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class="vector-toc-numb">2</span> <span>Other expressions</span> </div> </a> <button aria-controls="toc-Other_expressions-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 Other expressions subsection</span> </button> <ul id="toc-Other_expressions-sublist" class="vector-toc-list"> <li id="toc-Characteristic_functions_and_natural_state_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characteristic_functions_and_natural_state_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Characteristic functions and natural state variables</span> </div> </a> <ul id="toc-Characteristic_functions_and_natural_state_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physical_interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physical_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Physical interpretation</span> </div> </a> <ul id="toc-Physical_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_heat" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_heat"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relationship to heat</span> </div> </a> <ul id="toc-Relationship_to_heat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Heat_of_reaction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heat_of_reaction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Heat of reaction</span> </div> </a> <ul id="toc-Heat_of_reaction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_enthalpy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific_enthalpy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Specific enthalpy</span> </div> </a> <ul id="toc-Specific_enthalpy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enthalpy_changes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Enthalpy_changes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Enthalpy changes</span> </div> </a> <ul id="toc-Enthalpy_changes-sublist" class="vector-toc-list"> <li id="toc-Chemical_properties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Chemical_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1</span> <span>Chemical properties</span> </div> </a> <ul id="toc-Chemical_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physical_properties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Physical_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.2</span> <span>Physical properties</span> </div> </a> <ul id="toc-Physical_properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Open_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Open_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Open systems</span> </div> </a> <ul id="toc-Open_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Diagrams" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Diagrams</span> </div> </a> <button aria-controls="toc-Diagrams-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 Diagrams subsection</span> </button> <ul id="toc-Diagrams-sublist" class="vector-toc-list"> <li id="toc-Some_basic_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Some_basic_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Some basic applications</span> </div> </a> <ul id="toc-Some_basic_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Throttling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Throttling"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Throttling</span> </div> </a> <ul id="toc-Throttling-sublist" class="vector-toc-list"> <li id="toc-Example_1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Example 1</span> </div> </a> <ul id="toc-Example_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>Example 2</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Compressors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compressors"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Compressors</span> </div> </a> <ul id="toc-Compressors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History_and_etymology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History_and_etymology"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History and etymology</span> </div> </a> <ul id="toc-History_and_etymology-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> <ul id="toc-References-sublist" class="vector-toc-list"> </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> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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">Enthalpy</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 69 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-69" 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">69 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Entalpie" title="Entalpie – Afrikaans" lang="af" hreflang="af" data-title="Entalpie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AD%D8%AA%D9%88%D9%89_%D8%AD%D8%B1%D8%A7%D8%B1%D9%8A" title="محتوى حراري – Arabic" lang="ar" hreflang="ar" data-title="محتوى حراري" 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/Entalp%C3%ADa" title="Entalpía – Asturian" lang="ast" hreflang="ast" data-title="Entalpía" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Entalpiya" title="Entalpiya – Azerbaijani" lang="az" hreflang="az" data-title="Entalpiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%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%A4%E0%A6%BE%E0%A6%AA%E0%A6%97%E0%A6%A4%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%AC%E0%A6%BF%E0%A6%AD%E0%A6%AC" title="অভ্যন্তরীণ তাপগতীয় বিভব – Bangla" lang="bn" hreflang="bn" data-title="অভ্যন্তরীণ তাপগতীয় বিভব" 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/Enthalpy" title="Enthalpy – Minnan" lang="nan" hreflang="nan" data-title="Enthalpy" 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%AD%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BF%D1%96%D1%8F" title="Энтальпія – Belarusian" lang="be" hreflang="be" data-title="Энтальпія" 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%AD%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BF%D1%96%D1%8F" title="Энтальпія – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Энтальпія" 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Entalpiya" title="Entalpiya – Central Bikol" lang="bcl" hreflang="bcl" data-title="Entalpiya" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BD%D1%82%D0%B0%D0%BB%D0%BF%D0%B8%D1%8F" title="Енталпия – Bulgarian" lang="bg" hreflang="bg" data-title="Енталпия" 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/Entalpija" title="Entalpija – Bosnian" lang="bs" hreflang="bs" data-title="Entalpija" 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/Entalpia" title="Entalpia – Catalan" lang="ca" hreflang="ca" data-title="Entalpia" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Entalpie" title="Entalpie – Czech" lang="cs" hreflang="cs" data-title="Entalpie" 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/Entalpi" title="Entalpi – Danish" lang="da" hreflang="da" data-title="Entalpi" 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/Enthalpie" title="Enthalpie – German" lang="de" hreflang="de" data-title="Enthalpie" 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/Entalpia" title="Entalpia – Estonian" lang="et" hreflang="et" data-title="Entalpia" 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%CE%BD%CE%B8%CE%B1%CE%BB%CF%80%CE%AF%CE%B1" title="Ενθαλπία – Greek" lang="el" hreflang="el" data-title="Ενθαλπία" 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/Entalp%C3%ADa" title="Entalpía – Spanish" lang="es" hreflang="es" data-title="Entalpía" 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/Entalpio" title="Entalpio – Esperanto" lang="eo" hreflang="eo" data-title="Entalpio" 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/Entalpia" title="Entalpia – Basque" lang="eu" hreflang="eu" data-title="Entalpia" 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%A2%D9%86%D8%AA%D8%A7%D9%84%D9%BE%DB%8C" title="آنتالپی – Persian" lang="fa" hreflang="fa" data-title="آنتالپی" 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/Enthalpie" title="Enthalpie – French" lang="fr" hreflang="fr" data-title="Enthalpie" 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/Eantalpacht" title="Eantalpacht – Irish" lang="ga" hreflang="ga" data-title="Eantalpacht" 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/Entalp%C3%ADa" title="Entalpía – Galician" lang="gl" hreflang="gl" data-title="Entalpía" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%94%ED%83%88%ED%94%BC" title="엔탈피 – Korean" lang="ko" hreflang="ko" data-title="엔탈피" 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/%D4%B7%D5%B6%D5%BF%D5%A1%D5%AC%D5%BA%D5%AB%D5%A1" title="Էնտալպիա – Armenian" lang="hy" hreflang="hy" data-title="Էնտալպիա" 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%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3_%E0%A4%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE" title="पूर्ण ऊष्मा – Hindi" lang="hi" hreflang="hi" data-title="पूर्ण ऊष्मा" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Entalpija" title="Entalpija – Croatian" lang="hr" hreflang="hr" data-title="Entalpija" 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/Entalpi" title="Entalpi – Indonesian" lang="id" hreflang="id" data-title="Entalpi" 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/Entalpia" title="Entalpia – Italian" lang="it" hreflang="it" data-title="Entalpia" 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%AA%D7%9C%D7%A4%D7%99%D7%94" title="אנתלפיה – Hebrew" lang="he" hreflang="he" data-title="אנתלפיה" 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%AD%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BF%D0%B8%D1%8F" title="Энтальпия – Kazakh" lang="kk" hreflang="kk" data-title="Энтальпия" 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/Antalfi" title="Antalfi – Haitian Creole" lang="ht" hreflang="ht" data-title="Antalfi" 