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id="toc-18th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#18th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>18th century</span> </div> </a> <ul id="toc-18th_century-sublist" class="vector-toc-list"> <li id="toc-Heat_vs._temperature" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Heat_vs._temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Heat vs. temperature</span> </div> </a> <ul id="toc-Heat_vs._temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Evaporative_cooling" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Evaporative_cooling"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Evaporative cooling</span> </div> </a> <ul id="toc-Evaporative_cooling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discovery_of_specific_heat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discovery_of_specific_heat"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Discovery of specific heat</span> </div> </a> <ul id="toc-Discovery_of_specific_heat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Degrees_of_heat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Degrees_of_heat"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.4</span> <span>Degrees of heat</span> </div> </a> <ul id="toc-Degrees_of_heat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discovery_of_latent_heat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discovery_of_latent_heat"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.5</span> <span>Discovery of latent heat</span> </div> </a> <ul id="toc-Discovery_of_latent_heat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First_calorimeter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#First_calorimeter"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.6</span> <span>First calorimeter</span> </div> </a> <ul id="toc-First_calorimeter-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Classical_thermodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Classical thermodynamics</span> </div> </a> <ul id="toc-Classical_thermodynamics-sublist" class="vector-toc-list"> <li id="toc-Clausius_(1850)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Clausius_(1850)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Clausius (1850)</span> </div> </a> <ul id="toc-Clausius_(1850)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-James_Clerk_Maxwell_(1871)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#James_Clerk_Maxwell_(1871)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>James Clerk Maxwell (1871)</span> </div> </a> <ul id="toc-James_Clerk_Maxwell_(1871)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bryan_(1907)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bryan_(1907)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Bryan (1907)</span> </div> </a> <ul id="toc-Bryan_(1907)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carathéodory_(1909)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Carathéodory_(1909)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.4</span> <span>Carathéodory (1909)</span> </div> </a> <ul id="toc-Carathéodory_(1909)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Planck_(1926)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Planck_(1926)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.5</span> <span>Planck (1926)</span> </div> </a> <ul id="toc-Planck_(1926)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Heat_transfer" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Heat_transfer"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Heat transfer</span> </div> </a> <button aria-controls="toc-Heat_transfer-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 Heat transfer subsection</span> </button> <ul id="toc-Heat_transfer-sublist" class="vector-toc-list"> <li id="toc-Heat_transfer_between_two_bodies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_transfer_between_two_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Heat transfer between two bodies</span> </div> </a> <ul id="toc-Heat_transfer_between_two_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heat_engine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_engine"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Heat engine</span> </div> </a> <ul id="toc-Heat_engine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heat_pump_or_refrigerator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_pump_or_refrigerator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Heat pump or refrigerator</span> </div> </a> <ul id="toc-Heat_pump_or_refrigerator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Macroscopic_view" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Macroscopic_view"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Macroscopic view</span> </div> </a> <ul id="toc-Macroscopic_view-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Microscopic_view" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Microscopic_view"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Microscopic view</span> </div> </a> <ul id="toc-Microscopic_view-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calorimetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calorimetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Calorimetry</span> </div> </a> <ul id="toc-Calorimetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Engineering</span> </div> </a> <ul id="toc-Engineering-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Latent_and_sensible_heat" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Latent_and_sensible_heat"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Latent and sensible heat</span> </div> </a> <ul id="toc-Latent_and_sensible_heat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heat_capacity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Heat_capacity"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Heat capacity</span> </div> </a> <ul id="toc-Heat_capacity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-"Hotness"" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#"Hotness""> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>"Hotness"</span> </div> </a> <ul id="toc-"Hotness"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_thermodynamics_2" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Classical_thermodynamics_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Classical thermodynamics</span> </div> </a> <button aria-controls="toc-Classical_thermodynamics_2-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 Classical thermodynamics subsection</span> </button> <ul id="toc-Classical_thermodynamics_2-sublist" class="vector-toc-list"> <li id="toc-Heat_and_enthalpy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_and_enthalpy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Heat and enthalpy</span> </div> </a> <ul id="toc-Heat_and_enthalpy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heat_and_entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_and_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Heat and entropy</span> </div> </a> <ul id="toc-Heat_and_entropy-sublist" class="vector-toc-list"> </ul> </li> </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">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">10</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-Quotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Quotations</span> </div> </a> <ul id="toc-Quotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography_of_cited_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography_of_cited_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Bibliography of cited references</span> </div> </a> <ul id="toc-Bibliography_of_cited_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Further bibliography</span> </div> </a> <ul id="toc-Further_bibliography-sublist" class="vector-toc-list"> </ul> </li> </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">11</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 id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Heat</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 114 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-114" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">114 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Warmte" title="Warmte – Afrikaans" lang="af" hreflang="af" data-title="Warmte" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%99%E1%89%80%E1%89%B5" title="ሙቀት – Amharic" lang="am" hreflang="am" data-title="ሙቀት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%89%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE" title="उष्मा – Angika" lang="anp" hreflang="anp" data-title="उष्मा" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/H%C4%81t" title="Hāt – Old English" lang="ang" hreflang="ang" data-title="Hāt" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target"><span>Ænglisc</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B1%D8%A7%D8%B1%D8%A9" title="حرارة – Arabic" lang="ar" hreflang="ar" data-title="حرارة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Calor" title="Calor – Aragonese" lang="an" hreflang="an" data-title="Calor" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/C%C3%A2ldur%C3%A2" title="Câldurâ – Aromanian" lang="rup" hreflang="rup" data-title="Câldurâ" data-language-autonym="Armãneashti" data-language-local-name="Aromanian" class="interlanguage-link-target"><span>Armãneashti</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Calor" title="Calor – Asturian" lang="ast" hreflang="ast" data-title="Calor" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Lupi" title="Lupi – Aymara" lang="ay" hreflang="ay" data-title="Lupi" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0stilik" title="İstilik – Azerbaijani" lang="az" hreflang="az" data-title="İstilik" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%DB%8C%D8%B3%D8%AA%DB%8C%E2%80%8C%D9%84%DB%8C%DA%A9" title="ایستی‌لیک – South Azerbaijani" lang="azb" hreflang="azb" data-title="ایستی‌لیک" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</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" title="তাপ – Bangla" lang="bn" hreflang="bn" data-title="তাপ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Jia%CC%8Dt" title="Jia̍t – Minnan" lang="nan" hreflang="nan" data-title="Jia̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BF%D0%BB%D1%8B%D0%BD%D1%8F" title="Цеплыня – Belarusian" lang="be" hreflang="be" data-title="Цеплыня" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Init" title="Init – Central Bikol" lang="bcl" hreflang="bcl" data-title="Init" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D0%B0" title="Топлина – Bulgarian" lang="bg" hreflang="bg" data-title="Топлина" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/W%C3%A4rme" title="Wärme – Bavarian" lang="bar" hreflang="bar" data-title="Wärme" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Toplota" title="Toplota – Bosnian" lang="bs" hreflang="bs" data-title="Toplota" 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/Calor" title="Calor – Catalan" lang="ca" hreflang="ca" data-title="Calor" 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/%C4%82%D1%88%C4%83" title="Ăшă – Chuvash" lang="cv" hreflang="cv" data-title="Ăшă" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Kainit" title="Kainit – Cebuano" lang="ceb" hreflang="ceb" data-title="Kainit" data-language-autonym="Cebuano" data-language-local-name="Cebuano" class="interlanguage-link-target"><span>Cebuano</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teplo" title="Teplo – Czech" lang="cs" hreflang="cs" data-title="Teplo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Madziya" title="Madziya – Shona" lang="sn" hreflang="sn" data-title="Madziya" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gwres" title="Gwres – Welsh" lang="cy" hreflang="cy" data-title="Gwres" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Varme" title="Varme – Danish" lang="da" hreflang="da" data-title="Varme" 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/W%C3%A4rme" title="Wärme – German" lang="de" hreflang="de" data-title="Wärme" 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/Soojushulk" title="Soojushulk – Estonian" lang="et" hreflang="et" data-title="Soojushulk" 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%98%CE%B5%CF%81%CE%BC%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Θερμότητα – Greek" lang="el" hreflang="el" data-title="Θερμότητα" 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/Calor" title="Calor – Spanish" lang="es" hreflang="es" data-title="Calor" 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/Varmo" title="Varmo – Esperanto" lang="eo" hreflang="eo" data-title="Varmo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Calol" title="Calol – Extremaduran" lang="ext" hreflang="ext" data-title="Calol" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bero" title="Bero – Basque" lang="eu" hreflang="eu" data-title="Bero" 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/%DA%AF%D8%B1%D9%85%D8%A7" title="گرما – Persian" lang="fa" hreflang="fa" data-title="گرما" 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/Chaleur_(thermodynamique)" title="Chaleur (thermodynamique) – French" lang="fr" hreflang="fr" data-title="Chaleur (thermodynamique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Waarmte" title="Waarmte – Western Frisian" lang="fy" hreflang="fy" data-title="Waarmte" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teas" title="Teas – Irish" lang="ga" hreflang="ga" data-title="Teas" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/%C3%87hiass" title="Çhiass – Manx" lang="gv" hreflang="gv" data-title="Çhiass" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Calor" title="Calor – Galician" lang="gl" hreflang="gl" data-title="Calor" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%89%E0%AA%B7%E0%AB%8D%E0%AA%AE%E0%AA%BE" title="ઉષ્મા – Gujarati" lang="gu" hreflang="gu" data-title="ઉષ્મા" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B4" title="열 – Korean" lang="ko" hreflang="ko" data-title="열" 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%D6%84%D5%A1%D5%B6%D5%A1%D5%AF" title="Ջերմաքանակ – Armenian" lang="hy" hreflang="hy" data-title="Ջերմաքանակ" 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" title="ऊष्मा – Hindi" lang="hi" hreflang="hi" data-title="ऊष्मा" 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/Toplina" title="Toplina – Croatian" lang="hr" hreflang="hr" data-title="Toplina" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Kaloro" title="Kaloro – Ido" lang="io" hreflang="io" data-title="Kaloro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Panas" title="Panas – Indonesian" lang="id" hreflang="id" data-title="Panas" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Varmi" title="Varmi – Icelandic" lang="is" hreflang="is" data-title="Varmi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Calore" title="Calore – Italian" lang="it" hreflang="it" data-title="Calore" 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%97%D7%95%D7%9D_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="חום (פיזיקה) – Hebrew" lang="he" hreflang="he" data-title="חום (פיזיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Kalor" title="Kalor – Javanese" lang="jv" hreflang="jv" data-title="Kalor" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Fefeku" title="Fefeku – Kabiye" lang="kbp" hreflang="kbp" data-title="Fefeku" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%89%E0%B2%B7%E0%B3%8D%E0%B2%A3%E0%B2%A4%E0%B3%86" title="ಉಷ್ಣತೆ – Kannada" lang="kn" hreflang="kn" data-title="ಉಷ್ಣತೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%97%E1%83%91%E1%83%9D" title="სითბო – Georgian" lang="ka" hreflang="ka" data-title="სითბო" 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%96%D1%8B%D0%BB%D1%83%D0%BB%D1%8B%D2%9B" title="Жылулық – Kazakh" lang="kk" hreflang="kk" data-title="Жылулық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Joto" title="Joto – Swahili" lang="sw" hreflang="sw" data-title="Joto" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Chal%C3%A8" title="Chalè – Haitian Creole" lang="ht" hreflang="ht" data-title="Chalè" 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-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/T%C3%AAhn_(fiz%C3%AEk)" title="Têhn (fizîk) – Kurdish" lang="ku" hreflang="ku" data-title="Têhn (fizîk)" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%96%D1%8B%D0%BB%D1%83%D1%83%D0%BB%D1%83%D0%BA" title="Жылуулук – Kyrgyz" lang="ky" hreflang="ky" data-title="Жылуулук" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Calor" title="Calor – Latin" lang="la" hreflang="la" data-title="Calor" 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/Siltums" title="Siltums – Latvian" lang="lv" hreflang="lv" data-title="Siltums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/%C5%A0iluma" title="Šiluma – Lithuanian" lang="lt" hreflang="lt" data-title="Šiluma" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Wermte" title="Wermte – Limburgish" lang="li" hreflang="li" data-title="Wermte" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Ey%C3%A1ngala" title="Eyángala – Lingala" lang="ln" hreflang="ln" data-title="Eyángala" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C5%91" title="Hő – Hungarian" lang="hu" hreflang="hu" data-title="Hő" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%A4%E0%A4%BE%E0%A4%AA" title="ताप – Maithili" lang="mai" hreflang="mai" data-title="ताप" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target"><span>मैथिली</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D0%B0" title="Топлина – Macedonian" lang="mk" hreflang="mk" data-title="Топлина" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B4%BE%E0%B4%AA%E0%B4%82" title="താപം – Malayalam" lang="ml" hreflang="ml" data-title="താപം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AD%D8%B1%D8%A7%D8%B1%D8%A9" title="حرارة – Egyptian Arabic" lang="arz" hreflang="arz" data-title="حرارة" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Haba" title="Haba – Malay" lang="ms" hreflang="ms" data-title="Haba" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D1%83%D0%BB%D0%B0%D0%B0%D0%BD" title="Дулаан – Mongolian" lang="mn" hreflang="mn" data-title="Дулаан" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%95%E1%80%B0" title="အပူ – Burmese" lang="my" hreflang="my" data-title="အပူ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Katakata" title="Katakata – Fijian" lang="fj" hreflang="fj" data-title="Katakata" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Warmte" title="Warmte – Dutch" lang="nl" hreflang="nl" data-title="Warmte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%A4%E0%A4%BE%E0%A4%AA" title="ताप – Nepali" lang="ne" hreflang="ne" data-title="ताप" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%86%B1" title="熱 – Japanese" lang="ja" hreflang="ja" data-title="熱" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Waremk" title="Waremk – Northern Frisian" lang="frr" hreflang="frr" data-title="Waremk" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Varme" title="Varme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Varme" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Varme" title="Varme – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Varme" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Ho%27a" title="Ho'a – Oromo" lang="om" hreflang="om" data-title="Ho'a" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Issiqlik_miqdori" title="Issiqlik miqdori – Uzbek" lang="uz" hreflang="uz" data-title="Issiqlik miqdori" 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%97%E0%A8%B0%E0%A8%AE%E0%A9%80" title="ਗਰਮੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਰਮੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Iit" title="Iit – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Iit" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ciep%C5%82o" title="Ciepło – Polish" lang="pl" hreflang="pl" data-title="Ciepło" 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/Calor" title="Calor – Portuguese" lang="pt" hreflang="pt" data-title="Calor" 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/C%C4%83ldur%C4%83" title="Căldură – Romanian" lang="ro" hreflang="ro" data-title="Căldură" 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%A2%D0%B5%D0%BF%D0%BB%D0%BE%D1%82%D0%B0" title="Теплота – Russian" lang="ru" hreflang="ru" data-title="Теплота" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Nxeht%C3%ABsia" title="Nxehtësia – Albanian" lang="sq" hreflang="sq" data-title="Nxehtësia" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Caluri" title="Caluri – Sicilian" lang="scn" hreflang="scn" data-title="Caluri" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Heat" title="Heat – Simple English" lang="en-simple" hreflang="en-simple" data-title="Heat" 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/Teplo" title="Teplo – Slovak" lang="sk" hreflang="sk" data-title="Teplo" 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/Toplota" title="Toplota – Slovenian" lang="sl" hreflang="sl" data-title="Toplota" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Kul" title="Kul – Somali" lang="so" hreflang="so" data-title="Kul" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%DB%95%D8%B1%D9%85%DB%8C" title="گەرمی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="گەرمی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BB%D0%BE%D1%82%D0%B0" title="Топлота – Serbian" lang="sr" hreflang="sr" data-title="Топлота" 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/Toplina" title="Toplina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Toplina" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Panas" title="Panas – Sundanese" lang="su" hreflang="su" data-title="Panas" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/L%C3%A4mp%C3%B6_(fysiikka)" title="Lämpö (fysiikka) – Finnish" lang="fi" hreflang="fi" data-title="Lämpö (fysiikka)" 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/V%C3%A4rme" title="Värme – Swedish" lang="sv" hreflang="sv" data-title="Värme" 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%AE%E0%AF%8D_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="வெப்பம் (இயற்பியல்) – Tamil" lang="ta" hreflang="ta" data-title="வெப்பம் (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/A%E1%BA%93%C9%A3al" title="Aẓɣal – Kabyle" lang="kab" hreflang="kab" data-title="Aẓɣal" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D2%96%D1%8B%D0%BB%D1%8B%D0%BB%D1%8B%D0%BA" title="Җылылык – Tatar" lang="tt" hreflang="tt" data-title="Җылылык" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%89%E0%B0%B7%E0%B1%8D%E0%B0%A3%E0%B0%AE%E0%B1%81" title="ఉష్ణము – Telugu" lang="te" hreflang="te" data-title="ఉష్ణము" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A3%E0%B9%89%E0%B8%AD%E0%B8%99" title="ความร้อน – Thai" lang="th" hreflang="th" data-title="ความร้อน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE" title="Гармо – Tajik" lang="tg" hreflang="tg" data-title="Гармо" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Is%C4%B1" title="Isı – Turkish" lang="tr" hreflang="tr" data-title="Isı" 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%9A%D1%96%D0%BB%D1%8C%D0%BA%D1%96%D1%81%D1%82%D1%8C_%D1%82%D0%B5%D0%BF%D0%BB%D0%BE%D1%82%D0%B8" title="Кількість теплоти – Ukrainian" lang="uk" hreflang="uk" data-title="Кількість теплоти" 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%A7%D8%B1%D8%AA" title="حرارت – Urdu" lang="ur" hreflang="ur" data-title="حرارت" 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/Nhi%E1%BB%87t" title="Nhiệt – Vietnamese" lang="vi" hreflang="vi" data-title="Nhiệt" 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-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/L%C3%A4mm%C3%BCs" title="Lämmüs – Võro" lang="vro" hreflang="vro" data-title="Lämmüs" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-guc mw-list-item"><a href="https://guc.wikipedia.org/wiki/Walatshi" title="Walatshi – Wayuu" lang="guc" hreflang="guc" data-title="Walatshi" data-language-autonym="Wayuunaiki" data-language-local-name="Wayuu" class="interlanguage-link-target"><span>Wayuunaiki</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%86%B1" title="熱 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="熱" 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%E9%87%8F" title="热量 – Wu" lang="wuu" hreflang="wuu" data-title="热量" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%94%D7%99%D7%A5" title="היץ – Yiddish" lang="yi" hreflang="yi" data-title="היץ" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%86%B1" title="熱 – Cantonese" lang="yue" hreflang="yue" data-title="熱" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of energy transfer</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"><span>"Heating" redirects here. Not to be confused with <a href="/wiki/Home_heating" class="mw-redirect" title="Home heating">Home heating</a>.