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href="#Internal_energy_of_the_ideal_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Internal energy of the ideal gas</span> </div> </a> <ul id="toc-Internal_energy_of_the_ideal_gas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_energy_of_a_closed_thermodynamic_system" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Internal_energy_of_a_closed_thermodynamic_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Internal energy of a closed thermodynamic system</span> </div> </a> <button aria-controls="toc-Internal_energy_of_a_closed_thermodynamic_system-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 Internal energy of a closed thermodynamic system subsection</span> </button> <ul id="toc-Internal_energy_of_a_closed_thermodynamic_system-sublist" class="vector-toc-list"> <li id="toc-Changes_due_to_temperature_and_volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Changes_due_to_temperature_and_volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Changes due to temperature and volume</span> </div> </a> <ul id="toc-Changes_due_to_temperature_and_volume-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Changes_due_to_temperature_and_pressure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Changes_due_to_temperature_and_pressure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Changes due to temperature and pressure</span> </div> </a> <ul id="toc-Changes_due_to_temperature_and_pressure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Changes_due_to_volume_at_constant_temperature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Changes_due_to_volume_at_constant_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Changes due to volume at constant temperature</span> </div> </a> <ul id="toc-Changes_due_to_volume_at_constant_temperature-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Internal_energy_of_multi-component_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Internal_energy_of_multi-component_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Internal energy of multi-component systems</span> </div> </a> <ul id="toc-Internal_energy_of_multi-component_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_energy_in_an_elastic_medium" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Internal_energy_in_an_elastic_medium"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Internal energy in an elastic medium</span> </div> </a> <ul id="toc-Internal_energy_in_an_elastic_medium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">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-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">10.1</span> <span>Bibliography of cited references</span> </div> </a> <ul id="toc-Bibliography_of_cited_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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" > <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">Internal energy</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 66 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-66" 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">66 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D8%A9_%D8%AF%D8%A7%D8%AE%D9%84%D9%8A%D8%A9" title="طاقة داخلية – Arabic" lang="ar" hreflang="ar" data-title="طاقة داخلية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Enerx%C3%ADa_interno" title="Enerxía interno – Asturian" lang="ast" hreflang="ast" data-title="Enerxía interno" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AD%E0%A7%8D%E0%A6%AF%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0%E0%A7%80%E0%A6%A3_%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="অভ্যন্তরীণ শক্তি – Bangla" lang="bn" hreflang="bn" data-title="অভ্যন্তরীণ শক্তি" 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/L%C5%8De-l%C3%AAng" title="Lōe-lêng – Minnan" lang="nan" hreflang="nan" data-title="Lōe-lêng" 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%A3%D0%BD%D1%83%D1%82%D1%80%D0%B0%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Унутраная энергія – Belarusian" lang="be" hreflang="be" data-title="Унутраная энергія" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9D%D1%83%D1%82%D1%80%D0%B0%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%8D%D1%80%D0%B3%D1%96%D1%8F" title="Нутраная энэргія – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Нутраная энэргія" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%8A%D1%82%D1%80%D0%B5%D1%88%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Вътрешна енергия – Bulgarian" lang="bg" hreflang="bg" data-title="Вътрешна енергия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Unutra%C5%A1nja_energija" title="Unutrašnja energija – Bosnian" lang="bs" hreflang="bs" data-title="Unutrašnja energija" 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/Energia_interna" title="Energia interna – Catalan" lang="ca" hreflang="ca" data-title="Energia interna" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%D0%B0%D0%BB%D1%82%D0%B8_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Шалти энерги – Chuvash" lang="cv" hreflang="cv" data-title="Шалти энерги" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vnit%C5%99n%C3%AD_energie" title="Vnitřní energie – Czech" lang="cs" hreflang="cs" data-title="Vnitřní energie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Indre_energi" title="Indre energi – Danish" lang="da" hreflang="da" data-title="Indre energi" 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/Innere_Energie" title="Innere Energie – German" lang="de" hreflang="de" data-title="Innere Energie" 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/Siseenergia" title="Siseenergia – Estonian" lang="et" hreflang="et" data-title="Siseenergia" 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%95%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BD%CE%AD%CF%81%CE%B3%CE%B5%CE%B9%CE%B1" title="Εσωτερική ενέργεια – Greek" lang="el" hreflang="el" data-title="Εσωτερική ενέργεια" 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/Energ%C3%ADa_interna" title="Energía interna – Spanish" lang="es" hreflang="es" data-title="Energía interna" 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/Interna_energio" title="Interna energio – Esperanto" lang="eo" hreflang="eo" data-title="Interna energio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Barne_energia" title="Barne energia – Basque" lang="eu" hreflang="eu" data-title="Barne energia" 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/%D8%A7%D9%86%D8%B1%DA%98%DB%8C_%D8%AF%D8%B1%D9%88%D9%86%DB%8C" title="انرژی درونی – Persian" lang="fa" hreflang="fa" data-title="انرژی درونی" 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/%C3%89nergie_interne" title="Énergie interne – French" lang="fr" hreflang="fr" data-title="Énergie interne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fuinneamh_inmhe%C3%A1nach" title="Fuinneamh inmheánach – Irish" lang="ga" hreflang="ga" data-title="Fuinneamh inmheánach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Enerx%C3%ADa_interna" title="Enerxía interna – Galician" lang="gl" hreflang="gl" data-title="Enerxía interna" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%B4%EB%B6%80_%EC%97%90%EB%84%88%EC%A7%80" title="내부 에너지 – Korean" lang="ko" hreflang="ko" data-title="내부 에너지" 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%86%D5%A5%D6%80%D6%84%D5%AB%D5%B6_%D5%A7%D5%B6%D5%A5%D6%80%D5%A3%D5%AB%D5%A1" title="Ներքին էներգիա – Armenian" lang="hy" hreflang="hy" data-title="Ներքին էներգիա" 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%86%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="आन्तरिक ऊर्जा – Hindi" lang="hi" hreflang="hi" data-title="आन्तरिक ऊर्जा" 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/Unutarnja_energija" title="Unutarnja energija – Croatian" lang="hr" hreflang="hr" data-title="Unutarnja energija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Energi_dalam" title="Energi dalam – Indonesian" lang="id" hreflang="id" data-title="Energi dalam" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Energia_interna" title="Energia interna – Italian" lang="it" hreflang="it" data-title="Energia interna" 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%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A4%D7%A0%D7%99%D7%9E%D7%99%D7%AA" title="אנרגיה פנימית – Hebrew" lang="he" hreflang="he" data-title="אנרגיה פנימית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%86%D1%88%D0%BA%D1%96_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Ішкі энергия – Kazakh" lang="kk" hreflang="kk" data-title="Ішкі энергия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/En%C3%A8ji_t%C3%A8mik" title="Enèji tèmik – Haitian Creole" lang="ht" hreflang="ht" data-title="Enèji tèmik" 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-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D1%87%D0%BA%D0%B8_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Ички энергия – Kyrgyz" lang="ky" hreflang="ky" data-title="Ички энергия" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Iek%C5%A1%C4%93j%C4%81_ener%C4%A3ija" title="Iekšējā enerģija – Latvian" lang="lv" hreflang="lv" data-title="Iekšējā enerģija" 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/Vidin%C4%97_energija" title="Vidinė energija – Lithuanian" lang="lt" hreflang="lt" data-title="Vidinė energija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bels%C5%91_energia" title="Belső energia – Hungarian" lang="hu" hreflang="hu" data-title="Belső energia" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%BD%D0%B0%D1%82%D1%80%D0%B5%D1%88%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Внатрешна енергија – Macedonian" lang="mk" hreflang="mk" data-title="Внатрешна енергија" 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%86%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%B0%E0%B4%BF%E0%B4%95_%E0%B4%8A%E0%B5%BC%E0%B4%9C%E0%B5%8D%E0%B4%9C%E0%B4%82" title="ആന്തരിക ഊർജ്ജം – Malayalam" lang="ml" hreflang="ml" data-title="ആന്തരിക ഊർജ്ജം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tenaga_dalaman" title="Tenaga dalaman – Malay" lang="ms" hreflang="ms" data-title="Tenaga dalaman" 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%D0%BE%D1%82%D0%BE%D0%BE%D0%B4_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Дотоод энерги – Mongolian" lang="mn" hreflang="mn" data-title="Дотоод энерги" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inwendige_energie" title="Inwendige energie – Dutch" lang="nl" hreflang="nl" data-title="Inwendige energie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%85%E9%83%A8%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC" title="内部エネルギー – Japanese" lang="ja" hreflang="ja" data-title="内部エネルギー" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Indre_energi" title="Indre energi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Indre energi" 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/Indre_energi" title="Indre energi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Indre energi" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Energia_int%C3%A8rna" title="Energia intèrna – Occitan" lang="oc" hreflang="oc" data-title="Energia intèrna" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ichki_energiya" title="Ichki energiya – Uzbek" lang="uz" hreflang="uz" data-title="Ichki energiya" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Energia_wewn%C4%99trzna" title="Energia wewnętrzna – Polish" lang="pl" hreflang="pl" data-title="Energia wewnętrzna" 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/Energia_interna" title="Energia interna – Portuguese" lang="pt" hreflang="pt" data-title="Energia interna" 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/Energie_intern%C4%83" title="Energie internă – Romanian" lang="ro" hreflang="ro" data-title="Energie internă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BD%D1%83%D1%82%D1%80%D0%B5%D0%BD%D0%BD%D1%8F%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Внутренняя энергия – Russian" lang="ru" hreflang="ru" data-title="Внутренняя энергия" 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/Energjia_e_brendshme" title="Energjia e brendshme – Albanian" lang="sq" hreflang="sq" data-title="Energjia e brendshme" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Internal_energy" title="Internal energy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Internal energy" 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/Vn%C3%BAtorn%C3%A1_energia" title="Vnútorná energia – Slovak" lang="sk" hreflang="sk" data-title="Vnútorná energia" 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/Notranja_energija" title="Notranja energija – Slovenian" lang="sl" hreflang="sl" data-title="Notranja energija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%88%D8%B2%DB%95%DB%8C_%D9%86%D8%A7%D9%88%DB%95%DA%A9%DB%8C" title="وزەی ناوەکی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="وزەی ناوەکی" 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/Unutra%C5%A1nja_energija" title="Unutrašnja energija – Serbian" lang="sr" hreflang="sr" data-title="Unutrašnja