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-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Entalpija" title="Entalpija – Latvian" lang="lv" hreflang="lv" data-title="Entalpija" 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/Entalpija" title="Entalpija – Lithuanian" lang="lt" hreflang="lt" data-title="Entalpija" 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/Entalpia" title="Entalpia – Hungarian" lang="hu" hreflang="hu" data-title="Entalpia" 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%95%D0%BD%D1%82%D0%B0%D0%BB%D0%BF%D0%B8%D1%98%D0%B0" title="Енталпија – Macedonian" lang="mk" hreflang="mk" data-title="Енталпија" 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%8E%E0%B5%BB%E0%B4%A4%E0%B4%BE%E0%B5%BD%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B4%BF" title="എൻതാൽപ്പി – Malayalam" lang="ml" hreflang="ml" data-title="എൻതാൽപ്പി" 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/Entalpi" title="Entalpi – Malay" lang="ms" hreflang="ms" data-title="Entalpi" 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%AD%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BF%D0%B8" title="Энтальпи – Mongolian" lang="mn" hreflang="mn" data-title="Энтальпи" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%84%E1%80%BA%E1%80%9E%E1%80%AC%E1%80%95%E1%80%AE" title="အင်သာပီ – Burmese" lang="my" hreflang="my" data-title="အင်သာပီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Enthalpie" title="Enthalpie – Dutch" lang="nl" hreflang="nl" data-title="Enthalpie" 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/%E3%82%A8%E3%83%B3%E3%82%BF%E3%83%AB%E3%83%94%E3%83%BC" title="エンタルピー – Japanese" lang="ja" hreflang="ja" data-title="エンタルピー" 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/Entalpi" title="Entalpi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Entalpi" 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/Entalpi" title="Entalpi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Entalpi" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Entalpiya" title="Entalpiya – Uzbek" lang="uz" hreflang="uz" data-title="Entalpiya" 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/Entalpia" title="Entalpia – Polish" lang="pl" hreflang="pl" data-title="Entalpia" 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/Entalpia" title="Entalpia – Portuguese" lang="pt" hreflang="pt" data-title="Entalpia" 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/Entalpie" title="Entalpie – Romanian" lang="ro" hreflang="ro" data-title="Entalpie" 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%AD%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BF%D0%B8%D1%8F" title="Энтальпия – Russian" lang="ru" hreflang="ru" data-title="Энтальпия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%AE%E1%B1%B1%E1%B1%9B%E1%B1%B7%E1%B1%9F%E1%B1%9E%E1%B1%AF%E1%B1%A4" title="ᱮᱱᱛᱷᱟᱞᱯᱤ – Santali" lang="sat" hreflang="sat" data-title="ᱮᱱᱛᱷᱟᱞᱯᱤ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Enthalpy" title="Enthalpy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Enthalpy" 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/Entalpia" title="Entalpia – Slovak" lang="sk" hreflang="sk" data-title="Entalpia" 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/Entalpija" title="Entalpija – Slovenian" lang="sl" hreflang="sl" data-title="Entalpija" 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/%D8%A6%DB%8E%D9%86%D8%AA%D8%A7%D9%84%D9%BE%DB%8C" title="ئێنتالپی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئێنتالپی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BD%D1%82%D0%B0%D0%BB%D0%BF%D0%B8%D1%98%D0%B0" title="Енталпија – Serbian" lang="sr" hreflang="sr" data-title="Енталпија" 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/Entalpija" title="Entalpija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Entalpija" 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/Entalpia" title="Entalpia – Finnish" lang="fi" hreflang="fi" data-title="Entalpia" 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/Entalpi" title="Entalpi – Swedish" lang="sv" hreflang="sv" data-title="Entalpi" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA_%E0%AE%85%E0%AE%9F%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D" title="வெப்ப அடக்கம் – Tamil" lang="ta" hreflang="ta" data-title="வெப்ப அடக்கம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AD%E0%B8%99%E0%B8%97%E0%B8%B2%E0%B8%A5%E0%B8%9B%E0%B8%B5" title="เอนทาลปี – Thai" lang="th" hreflang="th" data-title="เอนทาลปี" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Entalpi" title="Entalpi – Turkish" lang="tr" hreflang="tr" data-title="Entalpi" 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%95%D0%BD%D1%82%D0%B0%D0%BB%D1%8C%D0%BF%D1%96%D1%8F" title="Ентальпія – Ukrainian" lang="uk" hreflang="uk" data-title="Ентальпія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Entalp%C3%ACa" title="Entalpìa – Venetian" lang="vec" hreflang="vec" data-title="Entalpìa" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Enthalpy" title="Enthalpy – Vietnamese" lang="vi" hreflang="vi" data-title="Enthalpy" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%84%93" title="焓 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="焓" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%84%93" title="焓 – Wu" lang="wuu" hreflang="wuu" data-title="焓" 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/%E7%84%93" title="焓 – Cantonese" lang="yue" hreflang="yue" data-title="焓" 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/%E7%84%93" title="焓 – Chinese" lang="zh" hreflang="zh" data-title="焓" 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/Q161064#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Measure of energy in a thermodynamic system</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Entropy" title="Entropy">Entropy</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output 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nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">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 href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div 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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"><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 href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a class="mw-selflink selflink">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><b>Enthalpy</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɛ/: 'e' in 'dress'">ɛ</span><span title="'n' in 'nigh'">n</span><span title="/θ/: 'th' in 'thigh'">θ</span><span title="/əl/: 'le' in 'bottle'">əl</span><span title="'p' in 'pie'">p</span><span title="/i/: 'y' in 'happy'">i</span></span>/</a></span> <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/en\/transcoded\/9\/95\/En-US-enthalpy.ogg\/En-US-enthalpy.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"En-US-enthalpy.ogg"},"classes":["ext-phonos-PhonosButton","noexcerpt","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/en/transcoded/9/95/En-US-enthalpy.ogg/En-US-enthalpy.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label"></span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:En-US-enthalpy.ogg" title="File:En-US-enthalpy.ogg">ⓘ</a></sup></span></span></span>) is the sum of a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>'s <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> and the product of its <a href="/wiki/Pressure" title="Pressure">pressure</a> and <a href="/wiki/Volume" title="Volume">volume</a>.<sup id="cite_ref-GoldBk-enth_1-0" class="reference"><a href="#cite_note-GoldBk-enth-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is a <a href="/wiki/State_function" title="State function">state function</a> in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is conveniently provided by the large ambient atmosphere. The pressure–volume term expresses the <a href="/wiki/Work_(physics)" title="Work (physics)">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> that was done against constant external 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_{\text{ext}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ext</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{ext}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b10336e64007b8bef3510cca421475746fe94a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.962ex; height:2.509ex;" alt="{\displaystyle P_{\text{ext}}}"></span> to establish the system's physical dimensions from <span class="mwe-math-element"><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_{\text{system, initial}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>system, initial</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{system, initial}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d309fbe1d73df9a6037b9c83d2aefbee6c21e5f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.994ex; height:2.843ex;" alt="{\displaystyle V_{\text{system, initial}}=0}"></span> to some final 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_{\text{system, final}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>system, final</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{system, final}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfcea08809f3db6e55c48a7df8371d818363054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.682ex; height:2.843ex;" alt="{\displaystyle V_{\text{system, final}}}"></span> (as <span class="mwe-math-element"><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=P_{\text{ext}}\Delta V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ext</mtext> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=P_{\text{ext}}\Delta V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c1edfac36f711303b54958c524ead2bb2d856b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.219ex; height:2.509ex;" alt="{\displaystyle W=P_{\text{ext}}\Delta V}"></span>), i.e. to make room for it by displacing its surroundings.