</span> <span>For other uses, see <a href="/wiki/Heat_(disambiguation)" class="mw-disambig" title="Heat (disambiguation)">Heat (disambiguation)</a>.</span></div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Heat</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Glowing_metal.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Glowing_metal.jpg/220px-Glowing_metal.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Glowing_metal.jpg/330px-Glowing_metal.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Glowing_metal.jpg/440px-Glowing_metal.jpg 2x" data-file-width="1024" data-file-height="768" /></a></span><div class="infobox-caption">A glowing-hot metal bar showing <a href="/wiki/Incandescence" class="mw-redirect" title="Incandescence">incandescence</a>, the emission of light due to its temperature, is often recognized as a source of heat.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Joule" title="Joule">joule</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><a href="/wiki/British_thermal_unit" title="British thermal unit">British thermal unit</a>, <a href="/wiki/Calorie" title="Calorie">calorie</a></td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data"><a href="/wiki/Kilogram" title="Kilogram">kg</a>⋅<a href="/wiki/Metre" title="Metre">m</a><sup>2</sup>⋅<a href="/wiki/Second" title="Second">s</a><sup>−2</sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><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 {\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {T}}^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {T}}^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32eb0714136a2ff98e05402ba7a37a9a5e3d919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.263ex; height:2.676ex;" alt="{\displaystyle {\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {T}}^{-2}}"></span></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></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">Branches</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></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/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</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"><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 class="mw-selflink selflink">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>In <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <b>heat</b> is <a href="/wiki/Energy" title="Energy">energy</a> in transfer between a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a> and its surroundings by modes other than <a href="/wiki/Thermodynamic_work" class="mw-redirect" title="Thermodynamic work">thermodynamic work</a> and transfer of matter. Such modes are microscopic, mainly <a href="/wiki/Thermal_conduction" title="Thermal conduction">thermal conduction</a>, <a href="/wiki/Radiation" title="Radiation">radiation</a>, and <a href="/wiki/Friction" title="Friction">friction</a>, as distinct from the macroscopic modes, thermodynamic work and transfer of matter.<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> For a <a href="/wiki/Closed_system" title="Closed system">closed system</a> (transfer of matter excluded), the heat involved in a process is the difference in <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> between the final and initial states of a system, and subtracting the work done in the process.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For a closed system, this is the formulation of the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>. </p><p><a href="/wiki/Calorimetry" title="Calorimetry">Calorimetry</a> is measurement of quantity of energy transferred as heat by its effect on the states of interacting bodies, for example, by the amount of ice melted or by change in <a href="/wiki/Temperature" title="Temperature">temperature</a> of a body.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI), the unit of measurement for heat, as a form of energy, is the <a href="/wiki/Joule" title="Joule">joule</a> (J). </p><p>With various other meanings, the word 'heat' is also used in engineering, and it occurs also in ordinary language, but such are not the topic of the present article. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation_and_units">Notation and units</h2></div> <p>As a form of energy, heat has the unit <a href="/wiki/Joule" title="Joule">joule</a> (J) in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI). In addition, many applied branches of engineering use other, traditional units, such as the <a href="/wiki/British_thermal_unit" title="British thermal unit">British thermal unit</a> (BTU) and the <a href="/wiki/Calorie" title="Calorie">calorie</a>. The standard unit for the rate of heating is the <a href="/wiki/Watt" title="Watt">watt</a> (W), defined as one joule per second. </p><p>The symbol <span class="texhtml mvar" style="font-style:italic;">Q</span> for heat was introduced by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> and <a href="/wiki/Macquorn_Rankine" class="mw-redirect" title="Macquorn Rankine">Macquorn Rankine</a> in <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1859</span>.<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> </p><p>Heat released by a system into its surroundings is by convention, as a contributor to internal energy, a negative quantity (<span class="texhtml"><i>Q</i> < 0</span>); when a system absorbs heat from its surroundings, it is positive (<span class="texhtml"><i>Q</i> > 0</span>). Heat transfer rate, or heat flow per unit time, is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {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>, but it is not a time derivative of a <a href="/wiki/Functions_of_state" class="mw-redirect" title="Functions of state">function of state</a> (which can also be written with the dot notation) since heat is not a function of state.<sup id="cite_ref-Baierlein1999_5-0" class="reference"><a href="#cite_note-Baierlein1999-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Heat_flux" title="Heat flux">Heat flux</a> is defined as rate of heat transfer per unit cross-sectional area (watts per square metre). </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></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/History_of_thermodynamics" title="History of thermodynamics">History of thermodynamics</a></div> <p>In common language, English 'heat' or 'warmth', just as French <i>chaleur</i>, German <i>Hitze</i> or <i>Wärme</i>, <a href="/wiki/Latin" title="Latin">Latin</a> <i>calor</i>, <a href="/wiki/Greek_language" title="Greek language">Greek</a> θάλπος, etc. refers to either <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a> or <a href="/wiki/Temperature" title="Temperature">temperature</a>, or the human <a href="/wiki/Thermoception" title="Thermoception">perception of</a> these. Later, <i>chaleur</i> (as used by <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Sadi Carnot</a>), 'heat', and <i>Wärme</i> became equivalents also as specific scientific terms at an early stage of thermodynamics. Speculation on 'heat' as a separate form of matter has a long history, involving the <a href="/wiki/Phlogiston" class="mw-redirect" title="Phlogiston">phlogiston</a> theory, the <a href="/wiki/Caloric_theory" title="Caloric theory">caloric theory</a>, and <a href="/wiki/Fire_(classical_element)" title="Fire (classical element)">fire</a>. Many careful and accurate historical experiments practically exclude friction, mechanical and thermodynamic work and matter transfer, investigating transfer of energy only by thermal conduction and radiation. Such experiments give impressive rational support to the caloric theory of heat. To account also for changes of internal energy due to friction, and mechanical and thermodynamic work, the caloric theory was, around the end of the eighteenth century, replaced by the "mechanical" theory of heat, which is accepted today. </p> <div class="mw-heading mw-heading3"><h3 id="17th_century–early_18th_century"><span id="17th_century.E2.80.93early_18th_century"></span>17th century–early 18th century</h3></div> <div class="mw-heading mw-heading4"><h4 id=""Heat_is_motion""><span id=".22Heat_is_motion.22"></span>"Heat is motion"</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Portrait_of_Galileo_Galilei.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Portrait_of_Galileo_Galilei.jpg/150px-Portrait_of_Galileo_Galilei.jpg" decoding="async" width="150" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Portrait_of_Galileo_Galilei.jpg/225px-Portrait_of_Galileo_Galilei.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Portrait_of_Galileo_Galilei.jpg/300px-Portrait_of_Galileo_Galilei.jpg 2x" data-file-width="326" data-file-height="448" /></a><figcaption>Galileo Galilei</figcaption></figure> <p>As scientists of the early modern age began to adopt the view that matter consists of particles, a close relationship between heat and the motion of those particles was widely surmised, or even the equivalency of the concepts, boldly expressed by the English philosopher <a href="/wiki/Francis_Bacon" title="Francis Bacon">Francis Bacon</a> in 1620. "It must not be thought that heat generates motion, or motion heat (though in some respects this be true), but that the very essence of heat ... is motion and nothing else."<sup id="cite_ref-FOOTNOTEBacon1902153_6-0" class="reference"><a href="#cite_note-FOOTNOTEBacon1902153-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> "not a ... motion of the whole, but of the small particles of the body."<sup id="cite_ref-FOOTNOTEBacon1902156_7-0" class="reference"><a href="#cite_note-FOOTNOTEBacon1902156-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In <i><a href="/wiki/The_Assayer" title="The Assayer">The Assayer</a></i> (published 1623) <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>, in turn, described heat as an artifact of our minds. </p> <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>... about the proposition “motion is the cause of heat”... I suspect that people in general have a concept of this which is very remote from the truth. For they believe that heat is a real phenomenon, or property ... which actually resides in the material by which we feel ourselves warmed.<sup id="cite_ref-FOOTNOTEGalilei1957273–4_8-0" class="reference"><a href="#cite_note-FOOTNOTEGalilei1957273–4-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Galileo wrote that heat and pressure are apparent properties only, caused by the movement of particles, which is a real phenomenon.<sup id="cite_ref-FOOTNOTEAdriaans2024_9-0" class="reference"><a href="#cite_note-FOOTNOTEAdriaans2024-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In 1665,<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><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> and again in 1681,<sup id="cite_ref-FOOTNOTEHooke1705[httpsarchiveorgdetailsb30454621_0001page116mode1up_116]_12-0" class="reference"><a href="#cite_note-FOOTNOTEHooke1705[httpsarchiveorgdetailsb30454621_0001page116mode1up_116]-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> English polymath <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a> reiterated that heat is nothing but the motion of the constituent particles of objects, and in 1675, his colleague, Anglo-Irish scientist <a href="/wiki/Robert_Boyle" title="Robert Boyle">Robert Boyle</a> repeated that this motion is what heat consists of.<sup id="cite_ref-FOOTNOTEBoyle1675[httpsarchiveorgdetailsexperimentsnotes00boylpagen100mode1up_61-62]_13-0" class="reference"><a href="#cite_note-FOOTNOTEBoyle1675[httpsarchiveorgdetailsexperimentsnotes00boylpagen100mode1up_61-62]-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:John_Locke._Portrait_by_Herman_Verelst_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/John_Locke._Portrait_by_Herman_Verelst_%28cropped%29.jpg/150px-John_Locke._Portrait_by_Herman_Verelst_%28cropped%29.jpg" decoding="async" width="150" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/John_Locke._Portrait_by_Herman_Verelst_%28cropped%29.jpg/225px-John_Locke._Portrait_by_Herman_Verelst_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/John_Locke._Portrait_by_Herman_Verelst_%28cropped%29.jpg/300px-John_Locke._Portrait_by_Herman_Verelst_%28cropped%29.jpg 2x" data-file-width="820" data-file-height="1006" /></a><figcaption>John Locke</figcaption></figure> <p>Heat has been discussed in ordinary language by philosophers. An example is this 1720 quote from the English philosopher <a href="/wiki/John_Locke" title="John Locke">John Locke</a>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p><i>Heat</i>, is a very brisk agitation of the insensible parts of the object, which produces in us that sensation from whence we denominate the object <i>hot</i>; so what in our sensation is <i>heat</i>, in the object is nothing but <i>motion</i>. This appears by the way, whereby heat is produc’d: for we see that the rubbing of a brass nail upon a board, will make it very hot; and the axle-trees of carts and coaches are often hot, and sometimes to a degree, that it sets them on fire, by the rubbing of the nave of the wheel upon it.<sup id="cite_ref-FOOTNOTELocke1720[httpsplaygooglecombooksreaderidQqxsP-VKrpkCpgGBSPA224hlen_GB_224]_14-0" class="reference"><a href="#cite_note-FOOTNOTELocke1720[httpsplaygooglecombooksreaderidQqxsP-VKrpkCpgGBSPA224hlen_GB_224]-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>When Bacon, Galileo, Hooke, Boyle and Locke wrote “heat”, they might more have referred to what we would now call “temperature”. No clear distinction was made between heat and temperature until the mid-18th century, nor between the internal energy of a body and the transfer of energy as heat until the mid-19th century. </p><p>Locke's description of heat was repeatedly quoted by English physicist <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Prescott Joule</a>. Also the <a href="/wiki/Heat_transfer" title="Heat transfer">transfer of heat</a> was explained by the motion of particles. Scottish physicist and chemist <a href="/wiki/Joseph_Black" title="Joseph Black">Joseph Black</a> wrote: "Many have supposed that heat is a tremulous ... motion of the particles of matter, which ... motion they imagined to be communicated from one body to another."<sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen161mode1up_80]_15-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen161mode1up_80]-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <a href="/wiki/John_Tyndall" title="John Tyndall">John Tyndall</a>'s <i>Heat Considered as Mode of Motion</i> (1863) was instrumental in popularizing the idea of heat as motion to the English-speaking public. The theory was developed in academic publications in French, English and German. </p> <div class="mw-heading mw-heading3"><h3 id="18th_century">18th century</h3></div> <div class="mw-heading mw-heading4"><h4 id="Heat_vs._temperature">Heat vs. temperature</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Brook_Taylor_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Brook_Taylor_%28cropped%29.jpg/150px-Brook_Taylor_%28cropped%29.jpg" decoding="async" width="150" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Brook_Taylor_%28cropped%29.jpg/225px-Brook_Taylor_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4b/Brook_Taylor_%28cropped%29.jpg 2x" data-file-width="233" data-file-height="282" /></a><figcaption>Brook Taylor</figcaption></figure> <p>Unstated distinctions between heat and “hotness” may be very old, heat seen as something dependent on the <i>quantity</i> of a hot substance, “heat”, vaguely perhaps distinct from the <i>quality</i> of "hotness". In 1723, the English mathematician <a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a> measured the temperature—the expansion of the liquid in a thermometer—of mixtures of various amounts of hot water in cold water. As expected, the increase in temperature was in proportion to the proportion of hot water in the mixture. The distinction between heat and temperature is implicitly expressed in the last sentence of his report.<sup id="cite_ref-FOOTNOTETaylor1723[httpsroyalsocietypublishingorgdoiepdf101098rstl17220053_291]_16-0" class="reference"><a href="#cite_note-FOOTNOTETaylor1723[httpsroyalsocietypublishingorgdoiepdf101098rstl17220053_291]-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>I successively fill'd the Vessels with one, two, three, &c. Parts of hot boiling Water, and the rest cold ... And having first observed where the Thermometer stood in cold Water, I found that its rising from that Mark ... was accurately proportional to the Quantity of hot Water in the Mixture, that is, to the Degree of Heat.</p></blockquote> <div class="mw-heading mw-heading4"><h4 id="Evaporative_cooling">Evaporative cooling</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:William_Cullen_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/William_Cullen_%28cropped%29.jpg/150px-William_Cullen_%28cropped%29.jpg" decoding="async" width="150" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/William_Cullen_%28cropped%29.jpg/226px-William_Cullen_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/William_Cullen_%28cropped%29.jpg/301px-William_Cullen_%28cropped%29.jpg 2x" data-file-width="401" data-file-height="488" /></a><figcaption>William Cullen</figcaption></figure><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Latent_heat#History" title="Latent heat">Latent heat § History</a></div> <p>In 1748, an account was published in <i>The Edinburgh Physical and Literary Essays</i> of an experiment by the Scottish physician and chemist <a href="/wiki/William_Cullen" title="William Cullen">William Cullen</a>. Cullen had used an <a href="/wiki/Air_pump" title="Air pump">air pump</a> to lower the pressure in a container with <a href="/wiki/Diethyl_ether" title="Diethyl ether">diethyl ether</a>. The ether boiled, while no heat was withdrawn from it, and its temperature decreased.<sup id="cite_ref-FOOTNOTEWest2014_17-0" class="reference"><a href="#cite_note-FOOTNOTEWest2014-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]_18-0" class="reference"><a href="#cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> And in 1758 on a warm day in <a href="/wiki/Cambridge" title="Cambridge">Cambridge</a>, England, <a href="/wiki/Benjamin_Franklin" title="Benjamin Franklin">Benjamin Franklin</a> and fellow scientist <a href="/wiki/John_Hadley_(chemist)" title="John Hadley (chemist)">John Hadley</a> experimented by continually wetting the ball of a mercury <a href="/wiki/Thermometer" title="Thermometer">thermometer</a> with ether and using <a href="/wiki/Bellows" title="Bellows">bellows</a> to evaporate the ether.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> With each subsequent <a href="/wiki/Evaporation" title="Evaporation">evaporation</a>, the thermometer read a lower temperature, eventually reaching 7 °F (−14 °C). </p> <div class="mw-heading mw-heading4"><h4 id="Discovery_of_specific_heat">Discovery of specific heat</h4></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/Specific_heat_capacity#History" title="Specific heat capacity">Specific heat capacity § History</a></div> <p>In 1756 or soon thereafter, Joseph Black, Cullen’s friend and former assistant, began an extensive study of heat.<sup id="cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]_18-1" class="reference"><a href="#cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1760 Black realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by <a href="/wiki/Daniel_Gabriel_Fahrenheit" title="Daniel Gabriel Fahrenheit">Daniel Gabriel Fahrenheit</a> on behalf of Dutch physician <a href="/wiki/Herman_Boerhaave" title="Herman Boerhaave">Herman Boerhaave</a>. For clarity, he then described a hypothetical but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of <a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">specific heat capacity</a>, being different for different substances. Black wrote: "Quicksilver [mercury] ... has less capacity for the matter of heat than water."<sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen157mode1up_76-77]_20-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen157mode1up_76-77]-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Degrees_of_heat">Degrees of heat</h4></div> <p>In his investigations of specific heat, Black used a unit of heat he called "degrees of heat"—as opposed to just "degrees" [of temperature]. This unit was context-dependent and could only be used when circumstances were identical. It was based on change in temperature multiplied by the mass of the substance involved.<sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen162mode1up_81]_21-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen162mode1up_81]-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If the stone and water ... were equal in bulk ... the water was heated by 10 degrees, the stone ... cooled 20 degrees; but if ... the stone had only the fiftieth part of the bulk of the water, it must have been ... 1000 degrees hotter before it was plunged into the water than it is now, for otherwise it could not have communicated 10 degrees of heat to ... [the] water.</p></blockquote> <div class="mw-heading mw-heading4"><h4 id="Discovery_of_latent_heat">Discovery of latent heat</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Black_Joseph_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Black_Joseph_%28cropped%29.jpg/150px-Black_Joseph_%28cropped%29.jpg" decoding="async" width="150" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Black_Joseph_%28cropped%29.jpg/224px-Black_Joseph_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Black_Joseph_%28cropped%29.jpg/299px-Black_Joseph_%28cropped%29.jpg 2x" data-file-width="517" data-file-height="677" /></a><figcaption>Joseph Black</figcaption></figure><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Latent_heat#History" title="Latent heat">Latent heat § History</a></div> <p>It was known that when the air temperature rises above freezing—air then becoming the obvious heat source—snow melts very slowly and the temperature of the melted snow is close to its freezing point.<sup id="cite_ref-FOOTNOTEWest2014_17-1" class="reference"><a href="#cite_note-FOOTNOTEWest2014-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In 1757, Black started to investigate if heat, therefore, was required for the melting of a solid, independent of any rise in temperature. As far Black knew, the general view at that time was that melting was inevitably accompanied by a small increase in temperature, and that no additional heat was needed beyond what this increase in temperature would require in itself. Soon, however, Black was able to show that much more heat was required during melting than could be explained by any increase in temperature alone.<sup id="cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]_22-0" class="reference"><a href="#cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen192mode1up_111-112]_23-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen192mode1up_111-112]-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> He was also able to show that heat is <i>released</i> by a liquid during its freezing; again, much more than could be explained by the decrease of its temperature alone.<sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen193mode1up_112]_24-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen193mode1up_112]-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>In 1762, Black announced the following research and results to a society of professors at the University of Glasgow.<sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]_25-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Black had placed equal masses of ice at 32 °F (0 °C) and water at 33 °F (0.6 °C) respectively in two identical, well separated containers. The water and the ice were both evenly heated to 40 °F by the air in the room, which was at a constant 47 °F (8 °C). The water had therefore received 40 – 33 = 7 “degrees of heat”. The ice had been heated for 21 times longer and had therefore received 7 × 21 = 147 “degrees of heat”.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The temperature of the ice had increased by 8 °F. The ice had thus absorbed 8 “degrees of heat”, which Black called <i>sensible heat</i>, manifest as temperature change, which could be felt and measured. In addition to that, 147 – 8 = 139 “degrees of heat” were absorbed as <i>latent heat</i>, manifest as phase change rather than as temperature change.<sup id="cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]_22-1" class="reference"><a href="#cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen196mode1up_115-117]_27-0" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen196mode1up_115-117]-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Black next showed that a water temperature of 176 °F was needed to melt an equal mass of ice until it was all 32 °F. So now 176 – 32 = 144 “degrees of heat” seemed to be needed to melt the ice. The modern value for the heat of fusion of ice would be 143 “degrees of heat” on the same scale (79.5 “degrees of heat Celsius”).<sup id="cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45]_28-0" class="reference"><a href="#cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45]-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]_25-1" class="reference"><a href="#cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Finally, Black increased the temperature of a mass of water, then vaporized an equal mass of water by even heating. He showed that 830 “degrees of heat” was needed for the vaporization; again based on the time required. The modern value for the heat of vaporization of water would be 967 “degrees of heat” on the same scale.<sup id="cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45-46]_29-0" class="reference"><a href="#cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45-46]-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="First_calorimeter">First calorimeter</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Ice_calorimeter_(cropped).jpg" class="mw-file-description"><img alt="Diagram of Lavoisier's and Laplace's ice calorimeter" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Ice_calorimeter_%28cropped%29.jpg/150px-Ice_calorimeter_%28cropped%29.jpg" decoding="async" width="150" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Ice_calorimeter_%28cropped%29.jpg/224px-Ice_calorimeter_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/4/42/Ice_calorimeter_%28cropped%29.jpg 2x" data-file-width="270" data-file-height="440" /></a><figcaption>Lavoisier's and Laplace's ice calorimeter</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Calorimeter" title="Calorimeter">Calorimeter</a> and <a href="/wiki/Calorimetry" title="Calorimetry">Calorimetry</a></div> <p>A calorimeter is a device used for measuring <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a>, as well as the heat absorbed or released in <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reactions</a> or <a href="/wiki/Physical_change" title="Physical change">physical changes</a>. In 1780, French chemist <a href="/wiki/Antoine_Lavoisier" title="Antoine Lavoisier">Antoine Lavoisier</a> used such an apparatus—which he named 'calorimeter'—to investigate the heat released by <a href="/wiki/Respiration_(physiology)" title="Respiration (physiology)">respiration</a>, by observing how this heat melted snow surrounding his apparatus.<sup id="cite_ref-FOOTNOTELavoisier1790[httpswwwgutenbergorgcacheepub30775pg30775-imageshtmlPage_343_345]_30-0" class="reference"><a href="#cite_note-FOOTNOTELavoisier1790[httpswwwgutenbergorgcacheepub30775pg30775-imageshtmlPage_343_345]-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> A so called <i>ice calorimeter</i> was used 1782–83 by Lavoisier and his colleague <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> to measure the heat released in various chemical reactions. The heat so released melted a specific amount of ice, and the heat required for the melting of a certain amount of ice was known beforehand.<sup id="cite_ref-FOOTNOTEBuchholzSchoeller2004[httpwwwajcnorgcgicontentfull795899S_899S–906S]_32-0" class="reference"><a href="#cite_note-FOOTNOTEBuchholzSchoeller2004[httpwwwajcnorgcgicontentfull795899S_899S–906S]-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Classical_thermodynamics">Classical thermodynamics</h3></div> <p>The modern understanding of heat is often partly attributed to <a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a>'s 1798 <a href="/wiki/Mechanical_theory_of_heat" class="mw-redirect" title="Mechanical theory of heat">mechanical theory of heat</a> (<i><a href="/wiki/An_Experimental_Enquiry_Concerning_the_Source_of_the_Heat_which_is_Excited_by_Friction" class="mw-redirect" title="An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction">An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction</a></i>), postulating a <a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">mechanical equivalent of heat</a>. A collaboration between <a href="/wiki/Nicolas_Cl%C3%A9ment" title="Nicolas Clément">Nicolas Clément</a> and <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Sadi Carnot</a> (<i><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></i>) in the 1820s had some related thinking along similar lines.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> In 1842, <a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">Julius Robert Mayer</a> frictionally generated heat in paper pulp and measured the temperature rise.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> In 1845, Joule published a paper entitled <i>The Mechanical Equivalent of Heat</i>, in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on heat production by friction in the passage of electricity through a resistor and in the rotation of a paddle in a vat of water.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> The theory of classical thermodynamics matured in the 1850s to 1860s. </p> <div class="mw-heading mw-heading4"><h4 id="Clausius_(1850)"><span id="Clausius_.281850.29"></span>Clausius (1850)</h4></div> <p>In 1850, Clausius, responding to Joule's experimental demonstrations of heat production by friction, rejected the caloric doctrine of conservation of heat, writing: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If we assume that heat, like matter, cannot be lessened in quantity, we must also assume that it cannot be increased; but it is almost impossible to explain the ascension of temperature brought about by friction otherwise than by assuming an actual increase of heat. The careful experiments of Joule, who developed heat in various ways by the application of mechanical force, establish almost to a certainty, not only the possibility of increasing the quantity of heat, but also the fact that the newly-produced heat is proportional to the work expended in its production. It may be remarked further, that many facts have lately transpired which tend to overthrow the hypothesis that heat is itself a body, and to prove that it consists in a motion of the ultimate particles of bodies.<sup id="cite_ref-Clausius1850_36-0" class="reference"><a href="#cite_note-Clausius1850-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>The process function <span class="texhtml mvar" style="font-style:italic;">Q</span> was introduced by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> in 1850. Clausius described it with the German compound <i>Wärmemenge</i>, translated as "amount of heat".<sup id="cite_ref-Clausius1850_36-1" class="reference"><a href="#cite_note-Clausius1850-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="James_Clerk_Maxwell_(1871)"><span id="James_Clerk_Maxwell_.281871.29"></span>James Clerk Maxwell (1871)</h4></div> <p><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> in his 1871 <i>Theory of Heat</i> outlines four stipulations for the definition of heat: </p> <ul><li>It is <i>something which may be transferred from one body to another</i>, according to the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>.</li> <li>It is a <i>measurable quantity</i>, and so can be treated mathematically.</li> <li>It <i>cannot be treated as a material substance</i>, because it may be transformed into something that is not a material substance, e.g., <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">mechanical work</a>.</li> <li>Heat is <i>one of the forms of energy</i>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="Bryan_(1907)"><span id="Bryan_.281907.29"></span>Bryan (1907)</h4></div> <p>In 1907, G.H. Bryan published an investigation of the foundations of thermodynamics, <i>Thermodynamics: an Introductory Treatise dealing mainly with First Principles and their Direct Applications</i>, B.G. Teubner, Leipzig. </p><p>Bryan was writing when thermodynamics had been established empirically, but people were still interested to specify its logical structure. The 1909 work of Carathéodory also belongs to this historical era. Bryan was a physicist while Carathéodory was a mathematician. </p><p>Bryan started his treatise with an introductory chapter on the notions of heat and of temperature. He gives an example of where the notion of heating as raising a body's temperature contradicts the notion of heating as imparting a quantity of heat to that body. </p><p>He defined an adiabatic transformation as one in which the body neither gains nor loses heat. This is not quite the same as defining an adiabatic transformation as one that occurs to a body enclosed by walls impermeable to radiation and conduction. </p><p>He recognized calorimetry as a way of measuring quantity of heat. He recognized water as having a <i>temperature of maximum density</i>. This makes water unsuitable as a thermometric substance around that temperature. He intended to remind readers of why thermodynamicists preferred an absolute scale of temperature, independent of the properties of a particular thermometric substance. </p><p>His second chapter started with the recognition of friction as a source of heat, by <a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Benjamin Thompson</a>, by <a href="/wiki/Humphry_Davy" title="Humphry Davy">Humphry Davy</a>, by <a href="/wiki/Julius_Robert_von_Mayer" class="mw-redirect" title="Julius Robert von Mayer">Robert Mayer</a>, and by <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Prescott Joule</a>. </p><p>He stated the <i>First Law of Thermodynamics</i>, or <i>Mayer–Joule Principle</i> as follows: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>When heat is transformed into work or conversely work is transformed into heat, the quantity of heat gained or lost is proportional to the quantity of work lost or gained.<sup id="cite_ref-Bryan_1907_38-0" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>He wrote: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If heat be measured in dynamical units the mechanical equivalent becomes equal to unity, and the equations of thermodynamics assume a simpler and more symmetrical form.<sup id="cite_ref-Bryan_1907_38-1" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>He explained how the caloric theory of Lavoisier and Laplace made sense in terms of pure calorimetry, though it failed to account for conversion of work into heat by such mechanisms as friction and conduction of electricity. </p><p>Having rationally defined quantity of heat, he went on to consider the second law, including the Kelvin definition of absolute thermodynamic temperature. </p><p>In section 41, he wrote: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p><b>§41. Physical unreality of reversible processes.</b> In Nature all phenomena are irreversible in a greater or less degree. The motions of celestial bodies afford the closest approximations to reversible motions, but motions which occur on this earth are largely retarded by friction, viscosity, electric and other resistances, and if the relative velocities of moving bodies were reversed, these resistances would still retard the relative motions and would not accelerate them as they should if the motions were perfectly reversible.<sup id="cite_ref-Bryan_1907_38-2" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>He then stated the principle of conservation of energy. </p><p>He then wrote: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>In connection with irreversible phenomena the following axioms have to be assumed. </p><ol><li>If a system can undergo an irreversible change it will do so.</li> <li>A perfectly reversible change cannot take place of itself; such a change can only be regarded as the limiting form of an irreversible change.<sup id="cite_ref-Bryan_1907_38-3" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></li></ol></blockquote> <p>On page 46, thinking of closed systems in thermal connection, he wrote: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>We are thus led to postulate a system in which <i>energy can pass from one element to another otherwise than by the performance of mechanical work</i>.<sup id="cite_ref-Bryan_1907_38-4" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>On page 47, still thinking of closed systems in thermal connection, he wrote: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p><b>§58. Quantity of Heat. Definition.</b> When energy flows from one system or part of a system to another otherwise than by the performance of work, the energy so transferred i[s] called <i>heat</i>.<sup id="cite_ref-Bryan_1907_38-5" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>On page 48, he wrote: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p><b>§ 59.</b> When two bodies act thermically on one another the quantities of heat gained by one and lost by the other are not necessarily equal. </p><p>In the case of bodies at a distance, heat may be taken from or given to the intervening medium. </p><p>The quantity of heat received by any portion of the ether may be defined in the same way as that received by a material body. [He was thinking of thermal radiation.] </p><p> Another important exception occurs when sliding takes place between two rough bodies in contact. The algebraic sum of the works done is different from zero, because, although the action and reaction are equal and opposite the velocities of the parts of the bodies in contact are different. Moreover, the work lost in the process does not increase the mutual potential energy of the system and there is no intervening medium between the bodies. Unless the lost energy can be accounted for in other ways, (as when friction produces electrification), it follows from the Principle of Conservation of Energy that the algebraic sum of the quantities of heat gained by the two systems is equal to the quantity of work lost by friction. [This thought was echoed by Bridgman, as above.]<sup id="cite_ref-Bryan_1907_38-6" class="reference"><a href="#cite_note-Bryan_1907-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <div class="mw-heading mw-heading4"><h4 id="Carathéodory_(1909)"><span id="Carath.C3.A9odory_.281909.29"></span>Carathéodory (1909)</h4></div> <p>A celebrated and frequent definition of heat in thermodynamics is based on the work of <a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a> (1909), referring to processes in a closed system.<sup id="cite_ref-Carathéodory_1909_39-0" class="reference"><a href="#cite_note-Carathéodory_1909-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> Carathéodory was responding to a suggestion by Max Born that he examine the logical structure of thermodynamics. </p><p>The <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> <span class="texhtml"><i>U<sub>X</sub></i></span> of a body in an arbitrary state <span class="texhtml"><i>X</i></span> can be determined by amounts of work adiabatically performed by the body on its surroundings when it starts from a reference state <span class="texhtml"><i>O</i></span>. Such work is assessed through quantities defined in the surroundings of the body. It is supposed that such work can be assessed accurately, without error due to friction in the surroundings; friction in the body is not excluded by this definition. The adiabatic performance of work is defined in terms of adiabatic walls, which allow transfer of energy as work, but no other transfer, of energy or matter. In particular they do not allow the passage of energy as heat. According to this definition, work performed adiabatically is in general accompanied by friction within the thermodynamic system or body. On the other hand, according to Carathéodory (1909), there also exist non-adiabatic, <i>diathermal</i> walls, which are postulated to be permeable only to heat. </p><p>For the definition of quantity of energy transferred as heat, it is customarily envisaged that an arbitrary state of interest <span class="texhtml"><i>Y</i></span> is reached from state <span class="texhtml"><i>O</i></span> by a process with two components, one adiabatic and the other not adiabatic. For convenience one may say that the adiabatic component was the sum of work done by the body through volume change through movement of the walls while the non-adiabatic wall was temporarily rendered adiabatic, and of isochoric adiabatic work. Then the non-adiabatic component is a process of energy transfer through the wall that passes only heat, newly made accessible for the purpose of this transfer, from the surroundings to the body. The change in internal energy to reach the state <span class="texhtml"><i>Y</i></span> from the state <span class="texhtml"><i>O</i></span> is the difference of the two amounts of energy transferred. </p><p>Although Carathéodory himself did not state such a definition, following his work it is customary in theoretical studies to define heat, <span class="texhtml"><i>Q</i></span>, to the body from its surroundings, in the combined process of change to state <span class="texhtml"><i>Y</i></span> from the state <span class="texhtml"><i>O</i></span>, as the change in internal energy, <span class="texhtml">Δ<i>U<sub>Y</sub></i></span>, minus the amount of work, <span class="texhtml"><i>W</i></span>, done by the body on its surrounds by the adiabatic process, so that <span class="texhtml"><i>Q</i> = Δ<i>U<sub>Y</sub></i> − <i>W</i></span>. </p><p>In this definition, for the sake of conceptual rigour, the quantity of energy transferred as heat is not specified directly in terms of the non-adiabatic process. It is defined through knowledge of precisely two variables, the change of internal energy and the amount of adiabatic work done, for the combined process of change from the reference state <span class="texhtml"><i>O</i></span> to the arbitrary state <span class="texhtml"><i>Y</i></span>. It is important that this does not explicitly involve the amount of energy transferred in the non-adiabatic component of the combined process. It is assumed here that the amount of energy required to pass from state <span class="texhtml"><i>O</i></span> to state <span class="texhtml"><i>Y</i></span>, the change of internal energy, is known, independently of the combined process, by a determination through a purely adiabatic process, like that for the determination of the internal energy of state <span class="texhtml"><i>X</i></span> above. The rigour that is prized in this definition is that there is one and only one kind of energy transfer admitted as fundamental: energy transferred as work. Energy transfer as heat is considered as a derived quantity. The uniqueness of work in this scheme is considered to guarantee rigor and purity of conception. The conceptual purity of this definition, based on the concept of energy transferred as work as an ideal notion, relies on the idea that some frictionless and otherwise non-dissipative processes of energy transfer can be realized in physical actuality. The second law of thermodynamics, on the other hand, assures us that such processes are not found in nature. </p><p>Before the rigorous mathematical definition of heat based on Carathéodory's 1909 paper, historically, heat, temperature, and thermal equilibrium were presented in thermodynamics textbooks as jointly <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notions</a>.