energija" 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/Unutarnja_energija" title="Unutarnja energija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Unutarnja energija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Sis%C3%A4energia" title="Sisäenergia – Finnish" lang="fi" hreflang="fi" data-title="Sisäenergia" 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/Inre_energi" title="Inre energi – Swedish" lang="sv" hreflang="sv" data-title="Inre energi" 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%85%E0%AE%95_%E0%AE%86%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%B2%E0%AF%8D" title="அக ஆற்றல் – Tamil" lang="ta" hreflang="ta" data-title="அக ஆற்றல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0%C3%A7_enerji" title="İç enerji – Turkish" lang="tr" hreflang="tr" data-title="İç enerji" 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%92%D0%BD%D1%83%D1%82%D1%80%D1%96%D1%88%D0%BD%D1%8F_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Внутрішня енергія – Ukrainian" lang="uk" hreflang="uk" data-title="Внутрішня енергія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/N%E1%BB%99i_n%C4%83ng" title="Nội năng – Vietnamese" lang="vi" hreflang="vi" data-title="Nội năng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%85%A7%E8%83%BD" title="內能 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="內能" 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/%E5%86%85%E8%83%BD" title="内能 – Wu" lang="wuu" hreflang="wuu" data-title="内能" 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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%A7%E8%83%BD" title="內能 – Cantonese" lang="yue" hreflang="yue" data-title="內能" 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 href="https://zh.wikipedia.org/wiki/%E5%86%85%E8%83%BD" title="内能 – Chinese" lang="zh" hreflang="zh" data-title="内能" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180241#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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.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">Internal energy</th></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">U</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">J</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">m<sup>2</sup>⋅kg/s<sup>2</sup></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br />other quantities</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 \Delta U=\sum _{i}p_{i}E_{i}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=\sum _{i}p_{i}E_{i}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f263b7e7c8a1bf29c94676f80c516c0b05459e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.387ex; width:15.043ex; height:5.509ex;" alt="{\displaystyle \Delta U=\sum _{i}p_{i}E_{i}\!}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U=nC_{V}\Delta T\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <mi>n</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>T</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=nC_{V}\Delta T\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca2cd8924ccfee2b35b50fa810894714bcab484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:14.942ex; height:2.509ex;" alt="{\displaystyle \Delta U=nC_{V}\Delta T\!}"></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 li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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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)">Branches</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 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mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></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 ); 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)"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></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 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></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)"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></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)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></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 class="mw-selflink selflink">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="hlist"><ul><li>History</li><li>Culture</li></ul></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); 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)">Scientists</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)">Other</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>internal energy</b> of a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a> is the <a href="/wiki/Energy" title="Energy">energy</a> of the system as a <a href="/wiki/State_function" title="State function">state function</a>, measured as the quantity of energy necessary to bring the system from its <a href="/wiki/Standard_state" title="Standard state">standard</a> internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as <a href="/wiki/Magnetization" title="Magnetization">magnetization</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It excludes the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> of motion of the system as a whole and the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, <i>i.e.</i>, the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an <a href="/wiki/Thermodynamic_system#Isolated_system" title="Thermodynamic system">isolated</a> system cannot change, as expressed in the law of <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>, a foundation of the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>.<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> The notion has been introduced to describe the systems characterized by temperature variations, temperature being added to the set of state parameters, the position variables known in mechanics (and their conjugated generalized force parameters), in a similar way to <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> of the conservative fields of force, gravitational and electrostatic. Internal energy changes equal the algebraic sum of the heat transferred and the work done. In systems without temperature changes, potential energy changes equal the work done by/on the system. </p><p>The internal energy cannot be measured absolutely. Thermodynamics concerns <i>changes</i> in the internal energy, not its absolute value. The processes that change the internal energy are transfers, into or out of the system, of substance, or of energy, as <a href="/wiki/Heat" title="Heat">heat</a>, or by <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">thermodynamic work</a>.<sup id="cite_ref-Born_146_4-0" class="reference"><a href="#cite_note-Born_146-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> These processes are measured by changes in the system's properties, such as temperature, <a href="/wiki/Entropy" title="Entropy">entropy</a>, volume, electric polarization, and <a href="/wiki/Chemical_composition" title="Chemical composition">molar constitution</a>. The internal energy depends only on the internal state of the system and not on the particular choice from many possible processes by which energy may pass into or out of the system. It is a <a href="/wiki/State_function" title="State function">state variable</a>, a <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a>, and an <a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">extensive property</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Thermodynamics defines internal energy macroscopically, for the body as a whole. In <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical mechanics</a>, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from <a href="/wiki/Translation_(physics)" class="mw-redirect" title="Translation (physics)">translations</a>, <a href="/wiki/Rotation" title="Rotation">rotations</a>, and <a href="/wiki/Oscillation" title="Oscillation">vibrations</a>, and of the potential energies associated with microscopic forces, including <a href="/wiki/Chemical_bonds" class="mw-redirect" title="Chemical bonds">chemical bonds</a>. </p><p>The unit of <a href="/wiki/Energy" title="Energy">energy</a> in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI) is the <a href="/wiki/Joule" title="Joule">joule</a> (J). The internal energy relative to the <a href="/wiki/Mass" title="Mass">mass</a> with unit J/kg is the <i>specific internal energy</i>. The corresponding quantity relative to the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> with unit J/<a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a> is the <i>molar internal energy</i>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Cardinal_functions">Cardinal functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=1" title="Edit section: Cardinal functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The internal energy of a system depends on its entropy S, its volume V and its number of massive particles: <span class="texhtml"><i>U</i>(<i>S</i>,<i>V</i>,{<i>N<sub>j</sub></i>})</span>. It expresses the thermodynamics of a system in the <i>energy representation</i>. As a <a href="/wiki/State_function" title="State function">function of state</a>, its arguments are exclusively extensive variables of state. Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, <span class="texhtml"><i>S</i>(<i>U</i>,<i>V</i>,{<i>N<sub>j</sub></i>})</span>, of the same list of extensive variables of state, except that the entropy, <span class="texhtml"><i>S</i></span>, is replaced in the list by the internal energy, <span class="texhtml"><i>U</i></span>. It expresses the <i>entropy representation</i>.<sup id="cite_ref-Tschoegl_17_7-0" class="reference"><a href="#cite_note-Tschoegl_17-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Callen_Ch_5_8-0" class="reference"><a href="#cite_note-Callen_Ch_5-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Each cardinal function is a monotonic function of each of its <i>natural</i> or <i>canonical</i> variables. Each provides its <i>characteristic</i> or <i>fundamental</i> equation, for example <span class="texhtml"><i>U</i> = <i>U</i>(<i>S</i>,<i>V</i>,{<i>N<sub>j</sub></i>})</span>, that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, <span class="texhtml"><i>U</i> = <i>U</i>(<i>S</i>,<i>V</i>,{<i>N<sub>j</sub></i>})</span> for <span class="texhtml"><i>S</i></span>, to get <span class="texhtml"><i>S</i> = <i>S</i>(<i>U</i>,<i>V</i>,{<i>N<sub>j</sub></i>})</span>. </p><p>In contrast, <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformations</a> are necessary to derive fundamental equations for other thermodynamic potentials and <a href="/wiki/Massieu_function" title="Massieu function">Massieu functions</a>. The entropy as a function only of extensive state variables is the one and only <i>cardinal function</i> of state for the generation of Massieu functions. It is not itself customarily designated a 'Massieu function', though rationally it might be thought of as such, corresponding to the term 'thermodynamic potential', which includes the internal energy.<sup id="cite_ref-Callen_Ch_5_8-1" class="reference"><a href="#cite_note-Callen_Ch_5-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><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> </p><p>For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle. Formal, in principle, manipulations of them are valuable for the understanding of thermodynamics. </p> <div class="mw-heading mw-heading2"><h2 id="Description_and_definition">Description and definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=2" title="Edit section: Description and definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The internal energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> of a given state of the system is determined relative to that of a standard state of the system, by adding up the macroscopic transfers of energy that accompany a change of state from the reference state to the given state: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U=\sum _{i}E_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=\sum _{i}E_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bce7012333053090c3342f89dd891ce29d5a4a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.721ex; height:5.509ex;" alt="{\displaystyle \Delta U=\sum _{i}E_{i},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4716a2c49bbbe155e8b399117ca78342e802cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.718ex; height:2.176ex;" alt="{\displaystyle \Delta U}"></span> denotes the difference between the internal energy of the given state and that of the reference state, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> are the various energies transferred to the system in the steps from the reference state to the given state. It is the energy needed to create the given state of the system from the reference state. From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{micro,pot}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>micro,pot</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\text{micro,pot}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca984ae3959cc68c5e4b8e18905333a125b57614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.676ex; height:2.843ex;" alt="{\displaystyle U_{\text{micro,pot}}}"></span>, and microscopic kinetic energy, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{micro,kin}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>micro,kin</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\text{micro,kin}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b55e7709b7f2809a85cdd2ad5d093b6906dd544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.539ex; height:2.843ex;" alt="{\displaystyle U_{\text{micro,kin}}}"></span>, components: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=U_{\text{micro,pot}}+U_{\text{micro,kin}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>micro,pot</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>micro,kin</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=U_{\text{micro,pot}}+U_{\text{micro,kin}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9fc65c2ede3aabfa1d9fae4fb82ee35d819dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.583ex; height:2.843ex;" alt="{\displaystyle U=U_{\text{micro,pot}}+U_{\text{micro,kin}}.