<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><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 pressure-volume term is very small for solids and liquids at common conditions, and fairly small for gases. Therefore, enthalpy is a stand-in for <a href="/wiki/Energy" title="Energy">energy</a> in chemical systems; <a href="/wiki/Bond_energy" title="Bond energy">bond</a>, <a href="/wiki/Lattice_energy" title="Lattice energy">lattice</a>, <a href="/wiki/Solvation" title="Solvation">solvation</a>, and other chemical "energies" are actually enthalpy differences. As a state function, enthalpy depends only on the final configuration of internal energy, pressure, and volume, not on the path taken to achieve it. </p><p>In the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI), the unit of measurement for enthalpy is the <a href="/wiki/Joule" title="Joule">joule</a>. Other historical conventional units still in use include the <a href="/wiki/Calorie" title="Calorie">calorie</a> and the <a href="/wiki/British_thermal_unit" title="British thermal unit">British thermal unit</a> (BTU). </p><p>The total enthalpy of a system cannot be measured directly because the internal energy contains components that are unknown, not easily accessible, or are not of interest for the thermodynamic problem at hand. In practice, a change in enthalpy is the preferred expression for measurements at constant pressure, because it simplifies the description of <a href="/wiki/Energy_transfer" class="mw-redirect" title="Energy transfer">energy transfer</a>. When transfer of matter into or out of the system is also prevented and no electrical or mechanical (stirring shaft or lift pumping) work is done, at constant pressure the enthalpy change equals the energy exchanged with the environment by <a href="/wiki/Heat" title="Heat">heat</a>. </p><p>In chemistry, the standard <a href="/wiki/Enthalpy_of_reaction" class="mw-redirect" title="Enthalpy of reaction">enthalpy of reaction</a> is the enthalpy change when reactants in their <a href="/wiki/Standard_state" title="Standard state">standard states</a> (<span class="nowrap"> <span class="texhtml"><i>p</i> = 1</span> <a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a> ;</span> usually <span class="nowrap"> <span class="texhtml"><i>T</i> = 298</span> <a href="/wiki/Kelvin_(unit)" class="mw-redirect" title="Kelvin (unit)">K</a> </span>) change to products in their standard states.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This quantity is the <a href="/wiki/Standard_enthalpy_of_reaction" title="Standard enthalpy of reaction">standard heat of reaction</a> at constant pressure and temperature, but it can be measured by <a href="/wiki/Calorimeter" title="Calorimeter">calorimetric</a> methods even if the temperature does vary during the measurement, provided that the initial and final pressure and temperature correspond to the standard state. The value does not depend on the path from initial to final state because enthalpy is a <a href="/wiki/State_function" title="State function">state function</a>. </p><p>Enthalpies of chemical substances are usually listed for 1 bar (100 kPa) pressure as a standard state. Enthalpies and enthalpy changes for reactions vary as a function of temperature,<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> but tables generally list the standard heats of formation of substances at 25 °C (298 K). For <a href="/wiki/Endothermic" class="mw-redirect" title="Endothermic">endothermic</a> (heat-absorbing) processes, the change <span class="texhtml">Δ<i>H</i></span> is a positive value; for <a href="/wiki/Exothermic" class="mw-redirect" title="Exothermic">exothermic</a> (heat-releasing) processes it is negative. </p><p>The enthalpy of an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> is independent of its pressure or volume, and depends only on its temperature, which correlates to its thermal energy. Real gases at common temperatures and pressures often closely approximate this behavior, which simplifies practical thermodynamic design and analysis. </p><p>The word "enthalpy" is derived from the Greek word <i>enthalpein</i>, which means "to heat".<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><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The enthalpy <span class="texhtml"><i>H</i></span> of a thermodynamic system is defined as the sum of its internal energy and the product of its pressure and volume:<sup id="cite_ref-GoldBk-enth_1-1" class="reference"><a href="#cite_note-GoldBk-enth-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=U+pV,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=U+pV,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c514fbfcfb3cccea0d47af1efe44d27a3b06d065" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.389ex; height:2.509ex;" alt="{\displaystyle H=U+pV,}"></span> where <span class="texhtml"><i>U</i></span> is the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>, <span class="texhtml mvar" style="font-style:italic;">p</span> is <a href="/wiki/Pressure" title="Pressure">pressure</a>, and <span class="texhtml mvar" style="font-style:italic;">V</span> is the <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">volume</a> of the system; <span class="texhtml mvar" style="font-style:italic;">p V</span> is sometimes referred to as the pressure energy <span class="texhtml mvar" style="font-style:italic;">Ɛ</span><sub>p</sub> .<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Enthalpy is an <a href="/wiki/Extensive_property" class="mw-redirect" title="Extensive property">extensive property</a>; it is proportional to the size of the system (for homogeneous systems). As <a href="/wiki/Intensive_properties" class="mw-redirect" title="Intensive properties">intensive properties</a>, the <i><a href="/wiki/Specific_enthalpy" class="mw-redirect" title="Specific enthalpy">specific enthalpy</a></i>, <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">h = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"> H </span><span class="sr-only">/</span><span class="den">m</span></span>⁠</span> </span> ,</span> is referenced to a unit of <a href="/wiki/Mass" title="Mass">mass</a> <span class="texhtml mvar" style="font-style:italic;">m</span> of the system, and the <i>molar enthalpy</i>, <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">H</span><sub>m</sub> = <span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> H </span><span class="sr-only">/</span><span class="den">n</span></span>⁠</span></span> ,</span> where <span class="texhtml mvar" style="font-style:italic;">n</span> is the number of <a href="/wiki/Mole_(unit)" title="Mole (unit)">moles</a>. For inhomogeneous systems the enthalpy is the sum of the enthalpies of the component subsystems: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\sum _{k}H_{k}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\sum _{k}H_{k}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ffdde634af790621486a9a0b107b3ef4b004bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.216ex; height:5.509ex;" alt="{\displaystyle H=\sum _{k}H_{k}\;,}"></span> where </p> <dl><dd><dl><dd><span class="texhtml mvar" style="font-style:italic;">H</span> is the total enthalpy of all the subsystems,</dd> <dd><span class="texhtml mvar" style="font-style:italic;">k</span> refers to the various subsystems,</dd> <dd><span class="texhtml mvar" style="font-style:italic;">H<sub>k</sub></span> refers to the enthalpy of each subsystem.</dd></dl></dd></dl> <p>A closed system may lie in thermodynamic equilibrium in a static <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a>, so that its pressure <span class="texhtml mvar" style="font-style:italic;">p</span> varies continuously with <a href="/wiki/Altitude" title="Altitude">altitude</a>, while, because of the equilibrium requirement, its temperature <span class="texhtml mvar" style="font-style:italic;">T</span> is invariant with altitude. (Correspondingly, the system's <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a> density also varies with altitude.) Then the enthalpy summation becomes an <a href="/wiki/Integral" title="Integral">integral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\int \left(\rho \,h\right)\,\mathrm {d} V\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\int \left(\rho \,h\right)\,\mathrm {d} V\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77838669841340708869f76107d23fbf56ac124" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.626ex; height:5.676ex;" alt="{\displaystyle H=\int \left(\rho \,h\right)\,\mathrm {d} V\;,}"></span> where </p> <dl><dd><dl><dd><span class="texhtml mvar" style="font-style:italic;">ρ</span> ("<a href="/wiki/Rho" title="Rho">rho</a>") is <a href="/wiki/Density" title="Density">density</a> (mass per unit volume),</dd> <dd><span class="texhtml mvar" style="font-style:italic;">h</span> is the specific enthalpy (enthalpy per unit mass),</dd> <dd><span class="texhtml">(<i>ρh</i>)</span> represents the <a href="/wiki/Energy_density" title="Energy density">enthalpy density</a> (enthalpy per unit volume),</dd> <dd><span class="texhtml">d<i>V</i></span> denotes an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimally</a> small element of volume within the system, for example, the volume of an infinitesimally thin horizontal layer.