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Carathéodory introduced his 1909 paper thus: "The proposition that the discipline of thermodynamics can be justified without recourse to any hypothesis that cannot be verified experimentally must be regarded as one of the most noteworthy results of the research in thermodynamics that was accomplished during the last century." Referring to the "point of view adopted by most authors who were active in the last fifty years", Carathéodory wrote: "There exists a physical quantity called heat that is not identical with the mechanical quantities (mass, force, pressure, etc.) and whose variations can be determined by calorimetric measurements." <a href="/wiki/James_Serrin" title="James Serrin">James Serrin</a> introduces an account of the theory of thermodynamics thus: "In the following section, we shall use the classical notions of <i>heat</i>, <i>work</i>, and <i>hotness</i> as primitive elements, ... That heat is an appropriate and natural primitive for thermodynamics was already accepted by Carnot. Its continued validity as a primitive element of thermodynamical structure is due to the fact that it synthesizes an essential physical concept, as well as to its successful use in recent work to unify different constitutive theories."<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> This traditional kind of presentation of the basis of thermodynamics includes ideas that may be summarized by the statement that heat transfer is purely due to spatial non-uniformity of temperature, and is by conduction and radiation, from hotter to colder bodies. It is sometimes proposed that this traditional kind of presentation necessarily rests on "<a href="/wiki/Circular_reasoning" title="Circular reasoning">circular reasoning</a>". </p><p>This alternative approach to the definition of quantity of energy transferred as heat differs in logical structure from that of Carathéodory, recounted just above. </p><p>This alternative approach admits calorimetry as a primary or direct way to measure quantity of energy transferred as heat. It relies on temperature as one of its primitive concepts, and used in calorimetry.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> It is presupposed that enough processes exist physically to allow measurement of differences in internal energies. Such processes are not restricted to adiabatic transfers of energy as work. They include calorimetry, which is the commonest practical way of finding internal energy differences.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> The needed temperature can be either empirical or absolute thermodynamic. </p><p>In contrast, the Carathéodory way recounted just above does not use calorimetry or temperature in its primary definition of quantity of energy transferred as heat. The Carathéodory way regards calorimetry only as a secondary or indirect way of measuring quantity of energy transferred as heat. As recounted in more detail just above, the Carathéodory way regards quantity of energy transferred as heat in a process as primarily or directly defined as a residual quantity. It is calculated from the difference of the internal energies of the initial and final states of the system, and from the actual work done by the system during the process. That internal energy difference is supposed to have been measured in advance through processes of purely adiabatic transfer of energy as work, processes that take the system between the initial and final states. By the Carathéodory way it is presupposed as known from experiment that there actually physically exist enough such adiabatic processes, so that there need be no recourse to calorimetry for measurement of quantity of energy transferred as heat. This presupposition is essential but is explicitly labeled neither as a law of thermodynamics nor as an axiom of the Carathéodory way. In fact, the actual physical existence of such adiabatic processes is indeed mostly supposition, and those supposed processes have in most cases not been actually verified empirically to exist.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Planck_(1926)"><span id="Planck_.281926.29"></span>Planck (1926)</h4></div> <p>Over the years, for example in his 1879 thesis, but particularly in 1926, Planck advocated regarding the generation of heat by rubbing as the most specific way to define heat.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Planck criticised Carathéodory for not attending to this.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> Carathéodory was a mathematician who liked to think in terms of adiabatic processes, and perhaps found friction too tricky to think about, while Planck was a physicist. </p> <div class="mw-heading mw-heading2"><h2 id="Heat_transfer">Heat transfer</h2></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/Heat_transfer" title="Heat transfer">Heat transfer</a></div> <div class="mw-heading mw-heading3"><h3 id="Heat_transfer_between_two_bodies">Heat transfer between two bodies</h3></div> <p>Referring to conduction, <a href="/wiki/J.R._Partington" class="mw-redirect" title="J.R. Partington">Partington</a> writes: "If a hot body is brought in conducting contact with a cold body, the temperature of the hot body falls and that of the cold body rises, and it is said that a <i>quantity of heat</i> has passed from the hot body to the cold body."<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p>Referring to radiation, <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a> writes: "In Radiation, the hotter body loses heat, and the colder body receives heat by means of a process occurring in some intervening medium which does not itself thereby become hot."<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>Maxwell writes that convection as such "is not a purely thermal phenomenon".<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> In thermodynamics, convection in general is regarded as transport of <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>. If, however, the convection is enclosed and circulatory, then it may be regarded as an intermediary that transfers energy as heat between source and destination bodies, because it transfers only energy and not matter from the source to the destination body.<sup id="cite_ref-Chandrasekhar_1961_56-0" class="reference"><a href="#cite_note-Chandrasekhar_1961-56"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p>In accordance with the first law for closed systems, energy transferred solely as heat leaves one body and enters another, changing the internal energies of each. Transfer, between bodies, of energy as work is a complementary way of changing internal energies. Though it is not logically rigorous from the viewpoint of strict physical concepts, a common form of words that expresses this is to say that heat and work are interconvertible. </p><p>Cyclically operating engines that use only heat and work transfers have two thermal reservoirs, a hot and a cold one. They may be classified by the range of operating temperatures of the working body, relative to those reservoirs. In a heat engine, the working body is at all times colder than the hot reservoir and hotter than the cold reservoir. In a sense, it uses heat transfer to produce work. In a heat pump, the working body, at stages of the cycle, goes both hotter than the hot reservoir, and colder than the cold reservoir. In a sense, it uses work to produce heat transfer. </p> <div class="mw-heading mw-heading3"><h3 id="Heat_engine">Heat engine</h3></div> <p>In classical thermodynamics, a commonly considered model is the <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a>. It consists of four bodies: the working body, the hot reservoir, the cold reservoir, and the work reservoir. A cyclic process leaves the working body in an unchanged state, and is envisaged as being repeated indefinitely often. Work transfers between the working body and the work reservoir are envisaged as reversible, and thus only one work reservoir is needed. But two thermal reservoirs are needed, because transfer of energy as heat is irreversible. A single cycle sees energy taken by the working body from the hot reservoir and sent to the two other reservoirs, the work reservoir and the cold reservoir. The hot reservoir always and only supplies energy, and the cold reservoir always and only receives energy. The second law of thermodynamics requires that no cycle can occur in which no energy is received by the cold reservoir. Heat engines achieve higher efficiency when the ratio of the initial and final temperature is greater. </p> <div class="mw-heading mw-heading3"><h3 id="Heat_pump_or_refrigerator">Heat pump or refrigerator</h3></div> <p>Another commonly considered model is the <a href="/wiki/Heat_pump" title="Heat pump">heat pump</a> or refrigerator. Again there are four bodies: the working body, the hot reservoir, the cold reservoir, and the work reservoir. A single cycle starts with the working body colder than the cold reservoir, and then energy is taken in as heat by the working body from the cold reservoir. Then the work reservoir does work on the working body, adding more to its internal energy, making it hotter than the hot reservoir. The hot working body passes heat to the hot reservoir, but still remains hotter than the cold reservoir. Then, by allowing it to expand without passing heat to another body, the working body is made colder than the cold reservoir. It can now accept heat transfer from the cold reservoir to start another cycle. </p><p>The device has transported energy from a colder to a hotter reservoir, but this is not regarded as by an inanimate agency; rather, it is regarded as by the harnessing of work . This is because work is supplied from the work reservoir, not just by a simple thermodynamic process, but by a cycle of <a href="/wiki/Thermodynamic_operation" title="Thermodynamic operation">thermodynamic operations</a> and processes, which may be regarded as directed by an animate or harnessing agency. Accordingly, the cycle is still in accord with the second law of thermodynamics. The 'efficiency' of a heat pump (which exceeds unity) is best when the temperature difference between the hot and cold reservoirs is least. </p><p>Functionally, such engines are used in two ways, distinguishing a target reservoir and a resource or surrounding reservoir. A heat pump transfers heat to the hot reservoir as the target from the resource or surrounding reservoir. A refrigerator transfers heat, from the cold reservoir as the target, to the resource or surrounding reservoir. The target reservoir may be regarded as leaking: when the target leaks heat to the surroundings, heat pumping is used; when the target leaks coldness to the surroundings, refrigeration is used. The engines harness work to overcome the leaks. </p> <div class="mw-heading mw-heading3"><h3 id="Macroscopic_view">Macroscopic view</h3></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-Cleanup_rewrite plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Crystal_Clear_app_kedit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/40px-Crystal_Clear_app_kedit.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/60px-Crystal_Clear_app_kedit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/80px-Crystal_Clear_app_kedit.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>may need to be rewritten</b> to comply with Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>.<span class="hide-when-compact"> <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Heat&action=edit">You can help</a>. The <a href="/wiki/Talk:Heat" title="Talk:Heat">talk page</a> may contain suggestions.</span> <span class="date-container"><i>(<span class="date">May 2016</span>)</i></span></div></td></tr></tbody></table> <p>According to <a href="/wiki/Planck" class="mw-redirect" title="Planck">Planck</a>, there are three main conceptual approaches to heat.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> One is the microscopic or kinetic theory approach. The other two are macroscopic approaches. One of the macroscopic approaches is through the law of conservation of energy taken as prior to thermodynamics, with a mechanical analysis of processes, for example in the work of Helmholtz. This mechanical view is taken in this article as currently customary for thermodynamic theory. The other macroscopic approach is the thermodynamic one, which admits heat as a primitive concept, which contributes, by scientific induction<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> to knowledge of the law of conservation of energy. This view is widely taken as the practical one, quantity of heat being measured by calorimetry. </p><p>Bailyn also distinguishes the two macroscopic approaches as the mechanical and the thermodynamic.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> The thermodynamic view was taken by the founders of thermodynamics in the nineteenth century. It regards quantity of energy transferred as heat as a primitive concept coherent with a primitive concept of temperature, measured primarily by calorimetry. A calorimeter is a body in the surroundings of the system, with its own temperature and internal energy; when it is connected to the system by a path for heat transfer, changes in it measure heat transfer. The mechanical view was pioneered by Helmholtz and developed and used in the twentieth century, largely through the influence of <a href="/wiki/Max_Born" title="Max Born">Max Born</a>.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> It regards quantity of heat transferred as heat as a derived concept, defined for closed systems as quantity of heat transferred by mechanisms other than work transfer, the latter being regarded as primitive for thermodynamics, defined by macroscopic mechanics. According to Born, the transfer of internal energy between open systems that accompanies transfer of matter "cannot be reduced to mechanics".<sup id="cite_ref-Born,_M_1949_p._44_61-0" class="reference"><a href="#cite_note-Born,_M_1949_p._44-61"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> It follows that there is no well-founded definition of quantities of energy transferred as heat or as work associated with transfer of matter. </p><p>Nevertheless, for the thermodynamical description of non-equilibrium processes, it is desired to consider the effect of a temperature gradient established by the surroundings across the system of interest when there is no physical barrier or wall between system and surroundings, that is to say, when they are open with respect to one another. The impossibility of a mechanical definition in terms of work for this circumstance does not alter the physical fact that a temperature gradient causes a diffusive flux of internal energy, a process that, in the thermodynamic view, might be proposed as a candidate concept for transfer of energy as heat. </p><p>In this circumstance, it may be expected that there may also be active other drivers of diffusive flux of internal energy, such as gradient of chemical potential which drives transfer of matter, and gradient of electric potential which drives electric current and iontophoresis; such effects usually interact with diffusive flux of internal energy driven by temperature gradient, and such interactions are known as cross-effects.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p><p>If cross-effects that result in diffusive transfer of internal energy were also labeled as heat transfers, they would sometimes violate the rule that pure heat transfer occurs only down a temperature gradient, never up one. They would also contradict the principle that all heat transfer is of one and the same kind, a principle founded on the idea of heat conduction between closed systems. One might to try to think narrowly of heat flux driven purely by temperature gradient as a conceptual component of diffusive internal energy flux, in the thermodynamic view, the concept resting specifically on careful calculations based on detailed knowledge of the processes and being indirectly assessed. In these circumstances, if perchance it happens that no transfer of matter is actualized, and there are no cross-effects, then the thermodynamic concept and the mechanical concept coincide, as if one were dealing with closed systems. But when there is transfer of matter, the exact laws by which temperature gradient drives diffusive flux of internal energy, rather than being exactly knowable, mostly need to be assumed, and in many cases are practically unverifiable. Consequently, when there is transfer of matter, the calculation of the pure 'heat flux' component of the diffusive flux of internal energy rests on practically unverifiable assumptions.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>quotations 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> This is a reason to think of heat as a specialized concept that relates primarily and precisely to closed systems, and applicable only in a very restricted way to open systems. </p><p>In many writings in this context, the term "heat flux" is used when what is meant is therefore more accurately called diffusive flux of internal energy; such usage of the term "heat flux" is a residue of older and now obsolete language usage that allowed that a body may have a "heat content".<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Microscopic_view">Microscopic view</h3></div> <p>In the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory</a>, heat is explained in terms of the microscopic motions and interactions of constituent particles, such as electrons, atoms, and molecules.<sup id="cite_ref-Kittel_&_Kroemer_67-0" class="reference"><a href="#cite_note-Kittel_&_Kroemer-67"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> The immediate meaning of the kinetic energy of the constituent particles is not as heat. It is as a component of internal energy. In microscopic terms, heat is a transfer quantity, and is described by a transport theory, not as steadily localized kinetic energy of particles. Heat transfer arises from temperature gradients or differences, through the diffuse exchange of microscopic kinetic and potential particle energy, by particle collisions and other interactions. An early and vague expression of this was made by <a href="/wiki/Francis_Bacon" title="Francis Bacon">Francis Bacon</a>.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> Precise and detailed versions of it were developed in the nineteenth century.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, for a closed system (no transfer of matter), heat is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in occupation numbers of the energy levels of the system, without change in the values of the energy levels themselves.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> It is possible for macroscopic thermodynamic work to alter the occupation numbers without change in the values of the system energy levels themselves, but what distinguishes transfer as heat is that the transfer is entirely due to disordered, microscopic action, including radiative transfer. A <a href="/wiki/Microstate_(statistical_mechanics)#Microscopic_definitions_of_thermodynamic_concepts" title="Microstate (statistical mechanics)">mathematical definition</a> can be formulated for small increments of quasi-static adiabatic work in terms of the statistical distribution of an ensemble of microstates. </p> <div class="mw-heading mw-heading3"><h3 id="Calorimetry">Calorimetry</h3></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/Calorimetry" title="Calorimetry">Calorimetry</a></div> <p>Quantity of heat transferred can be measured by calorimetry, or determined through calculations based on other quantities. </p><p>Calorimetry is the empirical basis of the idea of quantity of heat transferred in a process. The transferred heat is measured by changes in a body of known properties, for example, temperature rise, change in volume or length, or phase change, such as melting of ice.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> </p><p>A calculation of quantity of heat transferred can rely on a hypothetical quantity of energy transferred as <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic</a> work and on the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>. Such calculation is the primary approach of many theoretical studies of quantity of heat transferred.<sup id="cite_ref-Carathéodory_1909_39-1" class="reference"><a href="#cite_note-Carathéodory_1909-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bryan_47_74-0" class="reference"><a href="#cite_note-Bryan_47-74"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Engineering">Engineering</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section 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/Heat" title="Special:EditPage/Heat">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>.