}"></span></dd></dl> <p>The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles. The microscopic potential energy algebraic summative components are those of the <a href="/wiki/Chemical_energy" title="Chemical energy">chemical</a> and <a href="/wiki/Nuclear_potential_energy" class="mw-redirect" title="Nuclear potential energy">nuclear</a> particle bonds, and the physical force fields within the system, such as due to internal <a href="/wiki/Electrostatic_induction" title="Electrostatic induction">induced</a> electric or <a href="/wiki/Magnetism" title="Magnetism">magnetic</a> <a href="/wiki/Dipole" title="Dipole">dipole</a> <a href="/wiki/Moment_(physics)" title="Moment (physics)">moment</a>, as well as the energy of <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a> of solids (<a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a>-<a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a>). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics. </p><p>Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitational</a>, <a href="/wiki/Electrostatics" title="Electrostatics">electrostatic</a>, or <a href="/wiki/Electromagnetics" class="mw-redirect" title="Electromagnetics">electromagnetic</a> <a href="/wiki/Field_(physics)" title="Field (physics)">fields</a>. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the object with the field. In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter. </p><p>For practical considerations in thermodynamics or engineering, it is rarely necessary, convenient, nor even possible, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass. Typically, descriptions only include components relevant to the system under study. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.<sup id="cite_ref-klotz_12-0" class="reference"><a href="#cite_note-klotz-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Therefore, a convenient null reference point may be chosen for the internal energy. </p><p>The internal energy is an <a href="/wiki/Extensive_variable" class="mw-redirect" title="Extensive variable">extensive property</a>: it depends on the size of the system, or on the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> it contains. </p><p>At any temperature greater than <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an <a href="/wiki/Isolated_system" title="Isolated system">isolated system</a> (cf. table). In the classical picture of thermodynamics, kinetic energy vanishes at zero temperature and the internal energy is purely potential energy. However, quantum mechanics has demonstrated that even at zero temperature particles maintain a residual energy of motion, the <a href="/wiki/Zero_point_energy" class="mw-redirect" title="Zero point energy">zero point energy</a>. A system at absolute zero is merely in its quantum-mechanical ground state, the lowest energy state available. At absolute zero a system of given composition has attained its minimum attainable <a href="/wiki/Entropy" title="Entropy">entropy</a>. </p><p>The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a> relates the pseudo-random kinetic energy of individual particles to the mean kinetic energy of the entire ensemble of particles comprising a system. Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. While temperature is an intensive measure, this energy expresses the concept as an extensive property of the system, often referred to as the <i>thermal energy</i>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hyperphysics_14-0" class="reference"><a href="#cite_note-hyperphysics-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The scaling property between temperature and thermal energy is the entropy change of the system. </p><p>Statistical mechanics considers any system to be statistically distributed across an ensemble of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> <a href="/wiki/Microstate_(statistical_mechanics)" title="Microstate (statistical mechanics)">microstates</a>. In a system that is in thermodynamic contact equilibrium with a heat reservoir, each microstate has an energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> and is associated with a probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>. The internal energy is the <a href="/wiki/Mean" title="Mean">mean</a> value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\sum _{i=1}^{N}p_{i}\,E_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\sum _{i=1}^{N}p_{i}\,E_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084963864682f3f51caf73cfc24a0cd19efe3153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.141ex; height:7.343ex;" alt="{\displaystyle U=\sum _{i=1}^{N}p_{i}\,E_{i}.}"></span></dd></dl> <p>This is the statistical expression of the law of <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>. </p> <table align="right" border="0" cellpadding="5" cellspacing="0"> <tbody><tr> <td> <table class="wikitable" style="text-align: center;"> <caption>Interactions of thermodynamic systems </caption> <tbody><tr> <th>Type of system </th> <th><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow</a> </th> <th><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a> </th> <th><a href="/wiki/Heat" title="Heat">Heat</a> </th></tr> <tr> <td><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open</a> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td></tr> <tr> <td><a href="/wiki/Thermodynamic_system#Closed_system" title="Thermodynamic system">Closed</a> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td></tr> <tr> <td><a href="/wiki/Thermally_isolated_system" title="Thermally isolated system">Thermally isolated</a> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td></tr> <tr> <td><a href="/wiki/Mechanically_isolated_system" title="Mechanically isolated system">Mechanically isolated</a> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td> <td><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> </td></tr> <tr> <td><a href="/wiki/Isolated_system" title="Isolated system">Isolated</a> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td> <td><span typeof="mw:File"><span><img alt="Red X" src="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/13px-Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/20px-Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/ba/Red_x.svg/26px-Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display: none;">N</span> </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Internal_energy_changes">Internal energy changes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=3" title="Edit section: Internal energy changes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thermodynamics is chiefly concerned with the changes in internal energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4716a2c49bbbe155e8b399117ca78342e802cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.718ex; height:2.176ex;" alt="{\displaystyle \Delta U}"></span>. </p><p>For a closed system, with mass transfer excluded, the changes in internal energy are due to heat transfer <span class="mwe-math-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> and due to <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">thermodynamic work</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> done <i>by</i> the system on its surroundings.<sup id="cite_ref-signconvention_15-0" class="reference"><a href="#cite_note-signconvention-15"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> Accordingly, the internal energy change <span class="mwe-math-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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4716a2c49bbbe155e8b399117ca78342e802cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.718ex; height:2.176ex;" alt="{\displaystyle \Delta U}"></span> for a process may be written <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\quad {\text{(closed system, no transfer of substance)}}.}"> <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="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(closed system, no transfer of substance)</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=Q-W\quad {\text{(closed system, no transfer of substance)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa8da3aabf6a89c2920bfef0a94c64b06f8291b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.29ex; height:2.843ex;" alt="{\displaystyle \Delta U=Q-W\quad {\text{(closed system, no transfer of substance)}}.}"></span> </p><p>When a closed system receives energy as heat, this energy increases the internal energy. It is distributed between microscopic kinetic and microscopic potential energies. In general, thermodynamics does not trace this distribution. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be <i><a href="/wiki/Sensible_heat" title="Sensible heat">sensible</a></i>. </p><p>A second kind of mechanism of change in the internal energy of a closed system changed is in its doing of <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> on its surroundings. Such work may be simply mechanical, as when the system expands to drive a piston, or, for example, when the system changes its electric polarization so as to drive a change in the electric field in the surroundings. </p><p>If the system is not closed, the third mechanism that can increase the internal energy is transfer of substance into the system. This increase, <span class="mwe-math-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 U_{\mathrm {matter} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U_{\mathrm {matter} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db03d811af97718fed17c4347d4b1c82e132e3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.6ex; height:2.509ex;" alt="{\displaystyle \Delta U_{\mathrm {matter} }}"></span> cannot be split into heat and work components.<sup id="cite_ref-Born_146_4-1" class="reference"><a href="#cite_note-Born_146-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> If the system is so set up physically that heat transfer and work that it does are by pathways separate from and independent of matter transfer, then the transfers of energy add to change the internal energy: <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+\Delta U_{\text{matter}}\quad {\text{(matter transfer pathway separate from heat and work transfer pathways)}}.}"> <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 mathvariant="normal">Δ</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>matter</mtext> </mrow> </msub> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(matter transfer pathway separate from heat and work transfer pathways)</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=Q-W+\Delta U_{\text{matter}}\quad {\text{(matter transfer pathway separate from heat and work transfer pathways)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0e22a2c156e3660711da651e99a342d0382db3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:102.527ex; height:2.843ex;" alt="{\displaystyle \Delta U=Q-W+\Delta U_{\text{matter}}\quad {\text{(matter transfer pathway separate from heat and work transfer pathways)}}.