</dd></dl></dd></dl> <p>The integral therefore represents the sum of the enthalpies of all the elements of the volume. </p><p>The enthalpy of a closed homogeneous system is its energy function <span class="nowrap"> <span class="texhtml"><i>H</i>( <i>S</i>, <i>p</i> ) </span> ,</span> with its entropy <span class="texhtml"><i>S</i>[ <i>p</i> ]</span> and its pressure <span class="texhtml mvar" style="font-style:italic;">p</span> as <a href="/wiki/Thermodynamic_potential#Natural_variables" title="Thermodynamic potential">natural state variables</a> which provide a differential relation for <span class="texhtml">d<i>H</i></span> of the simplest form, derived as follows. We start from the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> for closed systems for an infinitesimal process: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q-\mathrm {\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> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>Q</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>W</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q-\mathrm {\delta } \,W\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e9b5194adf9d54a4f57162ce9cd9f731e063f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.451ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q-\mathrm {\delta } \,W\;,}"></span> </p><p>where </p> <dl><dd><dl><dd><span class="texhtml">δ<i>Q</i></span> is a small amount of heat added to the system,</dd> <dd><span class="texhtml">δ<i>W</i></span> is a small amount of work performed by the system.</dd></dl></dd></dl> <p>In a homogeneous system in which only <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible</a> processes or pure heat transfer are considered, the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> gives <span class="nowrap"><span class="texhtml"> δ<i>Q</i> = <i>T</i> d<i>S</i></span> ,</span> with <span class="texhtml mvar" style="font-style:italic;">T</span> the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> and <span class="texhtml">d<i>S</i></span> the infinitesimal change in <a href="/wiki/Entropy" title="Entropy">entropy</a> <span class="texhtml mvar" style="font-style:italic;">S</span> of the system. Furthermore, if only <span class="texhtml mvar" style="font-style:italic;">p V</span> work is done, <span class="nowrap"><span class="texhtml"> δ<i>W</i> = <i>p</i> d<i>V</i></span> .</span> As a result, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} 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> <mtext> </mtext> <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/b68e93b1ef988fb5b4c3d6537cd835318216318a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.693ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V~.}"></span> </p><p>Adding <span class="texhtml">d(<i>p V</i>)</span> to both sides of this expression gives </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U+\mathrm {d} (p\,V)=T\,\mathrm {d} S-p\,\mathrm {d} V+\mathrm {d} (p\,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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mspace width="thinmathspace" /> <mi>V</mi> <mo stretchy="false">)</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mspace width="thinmathspace" /> <mi>V</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U+\mathrm {d} (p\,V)=T\,\mathrm {d} S-p\,\mathrm {d} V+\mathrm {d} (p\,V)\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61666bc5e84bc414aad3aa288a6af7c039dad389" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.329ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} U+\mathrm {d} (p\,V)=T\,\mathrm {d} S-p\,\mathrm {d} V+\mathrm {d} (p\,V)\;,}"></span> </p><p>or </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} (U+p\,V)=T\,\mathrm {d} S+V\,\mathrm {d} p~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mspace width="thinmathspace" /> <mi>V</mi> <mo stretchy="false">)</mo> <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>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (U+p\,V)=T\,\mathrm {d} S+V\,\mathrm {d} p~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8a9fab0d2634fda42c38e0e91d82a4b7904d88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.686ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} (U+p\,V)=T\,\mathrm {d} S+V\,\mathrm {d} p~.}"></span> </p><p>So </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} H(S,\,p)=T\,\mathrm {d} S+V\,\mathrm {d} p~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">)</mo> <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>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H(S,\,p)=T\,\mathrm {d} S+V\,\mathrm {d} p~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc5b1fcd21bf3d86f2d275377c197c57fe7a15e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.226ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} H(S,\,p)=T\,\mathrm {d} S+V\,\mathrm {d} p~}"></span> </p><p>and the coefficients of the natural variable differentials <span class="texhtml">d<i>S</i></span> and <span class="texhtml">d<i>p</i></span> are just the single variables <span class="texhtml mvar" style="font-style:italic;">T</span> and <span class="texhtml mvar" style="font-style:italic;">V</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_expressions">Other expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=2" title="Edit section: Other expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above expression of <span class="texhtml">d<i>H</i></span> in terms of entropy and pressure may be unfamiliar to some readers. There are also expressions in terms of more directly measurable variables such as temperature and pressure:<sup id="cite_ref-Guggenheim_9-0" class="reference"><a href="#cite_note-Guggenheim-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page:  88">(p 88)</span></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> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T+V\,(1-\alpha T)\,\mathrm {d} p~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>+</mo> <mi>V</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T+V\,(1-\alpha T)\,\mathrm {d} p~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536037426acc49e9951bb6185ec30937c207f2b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.542ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T+V\,(1-\alpha T)\,\mathrm {d} p~.}"></span> </p><p>Here <span class="texhtml">C</span><sub>p</sub> is the <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> at <i>constant <a href="/wiki/Pressure" title="Pressure">pressure</a></i> and <span class="texhtml mvar" style="font-style:italic;">α</span> is the <a href="/wiki/Coefficient_of_thermal_expansion" class="mw-redirect" title="Coefficient of thermal expansion">coefficient of (cubic) thermal expansion</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {\,1\,}{V}}\left({\frac {\partial V}{\,\partial T\,}}\right)_{\mathsf {p}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mspace width="thinmathspace" /> </mrow> <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> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>T</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {\,1\,}{V}}\left({\frac {\partial V}{\,\partial T\,}}\right)_{\mathsf {p}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef6d3b4bea2565456acb9cb25c5102408e6b879" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.654ex; height:6.509ex;" alt="{\displaystyle \alpha ={\frac {\,1\,}{V}}\left({\frac {\partial V}{\,\partial T\,}}\right)_{\mathsf {p}}~.}"></span> </p><p>With this expression one can, in principle, determine the enthalpy if <span class="texhtml mvar" style="font-style:italic;">C</span><sub>p</sub> and <span class="texhtml mvar" style="font-style:italic;">V</span> are known as functions of <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">T</span> . However the expression is more complicated than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</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>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f793847f6aae2f46c6ce47004e8f6c03e509aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.037ex; height:2.509ex;" alt="{\displaystyle \;\mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p\;}"></span> because <span class="texhtml mvar" style="font-style:italic;">T</span> is not a natural variable for the enthalpy <span class="texhtml mvar" style="font-style:italic;">H</span>. </p><p>At constant 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 \;\mathrm {d} P=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathrm {d} P=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98043e5626fa1761e76a4f40653638a0e863bb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.589ex; height:2.176ex;" alt="{\displaystyle \;\mathrm {d} P=0\;}"></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e03d0cd1ca936311cc7f86ffb8c837105f8ff46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.387ex; height:2.843ex;" alt="{\displaystyle \;\mathrm {d} H=C_{\mathsf {p}}\,\mathrm {d} T~.}"></span> For an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</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 \;\mathrm {d} H\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\mathrm {d} H\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103877400911496bc31d41b0c50f44c30617ed53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.646ex; height:2.176ex;" alt="{\displaystyle \;\mathrm {d} H\;}"></span> reduces to this form even if the process involves a pressure change, because <span class="nowrap"><span class="texhtml"> <i>α T</i> = 1 </span> .