</span> <span class="date-container"><i>(<span class="date">May 2016</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> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hot_metalwork.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Hot_metalwork.jpg/220px-Hot_metalwork.jpg" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Hot_metalwork.jpg/330px-Hot_metalwork.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Hot_metalwork.jpg/440px-Hot_metalwork.jpg 2x" data-file-width="1600" data-file-height="1163" /></a><figcaption>A red-hot iron rod from which <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a> to the surrounding environment will be primarily through <a href="/wiki/Thermal_radiation" title="Thermal radiation">radiation</a></figcaption></figure> <p>The discipline of <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>, typically considered an aspect of <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a> and <a href="/wiki/Chemical_engineering" title="Chemical engineering">chemical engineering</a>, deals with specific applied methods by which thermal energy in a system is generated, or converted, or transferred to another system. Although the definition of heat implicitly means the transfer of energy, the term <i>heat transfer</i> encompasses this traditional usage in many engineering disciplines and laymen language. </p><p><i>Heat transfer</i> is generally described as including the mechanisms of <a href="/wiki/Heat_conduction" class="mw-redirect" title="Heat conduction">heat conduction</a>, <a href="/wiki/Heat_convection" class="mw-redirect" title="Heat convection">heat convection</a>, <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a>, but may include <a href="/wiki/Mass_transfer" title="Mass transfer">mass transfer</a> and heat in processes of <a href="/wiki/Phase_changes" class="mw-redirect" title="Phase changes">phase changes</a>. </p><p>Convection may be described as the combined effects of conduction and fluid flow. From the thermodynamic point of view, heat flows into a fluid by diffusion to increase its energy, the fluid then transfers (<a href="/wiki/Advection" title="Advection">advects</a>) this increased internal energy (not heat) from one location to another, and this is then followed by a second thermal interaction which transfers heat to a second body or system, again by diffusion. This entire process is often regarded as an additional mechanism of heat transfer, although technically, "heat transfer" and thus heating and cooling occurs only on either end of such a conductive flow, but not as a result of flow. Thus, conduction can be said to "transfer" heat only as a net result of the process, but may not do so at every time within the complicated convective process. </p> <div class="mw-heading mw-heading2"><h2 id="Latent_and_sensible_heat">Latent and sensible heat</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Black_Joseph.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Black_Joseph.jpg/170px-Black_Joseph.jpg" decoding="async" width="170" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Black_Joseph.jpg/255px-Black_Joseph.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Black_Joseph.jpg/340px-Black_Joseph.jpg 2x" data-file-width="1230" data-file-height="1545" /></a><figcaption>Joseph Black</figcaption></figure> <p>In an 1847 lecture entitled <i>On Matter, Living Force, and Heat</i>, <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Prescott Joule</a> characterized the terms <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a> and <a href="/wiki/Sensible_heat" title="Sensible heat">sensible heat</a> as components of heat each affecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>quotations 2<span class="cite-bracket">]</span></a></sup> He described latent energy as the energy possessed via a distancing of particles where attraction was over a greater distance, i.e. a form of <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>, and the sensible heat as an energy involving the motion of particles, i.e. <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>. </p><p>Latent heat is the heat released or absorbed by a <a href="/wiki/Chemical_substance" title="Chemical substance">chemical substance</a> or a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a> during a change of <a href="/wiki/State_of_matter" title="State of matter">state</a> that occurs without a change in temperature. Such a process may be a <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a>, such as the melting of ice or the boiling of water.<sup id="cite_ref-Perrot_78-0" class="reference"><a href="#cite_note-Perrot-78"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Heat_capacity">Heat capacity</h2></div> <p><a href="/wiki/Heat_capacity" title="Heat capacity">Heat capacity</a> is a <a href="/wiki/Measurement" title="Measurement">measurable</a> <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantity</a> equal to the ratio of the heat added to an object to the resulting <a href="/wiki/Temperature" title="Temperature">temperature</a> change.<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> The <i>molar heat capacity</i> is the heat capacity per unit amount (SI unit: <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a>) of a pure substance, and the <i>specific heat capacity</i>, often called simply <i>specific heat</i>, is the heat capacity per unit mass of a material. Heat capacity is a physical property of a substance, which means that it depends on the state and properties of the substance under consideration. </p><p>The specific heats of monatomic gases, such as helium, are nearly constant with temperature. Diatomic gases such as hydrogen display some temperature dependence, and triatomic gases (e.g., carbon dioxide) still more. </p><p>Before the development of the laws of thermodynamics, heat was measured by changes in the states of the participating bodies. </p><p>Some general rules, with important exceptions, can be stated as follows. </p><p>In general, most bodies expand on heating. In this circumstance, heating a body at a constant volume increases the pressure it exerts on its constraining walls, while heating at a constant pressure increases its volume. </p><p>Beyond this, most substances have three ordinarily recognized <a href="/wiki/States_of_matter" class="mw-redirect" title="States of matter">states of matter</a>, solid, liquid, and gas. Some can also exist in a <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a>. Many have further, more finely differentiated, states of matter, such as <a href="/wiki/Glass" title="Glass">glass</a> and <a href="/wiki/Liquid_crystal" title="Liquid crystal">liquid crystal</a>. In many cases, at fixed temperature and pressure, a substance can exist in several distinct states of matter in what might be viewed as the same 'body'. For example, ice may float in a glass of water. Then the ice and the water are said to constitute two <a href="/wiki/Phase_(matter)" title="Phase (matter)">phases</a> within the 'body'. Definite <a href="/wiki/Phase_rule" title="Phase rule">rules</a> are known, telling how distinct phases may coexist in a 'body'. Mostly, at a fixed pressure, there is a definite temperature at which heating causes a solid to melt or evaporate, and a definite temperature at which heating causes a liquid to evaporate. In such cases, cooling has the reverse effects. </p><p>All of these, the commonest cases, fit with a rule that heating can be measured by changes of state of a body. Such cases supply what are called <a href="/wiki/Thermometer" title="Thermometer"><i>thermometric bodies</i></a>, that allow the definition of empirical temperatures. Before 1848, all temperatures were defined in this way. There was thus a tight link, apparently logically determined, between heat and temperature, though they were recognized as conceptually thoroughly distinct, especially by <a href="/wiki/Joseph_Black" title="Joseph Black">Joseph Black</a> in the later eighteenth century. </p><p>There are important exceptions. They break the obviously apparent link between heat and temperature. They make it clear that empirical definitions of temperature are contingent on the peculiar properties of particular thermometric substances, and are thus precluded from the title 'absolute'. For example, <a href="/wiki/Properties_of_water#Density_of_water_and_ice" title="Properties of water">water contracts</a> on being heated near 277 K. It cannot be used as a thermometric substance near that temperature. Also, over a certain temperature range, ice contracts on heating. Moreover, many substances can exist in metastable states, such as with negative pressure, that survive only transiently and in very special conditions. Such facts, sometimes called 'anomalous', are some of the reasons for the thermodynamic definition of absolute temperature. </p><p>In the early days of measurement of high temperatures, another factor was important, and used by <a href="/wiki/Josiah_Wedgwood" title="Josiah Wedgwood">Josiah Wedgwood</a> in his <a href="/wiki/Pyrometer#History" title="Pyrometer">pyrometer</a>. The temperature reached in a process was estimated by the shrinkage of a sample of clay. The higher the temperature, the more the shrinkage. This was the only available more or less reliable method of measurement of temperatures above 1000 °C (1,832 °F). But such shrinkage is irreversible. The clay does not expand again on cooling. That is why it could be used for the measurement. But only once. It is not a thermometric material in the usual sense of the word. </p><p>Nevertheless, the <a href="/wiki/Temperature#Thermodynamic_approach_to_temperature" title="Temperature">thermodynamic definition</a> of absolute temperature does make essential use of the concept of heat, with proper circumspection. </p> <div class="mw-heading mw-heading2"><h2 id=""Hotness""><span id=".22Hotness.22"></span>"Hotness"</h2></div> <p>The property of hotness is a concern of thermodynamics that should be defined without reference to the concept of heat. Consideration of hotness leads to the concept of empirical temperature.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> All physical systems are capable of heating or cooling others.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> With reference to hotness, the comparative terms hotter and colder are defined by the rule that heat flows from the hotter body to the colder.<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> </p><p>If a physical system is inhomogeneous or very rapidly or irregularly changing, for example by turbulence, it may be impossible to characterize it by a temperature, but still there can be transfer of energy as heat between it and another system. If a system has a physical state that is regular enough, and persists long enough to allow it to reach thermal equilibrium with a specified thermometer, then it has a temperature according to that thermometer. An empirical thermometer registers degree of hotness for such a system. Such a temperature is called empirical.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> For example, Truesdell writes about classical thermodynamics: "At each time, the body is assigned a real number called the <i>temperature</i>. This number is a measure of how hot the body is."<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> </p><p>Physical systems that are too turbulent to have temperatures may still differ in hotness. A physical system that passes heat to another physical system is said to be the hotter of the two. More is required for the system to have a thermodynamic temperature. Its behavior must be so regular that its empirical temperature is the same for all suitably calibrated and scaled thermometers, and then its hotness is said to lie on the one-dimensional hotness manifold. This is part of the reason why heat is defined following Carathéodory and Born, solely as occurring other than by work or transfer of matter; temperature is advisedly and deliberately not mentioned in this now widely accepted definition. </p><p>This is also the reason that the <a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">zeroth law of thermodynamics</a> is stated explicitly. If three physical systems, <i>A</i>, <i>B</i>, and <i>C</i> are each not in their own states of internal thermodynamic equilibrium, it is possible that, with suitable physical connections being made between them, <i>A</i> can heat <i>B</i> and <i>B</i> can heat <i>C</i> and <i>C</i> can heat <i>A</i>. In non-equilibrium situations, cycles of flow are possible. It is the special and uniquely distinguishing characteristic of internal thermodynamic equilibrium that this possibility is not open to thermodynamic systems (as distinguished amongst physical systems) which are in their own states of internal thermodynamic equilibrium; this is the reason why the zeroth law of thermodynamics needs explicit statement. That is to say, the relation 'is not colder than' between general non-equilibrium physical systems is not transitive, whereas, in contrast, the relation 'has no lower a temperature than' between thermodynamic systems in their own states of internal thermodynamic equilibrium is transitive. It follows from this that the relation 'is in thermal equilibrium with' is transitive, which is one way of stating the zeroth law. </p><p>Just as temperature may be undefined for a sufficiently inhomogeneous system, so also may entropy be undefined for a system not in its own state of internal thermodynamic equilibrium. For example, 'the temperature of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>' is not a defined quantity. Likewise, 'the entropy of the Solar System' is not defined in classical thermodynamics. It has not been possible to define non-equilibrium entropy, as a simple number for a whole system, in a clearly satisfactory way.<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Classical_thermodynamics_2">Classical thermodynamics</h2></div> <div class="mw-heading mw-heading3"><h3 id="Heat_and_enthalpy">Heat and enthalpy</h3></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/Internal_energy" title="Internal energy">Internal energy</a> and <a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></div> <p>For a <a href="/wiki/Closed_system" title="Closed system">closed system</a> (a system from which no matter can enter or exit), one version of the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> states that the change in <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> <span class="texhtml">Δ<i>U</i></span> of the system is equal to the amount of heat <span class="texhtml"><i>Q</i></span> supplied to the system minus the amount of <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">thermodynamic work</a> <span class="texhtml"><i>W</i></span> done by system on its surroundings. The foregoing sign convention for work is used in the present article, but an alternate sign convention, followed by IUPAC, for work, is to consider the work performed on the system by its surroundings as positive. This is the convention adopted by many modern textbooks of physical chemistry, such as those by <a href="/wiki/Peter_Atkins" title="Peter Atkins">Peter Atkins</a> and Ira Levine, but many textbooks on physics define work as work done by the system. </p><p><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 \Delta U=Q-W\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <mi>Q</mi> <mo>−</mo> <mi>W</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=Q-W\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8391cac1bad629cc854b7be2305224c55c87651" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.965ex; height:2.509ex;" alt="{\displaystyle \Delta U=Q-W\,.}"></span> </p><p>This formula can be re-written so as to express a definition of quantity of energy transferred as heat, based purely on the concept of adiabatic work, if it is supposed that <span class="texhtml">Δ<i>U</i></span> is defined and measured solely by processes of adiabatic work: </p><p><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 Q=\Delta U+W.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>+</mo> <mi>W</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\Delta U+W.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413f01a50f1a486e3edbbecfb0d8cd939da4851a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.578ex; height:2.509ex;" alt="{\displaystyle Q=\Delta U+W.}"></span> </p><p>The thermodynamic work done by the system is through mechanisms defined by its thermodynamic state variables, for example, its volume <span class="texhtml">V</span>, not through variables that necessarily involve mechanisms in the surroundings. The latter are such as shaft work, and include isochoric work. </p><p>The internal energy, <span class="texhtml"><i>U</i></span>, is a <a href="/wiki/State_function" title="State function">state function</a>. In cyclical processes, such as the operation of a heat engine, state functions of the working substance return to their initial values upon completion of a cycle. </p><p>The differential, or infinitesimal increment, for the internal energy in an infinitesimal process is an <a href="/wiki/Exact_differential" title="Exact differential">exact differential</a> <span class="texhtml">d<i>U</i></span>. The symbol for <a href="/wiki/Exact_differential" title="Exact differential">exact differentials</a> is the lowercase letter <span class="texhtml">d</span>. </p><p>In contrast, neither of the infinitesimal increments <span class="texhtml">δ<i>Q</i></span> nor <span class="texhtml">δ<i>W</i></span> in an infinitesimal process represents the change in a state function of the system. Thus, infinitesimal increments of heat and work are inexact differentials. The lowercase Greek letter delta, <span class="texhtml">δ</span>, is the symbol for <a href="/wiki/Inexact_differential" title="Inexact differential">inexact differentials</a>. The integral of any inexact differential in a process where the system leaves and then returns to the same thermodynamic state does not necessarily equal zero. </p><p>As recounted above, in the section headed <i>heat and entropy</i>, the second law of thermodynamics observes that if heat is supplied to a system in a <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible process</a>, the increment of heat <span class="texhtml">δ<i>Q</i></span> and the temperature <span class="texhtml"><i>T</i></span> form the <a href="/wiki/Exact_differential" title="Exact differential">exact differential</a> </p><p><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 \mathrm {d} S={\frac {\delta Q}{T}},}"> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aff37c97b039049201168ef9c5ecaf7e0b916f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.26ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}},}"></span> </p><p>and that <span class="texhtml"><i>S</i></span>, the entropy of the working body, is a state function. Likewise, with a well-defined pressure, <span class="texhtml"><i>P</i></span>, behind a slowly moving (quasistatic) boundary, the work differential, <span class="texhtml">δ<i>W</i></span>, and the pressure, <span class="texhtml"><i>P</i></span>, combine to form the exact differential </p><p><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 \mathrm {d} V={\frac {\delta W}{P}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>W</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V={\frac {\delta W}{P}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97883005c7a65cc2673004f99d75326a2dd4ea96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.145ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} V={\frac {\delta W}{P}},}"></span> </p><p>with <span class="texhtml"><i>V</i></span> the volume of the system, which is a state variable. In general, for systems of uniform pressure and temperature without composition change, <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 \mathrm {d} U=T\mathrm {d} S-P\mathrm {d} V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=T\mathrm {d} S-P\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f187be0668a13a9193a590c1a9d6d87ae2e3dee1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.914ex; height:2.343ex;" alt="{\displaystyle \mathrm {d} U=T\mathrm {d} S-P\mathrm {d} V.}"></span> </p><p>Associated with this differential equation is the concept that the internal energy may be considered to be a function <span class="texhtml"><i>U</i> (<i>S</i>,<i>V</i>)</span> of its <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">natural variables</a> <span class="texhtml"><i>S</i></span> and <span class="texhtml"><i>V</i></span>. The internal energy representation of the <a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">fundamental thermodynamic relation</a> is written as<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Adkins_1983_101_93-0" class="reference"><a href="#cite_note-Adkins_1983_101-93"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> <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 U=U(S,V).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=U(S,V).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d912434de00b81c5e1f47cc604d9daf6a66b4065" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.44ex; height:2.843ex;" alt="{\displaystyle U=U(S,V).}"></span> </p><p>If <span class="texhtml"><i>V</i></span> is constant </p><p><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 T\mathrm {d} S=\mathrm {d} U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\,{\text{constant)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <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" /> <mo stretchy="false">(</mo> <mi>V</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\mathrm {d} S=\mathrm {d} U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\,{\text{constant)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c255f9a47e3e5f7fa1355a3be75022e634c404e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.285ex; height:2.843ex;" alt="{\displaystyle T\mathrm {d} S=\mathrm {d} U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\,{\text{constant)}}}"></span> </p><p>and if <span class="texhtml"><i>P</i></span> is constant </p><p><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 T\mathrm {d} S=\mathrm {d} H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\,{\text{constant)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <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" /> <mo stretchy="false">(</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\mathrm {d} S=\mathrm {d} H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\,{\text{constant)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc269f5d774762cd9a7506ee08b2e4e0f86293c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.525ex; height:2.843ex;" alt="{\displaystyle T\mathrm {d} S=\mathrm {d} H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\,{\text{constant)}}}"></span> </p><p>with the enthalpy <span class="texhtml"><i>H</i></span> defined by </p><p><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 H=U+PV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>P</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=U+PV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ce31bac0468878564a6e7d27d92461753e2431" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.965ex; height:2.343ex;" alt="{\displaystyle H=U+PV.