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span> Translated into English: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1856" class="citation journal cs1">Clausius, R. (July 1856). <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up">"On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat"</a>. <i>London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science</i>. 4th. <b>2</b>: 86<span class="reference-accessdate">. Retrieved <span class="nowrap">24 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=London%2C+Edinburgh%2C+and+Dublin+Philosophical+Magazine+and+Journal+of+Science&rft.atitle=On+a+Modified+Form+of+the+Second+Fundamental+Theorem+in+the+Mechanical+Theory+of+Heat&rft.volume=2&rft.pages=86&rft.date=1856-07&rft.aulast=Clausius&rft.aufirst=R.&rft_id=https%3A%2F%2Fwww.biodiversitylibrary.org%2Fitem%2F20044%23page%2F100%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span> Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1867" class="citation book cs1"><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1867). <a rel="nofollow" class="external text" href="https://archive.org/details/mechanicaltheor04claugoog"><i>The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies</i></a>. London: John van Voorst<span class="reference-accessdate">. Retrieved <span class="nowrap">19 June</span> 2012</span>. <q>editions:PwR_Sbkwa8IC.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mechanical+Theory+of+Heat+%E2%80%93+with+its+Applications+to+the+Steam+Engine+and+to+Physical+Properties+of+Bodies&rft.place=London&rft.pub=John+van+Voorst&rft.date=1867&rft.aulast=Clausius&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmechanicaltheor04claugoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></li> <li>Denbigh, K. (1954/1981). <i>The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering</i>, fourth edition, Cambridge University Press, Cambridge UK, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-23682-7" title="Special:BookSources/0-521-23682-7">0-521-23682-7</a>.</li> <li>Eu, B.C. (2002). <i>Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics</i>, Kluwer Academic Publishers, Dordrecht, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0788-4" title="Special:BookSources/1-4020-0788-4">1-4020-0788-4</a>.</li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs, J.W.</a> (1876/1878). On the equilibrium of heterogeneous substances, <i>Trans. Conn. Acad.</i>, <b>3</b>: 108–248, 343–524, reprinted in <i>The Collected Works of J. Willard Gibbs, Ph.D, LL. D.</i>, edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp. 55–353.</li> <li>Griem, H.R. (2005). <i>Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)</i>, Cambridge University Press, New York <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-61941-6" title="Special:BookSources/0-521-61941-6">0-521-61941-6</a>.</li> <li>Glansdorff, P., Prigogine, I. (1971). <i>Thermodynamic Theory of Structure, Stability, and Fluctuations</i>, Wiley-Interscience, London, 1971, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-30280-5" title="Special:BookSources/0-471-30280-5">0-471-30280-5</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrandy2008" class="citation book cs1">Grandy, Walter T. (2008). <a rel="nofollow" class="external text" href="http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176"><i>Entropy and the time evolution of macroscopic systems</i></a>. Oxford New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-954617-6" title="Special:BookSources/978-0-19-954617-6"><bdi>978-0-19-954617-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/190843367">190843367</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entropy+and+the+time+evolution+of+macroscopic+systems&rft.place=Oxford+New+York&rft.pub=Oxford+University+Press&rft.date=2008&rft_id=info%3Aoclcnum%2F190843367&rft.isbn=978-0-19-954617-6&rft.aulast=Grandy&rft.aufirst=Walter+T.&rft_id=http%3A%2F%2Fglobal.oup.com%2Facademic%2Fproduct%2Fentropy-and-the-time-evolution-of-macroscopic-systems-9780199546176&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></li> <li>Greven, A., Keller, G., Warnecke (editors) (2003). <i>Entropy</i>, Princeton University Press, Princeton NJ, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-11338-6" title="Special:BookSources/0-691-11338-6">0-691-11338-6</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuggenheim1949" class="citation journal cs1"><a href="/wiki/Edward_A._Guggenheim" title="Edward A. 