</span><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p><p>In a more general form, the first law describes the internal energy with additional terms involving the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> and the number of particles of various types. The differential statement for <span class="texhtml">d<i>H</i></span> then becomes </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p+\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>H</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>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</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> <mspace width="thinmathspace" /> <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> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae80b0cf9a7cc88b95841e7abcc92ae8cc83c40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.168ex; height:5.509ex;" alt="{\displaystyle \mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} p+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\;,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">μ<sub>i</sub></span> is the chemical potential per particle for a type <span class="texhtml mvar" style="font-style:italic;">i</span> particle, and <span class="texhtml mvar" style="font-style:italic;">N<sub>i</sub></span> is the number of such particles. The last term can also be written as <span class="texhtml"><i>μ<sub>i</sub></i> d<i>n<sub>i</sub></i></span> (with <span class="texhtml"> d<i>n</i><sub><i>i</i> 0</sub> </span> the number of moles of component <span class="texhtml mvar" style="font-style:italic;">i</span> added to the system and, in this case, <span class="texhtml mvar" style="font-style:italic;">μ<sub>i</sub></span> the molar chemical potential) or as <span class="texhtml"><i>μ<sub>i</sub></i> d<i>m<sub>i</sub></i></span> (with <span class="texhtml">d<i>m<sub>i</sub></i></span> the mass of component <span class="texhtml mvar" style="font-style:italic;">i</span> added to the system and, in this case, <span class="texhtml mvar" style="font-style:italic;">μ<sub>i</sub></span> the specific chemical potential). </p> <div class="mw-heading mw-heading3"><h3 id="Characteristic_functions_and_natural_state_variables">Characteristic functions and natural state variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=3" title="Edit section: Characteristic functions and natural state variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The enthalpy, <span class="nowrap"> <span class="texhtml"><i>H</i>( <i>S</i>[<i>p</i>], <i>p</i>, {<i>N<sub>i</sub></i>} ) </span> ,</span> 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 include both one intensive and several extensive <a href="/wiki/State_variable" title="State variable">state variables</a>. The state variables <span class="nowrap"> <span class="texhtml"><i>S</i>[<i>p</i>]</span> ,</span> <span class="texhtml"><i>p</i></span>, and <span class="texhtml">{<i>N<sub>i</sub></i>} </span> are said to be the <a href="/wiki/Thermodynamic_potential#Natural_variables" title="Thermodynamic potential"><i>natural state variables</i></a> in this representation. They are suitable for describing processes in which they are determined by factors in the surroundings. For example, when a virtual parcel of atmospheric air moves to a different altitude, the pressure surrounding it changes, and the process is often so rapid that there is too little time for heat transfer. This is the basis of the so-called <a href="/wiki/Adiabatic_approximation" class="mw-redirect" title="Adiabatic approximation">adiabatic approximation</a> that is used in <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Conjugate with the enthalpy, with these arguments, the other characteristic function of state of a thermodynamic system is its entropy, as a function, <span class="nowrap"> <span class="texhtml"><i>S</i>[<i>p</i>]( <i>H</i>, <i>p</i>, {N<sub>i</sub>} ) </span> ,</span> of the same list of variables of state, except that the entropy, <span class="texhtml"><i>S</i>[<i>p</i>]</span>, is replaced in the list by the enthalpy, <span class="texhtml mvar" style="font-style:italic;">H</span>. It expresses the <i>entropy representation</i>. The state variables <span class="texhtml mvar" style="font-style:italic;">H</span>, <span class="texhtml mvar" style="font-style:italic;">p</span>, and <span class="texhtml">{<i>N<sub>i</sub></i>}</span> are said to be the <i>natural state variables</i> in this representation. They are suitable for describing processes in which they are experimentally controlled. For example, <span class="texhtml mvar" style="font-style:italic;">H</span> and <span class="texhtml mvar" style="font-style:italic;">p</span> can be controlled by allowing heat transfer, and by varying only the external pressure on the piston that sets the volume of the system.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Physical_interpretation">Physical interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=4" title="Edit section: Physical interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <span class="texhtml mvar" style="font-style:italic;">U</span> term is the energy of the system, and the <span class="texhtml mvar" style="font-style:italic;">p V</span> term can be interpreted as the <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> that would be required to "make room" for the system if the pressure of the environment remained constant. When a system, for example, <span class="texhtml mvar" style="font-style:italic;">n</span> <a href="/wiki/Mole_(unit)" title="Mole (unit)">moles</a> of a gas of <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">volume</a> <span class="texhtml mvar" style="font-style:italic;">V</span> at <a href="/wiki/Pressure" title="Pressure">pressure</a> <span class="texhtml mvar" style="font-style:italic;">p</span> and <a href="/wiki/Temperature" title="Temperature">temperature</a> <span class="texhtml mvar" style="font-style:italic;">T</span>, is created or brought to its present state from <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, energy must be supplied equal to its internal energy <span class="texhtml mvar" style="font-style:italic;">U</span> plus <span class="texhtml mvar" style="font-style:italic;">p V</span>, where <span class="texhtml mvar" style="font-style:italic;">p V</span> is the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> done in pushing against the ambient (atmospheric) pressure. </p><p>In <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> it may be more interesting to study the internal properties of a constant-volume system and therefore the internal energy is used.<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><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, experiments are often conducted at constant <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a>, and the pressure–volume work represents a small, well-defined energy exchange with the atmosphere, so that <span class="texhtml">Δ<i>H</i></span> is the appropriate expression for the <a href="/wiki/Heat_of_reaction" class="mw-redirect" title="Heat of reaction">heat of reaction</a>. For a <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a>, the change in its enthalpy after a full cycle is equal to zero, since the final and initial state are equal. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_heat">Relationship to heat</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=5" title="Edit section: Relationship to heat"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to discuss the relation between the enthalpy increase and heat supply, we return to the first law for closed systems, with the physics sign convention: <span class="texhtml"> d<i>U</i> = δ<i>Q</i> − δ<i>W</i></span>, where the heat <span class="texhtml">δ<i>Q</i></span> is supplied by conduction, radiation, <a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a>. We apply it to the special case with a constant pressure at the surface. In this case the work is given by <span class="nowrap"> <span class="texhtml"><i>p</i> d<i>V</i></span> </span> (where <span class="texhtml mvar" style="font-style:italic;">p</span> is the pressure at the surface, <span class="texhtml">d<i>V</i></span> is the increase of the volume of the system). Cases of long range electromagnetic interaction require further state variables in their formulation, and are not considered here. In this case the first law reads: <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=\mathrm {\delta } \,Q-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> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>Q</mi> <mo>−</mo> <mi>p</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q-p\,\mathrm {d} V~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76e9caee96b1894594ca37ed695dd15462e7ef4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.152ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q-p\,\mathrm {d} V~.}"></span> </p><p>Now, <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} H=\mathrm {d} U+\mathrm {d} (p\,V)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mspace width="thinmathspace" /> <mi>V</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H=\mathrm {d} U+\mathrm {d} (p\,V)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d708e7ea3c9b293f8eb18710e7fa1d6fdd4647" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.043ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} H=\mathrm {d} U+\mathrm {d} (p\,V)~.}"></span> </p><p>So <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {d} H&=\mathrm {\delta } Q+V\,\mathrm {d} p+p\,\mathrm {d} V-p\,\mathrm {d} V\\&=\mathrm {\delta } Q+V\,\mathrm {d} p~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mi>Q</mi> <mo>+</mo> <mi>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</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> <mi>p</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mi>Q</mi> <mo>+</mo> <mi>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} H&=\mathrm {\delta } Q+V\,\mathrm {d} p+p\,\mathrm {d} V-p\,\mathrm {d} V\\&=\mathrm {\delta } Q+V\,\mathrm {d} p~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2425277e3f3c8910c98fb3498672f4d97fc2af05" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.523ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} H&=\mathrm {\delta } Q+V\,\mathrm {d} p+p\,\mathrm {d} V-p\,\mathrm {d} V\\&=\mathrm {\delta } Q+V\,\mathrm {d} p~.\end{aligned}}}"></span> </p><p>If the system is under <a href="/wiki/Isobaric_system" class="mw-redirect" title="Isobaric system">constant pressure</a>, <span class="nowrap"> <span class="texhtml"> d<i>p</i> = 0 </span> </span> and consequently, the increase in enthalpy of the system is equal to the <a href="/wiki/Heat" title="Heat">heat</a> added: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} H=\mathrm {\delta } \,Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H=\mathrm {\delta } \,Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c08cc372acd80165ac077b2e8c1fd36b5e90e3b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.729ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} H=\mathrm {\delta } \,Q}"></span> This is why the now-obsolete term <i>heat content</i> was used for enthalpy in the 19th century. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=6" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In thermodynamics, one can calculate enthalpy by determining the requirements for creating a system from "nothingness"; the mechanical work required, <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">p V</span> ,</span> differs based upon the conditions that obtain during the creation of the <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>. </p><p><a href="/wiki/Energy" title="Energy">Energy</a> must be supplied to remove particles from the surroundings to make space for the creation of the system, assuming that the pressure <span class="texhtml mvar" style="font-style:italic;">p</span> remains constant; this is the <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">p V</span> term.</span> The supplied energy must also provide the change in internal energy, <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">U</span> ,</span> which includes <a href="/wiki/Activation_energy" title="Activation energy">activation energies</a>, ionization energies, mixing energies, vaporization energies, chemical bond energies, and so forth. Together, these constitute the change in the enthalpy <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;"> U + p V </span> .</span> For systems at constant pressure, with no external work done other than the <span class="texhtml mvar" style="font-style:italic;">p V</span> work, the change in enthalpy is the heat received by the system. </p><p>For a simple system with a constant number of particles at constant pressure, the difference in enthalpy is the maximum amount of thermal energy derivable from an isobaric thermodynamic process.<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> <div class="mw-heading mw-heading3"><h3 id="Heat_of_reaction">Heat of reaction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=7" title="Edit section: Heat of reaction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Standard_enthalpy_of_reaction" title="Standard enthalpy of reaction">Standard enthalpy of reaction</a></div> <p>The total enthalpy of a system cannot be measured directly; the <i>enthalpy change</i> of a <a href="/wiki/System_(thermodynamics)" class="mw-redirect" title="System (thermodynamics)">system</a> is measured instead. Enthalpy change is defined by the following equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H=H_{\mathsf {f}}-H_{\mathsf {i}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>H</mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta H=H_{\mathsf {f}}-H_{\mathsf {i}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5840470dbd1ae8f8f4835ff632f90016f4025fb2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.263ex; height:2.509ex;" alt="{\displaystyle \Delta H=H_{\mathsf {f}}-H_{\mathsf {i}}\,,}"></span> where </p> <dl><dd><dl><dd><span class="texhtml"> Δ<i>H</i> </span> is the "enthalpy change",</dd> <dd><span class="texhtml mvar" style="font-style:italic;">H</span><sub>f</sub> is the final enthalpy of the system (in a chemical reaction, the enthalpy of the products or the system at equilibrium),</dd> <dd><span class="texhtml mvar" style="font-style:italic;">H</span><sub>i</sub> is the initial enthalpy of the system (in a chemical reaction, the enthalpy of the reactants).</dd></dl></dd></dl> <p>For an <a href="/wiki/Exothermic_reaction" title="Exothermic reaction">exothermic reaction</a> at constant <a href="/wiki/Pressure" title="Pressure">pressure</a>, the system's change in enthalpy, <span class="texhtml">Δ<i>H</i></span>, is negative due to the products of the reaction having a smaller enthalpy than the reactants, and equals the heat released in the reaction if no electrical or shaft work is done. In other words, the overall decrease in enthalpy is achieved by the generation of heat.<sup id="cite_ref-Laidler-Meiser-1982_19-0" class="reference"><a href="#cite_note-Laidler-Meiser-1982-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Conversely, for a constant-pressure <a href="/wiki/Endothermic" class="mw-redirect" title="Endothermic">endothermic</a> reaction, <span class="texhtml">Δ<i>H</i></span> is positive and equal to the heat <i>absorbed</i> in the reaction. </p><p>From the definition of enthalpy as <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;"> H = U + p V </span> ,</span> the enthalpy change at constant pressure is <span class="nowrap"> <span class="texhtml"> Δ<i>H</i> = Δ<i>U</i> + <i>p</i> Δ<i>V</i></span> .</span> However, for most chemical reactions, the work term <span class="texhtml"><i>p</i> Δ<i>V</i></span> is much smaller than the internal energy change <span class="texhtml">Δ<i>U</i></span>, which is approximately equal to <span class="texhtml">Δ<i>H</i></span>. As an example, for the combustion of carbon monoxide <span class="nowrap"> 2 CO(g) + O<sub>2</sub>(g) → 2 CO<sub>2</sub>(g) ,</span> <span class="nowrap"> <span class="texhtml"> Δ<i>H</i> = −566.0</span> kJ</span> and <span class="nowrap"> <span class="texhtml"> Δ<i>U</i> = −563.5</span> kJ.</span><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> Since the differences are so small, reaction enthalpies are often described as reaction energies and analyzed in terms of <a href="/wiki/Bond_energy" title="Bond energy">bond energies</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Specific_enthalpy">Specific enthalpy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=8" title="Edit section: Specific enthalpy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>specific enthalpy</i> of a uniform system is defined as <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;"> h = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">H</span><span class="sr-only">/</span><span class="den"> m </span></span>⁠</span> </span> </span>, where <span class="texhtml mvar" style="font-style:italic;">m</span> is the mass of the system. Its <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> is joule per kilogram. It can be expressed in other specific quantities by <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;"> h = u + p v </span> ,</span> where <span class="texhtml mvar" style="font-style:italic;">u</span> is the specific <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>, <span class="texhtml mvar" style="font-style:italic;">p</span> is the pressure, and <span class="texhtml mvar" style="font-style:italic;">v</span> is <a href="/wiki/Specific_volume" title="Specific volume">specific volume</a>, which is equal to <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"><i>ρ</i></span></span>⁠</span></span>, where <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the <a href="/wiki/Density" title="Density">density</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Enthalpy_changes">Enthalpy changes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=9" title="Edit section: Enthalpy changes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An enthalpy change describes the change in enthalpy observed in the constituents of a thermodynamic system when undergoing a transformation or chemical reaction. It is the difference between the enthalpy after the process has completed, i.e. the enthalpy of the <a href="/wiki/Product_(chemistry)" title="Product (chemistry)">products</a> assuming that the reaction goes to completion, and the initial enthalpy of the system, namely the reactants. These processes are specified solely by their initial and final states, so that the enthalpy change for the reverse is the negative of that for the forward process. </p><p>A common standard enthalpy change is the <a href="/wiki/Enthalpy_of_formation" class="mw-redirect" title="Enthalpy of formation">enthalpy of formation</a>, which has been determined for a large number of substances. Enthalpy changes are routinely measured and compiled in chemical and physical reference works, such as the <a href="/wiki/CRC_Handbook_of_Chemistry_and_Physics" title="CRC Handbook of Chemistry and Physics">CRC Handbook of Chemistry and Physics</a>. The following is a selection of enthalpy changes commonly recognized in thermodynamics. </p><p>When used in these recognized terms the qualifier <i>change</i> is usually dropped and the property is simply termed <i>enthalpy of 'process'</i>. Since these properties are often used as reference values it is very common to quote them for a standardized set of environmental parameters, or <a href="/wiki/Standard_conditions" class="mw-redirect" title="Standard conditions">standard conditions</a>, including: </p> <ul><li>A <a href="/wiki/Pressure" title="Pressure">pressure</a> of one atmosphere (1 atm or 1013.25 hPa) or 1 bar</li> <li>A <a href="/wiki/Temperature" title="Temperature">temperature</a> of 25 °C or 298.15 K</li> <li>A <a href="/wiki/Concentration" title="Concentration">concentration</a> of 1.0 M when the element or compound is present in solution</li> <li>Elements or compounds in their normal physical states, i.e. <a href="/wiki/Standard_state" title="Standard state">standard state</a></li></ul> <p>For such standardized values the name of the enthalpy is commonly prefixed with the term <i>standard</i>, e.g. <i>standard enthalpy of formation</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Chemical_properties">Chemical properties</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=10" title="Edit section: Chemical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Enthalpy_of_reaction" class="mw-redirect" title="Enthalpy of reaction">Enthalpy of