}"></span> </p><p>If a system undergoes certain phase transformations while being heated, such as melting and vaporization, it may be observed that the temperature of the system does not change until the entire sample has completed the transformation. The energy introduced into the system while the temperature does not change is called <i>latent energy</i> or <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a>, in contrast to sensible heat, which is associated with temperature change. </p> <div class="mw-heading mw-heading2"><h2 id="Internal_energy_of_the_ideal_gas">Internal energy of the ideal gas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=4" title="Edit section: Internal energy of the ideal gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thermodynamics often uses the concept of the <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> for teaching purposes, and as an approximation for working systems. The ideal gas consists of particles considered as point objects that interact only by elastic collisions and fill a volume such that their <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a> between collisions is much larger than their diameter. Such systems approximate <a href="/wiki/Monatomic" class="mw-redirect" title="Monatomic">monatomic</a> gases such as <a href="/wiki/Helium" title="Helium">helium</a> and other <a href="/wiki/Noble_gas" title="Noble gas">noble gases</a>. For an ideal gas the kinetic energy consists only of the <a href="/wiki/Translation_(physics)" class="mw-redirect" title="Translation (physics)">translational</a> energy of the individual atoms. Monatomic particles do not possess rotational or vibrational degrees of freedom, and are not <a href="/wiki/Energy_level" title="Energy level">electronically excited</a> to higher energies except at very high <a href="/wiki/Temperature" title="Temperature">temperatures</a>. </p><p>Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): <span class="mwe-math-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=U(N,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=U(N,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc9f55c567c6a8e6c0855bb1e9f53c58dfc0c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.207ex; height:2.843ex;" alt="{\displaystyle U=U(N,T)}"></span>. It is not dependent on other thermodynamic quantities such as pressure or density. </p><p>The internal energy of an ideal gas is proportional to its <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> (number of moles) <span class="mwe-math-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> and to its temperature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <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> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=c_{V}NT,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=c_{V}NT,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65321b236ba998886794e791024324c77e3e739e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.731ex; height:2.509ex;" alt="{\displaystyle U=c_{V}NT,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.503ex; height:2.009ex;" alt="{\displaystyle c_{V}}"></span> is the isochoric (at constant volume) <a href="/wiki/Molar_heat_capacity" title="Molar heat capacity">molar heat capacity</a> of the gas; <span class="mwe-math-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_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.503ex; height:2.009ex;" alt="{\displaystyle c_{V}}"></span> is constant for an ideal gas. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, <span class="mwe-math-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>, <span class="mwe-math-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> (entropy, volume, <a href="/wiki/Amount_of_substance" title="Amount of substance">number of moles</a>). In case of the ideal gas it is in the following way <sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V,N)=\mathrm {const} \cdot e^{\frac {S}{c_{V}N}}V^{\frac {-R}{c_{V}}}N^{\frac {R+c_{V}}{c_{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>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> </mrow> </mfrac> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>R</mi> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> </msup> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V,N)=\mathrm {const} \cdot e^{\frac {S}{c_{V}N}}V^{\frac {-R}{c_{V}}}N^{\frac {R+c_{V}}{c_{V}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1488822daa630308a960fcee683dbec19076a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.832ex; height:4.676ex;" alt="{\displaystyle U(S,V,N)=\mathrm {const} \cdot e^{\frac {S}{c_{V}N}}V^{\frac {-R}{c_{V}}}N^{\frac {R+c_{V}}{c_{V}}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {const} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {const} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b337bfdc18c1579eba37a95b5c9b28eec7cb187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.308ex; height:2.009ex;" alt="{\displaystyle \mathrm {const} }"></span> is an arbitrary positive constant and where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the <a href="/wiki/Gas_constant" title="Gas constant">universal gas constant</a>. It is easily seen that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is a linearly <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a> of the three variables (that is, it is <i>extensive</i> in these variables), and that it is weakly <a href="/wiki/Convex_function" title="Convex function">convex</a>. Knowing temperature and pressure to be the derivatives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {\partial U}{\partial S}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {\partial U}{\partial S}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99b88a80952c46255dbff179570937d0cb8c2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.318ex; height:5.509ex;" alt="{\displaystyle T={\frac {\partial U}{\partial S}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=-{\frac {\partial U}{\partial V}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=-{\frac {\partial U}{\partial V}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6205289e81344c9f3a6760bbf4f2f8fe914cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.24ex; height:5.509ex;" alt="{\displaystyle P=-{\frac {\partial U}{\partial V}},}"></span> the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV=NRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mi>N</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV=NRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b07e7d66ad0668041e492dbbaafff89e37f763d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.095ex; height:2.176ex;" alt="{\displaystyle PV=NRT}"></span> immediately follows as below: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {\partial U}{\partial S}}={\frac {U}{c_{V}N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {\partial U}{\partial S}}={\frac {U}{c_{V}N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09fb483bf4e5d8a58f244d779f67b10dce4de25a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.173ex; height:5.843ex;" alt="{\displaystyle T={\frac {\partial U}{\partial S}}={\frac {U}{c_{V}N}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=-{\frac {\partial U}{\partial V}}=U{\frac {R}{c_{V}V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>V</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=-{\frac {\partial U}{\partial V}}=U{\frac {R}{c_{V}V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9226bedfb8bf3cdc88c8fab4c08c9c58e49a8a49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.601ex; height:5.843ex;" alt="{\displaystyle P=-{\frac {\partial U}{\partial V}}=U{\frac {R}{c_{V}V}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {P}{T}}={\frac {\frac {UR}{c_{V}V}}{\frac {U}{c_{V}N}}}={\frac {NR}{V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi>U</mi> <mi>R</mi> </mrow> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>V</mi> </mrow> </mfrac> <mfrac> <mi>U</mi> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> </mrow> </mfrac> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mi>R</mi> </mrow> <mi>V</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P}{T}}={\frac {\frac {UR}{c_{V}V}}{\frac {U}{c_{V}N}}}={\frac {NR}{V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca1f231cc0c0a3eaa4e9fdc5c1d85db65753fa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:18.476ex; height:8.843ex;" alt="{\displaystyle {\frac {P}{T}}={\frac {\frac {UR}{c_{V}V}}{\frac {U}{c_{V}N}}}={\frac {NR}{V}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV=NRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mi>N</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV=NRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b07e7d66ad0668041e492dbbaafff89e37f763d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.095ex; height:2.176ex;" alt="{\displaystyle PV=NRT}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Internal_energy_of_a_closed_thermodynamic_system">Internal energy of a closed thermodynamic system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=5" title="Edit section: Internal energy of a closed thermodynamic system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above summation of all components of change in internal energy assumes that a positive energy denotes heat added to the system or the negative of work done by the system on its surroundings.<sup id="cite_ref-signconvention_15-1" class="reference"><a href="#cite_note-signconvention-15"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p><p>This relationship may be expressed in <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> terms using the differentials of each term, though only the internal energy is an <a href="/wiki/Exact_differential" title="Exact differential">exact differential</a>.<sup id="cite_ref-adkins1983_17-0" class="reference"><a href="#cite_note-adkins1983-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 33">: 33 </span></sup> For a closed system, with transfers only as heat and work, the change in the internal energy is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=\delta Q-\delta W,}"> <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>δ</mi> <mi>Q</mi> <mo>−</mo> <mi>δ</mi> <mi>W</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=\delta Q-\delta W,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60fada72885619f961416a97c87fdb93dc3e76fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.032ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} U=\delta Q-\delta W,}"></span></dd></dl> <p>expressing the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>. It may be expressed in terms of other thermodynamic parameters. Each term is composed of an <a href="/wiki/Intensive_variable" class="mw-redirect" title="Intensive variable">intensive variable</a> (a generalized force) and its <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">conjugate</a> infinitesimal <a href="/wiki/Extensive_variable" class="mw-redirect" title="Extensive variable">extensive variable</a> (a generalized displacement). </p><p>For example, the mechanical work done by the system may be related to the <a href="/wiki/Pressure" title="Pressure">pressure</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">volume</a> change <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} V}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80507190aa9d38a279909db47b63657f2b62ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.08ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} V}"></span>. The pressure is the intensive generalized force, while the volume change is the extensive generalized displacement: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=P\,\mathrm {d} V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=P\,\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b42d27cb93476cab9abfa91b1314377da90da90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.441ex; height:2.343ex;" alt="{\displaystyle \delta W=P\,\mathrm {d} V.}"></span></dd></dl> <p>This defines the direction of work, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>, to be energy transfer from the working system to the surroundings, indicated by a positive term.<sup id="cite_ref-signconvention_15-2" class="reference"><a href="#cite_note-signconvention-15"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> Taking the direction of heat transfer <span class="mwe-math-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> to be into the working fluid and assuming a <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible process</a>, the heat is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta Q=T\mathrm {d} S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q=T\mathrm {d} S,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842dd6d796f9f794136f9dfa26251e902e740b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.06ex; height:2.676ex;" alt="{\displaystyle \delta Q=T\mathrm {d} S,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> denotes the <a href="/wiki/Temperature" title="Temperature">temperature</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> denotes the <a href="/wiki/Entropy" title="Entropy">entropy</a>. </p><p>The change in internal energy becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <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/bedb28bf714d1055421d3c5002d581ad6d85fab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.688ex; height:2.343ex;" alt="{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Changes_due_to_temperature_and_volume">Changes due to temperature and volume</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=6" title="Edit section: Changes due to temperature and volume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The expression relating changes in internal energy to changes in temperature and volume is </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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=C_{V}\,\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-P\right]\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> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>−</mo> <mi>P</mi> </mrow> <mo>]</mo> </mrow> <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=C_{V}\,\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-P\right]\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed701e880407b580dd14a57cd80113a92b019e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.095ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} U=C_{V}\,\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-P\right]\mathrm {d} V.