}"></span> </p><p>The enthalpy may be considered to be a function <span class="texhtml"><i>H</i>(<i>S</i>, <i>P</i>)</span> of its natural variables <span class="texhtml"><i>S</i></span> and <span class="texhtml"><i>P</i></span>. The enthalpy representation of the fundamental thermodynamic relation is written<sup id="cite_ref-Adkins_1983_101_93-1" class="reference"><a href="#cite_note-Adkins_1983_101-93"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Callen_147_94-0" class="reference"><a href="#cite_note-Callen_147-94"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> <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 H=H(S,P).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=H(S,P).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee1311a3415d0426de358b55f142788b37d7c2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.96ex; height:2.843ex;" alt="{\displaystyle H=H(S,P).}"></span> </p><p>The internal energy representation and the enthalpy representation are <a href="/wiki/Legendre_transformation" title="Legendre transformation">partial Legendre transforms</a> of one another. They contain the same physical information, written in different ways. Like the internal energy, the enthalpy stated as a function of its natural variables is a thermodynamic potential and contains all thermodynamic information about a body.<sup id="cite_ref-Callen_147_94-1" class="reference"><a href="#cite_note-Callen_147-94"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> </p><p>If a quantity <span class="texhtml"><i>Q</i></span> of heat is added to a body while it does only expansion work <span class="texhtml"><i>W</i></span> on its surroundings, one has </p><p><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 \Delta H=\Delta U+\Delta (PV)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>H</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta H=\Delta U+\Delta (PV)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259e13f1f0184312f55ee2dd05db9626120065d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.969ex; height:2.843ex;" alt="{\displaystyle \Delta H=\Delta U+\Delta (PV)\,.}"></span> </p><p>If this is constrained to happen at constant pressure, i.e. with <span class="texhtml">Δ<i>P</i> = 0</span>, the expansion work <span class="texhtml"><i>W</i></span> done by the body is given by <span class="texhtml"><i>W</i> = <i>P</i> Δ<i>V</i></span>; recalling the first law of thermodynamics, one has </p><p><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 \Delta U=Q-W=Q-P\,\Delta V{\text{ and }}\Delta (PV)=P\,\Delta V\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <mi>Q</mi> <mo>−</mo> <mi>W</mi> <mo>=</mo> <mi>Q</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=Q-W=Q-P\,\Delta V{\text{ and }}\Delta (PV)=P\,\Delta V\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2fd4068bd65acb25fe01ccaec7d84b5f74a9ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.738ex; height:2.843ex;" alt="{\displaystyle \Delta U=Q-W=Q-P\,\Delta V{\text{ and }}\Delta (PV)=P\,\Delta V\,.}"></span> </p><p>Consequently, by substitution one has </p><p><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 {\begin{aligned}\Delta H&=Q-P\,\Delta V+P\,\Delta V\\&=Q\qquad \qquad \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}\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> <mi mathvariant="normal">Δ</mi> <mi>H</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Q</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mo>+</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>Q</mi> <mspace width="2em" /> <mspace width="2em" /> <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>at constant pressure without electrical work.</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta H&=Q-P\,\Delta V+P\,\Delta V\\&=Q\qquad \qquad \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faff5901f63c53adb188de22420cf3644f66e00e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:71.221ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\Delta H&=Q-P\,\Delta V+P\,\Delta V\\&=Q\qquad \qquad \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}\end{aligned}}}"></span> </p><p>In this scenario, the increase in enthalpy is equal to the quantity of heat added to the system. This is the basis of the determination of enthalpy changes in chemical reactions by calorimetry. Since many processes do take place at constant atmospheric pressure, the enthalpy is sometimes given the misleading name of 'heat content'<sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> or heat function,<sup id="cite_ref-97" class="reference"><a href="#cite_note-97"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> while it actually depends strongly on the energies of covalent bonds and intermolecular forces. </p><p>In terms of the natural variables <span class="texhtml"><i>S</i> and <i>P</i></span> of the state function <span class="texhtml"><i>H</i></span>, this process of change of state from state 1 to state 2 can be expressed as </p><p><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 {\begin{aligned}\Delta H&=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S+\int _{P_{1}}^{P_{2}}\left({\frac {\partial H}{\partial P}}\right)_{S}\mathrm {d} P\\&=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}\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> <mi mathvariant="normal">Δ</mi> <mi>H</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <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>at constant pressure without electrical work.</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta H&=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S+\int _{P_{1}}^{P_{2}}\left({\frac {\partial H}{\partial P}}\right)_{S}\mathrm {d} P\\&=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28621570e92fd7444447904d342dbc5383418fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:74.977ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\Delta H&=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S+\int _{P_{1}}^{P_{2}}\left({\frac {\partial H}{\partial P}}\right)_{S}\mathrm {d} P\\&=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}\end{aligned}}}"></span> </p><p>It is known that the temperature <span class="texhtml"><i>T</i>(<i>S</i>, <i>P</i>)</span> is identically stated by </p><p><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 \left({\frac {\partial H}{\partial S}}\right)_{P}\equiv T(S,P)\,.}"> <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>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>≡</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial H}{\partial S}}\right)_{P}\equiv T(S,P)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a24fc1e8c073e5f62cd58aefde0e6c2acb876b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.962ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial H}{\partial S}}\right)_{P}\equiv T(S,P)\,.}"></span> </p><p>Consequently, </p><p><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 \Delta H=\int _{S_{1}}^{S_{2}}T(S,P)\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>H</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>T</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <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>at constant pressure without electrical work.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta H=\int _{S_{1}}^{S_{2}}T(S,P)\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7f868908bb3a92acfe7c985772e242ab02be29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:74.779ex; height:6.676ex;" alt="{\displaystyle \Delta H=\int _{S_{1}}^{S_{2}}T(S,P)\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure without electrical work.}}}"></span> </p><p>In this case, the integral specifies a quantity of heat transferred at constant pressure. </p> <div class="mw-heading mw-heading3"><h3 id="Heat_and_entropy">Heat and entropy</h3></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" title="Entropy">Entropy</a></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/140px-Clausius-1.jpg" decoding="async" width="140" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Clausius-1.jpg/210px-Clausius-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Clausius-1.jpg/280px-Clausius-1.jpg 2x" data-file-width="578" data-file-height="900" /></a><figcaption>Rudolf Clausius</figcaption></figure> <p>In 1856, <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a>, referring to closed systems, in which transfers of matter do not occur, defined the <i>second fundamental theorem</i> (the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>) in the mechanical <a href="/wiki/Theory_of_heat" class="mw-redirect" title="Theory of heat">theory of heat</a> (<a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>): "if two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat <i>Q</i> from <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> at the temperature <i>T</i>, has the <i>equivalence-value</i>:"<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-99" class="reference"><a href="#cite_note-99"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> </p><p><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 {\frac {Q}{T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>T</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Q}{T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa294e6154fc589e44e2e05ed1a64a7b31f87131" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.321ex; height:5.343ex;" alt="{\displaystyle {\frac {Q}{T}}.}"></span> </p><p>In 1865, he came to define the <a href="/wiki/Entropy" title="Entropy">entropy</a> symbolized by <i>S</i>, such that, due to the supply of the amount of heat <i>Q</i> at temperature <i>T</i> the entropy of the system is increased by </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 \Delta S={\frac {Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S={\frac {Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04598211a13c3fba48a205e5b2e05313693ee4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.208ex; height:5.343ex;" alt="{\displaystyle \Delta S={\frac {Q}{T}}}"></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>In a transfer of energy as heat without work being done, there are changes of entropy in both the surroundings which lose heat and the system which gains it. The increase, <span class="texhtml">Δ<i>S</i></span>, of entropy in the system may be considered to consist of two parts, an increment, <span class="texhtml">Δ<i>S</i>′</span> that matches, or 'compensates', the change, <span class="texhtml">−Δ<i>S</i>′</span>, of entropy in the surroundings, and a further increment, <span class="texhtml">Δ<i>S</i>′′</span> that may be considered to be 'generated' or 'produced' in the system, and is said therefore to be 'uncompensated'. Thus </p><p><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 \Delta S=\Delta S'+\Delta S''.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mo>″</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\Delta S'+\Delta S''.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3572d907aed1964632ae5e7cf08b47afc8bf455" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.758ex; height:2.676ex;" alt="{\displaystyle \Delta S=\Delta S'+\Delta S''.}"></span> </p><p>This may also be written </p><p><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 \Delta S_{\mathrm {system} }=\Delta S_{\mathrm {compensated} }+\Delta S_{\mathrm {uncompensated} }\,\,\,\,{\text{with}}\,\,\,\,\Delta S_{\mathrm {compensated} }=-\Delta S_{\mathrm {surroundings} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>with</mtext> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\mathrm {system} }=\Delta S_{\mathrm {compensated} }+\Delta S_{\mathrm {uncompensated} }\,\,\,\,{\text{with}}\,\,\,\,\Delta S_{\mathrm {compensated} }=-\Delta S_{\mathrm {surroundings} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6ee3d938c6bcba0fd280d904d8ba1956d72a67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:80.76ex; height:2.843ex;" alt="{\displaystyle \Delta S_{\mathrm {system} }=\Delta S_{\mathrm {compensated} }+\Delta S_{\mathrm {uncompensated} }\,\,\,\,{\text{with}}\,\,\,\,\Delta S_{\mathrm {compensated} }=-\Delta S_{\mathrm {surroundings} }.}"></span> </p><p>The total change of entropy in the system and surroundings is thus </p><p><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 \Delta S_{\mathrm {overall} }=\Delta S^{\prime }+\Delta S^{\prime \prime }-\Delta S^{\prime }=\Delta S^{\prime \prime }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\mathrm {overall} }=\Delta S^{\prime }+\Delta S^{\prime \prime }-\Delta S^{\prime }=\Delta S^{\prime \prime }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df1d6e23dcf394d9fcfe7de10e00f0ed31fe3655" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.393ex; height:2.843ex;" alt="{\displaystyle \Delta S_{\mathrm {overall} }=\Delta S^{\prime }+\Delta S^{\prime \prime }-\Delta S^{\prime }=\Delta S^{\prime \prime }.}"></span> </p><p>This may also be written </p><p><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 \Delta S_{\mathrm {overall} }=\Delta S_{\mathrm {compensated} }+\Delta S_{\mathrm {uncompensated} }+\Delta S_{\mathrm {surroundings} }=\Delta S_{\mathrm {uncompensated} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\mathrm {overall} }=\Delta S_{\mathrm {compensated} }+\Delta S_{\mathrm {uncompensated} }+\Delta S_{\mathrm {surroundings} }=\Delta S_{\mathrm {uncompensated} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee4004b32d72fa0b3755df407371b87ca126d26" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:75.9ex; height:2.843ex;" alt="{\displaystyle \Delta S_{\mathrm {overall} }=\Delta S_{\mathrm {compensated} }+\Delta S_{\mathrm {uncompensated} }+\Delta S_{\mathrm {surroundings} }=\Delta S_{\mathrm {uncompensated} }.}"></span> </p><p>It is then said that an amount of entropy <span class="texhtml">Δ<i>S</i>′</span> has been transferred from the surroundings to the system. Because entropy is not a conserved quantity, this is an exception to the general way of speaking, in which an amount transferred is of a conserved quantity. </p><p>From the second law of thermodynamics it follows that in a spontaneous transfer of heat, in which the temperature of the system is different from that of the surroundings: </p><p><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 \Delta S_{\mathrm {overall} }>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S_{\mathrm {overall} }>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c43e4edf5a25eb0792cd9cbfc04935b97c1c2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.302ex; height:2.509ex;" alt="{\displaystyle \Delta S_{\mathrm {overall} }>0.}"></span> </p><p>For purposes of mathematical analysis of transfers, one thinks of fictive processes that are called <i>reversible</i>, with the temperature <span class="texhtml"><i>T</i></span> of the system being hardly less than that of the surroundings, and the transfer taking place at an imperceptibly slow rate. </p><p>Following the definition above in formula (<b><a href="#math_1">1</a></b>), for such a fictive reversible process, a quantity of transferred heat <span class="texhtml">δ<i>Q</i></span> (an <a href="/wiki/Inexact_differential" title="Inexact differential">inexact differential</a>) is analyzed as a quantity <span class="texhtml"><i>T</i> d<i>S</i></span>, with <span class="texhtml">d<i>S</i></span> (an <a href="/wiki/Exact_differential" title="Exact differential">exact differential</a>): </p><p><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 T\,\mathrm {d} S=\delta Q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mi>δ</mi> <mi>Q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,\mathrm {d} S=\delta Q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066fc5d3ecca4a70a6270a9596140b8c4f742939" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.447ex; height:2.676ex;" alt="{\displaystyle T\,\mathrm {d} S=\delta Q.}"></span> </p><p>This equality is only valid for a fictive transfer in which there is no production of entropy, that is to say, in which there is no uncompensated entropy. </p><p>If, in contrast, the process is natural, and can really occur, with irreversibility, then there is <a href="/wiki/Entropy_production" title="Entropy production">entropy production</a>, with <span class="texhtml">d<i>S</i><sub>uncompensated</sub> > 0</span>. The quantity <span class="texhtml"><i>T</i> d<i>S</i><sub>uncompensated</sub></span> was termed by Clausius the "uncompensated heat", though that does not accord with present-day terminology. Then one has </p><p><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 T_{surr}\,\mathrm {d} S=\delta Q+T\,\mathrm {d} S_{\mathrm {uncompensated} }>\delta Q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>u</mi> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mi>δ</mi> <mi>Q</mi> <mo>+</mo> <mi>T</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>></mo> <mi>δ</mi> <mi>Q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{surr}\,\mathrm {d} S=\delta Q+T\,\mathrm {d} S_{\mathrm {uncompensated} }>\delta Q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1241fb4c7f52e76f2033fbe9f6d46f9be352582" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.454ex; height:3.009ex;" alt="{\displaystyle T_{surr}\,\mathrm {d} S=\delta Q+T\,\mathrm {d} S_{\mathrm {uncompensated} }>\delta Q.}"></span> </p><p>This leads to the statement </p><p><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 T_{surr}\,\mathrm {d} S\geq \delta Q\quad {\text{(second law)}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>u</mi> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>≥</mo> <mi>δ</mi> <mi>Q</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(second law)</mtext> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{surr}\,\mathrm {d} S\geq \delta Q\quad {\text{(second law)}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a4a64001e8ed70baf3e168ad3e805a24d5ca90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.911ex; height:2.843ex;" alt="{\displaystyle T_{surr}\,\mathrm {d} S\geq \delta Q\quad {\text{(second law)}}\,.}"></span> which is the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> for closed systems. </p><p>In non-equilibrium thermodynamics that makes the approximation of assuming the hypothesis of local thermodynamic equilibrium, there is a special notation for this. The transfer of energy as heat is assumed to take place across an infinitesimal temperature difference, so that the system element and its surroundings have near enough the same temperature <span class="texhtml"><i>T</i></span>. Then one writes </p><p><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 \mathrm {d} S=\mathrm {d} S_{\mathrm {e} }+\mathrm {d} S_{\mathrm {i} }\,,}"> <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"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S=\mathrm {d} S_{\mathrm {e} }+\mathrm {d} S_{\mathrm {i} }\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa128ae6cfc6781aba011f16f6b4ceaf1660201c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.851ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} S=\mathrm {d} S_{\mathrm {e} }+\mathrm {d} S_{\mathrm {i} }\,,}"></span> </p><p>where by definition </p><p><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 \delta Q=T\,\mathrm {d} S_{\mathrm {e} }\,\,\,\,\,{\text{and}}\,\,\,\,\,\mathrm {d} S_{\mathrm {i} }\equiv \mathrm {d} S_{\mathrm {uncompensated} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mi>T</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q=T\,\mathrm {d} S_{\mathrm {e} }\,\,\,\,\,{\text{and}}\,\,\,\,\,\mathrm {d} S_{\mathrm {i} }\equiv \mathrm {d} S_{\mathrm {uncompensated} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5cb4e8e781f6ecca7286cb92e57a5710ae2be84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.469ex; height:3.009ex;" alt="{\displaystyle \delta Q=T\,\mathrm {d} S_{\mathrm {e} }\,\,\,\,\,{\text{and}}\,\,\,\,\,\mathrm {d} S_{\mathrm {i} }\equiv \mathrm {d} S_{\mathrm {uncompensated} }.}"></span> </p><p>The second law for a natural process asserts that<sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-101" class="reference"><a href="#cite_note-101"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-103" class="reference"><a href="#cite_note-103"><span class="cite-bracket">[</span>99<span class="cite-bracket">]</span></a></sup> <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 \mathrm {d} S_{\mathrm {i} }>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S_{\mathrm {i} }>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00811e40a618dc72ee3d466f18aca90bd030bf0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.315ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} S_{\mathrm {i} }>0.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style 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title="Heat recovery ventilation">Heat recovery ventilation</a></li> <li><a href="/wiki/Heat_transfer_coefficient" title="Heat transfer coefficient">Heat transfer coefficient</a></li> <li><a href="/wiki/Heat_wave" title="Heat wave">Heat wave</a></li> <li><a href="/wiki/History_of_heat" class="mw-redirect" title="History of heat">History of heat</a></li> <li><a href="/wiki/Orders_of_magnitude_(temperature)" title="Orders of magnitude (temperature)">Orders of magnitude (temperature)</a></li> <li><a href="/wiki/Relativistic_heat_conduction" title="Relativistic heat conduction">Relativistic heat conduction</a></li> <li><a href="/wiki/Renewable_heat" title="Renewable heat">Renewable heat</a></li> <li><a href="/wiki/Sigma_heat" title="Sigma heat">Sigma heat</a></li> <li><a href="/wiki/Thermal_energy_storage" title="Thermal energy storage">Thermal energy storage</a></li> <li><a href="/wiki/Thermal_management_of_electronic_devices_and_systems" class="mw-redirect" title="Thermal management of electronic devices and systems">Thermal management of electronic devices and systems</a></li> <li><a href="/wiki/Thermometer" title="Thermometer">Thermometer</a></li> <li><a href="/wiki/Waste_heat" title="Waste heat">Waste heat</a></li> <li><a href="/wiki/Waste_heat_recovery_unit" title="Waste heat recovery unit">Waste heat recovery unit</a></li> <li><a href="/wiki/Water_heat_recycling" title="Water heat recycling">Water heat recycling</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">These “degrees of heat” were context-dependent and could only be used when circumstances were identical—except for the one differing factor to be investigated. When Black investigated specific heat, the “degrees of heat” were based on change in temperature multiplied by mass. When Black investigated latent heat, they were based on change in temperature multiplied by time passed. Clearly these units were not equivalent.