reaction</a> - is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of substance reacts completely. </p><p><a href="/wiki/Enthalpy_of_formation" class="mw-redirect" title="Enthalpy of formation">Enthalpy of formation</a> - is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a compound is formed from its elementary antecedents. </p><p><a href="/wiki/Enthalpy_of_combustion" class="mw-redirect" title="Enthalpy of combustion">Enthalpy of combustion</a> - is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a substance burns completely with oxygen. </p><p><a href="/wiki/Enthalpy_of_hydrogenation" class="mw-redirect" title="Enthalpy of hydrogenation">Enthalpy of hydrogenation</a> - is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of an unsaturated compound reacts completely with an excess of hydrogen to form a <a href="/wiki/Saturated_and_unsaturated_compounds" title="Saturated and unsaturated compounds">saturated compound</a>. </p><p><a href="/wiki/Enthalpy_of_atomization" title="Enthalpy of atomization">Enthalpy of atomization</a> - is defined as the enthalpy change required to separate one mole of a substance into its constituent <a href="/wiki/Atom" title="Atom">atoms</a> completely. </p><p><a href="/wiki/Enthalpy_of_neutralization" title="Enthalpy of neutralization">Enthalpy of neutralization</a> - is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of water is formed when an acid and a base react. </p><p>Standard <a href="/wiki/Enthalpy_of_solution" class="mw-redirect" title="Enthalpy of solution">Enthalpy of solution</a> - is defined as the enthalpy change observed in a constituent of a thermodynamic system when one mole of a solute is dissolved completely in an excess of solvent, so that the solution is at infinite dilution. </p><p>Standard enthalpy of <a href="/wiki/Denaturation_(biochemistry)" title="Denaturation (biochemistry)">Denaturation (biochemistry)</a> - is defined as the enthalpy change required to denature one mole of compound. </p><p><a href="/wiki/Hydration_energy" title="Hydration energy">Enthalpy of hydration</a> - is defined as the enthalpy change observed when one mole of gaseous ions are completely dissolved in water forming one mole of aqueous ions. </p> <div class="mw-heading mw-heading4"><h4 id="Physical_properties">Physical properties</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=11" title="Edit section: Physical properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Enthalpy_of_fusion" title="Enthalpy of fusion">Enthalpy of fusion</a> - is defined as the enthalpy change required to completely change the state of one mole of substance from solid to liquid. </p><p><a href="/wiki/Enthalpy_of_vaporization" title="Enthalpy of vaporization">Enthalpy of vaporization</a> - is defined as the enthalpy change required to completely change the state of one mole of substance from liquid to gas. </p><p><a href="/wiki/Enthalpy_of_sublimation" title="Enthalpy of sublimation">Enthalpy of sublimation</a> - is defined as the enthalpy change required to completely change the state of one mole of substance from solid to gas. </p><p><a href="/wiki/Lattice_enthalpy" class="mw-redirect" title="Lattice enthalpy">Lattice enthalpy</a> - is defined as the energy required to separate one mole of an ionic compound into separated gaseous ions to an infinite distance apart (meaning no force of attraction). </p><p><a href="/wiki/Enthalpy_of_mixing" title="Enthalpy of mixing">Enthalpy of mixing</a> - is defined as the enthalpy change upon mixing of two (non-reacting) chemical substances. </p> <div class="mw-heading mw-heading3"><h3 id="Open_systems">Open systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=12" title="Edit section: Open systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Thermodynamic" class="mw-redirect" title="Thermodynamic">thermodynamic</a> <a href="/wiki/Open_system_(systems_theory)" title="Open system (systems theory)">open systems</a>, mass (of substances) may flow in and out of the system boundaries. The first law of thermodynamics for open systems states: The increase in the internal energy of a system is equal to the amount of energy added to the system by mass flowing in and by heating, minus the amount lost by mass flowing out and in the form of work done by the system: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q+\mathrm {d} U_{\mathsf {in}}-\mathrm {d} U_{\mathsf {out}}-\mathrm {\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> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>Q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>W</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q+\mathrm {d} U_{\mathsf {in}}-\mathrm {d} U_{\mathsf {out}}-\mathrm {\delta } \,W\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66933e99ee9bdbbb19c9d1aaab5b9f7482a85a41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.607ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} U=\mathrm {\delta } \,Q+\mathrm {d} U_{\mathsf {in}}-\mathrm {d} U_{\mathsf {out}}-\mathrm {\delta } \,W\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">U</span><sub>in</sub> is the average internal energy entering the system, and <span class="texhtml mvar" style="font-style:italic;">U</span><sub>out</sub> is the average internal energy leaving the system. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:First_law_open_system.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/First_law_open_system.svg/250px-First_law_open_system.svg.png" decoding="async" width="250" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/First_law_open_system.svg/375px-First_law_open_system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/First_law_open_system.svg/500px-First_law_open_system.svg.png 2x" data-file-width="374" data-file-height="257" /></a><figcaption>During <a href="/wiki/Steady-state_(chemical_engineering)" class="mw-redirect" title="Steady-state (chemical engineering)">steady, continuous</a> operation, an energy balance applied to an open system equates shaft work performed by the system to heat added plus net enthalpy added</figcaption></figure> <p>The region of space enclosed by the boundaries of the open system is usually called a <a href="/wiki/Control_volume" title="Control volume">control volume</a>, and it may or may not correspond to physical walls. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of mass into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of mass out as if it were driving a piston of fluid. There are then two types of work performed: <i>Flow work</i> described above, which is performed on the fluid (this is also often called <i><span class="texhtml mvar" style="font-style:italic;">p V</span> work</i>), and <i>mechanical work</i> (<i>shaft work</i>), which may be performed on some mechanical device such as a turbine or pump. </p><p>These two types of work are expressed in the equation <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 {\delta } \,W=\mathrm {d} (p_{\mathsf {out}}V_{\mathsf {out}})-\mathrm {d} (p_{\mathsf {in}}V_{\mathsf {in}})+\mathrm {\delta } \,W_{\mathsf {shaft}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">h</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {\delta } \,W=\mathrm {d} (p_{\mathsf {out}}V_{\mathsf {out}})-\mathrm {d} (p_{\mathsf {in}}V_{\mathsf {in}})+\mathrm {\delta } \,W_{\mathsf {shaft}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ba21fb75a7fa81b385a08e6af1a7f2f7a8fb30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.374ex; height:2.843ex;" alt="{\displaystyle \mathrm {\delta } \,W=\mathrm {d} (p_{\mathsf {out}}V_{\mathsf {out}})-\mathrm {d} (p_{\mathsf {in}}V_{\mathsf {in}})+\mathrm {\delta } \,W_{\mathsf {shaft}}~.}"></span> </p><p>Substitution into the equation above for the control volume (cv) yields: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U_{\mathsf {cv}}=\mathrm {\delta } \,Q+\mathrm {d} U_{\mathsf {in}}+\mathrm {d} \left(p_{\mathsf {in}}V_{\mathsf {in}}\right)-\mathrm {d} U_{\mathsf {out}}-\mathrm {d} \left(p_{\mathsf {out}}V_{\mathsf {out}}\right)-\mathrm {\delta } W_{\mathsf {shaft}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">v</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>Q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">h</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U_{\mathsf {cv}}=\mathrm {\delta } \,Q+\mathrm {d} U_{\mathsf {in}}+\mathrm {d} \left(p_{\mathsf {in}}V_{\mathsf {in}}\right)-\mathrm {d} U_{\mathsf {out}}-\mathrm {d} \left(p_{\mathsf {out}}V_{\mathsf {out}}\right)-\mathrm {\delta } W_{\mathsf {shaft}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1751c09c77c69adabcaf9eac61c51fd28091528" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.019ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} U_{\mathsf {cv}}=\mathrm {\delta } \,Q+\mathrm {d} U_{\mathsf {in}}+\mathrm {d} \left(p_{\mathsf {in}}V_{\mathsf {in}}\right)-\mathrm {d} U_{\mathsf {out}}-\mathrm {d} \left(p_{\mathsf {out}}V_{\mathsf {out}}\right)-\mathrm {\delta } W_{\mathsf {shaft}}~.}"></span> </p><p>The definition of enthalpy, <span class="texhtml mvar" style="font-style:italic;">H</span>, permits us to use this <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a> to account for both internal energy and <span class="texhtml mvar" style="font-style:italic;">p V</span> work in fluids for open systems: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} U_{\mathsf {cv}}=\mathrm {\delta } \,Q+\mathrm {d} H_{\mathsf {in}}-\mathrm {d} H_{\mathsf {out}}-\mathrm {\delta } \,W_{\mathsf {shaft}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">v</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <mi>Q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mspace width="thinmathspace" /> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> <mi mathvariant="sans-serif">h</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">f</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U_{\mathsf {cv}}=\mathrm {\delta } \,Q+\mathrm {d} H_{\mathsf {in}}-\mathrm {d} H_{\mathsf {out}}-\mathrm {\delta } \,W_{\mathsf {shaft}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97909f5c7cd7697f00b485140f101a7582a15ef6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.439ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} U_{\mathsf {cv}}=\mathrm {\delta } \,Q+\mathrm {d} H_{\mathsf {in}}-\mathrm {d} H_{\mathsf {out}}-\mathrm {\delta } \,W_{\mathsf {shaft}}~.