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>This is useful if the <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a> is known. </p><p>In case of an ideal gas, we can derive that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU=C_{V}\,dT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=C_{V}\,dT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd53ce58fc85c66bc0fca9c1187f4d02b902594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.494ex; height:2.509ex;" alt="{\displaystyle dU=C_{V}\,dT}"></span>, i.e. the internal energy of an ideal gas can be written as a function that depends only on the temperature. </p> <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px solid lightgray; width: 60%;;"><div class="hidden-title skin-nightmode-reset-color" style="">Proof of pressure independence for an ideal gas</div><div class="hidden-content mw-collapsible-content" style=""> <p>The expression relating changes in internal energy to changes in temperature and volume is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=C_{V}\,\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-P\right]\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> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>−</mo> <mi>P</mi> </mrow> <mo>]</mo> </mrow> <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=C_{V}\,\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-P\right]\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed701e880407b580dd14a57cd80113a92b019e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.095ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} U=C_{V}\,\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-P\right]\mathrm {d} V.}"></span></dd></dl> <p>The equation of state is the ideal gas law </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV=nRT.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV=nRT.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d6645984bb01105369da5d6d0763be8908d070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.073ex; height:2.176ex;" alt="{\displaystyle PV=nRT.}"></span></dd></dl> <p>Solve for pressure: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {nRT}{V}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {nRT}{V}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3a8858322768e827a28433eed4c1902b98571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.122ex; height:5.343ex;" alt="{\displaystyle P={\frac {nRT}{V}}.}"></span></dd></dl> <p>Substitute in to internal energy expression: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU=C_{V}\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-{\frac {nRT}{V}}\right]\mathrm {d} V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mrow> <mi>V</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=C_{V}\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-{\frac {nRT}{V}}\right]\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d332b8673ecdd8b67afab181dc253900e6770c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.517ex; height:6.176ex;" alt="{\displaystyle dU=C_{V}\mathrm {d} T+\left[T\left({\frac {\partial P}{\partial T}}\right)_{V}-{\frac {nRT}{V}}\right]\mathrm {d} V.}"></span></dd></dl> <p>Take the derivative of pressure with respect to temperature: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial P}{\partial T}}\right)_{V}={\frac {nR}{V}}.}"> <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>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial P}{\partial T}}\right)_{V}={\frac {nR}{V}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dadcfa7ec1f2d96049f22c1ed9cb649e7566b1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.557ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial P}{\partial T}}\right)_{V}={\frac {nR}{V}}.}"></span></dd></dl> <p>Replace: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU=C_{V}\,\mathrm {d} T+\left[{\frac {nRT}{V}}-{\frac {nRT}{V}}\right]\mathrm {d} V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mrow> <mi>V</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=C_{V}\,\mathrm {d} T+\left[{\frac {nRT}{V}}-{\frac {nRT}{V}}\right]\mathrm {d} V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/933f6fb9f3e81dadbb115e0afd1a575fdc8219a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.082ex; height:6.176ex;" alt="{\displaystyle dU=C_{V}\,\mathrm {d} T+\left[{\frac {nRT}{V}}-{\frac {nRT}{V}}\right]\mathrm {d} V.}"></span></dd></dl> <p>And simplify: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=C_{V}\,\mathrm {d} T.}"> <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> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=C_{V}\,\mathrm {d} T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c14d35a561605ad14554bdd0bd844c16178e05c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.294ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} U=C_{V}\,\mathrm {d} T.}"></span></dd></dl> </div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px solid lightgray; width: 60%;;"><div class="hidden-title skin-nightmode-reset-color" style="">Derivation of d<i>U</i> in terms of d<i>T</i> and d<i>V</i></div><div class="hidden-content mw-collapsible-content" style=""> <p>To express <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d25429855f05fe5da1e4beadb321ed305633e4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.075ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} U}"></span> in terms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.929ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} T}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80507190aa9d38a279909db47b63657f2b62ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.08ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} V}"></span>, the term </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} S=\left({\frac {\partial S}{\partial T}}\right)_{V}\mathrm {d} T+\left({\frac {\partial S}{\partial V}}\right)_{T}\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>S</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>+</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S=\left({\frac {\partial S}{\partial T}}\right)_{V}\mathrm {d} T+\left({\frac {\partial S}{\partial V}}\right)_{T}\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77880b00b6cfa0bac9ad17572402b378bd8ff5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.199ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} S=\left({\frac {\partial S}{\partial T}}\right)_{V}\mathrm {d} T+\left({\frac {\partial S}{\partial V}}\right)_{T}\mathrm {d} V}"></span></dd></dl> <p>is substituted in the <a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">fundamental thermodynamic relation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <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/bedb28bf714d1055421d3c5002d581ad6d85fab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.688ex; height:2.343ex;" alt="{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V.}"></span></dd></dl> <p>This gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU=T\left({\frac {\partial S}{\partial T}}\right)_{V}\,dT+\left[T\left({\frac {\partial S}{\partial V}}\right)_{T}-P\right]dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>−</mo> <mi>P</mi> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=T\left({\frac {\partial S}{\partial T}}\right)_{V}\,dT+\left[T\left({\frac {\partial S}{\partial V}}\right)_{T}-P\right]dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df1835ced4c98a027f356bb12ca15c40ff930ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.986ex; height:6.176ex;" alt="{\displaystyle dU=T\left({\frac {\partial S}{\partial T}}\right)_{V}\,dT+\left[T\left({\frac {\partial S}{\partial V}}\right)_{T}-P\right]dV.}"></span></dd></dl> <p>The term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\left({\frac {\partial S}{\partial T}}\right)_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\left({\frac {\partial S}{\partial T}}\right)_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae822fd2e96c41f55289865dd191c08fd0918b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.344ex; height:6.176ex;" alt="{\displaystyle T\left({\frac {\partial S}{\partial T}}\right)_{V}}"></span> is the <a href="/wiki/Specific_heat_capacity#Heat_capacity_of_compressible_bodies" title="Specific heat capacity">heat capacity at constant volume</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{V}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ea177b41743e1df124dddd371e2cd5fd6bc031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.805ex; height:2.509ex;" alt="{\displaystyle C_{V}.}"></span> </p><p>The partial derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> can be evaluated if the equation of state is known. From the fundamental thermodynamic relation, it follows that the differential of the <a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dA=-S\,dT-P\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mo>−</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dA=-S\,dT-P\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bcc0184bdffe0238a692fa262142facaaa45718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.227ex; height:2.343ex;" alt="{\displaystyle dA=-S\,dT-P\,dV.}"></span></dd></dl> <p>The <a href="/wiki/Symmetry_of_second_derivatives" title="Symmetry of second derivatives">symmetry of second derivatives</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> yields the <a href="/wiki/Maxwell_relation" class="mw-redirect" title="Maxwell relation">Maxwell relation</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93ce0a2dbf296f0b586107503377d734fc1c91c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.314ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}.}"></span></dd></dl> <p>This gives the expression above. </p> </div></div> <div class="mw-heading mw-heading3"><h3 id="Changes_due_to_temperature_and_pressure">Changes due to temperature and pressure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=7" title="Edit section: Changes due to temperature and pressure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When considering fluids or solids, an expression in terms of the temperature and pressure is usually more useful: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU=\left(C_{P}-\alpha PV\right)\,dT+\left(\beta _{T}P-\alpha T\right)V\,dP,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>α</mi> <mi>P</mi> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mi>P</mi> <mo>−</mo> <mi>α</mi> <mi>T</mi> </mrow> <mo>)</mo> </mrow> <mi>V</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=\left(C_{P}-\alpha PV\right)\,dT+\left(\beta _{T}P-\alpha T\right)V\,dP,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf20f0deaf9f1733d3de49bfc396763b0c71c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.755ex; height:2.843ex;" alt="{\displaystyle dU=\left(C_{P}-\alpha PV\right)\,dT+\left(\beta _{T}P-\alpha T\right)V\,dP,}"></span></dd></dl> <p>where it is assumed that the heat capacity at constant pressure is <a href="/wiki/Relations_between_specific_heats" class="mw-redirect" title="Relations between specific heats">related</a> to the heat capacity at constant volume according to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{P}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>+</mo> <mi>V</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{P}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a462b026fec199ad2348ea3d0c2ae1a21adcd02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.836ex; height:6.176ex;" alt="{\displaystyle C_{P}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}.