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"Bluff your way in the second law of thermodynamics". <i>Stud. Hist. Phil. Mod. Phys</i>. <b>32</b> (3): <span class="nowrap">305–</span>394. <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/0005327">cond-mat/0005327</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/2001SHPMP..32..305U">2001SHPMP..32..305U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1355-2198%2801%2900016-8">10.1016/S1355-2198(01)00016-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Stud.+Hist.+Phil.+Mod.+Phys.&rft.atitle=Bluff+your+way+in+the+second+law+of+thermodynamics&rft.volume=32&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E305-%3C%2Fspan%3E394&rft.date=2001&rft_id=info%3Aarxiv%2Fcond-mat%2F0005327&rft_id=info%3Adoi%2F10.1016%2FS1355-2198%2801%2900016-8&rft_id=info%3Abibcode%2F2001SHPMP..32..305U&rft.aulast=Uffink&rft.aufirst=J&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span></li> <li>Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of <i>Entropy</i>, Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-11338-6" title="Special:BookSources/0-691-11338-6">0-691-11338-6</a>.</li> <li><a href="/wiki/George_Uhlenbeck" title="George Uhlenbeck">Uhlenbeck, G.E.</a>, Ford, G.W. (1963). <i>Lectures in Statistical Mechanics</i>, American Mathematical Society, Providence RI.</li> <li><a href="/wiki/Mark_Zemansky" title="Mark Zemansky">Zemansky, M.W.</a> (1968). <i>Heat and Thermodynamics. An Intermediate Textbook</i>, fifth edition, McGraw-Hill Book Company, New York.</li></ul> </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=Second_law_of_thermodynamics&action=edit&section=43" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Goldstein, Martin, and Inge F., 1993. <i>The Refrigerator and the Universe</i>. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the second law, one a bit less technical than this entry. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-75324-2" title="Special:BookSources/978-0-674-75324-2">978-0-674-75324-2</a></li> <li>Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. <i>Maxwell's Demon 2 : Entropy, classical and quantum information, computing</i>. Bristol UK; Philadelphia PA: <a href="/wiki/Institute_of_Physics" title="Institute of Physics">Institute of Physics</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-585-49237-7" title="Special:BookSources/978-0-585-49237-7">978-0-585-49237-7</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalliwell1994" class="citation book cs1">Halliwell, J.J. (1994). <i>Physical Origins of Time Asymmetry</i>. Cambridge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-56837-1" title="Special:BookSources/978-0-521-56837-1"><bdi>978-0-521-56837-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Origins+of+Time+Asymmetry&rft.pub=Cambridge&rft.date=1994&rft.isbn=978-0-521-56837-1&rft.aulast=Halliwell&rft.aufirst=J.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span>(technical).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarnot1890" class="citation book cs1">Carnot, Sadi (1890). <a href="/wiki/Robert_Henry_Thurston" title="Robert Henry Thurston">Thurston, Robert Henry</a> (ed.). <i>Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power</i>. New York: J. Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reflections+on+the+Motive+Power+of+Heat+and+on+Machines+Fitted+to+Develop+That+Power&rft.place=New+York&rft.pub=J.+Wiley+%26+Sons&rft.date=1890&rft.aulast=Carnot&rft.aufirst=Sadi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span> (<a rel="nofollow" class="external text" href="https://books.google.com/books?id=tgdJAAAAIAAJ">full text of 1897 ed.</a>) (<a rel="nofollow" class="external text" href="http://www.history.rochester.edu/steam/carnot/1943/">html</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070818073812/http://www.history.rochester.edu/steam/carnot/1943/">Archived</a> 2007-08-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>)</li> <li>Stephen Jay Kline (1999). <i>The Low-Down on Entropy and Interpretive Thermodynamics</i>, La Cañada, CA: DCW Industries. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-928729-01-0" title="Special:BookSources/1-928729-01-0">1-928729-01-0</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKostic2011" class="citation book cs1">Kostic, M (2011). <i>Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization</i>. AIP Conference Proceedings. Vol. 1411. pp. <span class="nowrap">327–</span>350. <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/2011AIPC.1411..327K">2011AIPC.1411..327K</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.405.1945">10.1.1.405.1945</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3665247">10.1063/1.3665247</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7354-0985-9" title="Special:BookSources/978-0-7354-0985-9"><bdi>978-0-7354-0985-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Revisiting+The+Second+Law+of+Energy+Degradation+and+Entropy+Generation%3A+From+Sadi+Carnot%27s+Ingenious+Reasoning+to+Holistic+Generalization&rft.series=AIP+Conference+Proceedings&rft.pages=%3Cspan+class%3D%22nowrap%22%3E327-%3C%2Fspan%3E350&rft.date=2011&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.405.1945%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1063%2F1.3665247&rft_id=info%3Abibcode%2F2011AIPC.1411..327K&rft.isbn=978-0-7354-0985-9&rft.aulast=Kostic&rft.aufirst=M&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecond+law+of+thermodynamics" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|journal=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span> also at <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf">[1]</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Second_law_of_thermodynamics&action=edit&section=44" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: "<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/statphys-statmech/">Philosophy of Statistical Mechanics</a>" – by Lawrence Sklar.</li> <li><a rel="nofollow" class="external text" href="http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html"><i>Second law of thermodynamics</i></a> in the MIT Course <a rel="nofollow" class="external text" href="http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html"><i>Unified Thermodynamics and Propulsion</i></a> from Prof. Z. S. Spakovszky</li> <li><a href="/wiki/E.T._Jaynes" class="mw-redirect" title="E.T. Jaynes">E.T. Jaynes</a>, 1988, "<a rel="nofollow" class="external text" href="http://bayes.wustl.edu/etj/articles/ccarnot.pdf">The evolution of Carnot's principle</a>," in G. J. Erickson and C. R. Smith (eds.)<i>Maximum-Entropy and Bayesian Methods in Science and Engineering</i>, Vol,.1: p. 267.</li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf">Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich</a></li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p004y2bm">The second law of Thermodynamics</a>, BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (<i>In Our Time</i>, December 16, 2004)</li> <li><a rel="nofollow" class="external text" href="https://www.journals.uchicago.edu/doi/abs/10.1086/663835">The Journal of the International Society for the History of Philosophy of Science, 2012</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=Second_law_of_thermodynamics&oldid=1276070893">https://en.wikipedia.org/w/index.php?title=Second_law_of_thermodynamics&oldid=1276070893</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:Equations_of_physics" title="Category:Equations of physics">Equations of physics</a></li><li><a href="/wiki/Category:Laws_of_thermodynamics" title="Category:Laws of thermodynamics">Laws of thermodynamics</a></li><li><a href="/wiki/Category:Non-equilibrium_thermodynamics" title="Category:Non-equilibrium thermodynamics">Non-equilibrium thermodynamics</a></li><li><a href="/wiki/Category:Philosophy_of_thermal_and_statistical_physics" title="Category:Philosophy of thermal and statistical physics">Philosophy of thermal and statistical physics</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_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_August_2018" title="Category:Wikipedia articles needing clarification from August 2018">Wikipedia articles needing clarification from August 2018</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_February_2014" title="Category:Wikipedia articles needing clarification from February 2014">Wikipedia articles needing clarification from February 2014</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_January_2025" title="Category:Articles needing additional references from January 2025">Articles needing additional references from January 2025</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_August_2012" title="Category:Articles with unsourced statements from August 2012">Articles with unsourced statements from August 2012</a></li><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:CS1_errors:_periodical_ignored" title="Category:CS1 errors: periodical ignored">CS1 errors: periodical ignored</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 16 February 2025, at 18:33<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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Second law of thermodynamics - Wikipedia