</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">"I acknowledge the name of Calorimeter, which I have given it, as derived partly from Greek and partly from Latin, is in some degree open to criticism; but, in matters of science, a slight deviation from strict etymology, for the sake of giving distinctness of idea, is excusable; and I could not derive the name entirely from Greek without approaching too near to the names of known instruments employed for other purposes."</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCallen1985" class="citation book cs1">Callen, H.B. (1985). <i>Thermodynamics and an Introduction to Thermostatics</i> (2nd ed.). John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+and+an+Introduction+to+Thermostatics&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.aulast=Callen&rft.aufirst=H.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span> <a rel="nofollow" class="external free" href="http://cvika.grimoar.cz/callen/">http://cvika.grimoar.cz/callen/</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181017152028/http://cvika.grimoar.cz/callen/">Archived</a> 17 October 2018 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> or <a rel="nofollow" class="external free" href="http://keszei.chem.elte.hu/1alapFizkem/H.B.Callen-Thermodynamics.pdf">http://keszei.chem.elte.hu/1alapFizkem/H.B.Callen-Thermodynamics.pdf</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161230040339/http://keszei.chem.elte.hu/1alapFizkem/H.B.Callen-Thermodynamics.pdf">Archived</a> 30 December 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, p. 8: "Energy may be transferred via ... work. But it is equally possible to transfer energy via the hidden atomic modes of motion as well as via those that happen to be macroscopically observable. An energy transfer via the hidden atomic modes is called <i>heat</i>."</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Callen, H.B. (1985). p.19</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J.C.</a> (1871), Chapter III.</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">Caneva, K.L. (2021). <i>Helmholtz and the Conservation of Energy: Contexts of Creation and Reception</i>. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XOs2EAAAQBAJ&pg=PA562">p. 562</a>. (<a href="/wiki/Macquorn_Rankine" class="mw-redirect" title="Macquorn Rankine">Macquorn Rankine</a> in the same year used the same symbol. The two physicists were in correspondence at the time, so that it is difficult to say which of the two first introduced the symbol.)</span> </li> <li id="cite_note-Baierlein1999-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Baierlein1999_5-0">^</a></b></span> <span class="reference-text">Baierlein, R. (1999), p. 21.</span> </li> <li id="cite_note-FOOTNOTEBacon1902153-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBacon1902153_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBacon1902">Bacon (1902)</a>, p. 153.</span> </li> <li id="cite_note-FOOTNOTEBacon1902156-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBacon1902156_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBacon1902">Bacon (1902)</a>, p. 156.</span> </li> <li id="cite_note-FOOTNOTEGalilei1957273–4-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGalilei1957273–4_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGalilei1957">Galilei (1957)</a>, pp. 273–4.</span> </li> <li id="cite_note-FOOTNOTEAdriaans2024-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAdriaans2024_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAdriaans2024">Adriaans (2024)</a>.</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"><a href="#CITEREFHooke1665">Hooke (1665)</a>, p. <a rel="nofollow" class="external text" href="https://ttp.royalsociety.org/ttp/ttp.html?id=a9c4863d-db77-42d1-b294-fe66c85958b3&type=book">12</a>: (Facsimile, with pagination)</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="#CITEREFHooke1665">Hooke (1665)</a>, p. 12: (Machine-readable, no pagination)</span> </li> <li id="cite_note-FOOTNOTEHooke1705[httpsarchiveorgdetailsb30454621_0001page116mode1up_116]-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHooke1705[httpsarchiveorgdetailsb30454621_0001page116mode1up_116]_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHooke1705">Hooke (1705)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/b30454621_0001/page/116/mode/1up">116</a>.</span> </li> <li id="cite_note-FOOTNOTEBoyle1675[httpsarchiveorgdetailsexperimentsnotes00boylpagen100mode1up_61-62]-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoyle1675[httpsarchiveorgdetailsexperimentsnotes00boylpagen100mode1up_61-62]_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoyle1675">Boyle (1675)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/experimentsnotes00boyl/page/n100/mode/1up">61-62</a>.</span> </li> <li id="cite_note-FOOTNOTELocke1720[httpsplaygooglecombooksreaderidQqxsP-VKrpkCpgGBSPA224hlen_GB_224]-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELocke1720[httpsplaygooglecombooksreaderidQqxsP-VKrpkCpgGBSPA224hlen_GB_224]_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLocke1720">Locke (1720)</a>, p. <a rel="nofollow" class="external text" href="https://play.google.com/books/reader?id=QqxsP-VKrpkC&pg=GBS.PA224&hl=en_GB">224</a>.</span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen161mode1up_80]-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen161mode1up_80]_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n161/mode/1up">80</a>.</span> </li> <li id="cite_note-FOOTNOTETaylor1723[httpsroyalsocietypublishingorgdoiepdf101098rstl17220053_291]-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETaylor1723[httpsroyalsocietypublishingorgdoiepdf101098rstl17220053_291]_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTaylor1723">Taylor (1723)</a>, p. <a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/epdf/10.1098/rstl.1722.0053">291</a>.</span> </li> <li id="cite_note-FOOTNOTEWest2014-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEWest2014_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEWest2014_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFWest2014">West (2014)</a>.</span> </li> <li id="cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage38mode1up_38-39]_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRamsay1918">Ramsay (1918)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/lifelettersofjos00ramsrich/page/38/mode/1up">38-39</a>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</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="https://web.archive.org/web/20110128075748/http://www.historycarper.com/resources/twobf3/letter1.htm">"The Writings of Benjamin Franklin: London, 1757–1775"</a>. Historycarper.com. Archived from <a rel="nofollow" class="external text" href="http://www.historycarper.com/resources/twobf3/letter1.htm">the original</a> on 28 January 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Writings+of+Benjamin+Franklin%3A+London%2C+1757%E2%80%931775&rft.pub=Historycarper.com&rft_id=http%3A%2F%2Fwww.historycarper.com%2Fresources%2Ftwobf3%2Fletter1.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen157mode1up_76-77]-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen157mode1up_76-77]_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n157/mode/1up">76-77</a>.</span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen162mode1up_81]-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen162mode1up_81]_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n162/mode/1up">81</a>.</span> </li> <li id="cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage44mode1up_44]_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRamsay1918">Ramsay (1918)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/lifelettersofjos00ramsrich/page/44/mode/1up">44</a>.</span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen192mode1up_111-112]-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen192mode1up_111-112]_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n192/mode/1up">111-112</a>.</span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen193mode1up_112]-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen193mode1up_112]_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n193/mode/1up">112</a>.</span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen201mode1up_120]_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n201/mode/1up">120</a>.</span> </li> <li id="cite_note-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen196mode1up_115-117]-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBlack1807[httpsarchiveorgdetails2543060RX1nlmnihgovpagen196mode1up_115-117]_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBlack1807">Black (1807)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX1.nlm.nih.gov/page/n196/mode/1up">115-117</a>.</span> </li> <li id="cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45]-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45]_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRamsay1918">Ramsay (1918)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/lifelettersofjos00ramsrich/page/45/mode/1up">45</a>.</span> </li> <li id="cite_note-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45-46]-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERamsay1918[httpsarchiveorgdetailslifelettersofjos00ramsrichpage45mode1up_45-46]_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRamsay1918">Ramsay (1918)</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/lifelettersofjos00ramsrich/page/45/mode/1up">45-46</a>.</span> </li> <li id="cite_note-FOOTNOTELavoisier1790[httpswwwgutenbergorgcacheepub30775pg30775-imageshtmlPage_343_345]-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELavoisier1790[httpswwwgutenbergorgcacheepub30775pg30775-imageshtmlPage_343_345]_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLavoisier1790">Lavoisier (1790)</a>, p. <a rel="nofollow" class="external text" href="https://www.gutenberg.org/cache/epub/30775/pg30775-images.html#Page_343">345</a>.</span> </li> <li id="cite_note-FOOTNOTEBuchholzSchoeller2004[httpwwwajcnorgcgicontentfull795899S_899S–906S]-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBuchholzSchoeller2004[httpwwwajcnorgcgicontentfull795899S_899S–906S]_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBuchholzSchoeller2004">Buchholz & Schoeller (2004)</a>, pp. <a rel="nofollow" class="external text" href="http://www.ajcn.org/cgi/content/full/79/5/899S">899S–906S</a>.</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">Lervig, P. Sadi Carnot and the steam engine: Nicolas Clément's lectures on industrial chemistry, 1823–28. Br. J Hist. Sci. 18:147, 1985.</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">Blundell, S.J., Blundell, K.M. (2006). <i>Concepts in Thermal Physics</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/9780198567691" title="Special:BookSources/9780198567691">9780198567691</a>, p. 106.</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"><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule, J.P.</a> (1845).<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoule1850" class="citation journal cs1"><a rel="nofollow" class="external text" href="https://archive.org/stream/philtrans00608634/00608634#page/n0/mode/2up">"On the Mechanical Equivalent of Heat"</a>. <i>Philosophical Transactions of the Royal Society of London</i>. <b>140</b>: <span class="nowrap">61–</span>82. 1850. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1850.0004">10.1098/rstl.1850.0004</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=On+the+Mechanical+Equivalent+of+Heat&rft.volume=140&rft.pages=%3Cspan+class%3D%22nowrap%22%3E61-%3C%2Fspan%3E82&rft.date=1850&rft_id=info%3Adoi%2F10.1098%2Frstl.1850.0004&rft.aulast=Joule&rft.aufirst=J.+P.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fphiltrans00608634%2F00608634%23page%2Fn0%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></span> </li> <li id="cite_note-Clausius1850-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-Clausius1850_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Clausius1850_36-1"><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="CITEREFClausius1898" class="citation book cs1 cs1-prop-foreign-lang-source">Clausius, R. (1898) [1850]. Poggendorff, Johann Christian; <a href="/wiki/Max_Planck" title="Max Planck">Planck, Max</a> (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/ueberdiebewegen00claugoog/page/n17"><i>Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen</i></a>. Ostwald's Klassiker der exakten Wissenschaften (in German). Vol. 99. Leipzig: Wilhelm Engelmann. pp. <span class="nowrap">4–</span>5. <q>Wenn man annimmt, die Wärme könne, ebenso wie ein Stoff, nicht an Quantität geringer werden, so muss man auch annehmen, dass sie sich nicht vermehren könne. Es ist aber fast unmöglich z. B. die durch Reibung verursachte Erwärmung ohne eine Vermehrung der Wärmequantität zu erklären, und durch die sorgfältigen Versuche von Joule, bei welchen auf sehr verschiedene Weisen unter Anwendung von mechanischer Arbeit Erwärmung hervorgerufen wurde, ist ausser der Möglichkeit, die Wärmequantität überhaupt zu vermehren, auch der Satz, dass die Menge der neu erzeugten Wärme der dazu angewandten Arbeit proportional sei, fast zur Gewissheit geworden. Dazu kommt noch, dass in neuerer Zeit immer noch mehr Thatsachen bekannt...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=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.place=Leipzig&rft.series=Ostwald%27s+Klassiker+der+exakten+Wissenschaften&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4-%3C%2Fspan%3E5&rft.pub=Wilhelm+Engelmann&rft.date=1898&rft.aulast=Clausius&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fueberdiebewegen00claugoog%2Fpage%2Fn17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span><sup class="noprint Inline-Template noprint noexcerpt Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS"><span title="archive.org's scan of this page has the quotation partially obscured, hence why the German text quoted here is incomplete (January 2025)">better source needed</span></a></i>]</sup> Originally published in Poggendorff's <span title="German-language text"><i lang="de">Annalen</i></span>, vol. 79. Translated into English as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClausius1867" class="citation book cs1">Clausius, Rudolf (1867). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8LIEAAAAYAAJ&pg=PA25">"On the moving force of heat and the laws of heat which may be deduced therefrom"</a>. <i>The Mechanical Theory of Heat, with its Applications to the Steam-Engine and to the Physical Properties of Bodies</i>. Translated by Tyndall, John. London: J. Van Voorst. p. 25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+moving+force+of+heat+and+the+laws+of+heat+which+may+be+deduced+therefrom&rft.btitle=The+Mechanical+Theory+of+Heat%2C+with+its+Applications+to+the+Steam-Engine+and+to+the+Physical+Properties+of+Bodies&rft.place=London&rft.pages=25&rft.pub=J.+Van+Voorst&rft.date=1867&rft.aulast=Clausius&rft.aufirst=Rudolf&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8LIEAAAAYAAJ%26pg%3DPA25&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J.C.</a> (1871), p. 7.</span> </li> <li id="cite_note-Bryan_1907-38"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bryan_1907_38-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bryan_1907_38-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Bryan_1907_38-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Bryan_1907_38-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Bryan_1907_38-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Bryan_1907_38-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Bryan_1907_38-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryan1907" class="citation web cs1">Bryan, G.H. (1907). <a rel="nofollow" class="external text" href="https://archive.org/stream/thermodynamicsin00bryauoft/thermodynamicsin00bryauoft_djvu.txt">"Thermodynamics, an introductory treatise dealing mainly with first principles and their direct applications"</a>. Leipzig, Teubner<span class="reference-accessdate">. Retrieved <span class="nowrap">23 June</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Thermodynamics%2C+an+introductory+treatise+dealing+mainly+with+first+principles+and+their+direct+applications&rft.pub=Leipzig%2C+Teubner&rft.date=1907&rft.aulast=Bryan&rft.aufirst=G.H.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fthermodynamicsin00bryauoft%2Fthermodynamicsin00bryauoft_djvu.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/18px-PD-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/24px-PD-icon.svg.png 2x" data-file-width="196" data-file-height="196" /></span></span> This article incorporates text from this source, which is in the <a href="/wiki/Public_domain" title="Public domain">public domain</a>.</span> </li> <li id="cite_note-Carathéodory_1909-39"><span class="mw-cite-backlink">^ <a href="#cite_ref-Carathéodory_1909_39-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Carathéodory_1909_39-1"><sup><i><b>b</b></i></sup></a></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-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983).</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">Münster, A. (1970).</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">Pippard, A.B. (1957).</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Fowler, R., Guggenheim, E.A. (1939).</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">Buchdahl, H.A. (1966).</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">Lieb, E.H., Yngvason, J. (1999), p. 10.</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">Serrin, J. (1986), p. 5.</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">Owen, D.R. (1984), pp. 43–45.</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J.C.</a> (1871), p.v.</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><a href="/wiki/Peter_Atkins" title="Peter Atkins">Atkins, P.</a>, de Paula, J. (1978/2010), p. 54.</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><a href="/wiki/Brian_Pippard" title="Brian Pippard">Pippard, A.B.</a> (1957/1966), p. 15.</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text">Planck, M. (1926). 'Über die Begründung des zweiten Hauptsatzes der Thermodynamik', <i>Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl.</i>, 453—463.</span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text">Lieb & Yngvason (1999).</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><a href="/wiki/J.R._Partington" class="mw-redirect" title="J.R. Partington">Partington, J.R.</a> (1949), p. 118.</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J.C.</a> (1871), p. 10.</span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J.C.</a> (1871), p. 11.</span> </li> <li id="cite_note-Chandrasekhar_1961-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chandrasekhar_1961_56-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Subrahamanyan_Chandrasekhar" class="mw-redirect" title="Subrahamanyan Chandrasekhar">Chandrasekhar, S.</a> (1961).</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1897/1903), p. viii.</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><a href="/wiki/Jaakko_Hintikka" title="Jaakko Hintikka">Hintikka, J.</a> (1988), p. 180.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text">Bailyn, M. (1994), pp. 65, 79.</span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Born" title="Max Born">Born, M.</a>(1949), Lecture <span class="texhtml"><i>V</i></span>.</span> </li> <li id="cite_note-Born,_M_1949_p._44-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-Born,_M_1949_p._44_61-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Born" title="Max Born">Born, M.</a> (1949), p. 44.</span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text">De Groot, S.R., Mazur, P. (1962), p. 30.</span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text">Denbigh, K.G. (1951), p. 56.</span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text">Fitts, D.D. (1962), p. 28.</span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text">Gyarmati, I. (1970), p. 68.</span> </li> <li id="cite_note-Kittel_&_Kroemer-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kittel_&_Kroemer_67-0">^</a></b></span> <span class="reference-text">Kittel, C. Kroemer, H. (1980).</span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text">Bacon, F. (1620).</span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text">Partington, J.R. (1949), p. 131.</span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text">Partington, J.R. (1949), pp. 132–136.</span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text">Reif (1965), pp. 67–68</span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text">Maxwell J.C. (1872), p. 54.</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text">Planck (1927), Chapter 3.</span> </li> <li id="cite_note-Bryan_47-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bryan_47_74-0">^</a></b></span> <span class="reference-text">Bryan, G.H. (1907), p. 47.</span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><a href="/wiki/Herbert_Callen" title="Herbert Callen">Callen, H.B.</a> (1985), Section 1-8.</span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text"><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule J.P.</a> (1884).</span> </li> <li id="cite_note-Perrot-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-Perrot_78-0">^</a></b></span> <span class="reference-text">Perrot, P. (1998).