}"></span></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px solid lightgray; width: 60%;;"><div class="hidden-title skin-nightmode-reset-color" style="">Derivation of d<i>U</i> in terms of d<i>T</i> and d<i>P</i></div><div class="hidden-content mw-collapsible-content" style=""> <p>The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the <a href="/wiki/Coefficient_of_thermal_expansion" class="mw-redirect" title="Coefficient of thermal expansion">coefficient of thermal expansion</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>V</mi> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93addfd87f7a1f84c3309ab4eb62b163219649c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.039ex; height:6.176ex;" alt="{\displaystyle \alpha \equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}}"></span></dd></dl> <p>and the isothermal <a href="/wiki/Compressibility" title="Compressibility">compressibility</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{T}\equiv -{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>≡</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>V</mi> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{T}\equiv -{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f0f3989eedf20c293a4f0756671294c4e8f5ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.987ex; height:6.176ex;" alt="{\displaystyle \beta _{T}\equiv -{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}}"></span></dd></dl> <p>by writing </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" 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="dV=\left({\frac {\partial V}{\partial p}}\right)_{T}dP+\left({\frac {\partial V}{\partial T}}\right)_{P}dT=V\left(\alpha dT-\beta _{T}\,dP\right)"> <semantics> <mrow> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mi>d</mi> <mi>P</mi> <mo>+</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>d</mi> <mi>T</mi> <mo>=</mo> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>d</mi> <mi>T</mi> <mo>−</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation encoding="application/x-tex">dV=\left({\frac {\partial V}{\partial p}}\right)_{T}dP+\left({\frac {\partial V}{\partial T}}\right)_{P}dT=V\left(\alpha dT-\beta _{T}\,dP\right)</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2047a954988f6f1d899f7d827b4ed2f0ad18f4a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.652ex; height:6.176ex;" alt="dV=\left({\frac {\partial V}{\partial p}}\right)_{T}dP+\left({\frac {\partial V}{\partial T}}\right)_{P}dT=V\left(\alpha dT-\beta _{T}\,dP\right)"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>and equating d<i>V</i> to zero and solving for the ratio d<i>P</i>/d<i>T</i>. This gives </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" 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="\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}={\frac {\alpha }{\beta _{T}}}."> <semantics> <mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mrow> <annotation encoding="application/x-tex">\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}={\frac {\alpha }{\beta _{T}}}.</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7648f69e1b60021259129b28f8b86cf297e2cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:29.12ex; height:10.176ex;" alt="\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}={\frac {\alpha }{\beta _{T}}}."></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>Substituting (<b><a href="#math_2">2</a></b>) and (<b><a href="#math_3">3</a></b>) in (<b><a href="#math_1">1</a></b>) gives the above expression. </p> </div></div> <div class="mw-heading mw-heading3"><h3 id="Changes_due_to_volume_at_constant_temperature">Changes due to volume at constant temperature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=8" title="Edit section: Changes due to volume at constant temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Internal_pressure" title="Internal pressure">internal pressure</a> is defined as a <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of the internal energy with respect to the volume at constant temperature: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ccbe77835c6f1ae60ccfbdadd63e899a20f492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.211ex; height:6.176ex;" alt="{\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Internal_energy_of_multi-component_systems">Internal energy of multi-component systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=9" title="Edit section: Internal energy of multi-component systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_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/Internal_energy" title="Special:EditPage/Internal energy">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">November 2015</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In addition to including the entropy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and volume <span class="mwe-math-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> terms in the internal energy, a system is often described also in terms of the number of particles or chemical species it contains: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=U(S,V,N_{1},\ldots ,N_{n}),}"> <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>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=U(S,V,N_{1},\ldots ,N_{n}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60eb593b561705814a50ed65af00c5b537f5dee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.657ex; height:2.843ex;" alt="{\displaystyle U=U(S,V,N_{1},\ldots ,N_{n}),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e42efe207e17a74b4c17b1aa0cced7a84501bec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.776ex; height:2.843ex;" alt="{\displaystyle N_{j}}"></span> are the molar amounts of constituents of type <span class="mwe-math-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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> in the system. The internal energy is an <a href="/wiki/Extensive_variable" class="mw-redirect" title="Extensive variable">extensive</a> function of the extensive variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, <span class="mwe-math-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>, and the amounts <span class="mwe-math-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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e42efe207e17a74b4c17b1aa0cced7a84501bec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.776ex; height:2.843ex;" alt="{\displaystyle N_{j}}"></span>, the internal energy may be written as a linearly <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a> of first degree:<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>S</mi> <mo>,</mo> <mi>α</mi> <mi>V</mi> <mo>,</mo> <mi>α</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>α</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d28dc94d8ca893db586508c728d7ef37fff602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.34ex; height:2.843ex;" alt="{\displaystyle U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots ),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </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/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is a factor describing the growth of the system. The differential internal energy may be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U={\frac {\partial U}{\partial S}}\mathrm {d} S+{\frac {\partial U}{\partial V}}\mathrm {d} V+\sum _{i}\ {\frac {\partial U}{\partial N_{i}}}\mathrm {d} N_{i}\ =T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i},}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mi>T</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U={\frac {\partial U}{\partial S}}\mathrm {d} S+{\frac {\partial U}{\partial V}}\mathrm {d} V+\sum _{i}\ {\frac {\partial U}{\partial N_{i}}}\mathrm {d} N_{i}\ =T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97fac07fd2bf10fa460e6a1a309585127686317d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.641ex; height:6.509ex;" alt="{\displaystyle \mathrm {d} U={\frac {\partial U}{\partial S}}\mathrm {d} S+{\frac {\partial U}{\partial V}}\mathrm {d} V+\sum _{i}\ {\frac {\partial U}{\partial N_{i}}}\mathrm {d} N_{i}\ =T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i},}"></span></dd></dl> <p>which shows (or defines) temperature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <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> to be the partial derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> with respect to entropy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and pressure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> to be the negative of the similar derivative with respect to volume <span class="mwe-math-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>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {\partial U}{\partial S}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {\partial U}{\partial S}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99b88a80952c46255dbff179570937d0cb8c2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.318ex; height:5.509ex;" alt="{\displaystyle T={\frac {\partial U}{\partial S}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=-{\frac {\partial U}{\partial V}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=-{\frac {\partial U}{\partial V}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6205289e81344c9f3a6760bbf4f2f8fe914cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.24ex; height:5.509ex;" alt="{\displaystyle P=-{\frac {\partial U}{\partial V}},}"></span></dd></dl> <p>and where the coefficients <span class="mwe-math-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 \mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0a0293841cce9eef98b55e53a92b82ae59ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}"></span> are the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potentials</a> for the components of type <span class="mwe-math-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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in the system. The chemical potentials are defined as the partial derivatives of the internal energy with respect to the variations in composition: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd3be9488f998c605acdd32b156a3eb2c5cb956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.191ex; height:6.843ex;" alt="{\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}.}"></span></dd></dl> <p>As conjugate variables to the composition <span class="mwe-math-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 \lbrace N_{j}\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace N_{j}\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6276166bc9f1831ff5be6d1d6880c6162d5811d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.101ex; height:3.009ex;" alt="{\displaystyle \lbrace N_{j}\rbrace }"></span>, the chemical potentials are <a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">intensive properties</a>, intrinsically characteristic of the qualitative nature of the system, and not proportional to its extent. Under conditions of constant <span class="mwe-math-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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, because of the extensive nature of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> and its independent variables, using <a href="/wiki/Homogeneous_function" title="Homogeneous function">Euler's homogeneous function theorem</a>, the differential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d25429855f05fe5da1e4beadb321ed305633e4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.075ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} U}"></span> may be integrated and yields an expression for the internal energy: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=TS-PV+\sum _{i}\mu _{i}N_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mi>V</mi> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=TS-PV+\sum _{i}\mu _{i}N_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b59e86e604d4a92c1a6c36ed85dbdd777c76fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.486ex; height:5.509ex;" alt="{\displaystyle U=TS-PV+\sum _{i}\mu _{i}N_{i}.}"></span></dd></dl> <p>The sum over the composition of the system is the <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\sum _{i}\mu _{i}N_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\sum _{i}\mu _{i}N_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fca4781e9a4e6f530cdb46017b8b58c83e33ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.534ex; height:5.509ex;" alt="{\displaystyle G=\sum _{i}\mu _{i}N_{i}}"></span></dd></dl> <p>that arises from changing the composition of the system at constant temperature and pressure. For a single component system, the chemical potential equals the Gibbs energy per amount of substance, i.e. particles or moles according to the original definition of the unit for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace N_{j}\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace N_{j}\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6276166bc9f1831ff5be6d1d6880c6162d5811d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.101ex; height:3.009ex;" alt="{\displaystyle \lbrace N_{j}\rbrace }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Internal_energy_in_an_elastic_medium">Internal energy in an elastic medium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=10" title="Edit section: Internal energy in an elastic medium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> medium the potential energy component of the internal energy has an elastic nature expressed in terms of the <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acbf52cc4d4f83f187ceaa49f045114b71772e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.804ex; height:2.343ex;" alt="{\displaystyle \sigma _{ij}}"></span> and strain <span class="mwe-math-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 \varepsilon _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71e2079cee1685c2402d4d4ef48d75db18b4a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.561ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{ij}}"></span> involved in elastic processes. In <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a> for tensors, with summation over repeated indices, for unit volume, the infinitesimal statement is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=T\mathrm {d} S+\sigma _{ij}\mathrm {d} \varepsilon _{ij}.}"> <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> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=T\mathrm {d} S+\sigma _{ij}\mathrm {d} \varepsilon _{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38750019a910de15a77c7db591e6314dedf8d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.746ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} U=T\mathrm {d} S+\sigma _{ij}\mathrm {d} \varepsilon _{ij}.}"></span></dd></dl> <p><a href="/wiki/Homogeneous_function#Euler's_theorem" title="Homogeneous function">Euler's theorem</a> yields for the internal energy:<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mi>S</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db41be27ceaacbb94719691039e8d804a511cf41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.867ex; height:5.176ex;" alt="{\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}.}"></span></dd></dl> <p>For a linearly elastic material, the stress is related to the strain by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be431267e61d6d33045a6e8ffb7cc8929b38bb02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.698ex; height:2.843ex;" alt="{\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl},}"></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ijkl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ijkl}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94cb5780b92c6b4cec637a215f2f467f20f67927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.486ex; height:2.843ex;" alt="{\displaystyle C_{ijkl}}"></span> are the components of the 4th-rank elastic constant tensor of the medium. </p><p>Elastic deformations, such as <a href="/wiki/Sound" title="Sound">sound</a>, passing through a body, or other forms of macroscopic internal agitation or turbulent motion create states when the system is not in thermodynamic equilibrium. While such energies of motion continue, they contribute to the total energy of the system; thermodynamic internal energy pertains only when such motions have ceased. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=11" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Joule</a> studied the relationship between heat, work, and temperature. He observed that friction in a liquid, such as caused by its agitation with work by a paddle wheel, caused an increase in its temperature, which he described as producing a <i>quantity of heat</i>. Expressed in modern units, he found that c. 4186 joules of energy were needed to raise the temperature of one kilogram of water by one degree Celsius.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-signconvention-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-signconvention_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-signconvention_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-signconvention_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">This article uses the sign convention of the mechanical work as often defined in engineering, which is different from the convention used in physics and chemistry; in engineering, work performed by the system against the environment, e.g., a system expansion, is taken to be positive, while in physics and chemistry, it is taken to be negative.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Calorimetry" title="Calorimetry">Calorimetry</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Exergy" title="Exergy">Exergy</a></li> <li><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Thermodynamic equations</a></li> <li><a href="/wiki/Thermodynamic_potentials" class="mw-redirect" title="Thermodynamic potentials">Thermodynamic potentials</a></li> <li><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a></li> <li><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Crawford, F. H. (1963), pp. 106–107.</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">Haase, R. (1971), pp. 24–28.</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"><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="CITEREFE.I._Franses2014" class="citation cs2">E.I. 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A., pp. 15, 16.</span> </li> <li id="cite_note-hyperphysics-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-hyperphysics_14-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/eqpar.html#c2">Thermal energy</a> – Hyperphysics.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrubbström1985" class="citation book cs1">Grubbström, Robert W. (1985). "Towards a Generalized Exergy Concept". In van Gool, W.; Bruggink, J.J.C. (eds.). <i>Energy and time in the economic and physical sciences</i>. North-Holland. pp. 41–56. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0444877482" title="Special:BookSources/978-0444877482"><bdi>978-0444877482</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Towards+a+Generalized+Exergy+Concept&rft.btitle=Energy+and+time+in+the+economic+and+physical+sciences&rft.pages=41-56&rft.pub=North-Holland&rft.date=1985&rft.isbn=978-0444877482&rft.aulast=Grubbstr%C3%B6m&rft.aufirst=Robert+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span></span> </li> <li id="cite_note-adkins1983-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-adkins1983_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdkins,_C._J._(Clement_John)1983" class="citation book cs1">Adkins, C. J. 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Cambridge [Cambridgeshire]: Cambridge University Press. <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"><bdi>0-521-25445-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/9132054">9132054</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Equilibrium+thermodynamics&rft.place=Cambridge+%5BCambridgeshire%5D&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=1983&rft_id=info%3Aoclcnum%2F9132054&rft.isbn=0-521-25445-0&rft.au=Adkins%2C+C.+J.+%28Clement+John%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshit︠s︡PitaevskiĭSykes1980" class="citation book cs1">Landau, Lev Davidovich; Lifshit︠s︡, Evgeniĭ Mikhaĭlovich; Pitaevskiĭ, Lev Petrovich; Sykes, John Bradbury; Kearsley, M. J. (1980). <i>Statistical physics</i>. Oxford. p. 70. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-08-023039-3" title="Special:BookSources/0-08-023039-3"><bdi>0-08-023039-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/3932994">3932994</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+physics&rft.place=Oxford&rft.pages=70&rft.date=1980&rft_id=info%3Aoclcnum%2F3932994&rft.isbn=0-08-023039-3&rft.aulast=Landau&rft.aufirst=Lev+Davidovich&rft.au=Lifshit%EF%B8%A0s%EF%B8%A1%2C+Evgeni%C4%AD+Mikha%C4%ADlovich&rft.au=Pitaevski%C4%AD%2C+Lev+Petrovich&rft.au=Sykes%2C+John+Bradbury&rft.au=Kearsley%2C+M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></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"><a href="#CITEREFLandauLifshitz1986">Landau & Lifshitz 1986</a>, p. 8.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoule1850" class="citation journal cs1"><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule, J.P.</a> (1850). "On the Mechanical Equivalent of Heat". <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society" title="Philosophical Transactions of the Royal Society">Philosophical Transactions of the Royal Society</a></i>. <b>140</b>: 61–82. <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.1850.0004">10.1098/rstl.1850.0004</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:186209447">186209447</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&rft.atitle=On+the+Mechanical+Equivalent+of+Heat&rft.volume=140&rft.pages=61-82&rft.date=1850&rft_id=info%3Adoi%2F10.1098%2Frstl.1850.0004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186209447%23id-name%3DS2CID&rft.aulast=Joule&rft.aufirst=J.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography_of_cited_references">Bibliography of cited references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=15" title="Edit section: Bibliography of cited references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Adkins, C. J. (1968/1975). <i>Equilibrium Thermodynamics</i>, second edition, McGraw-Hill, 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/0-07-084057-1" title="Special:BookSources/0-07-084057-1">0-07-084057-1</a>.</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><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>Callen, H. B. (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, 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>Crawford, F. H. (1963). <i>Heat, Thermodynamics, and Statistical Physics</i>, Rupert Hart-Davis, London, Harcourt, Brace & World, Inc.</li> <li>Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of <i>Thermodynamics</i>, pages 1–97 of volume 1, ed. W. Jost, of <i>Physical Chemistry. An Advanced Treatise</i>, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_W._Leland_Jr." class="citation cs2">Thomas W. Leland Jr., G. A. Mansoori (ed.), <a rel="nofollow" class="external text" href="http://www.uic.edu/labs/trl/1.OnlineMaterials/BasicPrinciplesByTWLeland.pdf"><i>Basic Principles of Classical and Statistical Thermodynamics</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Principles+of+Classical+and+Statistical+Thermodynamics&rft.au=Thomas+W.+Leland+Jr.&rft_id=http%3A%2F%2Fwww.uic.edu%2Flabs%2Ftrl%2F1.OnlineMaterials%2FBasicPrinciplesByTWLeland.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1986" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, L. D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E. M.</a> (1986). <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics"><i>Theory of Elasticity (Course of Theoretical Physics Volume 7)</i></a>. (Translated from Russian by J. B. Sykes and W. H. Reid) (Third ed.). Boston, MA: Butterworth Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2633-0" title="Special:BookSources/978-0-7506-2633-0"><bdi>978-0-7506-2633-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Elasticity+%28Course+of+Theoretical+Physics+Volume+7%29&rft.place=Boston%2C+MA&rft.edition=Third&rft.pub=Butterworth+Heinemann&rft.date=1986&rft.isbn=978-0-7506-2633-0&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span></li> <li>Münster, A. (1970), Classical Thermodynamics, translated by E. S. Halberstadt, Wiley–Interscience, 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/0-471-62430-6" title="Special:BookSources/0-471-62430-6">0-471-62430-6</a>.</li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a>, (1923/1927). <i>Treatise on Thermodynamics</i>, translated by A. Ogg, third English edition, <a href="/wiki/Longman" title="Longman">Longmans, Green and Co.</a>, London.</li> <li>Tschoegl, N. W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam, <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/0-444-50426-5" title="Special:BookSources/0-444-50426-5">0-444-50426-5</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Internal_energy&action=edit&section=16" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlberty,_R._A.2001" class="citation journal cs1">Alberty, R. A. (2001). <a rel="nofollow" class="external text" href="http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf">"Use of Legendre transforms in chemical thermodynamics"</a> <span class="cs1-format">(PDF)</span>. <i>Pure Appl. Chem</i>. <b>73</b> (8): 1349–1380. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fpac200173081349">10.1351/pac200173081349</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:98264934">98264934</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pure+Appl.+Chem.&rft.atitle=Use+of+Legendre+transforms+in+chemical+thermodynamics&rft.volume=73&rft.issue=8&rft.pages=1349-1380&rft.date=2001&rft_id=info%3Adoi%2F10.1351%2Fpac200173081349&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A98264934%23id-name%3DS2CID&rft.au=Alberty%2C+R.+A.&rft_id=http%3A%2F%2Fwww.iupac.org%2Fpublications%2Fpac%2F2001%2Fpdf%2F7308x1349.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLewis,_Gilbert_Newton;_Randall,_Merle:_Revised_by_Pitzer,_Kenneth_S._&_Brewer,_Leo1961" class="citation book cs1">Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). <i>Thermodynamics</i> (2nd ed.). New York, NY USA: McGraw-Hill Book Co. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-113809-3" title="Special:BookSources/978-0-07-113809-3"><bdi>978-0-07-113809-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics&rft.place=New+York%2C+NY+USA&rft.edition=2nd&rft.pub=McGraw-Hill+Book+Co.&rft.date=1961&rft.isbn=978-0-07-113809-3&rft.au=Lewis%2C+Gilbert+Newton%3B+Randall%2C+Merle%3A+Revised+by+Pitzer%2C+Kenneth+S.+%26+Brewer%2C+Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInternal+energy" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></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 .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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title="Template talk:Energy footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Energy_footer" title="Special:EditPage/Template:Energy footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Energy" style="font-size:114%;margin:0 4em"><a href="/wiki/Energy" title="Energy">Energy</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_energy" title="History of energy">History</a></li> <li><a href="/wiki/Index_of_energy_articles" title="Index of energy articles">Index</a></li> <li><a href="/wiki/Outline_of_energy" title="Outline of energy">Outline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamental <br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Conservation of energy</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Energetics</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Units_of_energy" title="Units of energy">Units</a></li></ul></li> <li><a href="/wiki/Energy_condition" title="Energy condition">Energy condition</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Energy_system" title="Energy system">Energy system</a></li> <li><a href="/wiki/Energy_transformation" title="Energy transformation">Energy transformation</a></li> <li><a href="/wiki/Energy_transition" title="Energy transition">Energy transition</a></li> <li><a href="/wiki/Mass" title="Mass">Mass</a> <ul><li><a href="/wiki/Negative_mass" title="Negative mass">Negative mass</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a></li></ul></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Power</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Entropic_force" title="Entropic force">Entropic force</a></li> <li><a href="/wiki/Entropy" title="Entropy">Entropy</a></li> <li><a href="/wiki/Exergy" title="Exergy">Exergy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li> <li><a href="/wiki/Heat_capacity" title="Heat capacity">Heat capacity</a></li> <li><a href="/wiki/Heat_transfer" title="Heat transfer">Heat transfer</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversible process</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li> <li><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws of thermodynamics</a></li> <li><a href="/wiki/Negentropy" title="Negentropy">Negentropy</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li> <li><a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">Thermal equilibrium</a></li> <li><a href="/wiki/Thermal_reservoir" title="Thermal reservoir">Thermal reservoir</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li> <li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Thermodynamic free energy</a></li> <li><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Thermodynamic potential</a></li> <li><a href="/wiki/Thermodynamic_state" title="Thermodynamic state">Thermodynamic state</a></li> <li><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Thermodynamic system</a></li> <li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Thermodynamic temperature</a></li> <li><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume (thermodynamics)</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Work</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binding_energy" title="Binding energy">Binding</a> <ul><li><a href="/wiki/Nuclear_binding_energy" title="Nuclear binding energy">Nuclear</a></li></ul></li> <li><a href="/wiki/Chemical_energy" title="Chemical energy">Chemical</a></li> <li><a href="/wiki/Dark_energy" title="Dark energy">Dark</a></li> <li><a href="/wiki/Elastic_energy" title="Elastic energy">Elastic</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Electric potential energy</a></li> <li><a href="/wiki/Electrical_energy" title="Electrical energy">Electrical</a></li> <li><a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational</a> <ul><li><a href="/wiki/Gravitational_binding_energy" title="Gravitational binding energy">Binding</a></li></ul></li> <li><a href="/wiki/Interatomic_potential" title="Interatomic potential">Interatomic potential</a></li> <li><a class="mw-selflink selflink">Internal</a></li> <li><a href="/wiki/Ionization_energy" title="Ionization energy">Ionization</a></li> <li><a href="/wiki/Kinetic_energy" title="Kinetic energy">Kinetic</a></li> <li><a href="/wiki/Magnetic_energy" title="Magnetic energy">Magnetic</a></li> <li><a href="/wiki/Mechanical_energy" title="Mechanical energy">Mechanical</a></li> <li><a href="/wiki/Negative_energy" title="Negative energy">Negative</a></li> <li><a href="/wiki/Phantom_energy" title="Phantom energy">Phantom</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">Potential</a></li> <li><a href="/wiki/Quantum_chromodynamics_binding_energy" title="Quantum chromodynamics binding energy">Quantum chromodynamics binding energy</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Quantum_potential" title="Quantum potential">Quantum potential</a></li> <li><a href="/wiki/Quintessence_(physics)" title="Quintessence (physics)">Quintessence</a></li> <li><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant</a></li> <li><a href="/wiki/Rest_energy" class="mw-redirect" title="Rest energy">Rest</a></li> <li><a href="/wiki/Sound_energy" title="Sound energy">Sound</a></li> <li><a href="/wiki/Surface_energy" title="Surface energy">Surface</a></li> <li><a href="/wiki/Thermal_energy" title="Thermal energy">Thermal</a></li> <li><a href="/wiki/Vacuum_energy" title="Vacuum energy">Vacuum</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Energy_carrier" title="Energy carrier">Energy carriers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electric_battery" title="Electric battery">Battery</a></li> <li><a href="/wiki/Capacitor" title="Capacitor">Capacitor</a></li> <li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Fuel" title="Fuel">Fuel</a> <ul><li><a href="/wiki/Fossil_fuel" title="Fossil fuel">Fossil</a></li> <li><a href="/wiki/Fuel_oil" title="Fuel oil">Oil</a></li></ul></li> <li><a href="/wiki/Heat" title="Heat">Heat</a> <ul><li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li></ul></li> <li><a href="/wiki/Hydrogen" title="Hydrogen">Hydrogen</a> <ul><li><a href="/wiki/Hydrogen_fuel" class="mw-redirect" title="Hydrogen fuel">Hydrogen fuel</a></li></ul></li> <li><a href="/wiki/Mechanical_wave" title="Mechanical wave">Mechanical wave</a></li> <li><a href="/wiki/Radiation" title="Radiation">Radiation</a></li> <li><a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">Sound wave</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Work</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primary_energy" title="Primary energy">Primary energy</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioenergy" title="Bioenergy">Bioenergy</a></li> <li><a href="/wiki/Fossil_fuel" title="Fossil fuel">Fossil fuel</a> <ul><li><a href="/wiki/Coal" title="Coal">Coal</a></li> <li><a href="/wiki/Natural_gas" title="Natural gas">Natural gas</a></li> <li><a href="/wiki/Petroleum" title="Petroleum">Petroleum</a></li></ul></li> <li><a href="/wiki/Geothermal_energy" title="Geothermal energy">Geothermal</a></li> <li><a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational</a></li> <li><a href="/wiki/Hydropower" title="Hydropower">Hydropower</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine</a></li> <li><a href="/wiki/Nuclear_fuel" title="Nuclear fuel">Nuclear fuel</a> <ul><li><a href="/wiki/Natural_uranium" title="Natural uranium">Natural uranium</a></li></ul></li> <li><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant</a></li> <li><a href="/wiki/Solar_energy" title="Solar energy">Solar</a></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Energy_system" title="Energy system">Energy system</a><br />components</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biomass" title="Biomass">Biomass</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electricity_delivery" title="Electricity delivery">Electricity delivery</a></li> <li><a href="/wiki/Energy_engineering" title="Energy engineering">Energy engineering</a></li> <li><a href="/wiki/Fossil_fuel_power_station" title="Fossil fuel power station">Fossil fuel power station</a> <ul><li><a href="/wiki/Cogeneration" title="Cogeneration">Cogeneration</a></li> <li><a href="/wiki/Integrated_gasification_combined_cycle" title="Integrated gasification combined cycle">Integrated gasification combined cycle</a></li></ul></li> <li><a href="/wiki/Geothermal_power" title="Geothermal power">Geothermal power</a></li> <li><a href="/wiki/Hydropower" title="Hydropower">Hydropower</a> <ul><li><a href="/wiki/Hydroelectricity" title="Hydroelectricity">Hydroelectricity</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Wave_farm" class="mw-redirect" title="Wave farm">Wave farm</a></li></ul></li> <li><a href="/wiki/Nuclear_power" title="Nuclear power">Nuclear power</a> <ul><li><a href="/wiki/Nuclear_power_plant" title="Nuclear power plant">Nuclear power plant</a></li> <li><a href="/wiki/Radioisotope_thermoelectric_generator" title="Radioisotope thermoelectric generator">Radioisotope thermoelectric generator</a></li></ul></li> <li><a href="/wiki/Oil_refinery" title="Oil refinery">Oil refinery</a></li> <li><a href="/wiki/Solar_power" title="Solar power">Solar power</a> <ul><li><a href="/wiki/Concentrated_solar_power" title="Concentrated solar power">Concentrated solar power</a></li> <li><a href="/wiki/Photovoltaic_system" title="Photovoltaic system">Photovoltaic system</a></li></ul></li> <li><a href="/wiki/Solar_thermal_energy" title="Solar thermal energy">Solar thermal energy</a> <ul><li><a href="/wiki/Solar_furnace" title="Solar furnace">Solar furnace</a></li> <li><a href="/wiki/Solar_power_tower" title="Solar power tower">Solar power tower</a></li></ul></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind power</a> <ul><li><a href="/wiki/Airborne_wind_energy" title="Airborne wind energy">Airborne wind energy</a></li> <li><a href="/wiki/Wind_farm" title="Wind farm">Wind farm</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Use and<br /><a href="/wiki/Energy_supply" title="Energy supply">supply</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Efficient_energy_use" title="Efficient energy use">Efficient energy use</a> <ul><li><a href="/wiki/Energy_efficiency_in_agriculture" title="Energy efficiency in agriculture">Agriculture</a></li> <li><a href="/wiki/Power_usage_effectiveness" title="Power usage effectiveness">Computing</a></li> <li><a href="/wiki/Energy_efficiency_in_transport" title="Energy efficiency in transport">Transport</a></li></ul></li> <li><a href="/wiki/Energy_conservation" title="Energy conservation">Energy conservation</a></li> <li><a href="/wiki/Energy_consumption" title="Energy consumption">Energy consumption</a></li> <li><a href="/wiki/Energy_policy" title="Energy policy">Energy policy</a> <ul><li><a href="/wiki/Energy_development" title="Energy development">Energy development</a></li></ul></li> <li><a href="/wiki/Energy_security" title="Energy security">Energy security</a></li> <li><a href="/wiki/Energy_storage" title="Energy storage">Energy storage</a></li> <li><a href="/wiki/Renewable_energy" title="Renewable energy">Renewable energy</a></li> <li><a href="/wiki/Sustainable_energy" title="Sustainable energy">Sustainable energy</a></li> <li><a href="/wiki/World_energy_supply_and_consumption" title="World energy supply and consumption">World energy supply and consumption</a></li> <li><a href="/wiki/Energy_in_Africa" title="Energy in Africa">Africa</a></li> <li><a href="/wiki/Energy_in_Asia" class="mw-redirect" title="Energy in Asia">Asia</a></li> <li><a href="/wiki/Energy_in_Australia" title="Energy in Australia">Australia</a></li> <li><a href="/wiki/Energy_policy_of_Canada" title="Energy policy of Canada">Canada</a></li> <li><a href="/wiki/Energy_in_Europe" title="Energy in Europe">Europe</a></li> <li><a href="/wiki/Energy_in_Mexico" title="Energy in Mexico">Mexico</a></li> <li><a href="/wiki/Energy_in_South_America" title="Energy in South America">South America</a></li> <li><a href="/wiki/Energy_in_the_United_States" title="Energy in the United States">United States</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Misc.</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carbon_footprint" title="Carbon footprint">Carbon footprint</a></li> <li><a href="/wiki/Energy_democracy" title="Energy democracy">Energy democracy</a></li> <li><a href="/wiki/Energy_recovery" title="Energy recovery">Energy 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