</span> </li> <li id="cite_note-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-79">^</a></b></span> <span class="reference-text">Clark, J.O.E. (2004).</span> </li> <li id="cite_note-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-80">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallidayResnick2013" class="citation book cs1">Halliday, David; Resnick, Robert (2013). <i>Fundamentals of Physics</i>. Wiley. p. 524.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Physics&rft.pages=524&rft.pub=Wiley&rft.date=2013&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></span> </li> <li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text">Denbigh, K. (1981), p. 9.</span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983), p. 55.</span> </li> <li id="cite_note-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-83">^</a></b></span> <span class="reference-text">Baierlein, R. (1999), p. 349.</span> </li> <li id="cite_note-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-84">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983), p. 34.</span> </li> <li id="cite_note-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-85">^</a></b></span> <span class="reference-text">Pippard, A.B. (1957/1966), p. 18.</span> </li> <li id="cite_note-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-86">^</a></b></span> <span class="reference-text">Haase, R. (1971), p. 7.</span> </li> <li id="cite_note-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-87">^</a></b></span> <span class="reference-text"><a href="/wiki/Ernst_Mach" title="Ernst Mach">Mach, E.</a> (1900), section 5, pp. 48–49, section 22, pp. 60–61.</span> </li> <li id="cite_note-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-88">^</a></b></span> <span class="reference-text"><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a> (1980).</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"><a href="/wiki/James_Serrin" title="James Serrin">Serrin, J.</a> (1986), especially p. 6.</span> </li> <li id="cite_note-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-90">^</a></b></span> <span class="reference-text"><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a> (1969), p. 6.</span> </li> <li id="cite_note-91"><span class="mw-cite-backlink"><b><a href="#cite_ref-91">^</a></b></span> <span class="reference-text">Lieb, E.H., Yngvason, J. (2003), p. 190.</span> </li> <li id="cite_note-92"><span class="mw-cite-backlink"><b><a href="#cite_ref-92">^</a></b></span> <span class="reference-text"><a href="/wiki/Herbert_Callen" title="Herbert Callen">Callen, H.B.</a>, (1985), Section 2-3, pp. 40–42.</span> </li> <li id="cite_note-Adkins_1983_101-93"><span class="mw-cite-backlink">^ <a href="#cite_ref-Adkins_1983_101_93-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Adkins_1983_101_93-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Adkins, C.J. (1983), p. 101.</span> </li> <li id="cite_note-Callen_147-94"><span class="mw-cite-backlink">^ <a href="#cite_ref-Callen_147_94-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Callen_147_94-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Herbert_Callen" title="Herbert Callen">Callen, H.B.</a> (1985), p. 147.</span> </li> <li id="cite_note-95"><span class="mw-cite-backlink"><b><a href="#cite_ref-95">^</a></b></span> <span class="reference-text">Adkins, C.J. (1983), pp. 100–104.</span> </li> <li id="cite_note-96"><span class="mw-cite-backlink"><b><a href="#cite_ref-96">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983), p. 46.</span> </li> <li id="cite_note-97"><span class="mw-cite-backlink"><b><a href="#cite_ref-97">^</a></b></span> <span class="reference-text">Bailyn, M. (1994), p. 208.</span> </li> <li id="cite_note-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-98">^</a></b></span> <span class="reference-text"><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1854).</span> </li> <li id="cite_note-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-99">^</a></b></span> <span class="reference-text"><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1865), pp. 125–126.</span> </li> <li id="cite_note-100"><span class="mw-cite-backlink"><b><a href="#cite_ref-100">^</a></b></span> <span class="reference-text">De Groot, S.R., <a href="/wiki/Peter_Mazur" title="Peter Mazur">Mazur, P.</a> (1962), p. 20.</span> </li> <li id="cite_note-101"><span class="mw-cite-backlink"><b><a href="#cite_ref-101">^</a></b></span> <span class="reference-text">Kondepudi, D, <a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Prigogine, I.</a> (1998), p. 82.</span> </li> <li id="cite_note-102"><span class="mw-cite-backlink"><b><a href="#cite_ref-102">^</a></b></span> <span class="reference-text">Kondepudi, D. (2008), p. 114.</span> </li> <li id="cite_note-103"><span class="mw-cite-backlink"><b><a href="#cite_ref-103">^</a></b></span> <span class="reference-text">Lebon, g., Jou, D., Casas-Vásquez, J. (2008), p. 41.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Quotations">Quotations</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text">Denbigh states in a footnote that he is indebted to correspondence with <a href="/wiki/Edward_A._Guggenheim" title="Edward A. Guggenheim">Professor E.A. Guggenheim</a> and with Professor N.K. Adam. From this, Denbigh concludes "It seems, however, that when a system is able to exchange both heat and matter with its environment, it is impossible to make an unambiguous distinction between energy transported as heat and by the migration of matter, without already assuming the existence of the 'heat of transport'." Denbigh K.G. (1951), p. 56.</span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text">"Heat must therefore consist of either living force or of attraction through space. In the former case we can conceive the constituent particles of heated bodies to be, either in whole or in part, in a state of motion. In the latter we may suppose the particles to be removed by the process of heating, so as to exert attraction through greater space. I am inclined to believe that both of these hypotheses will be found to hold good,—that in some instances, particularly in the case of sensible heat, or such as is indicated by the thermometer, heat will be found to consist in the living force of the particles of the bodies in which it is induced; whilst in others, particularly in the case of latent heat, the phenomena are produced by the separation of particle from particle, so as to cause them to attract one another through a greater space." <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule, J.P.</a> (1884).</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography_of_cited_references">Bibliography of cited references</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 30em;"> <ul><li>Adkins, C.J. (1968/1983). <i>Equilibrium Thermodynamics</i>, (1st edition 1968), third edition 1983, 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/0-521-25445-0" title="Special:BookSources/0-521-25445-0">0-521-25445-0</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdriaans2024" class="citation cs2">Adriaans, P. (2024), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/sum2024/entries/information/#Phys">"Information"</a>, in Zalta, E.N.; Nodelman, U. (eds.), <i>The Stanford Encyclopedia of Philosophy</i> (Summer 2024 ed.), Metaphysics Research Lab, Stanford University</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Information&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Summer+2024&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2024&rft.aulast=Adriaans&rft.aufirst=P.&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fsum2024%2Fentries%2Finformation%2F%23Phys&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li><a href="/wiki/Peter_Atkins" title="Peter Atkins">Atkins, P.</a>, de Paula, J. (1978/2010). <i>Physical Chemistry</i>, (first edition 1978), ninth edition 2010, 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/978-0-19-954337-3" title="Special:BookSources/978-0-19-954337-3">978-0-19-954337-3</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBacon1902" class="citation book cs1"><a href="/wiki/Francis_Bacon" title="Francis Bacon">Bacon, F.</a> (1902) [1620]. Dewey, J. (ed.). <a rel="nofollow" class="external text" href="https://www.gutenberg.org/cache/epub/45988/pg45988-images.html"><i>Novum Organum: Or True Suggestions for the Interpretation of Nature</i></a>. P. F. Collier & son.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Novum+Organum%3A+Or+True+Suggestions+for+the+Interpretation+of+Nature&rft.pub=P.+F.+Collier+%26+son&rft.date=1902&rft.aulast=Bacon&rft.aufirst=F.&rft_id=https%3A%2F%2Fwww.gutenberg.org%2Fcache%2Fepub%2F45988%2Fpg45988-images.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaierlein1999" class="citation book cs1">Baierlein, R. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fqUU71spbZYC&pg=PA21"><i>Thermal Physics</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-65838-6" title="Special:BookSources/978-0-521-65838-6"><bdi>978-0-521-65838-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermal+Physics&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-65838-6&rft.aulast=Baierlein&rft.aufirst=R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfqUU71spbZYC%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li>Bailyn, M. (1994). <i>A Survey of Thermodynamics</i>, American Institute of Physics 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/0-88318-797-3" 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="CITEREFBlack1807" class="citation book cs1"><a href="/wiki/Joseph_Black" title="Joseph Black">Black, J.</a> (1807). Robison, J. (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/2543060RX2.nlm.nih.gov/page/n4/mode/1up"><i>Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh</i></a>. Vol. 1. Mathew Carey.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+the+Elements+of+Chemistry%3A+Delivered+in+the+University+of+Edinburgh&rft.pub=Mathew+Carey&rft.date=1807&rft.aulast=Black&rft.aufirst=J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2F2543060RX2.nlm.nih.gov%2Fpage%2Fn4%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyle1675" class="citation book cs1">Boyle, R. (1675). <a rel="nofollow" class="external text" href="https://archive.org/details/experimentsnotes00boyl/page/n8/mode/1up"><i>Experiments, notes, &c., about the mechanical origine or production of divers particular qualities: Among which is inserted a discourse of the imperfection of the chymist's doctrine of qualities; together with some reflections upon the hypothesis of alcali and acidum</i></a>. Printed by E. Flesher, for R. Davis.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experiments%2C+notes%2C+%26c.%2C+about+the+mechanical+origine+or+production+of+divers+particular+qualities%3A+Among+which+is+inserted+a+discourse+of+the+imperfection+of+the+chymist%27s+doctrine+of+qualities%3B+together+with+some+reflections+upon+the+hypothesis+of+alcali+and+acidum&rft.pub=Printed+by+E.+Flesher%2C+for+R.+Davis&rft.date=1675&rft.aulast=Boyle&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fexperimentsnotes00boyl%2Fpage%2Fn8%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li><a href="/wiki/Max_Born" title="Max Born">Born, M.</a> (1949). <a rel="nofollow" class="external text" href="https://archive.org/details/naturalphilosoph032159mbp"><i>Natural Philosophy of Cause and Chance</i></a>, Oxford University Press, London.</li> <li><a href="/wiki/George_H._Bryan" title="George H. Bryan">Bryan, G.H.</a> (1907). <a rel="nofollow" class="external text" href="https://archive.org/details/ost-physics-thermodynamicsin00bryauoft"><i>Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications</i></a>, B.G. Teubner, Leipzig.</li> <li><a href="/wiki/Hans_Adolf_Buchdahl" title="Hans Adolf Buchdahl">Buchdahl, H.A.</a> (1966). <i>The Concepts of Classical Thermodynamics</i>, Cambridge University Press, Cambridge UK.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuchholzSchoeller2004" class="citation journal cs1">Buchholz, A.C.; Schoeller, D.A. (2004). <a rel="nofollow" class="external text" href="http://www.ajcn.org/cgi/content/full/79/5/899S">"Is a Calorie a Calorie?"</a>. <i>American Journal of Clinical Nutrition</i>. <b>79</b> (5): <span class="nowrap">899S –</span> <span class="nowrap">906S</span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fajcn%2F79.5.899S">10.1093/ajcn/79.5.899S</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15113737">15113737</a><span class="reference-accessdate">. Retrieved <span class="nowrap">12 March</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Clinical+Nutrition&rft.atitle=Is+a+Calorie+a+Calorie%3F&rft.volume=79&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E899S+-%3C%2Fspan%3E+%3Cspan+class%3D%22nowrap%22%3E906S%3C%2Fspan%3E&rft.date=2004&rft_id=info%3Adoi%2F10.1093%2Fajcn%2F79.5.899S&rft_id=info%3Apmid%2F15113737&rft.aulast=Buchholz&rft.aufirst=A.C.&rft.au=Schoeller%2C+D.A.&rft_id=http%3A%2F%2Fwww.ajcn.org%2Fcgi%2Fcontent%2Ffull%2F79%2F5%2F899S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></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="CITEREFCarathéodory1909" class="citation journal cs1"><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory, C.</a> (1909). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428268">"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>.</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.aulast=Carath%C3%A9odory&rft.aufirst=C.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428268&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" 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>. A mostly reliable translation is to be found at Kestin, J. (1976). <i>The Second Law of Thermodynamics</i>, Dowden, Hutchinson & Ross, Stroudsburg PA.</li> <li><a href="/wiki/Subrahamanyan_Chandrasekhar" class="mw-redirect" title="Subrahamanyan Chandrasekhar">Chandrasekhar, S.</a> (1961). <i>Hydrodynamic and Hydromagnetic Stability</i>, Oxford University Press, Oxford UK.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClark2004" class="citation book cs1">Clark, J.O.E. (2004). <i>The Essential Dictionary of Science</i>. Barnes & Noble Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7607-4616-5" title="Special:BookSources/978-0-7607-4616-5"><bdi>978-0-7607-4616-5</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+Essential+Dictionary+of+Science&rft.pub=Barnes+%26+Noble+Books&rft.date=2004&rft.isbn=978-0-7607-4616-5&rft.aulast=Clark&rft.aufirst=J.O.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1854). <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></i> (<i>Poggendoff's Annalen</i>), Dec. 1854, vol. xciii. p. 481; translated in the <i>Journal de Mathematiques</i>, vol. xx. Paris, 1855, and in the <i>Philosophical Magazine</i>, August 1856, s. 4. vol. xii, p. 81.</li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius, R.</a> (1865/1867). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8LIEAAAAYAAJ&q=necessitating+any+other"><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. 1867. Also the second edition translated into English by W.R. Browne (1879) <a rel="nofollow" class="external text" href="https://archive.org/details/cu31924101120883">here</a> and <a rel="nofollow" class="external text" href="http://www3.nd.edu/~powers/ame.20231/clausius1879.pdf">here</a>.</li> <li>De Groot, S.R., <a href="/wiki/Peter_Mazur" title="Peter Mazur">Mazur, P.</a> (1962). <i>Non-equilibrium Thermodynamics</i>, North-Holland, Amsterdam. Reprinted (1984), Dover Publications Inc., 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/0486647412" title="Special:BookSources/0486647412">0486647412</a>.</li> <li>Denbigh, K. (1955/1981). <i>The Principles of Chemical Equilibrium</i>, Cambridge University Press, Cambridge <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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalilei1957" class="citation book cs1">Galilei, G. (1957) [1623]. "The Assayer". In Drake, S. (ed.). <a rel="nofollow" class="external text" href="https://www.mercaba.es/renacimiento/escritos_menores_de_galileo.pdf"><i>Discoveries and Opinions of Galileo</i></a> <span class="cs1-format">(PDF)</span>. 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From Concepts to Applications</i>, Prentice Hall, London, <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/0-13-288267-1" title="Special:BookSources/0-13-288267-1">0-13-288267-1</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1723" class="citation journal cs1"><a href="/wiki/Brook_Taylor" title="Brook Taylor">Taylor, B.</a> (31 December 1723). <a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/10.1098/rstl.1722.0053">"III. An account of the experiment, made to ascertain the proportion of the expansion of the liquor in the thermometer, with regard to the degrees of heat"</a>. <i>Philosophical Transactions of the Royal Society of London</i>. <b>32</b> (376): 291. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1722.0053">10.1098/rstl.1722.0053</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0261-0523">0261-0523</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=III.+An+account+of+the+experiment%2C+made+to+ascertain+the+proportion+of+the+expansion+of+the+liquor+in+the+thermometer%2C+with+regard+to+the+degrees+of+heat&rft.volume=32&rft.issue=376&rft.pages=291&rft.date=1723-12-31&rft_id=info%3Adoi%2F10.1098%2Frstl.1722.0053&rft.issn=0261-0523&rft.aulast=Taylor&rft.aufirst=B.&rft_id=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2F10.1098%2Frstl.1722.0053&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a> (1969). <i>Rational Thermodynamics: a Course of Lectures on Selected Topics</i>, McGraw-Hill Book Company, New York.</li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a> (1980). <i>The Tragicomical History of Thermodynamics 1822–1854</i>, Springer, 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/0-387-90403-4" title="Special:BookSources/0-387-90403-4">0-387-90403-4</a>.</li></ul> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWest2014" class="citation journal cs1"><a href="/wiki/John_B._West" title="John B. West">West, J.B.</a> (15 June 2014). <a rel="nofollow" class="external text" href="https://www.physiology.org/doi/10.1152/ajplung.00020.2014">"Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases"</a>. <i>American Journal of Physiology-Lung Cellular and Molecular Physiology</i>. <b>306</b> (12): <span class="nowrap">L1057 –</span> <span class="nowrap">L1063</span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1152%2Fajplung.00020.2014">10.1152/ajplung.00020.2014</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1040-0605">1040-0605</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24682452">24682452</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physiology-Lung+Cellular+and+Molecular+Physiology&rft.atitle=Joseph+Black%2C+carbon+dioxide%2C+latent+heat%2C+and+the+beginnings+of+the+discovery+of+the+respiratory+gases&rft.volume=306&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3EL1057+-%3C%2Fspan%3E+%3Cspan+class%3D%22nowrap%22%3EL1063%3C%2Fspan%3E&rft.date=2014-06-15&rft.issn=1040-0605&rft_id=info%3Apmid%2F24682452&rft_id=info%3Adoi%2F10.1152%2Fajplung.00020.2014&rft.aulast=West&rft.aufirst=J.B.&rft_id=https%3A%2F%2Fwww.physiology.org%2Fdoi%2F10.1152%2Fajplung.00020.2014&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Further_bibliography">Further bibliography</h3></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerettaE.P._Gyftopoulos2015" class="citation journal cs1">Beretta, G.P.; <a href="/wiki/Elias_Gyftopoulos" title="Elias Gyftopoulos">E.P. Gyftopoulos</a> (2015). <a rel="nofollow" class="external text" href="https://gianpaolo-beretta.unibs.it/Beretta-papers-online/m49-BerettaGyftopoulos-JERT-137-021006-2015.pdf">"What is heat?"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Energy Resources Technology</i>. ASME. <b>137</b> (2). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.4026382">10.1115/1.4026382</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Energy+Resources+Technology&rft.atitle=What+is+heat%3F&rft.volume=137&rft.issue=2&rft.date=2015&rft_id=info%3Adoi%2F10.1115%2F1.4026382&rft.aulast=Beretta&rft.aufirst=G.P.&rft.au=E.P.+Gyftopoulos&rft_id=https%3A%2F%2Fgianpaolo-beretta.unibs.it%2FBeretta-papers-online%2Fm49-BerettaGyftopoulos-JERT-137-021006-2015.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat" class="Z3988"></span></li> <li>Gyftopoulos, E.P., & Beretta, G.P. (1991). <i>Thermodynamics: foundations and applications.</i> (Dover Publications)</li> <li>Hatsopoulos, G.N., & Keenan, J.H. (1981). Principles of general thermodynamics. RE Krieger Publishing Company.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/b00fq3d4">Heat</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)"><i>In Our Time</i></a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><a rel="nofollow" class="external text" href="http://www.foxnews.com/story/0,2933,187464,00.html">Plasma heat at 2 gigakelvins</a> – Article about extremely high temperature generated by scientists (Foxnews.com)</li> <li><a rel="nofollow" class="external text" href="http://www.cheresources.com/convection.shtml">Correlations for Convective Heat Transfer</a> – ChE Online Resources</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output 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href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>114 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-scsbh","wgBackendResponseTime":147,"wgPageParseReport":{"limitreport":{"cputime":"1.205","walltime":"1.501","ppvisitednodes":{"value":13512,"limit":1000000},"postexpandincludesize":{"value":174466,"limit":2097152},"templateargumentsize":{"value":18641,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":21,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":279510,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1187.943 1 -total"," 22.23% 264.126 3 Template:Reflist"," 11.58% 137.531 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type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Heat","url":"https:\/\/en.wikipedia.org\/wiki\/Heat","sameAs":"http:\/\/www.wikidata.org\/entity\/Q44432","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q44432","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-08-08T22:48:59Z","dateModified":"2025-02-10T09:47:49Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d1\/Glowing_metal.jpg","headline":"energy that is transferred from one body to another as the result of a difference in temperature"}</script> </body> </html>