}"></span> </p><p>If we allow also the system boundary to move (e.g. due to moving pistons), we get a rather general form of the first law for open systems.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In terms of time derivatives, using <a href="/wiki/Notation_for_differentiation#Newton's_notation" title="Notation for differentiation">Newton's dot notation</a> for time derivatives, it reads: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} U}{\,\mathrm {d} \,t\,}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\,\mathrm {d} V_{k}}{\,\mathrm {d} t\,}}-P\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <mi>t</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} U}{\,\mathrm {d} \,t\,}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\,\mathrm {d} V_{k}}{\,\mathrm {d} t\,}}-P\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8967bb65ea648bc8c4a4e2a2c47f51b1f7d6cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.052ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {d} U}{\,\mathrm {d} \,t\,}}=\sum _{k}{\dot {Q}}_{k}+\sum _{k}{\dot {H}}_{k}-\sum _{k}p_{k}{\frac {\,\mathrm {d} V_{k}}{\,\mathrm {d} t\,}}-P\,,}"></span> </p><p>with sums over the various places <span class="texhtml mvar" style="font-style:italic;">k</span> where heat is supplied, mass flows into the system, and boundaries are moving. The <span class="texhtml mvar" style="font-style:italic;"> <span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;"><b>.</b></span><span style="display:block; line-height:1em;">H</span></span></span><sub>k</sub> </span> terms represent enthalpy flows, which can be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {H}}_{k}=h_{k}{\dot {m}}_{k}=H_{\mathsf {m}}{\dot {n}}_{k}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">m</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {H}}_{k}=h_{k}{\dot {m}}_{k}=H_{\mathsf {m}}{\dot {n}}_{k}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b69845a68b2e3f271669b1909576908b074d822" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.892ex; height:3.009ex;" alt="{\displaystyle {\dot {H}}_{k}=h_{k}{\dot {m}}_{k}=H_{\mathsf {m}}{\dot {n}}_{k}\,,}"></span> </p><p>with <span class="texhtml mvar" style="font-style:italic;"> <span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;"><b>.</b></span><span style="display:block; line-height:1em;">m</span></span></span><sub>k</sub> </span> the mass flow and <span class="texhtml mvar" style="font-style:italic;"> <span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;"><b>.</b></span><span style="display:block; line-height:1em;">n</span></span></span><sub>k</sub> </span> the molar flow at position <span class="texhtml mvar" style="font-style:italic;">k</span> respectively. The term <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> d <i>V</i><sub><i>k</i></sub> </span><span class="sr-only">/</span><span class="den"> d <i>t</i> </span></span>⁠</span></span> represents the rate of change of the system volume at position <span class="texhtml mvar" style="font-style:italic;">k</span> that results in <span class="texhtml mvar" style="font-style:italic;">p V</span> power done by the system. The parameter <span class="texhtml mvar" style="font-style:italic;">P</span> represents all other forms of power done by the system such as shaft power, but it can also be, say, electric power produced by an electrical power plant. </p><p>Note that the previous expression holds true only if the kinetic energy flow rate is conserved between system inlet and outlet.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This new type of energy, kinetic energy, was not mentioned before. Is it part of <span class="texhtml mvar" style="font-style:italic;">U</span> ? Does it need to be conserved, or just the net flow across boundary be zero? (March 2015)">clarification needed</span></a></i>]</sup> Otherwise, it has to be included in the enthalpy balance. During <a href="/wiki/Steady-state_(chemical_engineering)" class="mw-redirect" title="Steady-state (chemical engineering)">steady-state</a> operation of a device (<i>see <a href="/wiki/Turbine" title="Turbine">turbine</a>, <a href="/wiki/Pump" title="Pump">pump</a>, and <a href="/wiki/Engine" title="Engine">engine</a></i>), the average <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> d <i>U</i> </span><span class="sr-only">/</span><span class="den">d <i>t</i></span></span>⁠</span></span> may be set equal to zero. This yields a useful expression for the average <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> generation for these devices in the absence of chemical reactions: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\sum _{k}\left\langle {\dot {Q}}_{k}\right\rangle +\sum _{k}\left\langle {\dot {H}}_{k}\right\rangle -\sum _{k}\left\langle p_{k}{\frac {\,\mathrm {d} V_{k}}{\,\mathrm {d} t\,}}\right\rangle \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\sum _{k}\left\langle {\dot {Q}}_{k}\right\rangle +\sum _{k}\left\langle {\dot {H}}_{k}\right\rangle -\sum _{k}\left\langle p_{k}{\frac {\,\mathrm {d} V_{k}}{\,\mathrm {d} t\,}}\right\rangle \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03de1a4ba38c542952bbded34c1259a753dbb28b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.991ex; height:6.676ex;" alt="{\displaystyle P=\sum _{k}\left\langle {\dot {Q}}_{k}\right\rangle +\sum _{k}\left\langle {\dot {H}}_{k}\right\rangle -\sum _{k}\left\langle p_{k}{\frac {\,\mathrm {d} V_{k}}{\,\mathrm {d} t\,}}\right\rangle \,,}"></span> </p><p>where the <a href="/wiki/Angle_bracket" class="mw-redirect" title="Angle bracket">angle brackets</a> denote time averages. The technical importance of the enthalpy is directly related to its presence in the first law for open systems, as formulated above. </p> <div class="mw-heading mw-heading2"><h2 id="Diagrams">Diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=13" title="Edit section: Diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ts_diagram_of_N2_02.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Ts_diagram_of_N2_02.jpg/500px-Ts_diagram_of_N2_02.jpg" decoding="async" width="500" height="345" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Ts_diagram_of_N2_02.jpg/750px-Ts_diagram_of_N2_02.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Ts_diagram_of_N2_02.jpg/1000px-Ts_diagram_of_N2_02.jpg 2x" data-file-width="1013" data-file-height="698" /></a><figcaption><span class="nowrap"> <span class="texhtml"><i>T</i> – <i>s</i></span> </span> diagram of nitrogen.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The red curve at the left is the melting curve. The red dome represents the two-phase region with the low-entropy side the saturated liquid and the high-entropy side the saturated gas. The black curves give the <span class="nowrap"> <span class="texhtml"> <i>T</i> – <i>s</i> </span> </span> relation along isobars. The pressures are indicated in bar. The blue curves are isenthalps (curves of constant enthalpy). The values are indicated in blue in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den">kg</span></span>⁠</span>. The specific points <b>a</b>, <b>b</b>, etc., are treated in the main text.</figcaption></figure> <p>The enthalpy values of important substances can be obtained using commercial software. Practically all relevant material properties can be obtained either in tabular or in graphical form. There are many types of diagrams, such as <span class="nowrap"><span class="texhtml"><i>h</i> – <i>T</i></span> </span> diagrams, which give the specific enthalpy as function of temperature for various pressures, and <span class="nowrap"> <span class="texhtml"><i>h</i> – <i>p</i></span> </span> diagrams, which give <span class="texhtml mvar" style="font-style:italic;">h</span> as function of <span class="texhtml mvar" style="font-style:italic;">p</span> for various <span class="texhtml mvar" style="font-style:italic;">T</span>. One of the most common diagrams is the temperature–specific entropy diagram (<span class="nowrap"> <span class="texhtml"><i>T</i> – <i>s</i></span> </span> diagram). It gives the melting curve and saturated liquid and vapor values together with isobars and isenthalps. These diagrams are powerful tools in the hands of the thermal engineer. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Some_basic_applications">Some basic applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=14" title="Edit section: Some basic applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The points <span class="texhtml"><b>a</b></span> through <span class="texhtml"><b>h</b></span> in the figure play a role in the discussion in this section. </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr> <td>Point </td> <th><span class="texhtml mvar" style="font-style:italic;">T</span></th> <th><span class="texhtml mvar" style="font-style:italic;">p</span></th> <th><span class="texhtml mvar" style="font-style:italic;">s</span></th> <th><span class="texhtml mvar" style="font-style:italic;">h</span> </th></tr> <tr style="background:#EEEEEE;"> <td>Unit</td> <td><a href="/wiki/Kelvin_(unit)" class="mw-redirect" title="Kelvin (unit)">K</a></td> <td><a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den"> kg K </span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den"> kg </span></span>⁠</span> </td></tr> <tr> <td><span class="texhtml"><b>a</b></span></td> <td>300</td> <td>1</td> <td>6.85</td> <td>461 </td></tr> <tr> <td><span class="texhtml"><b>b</b></span></td> <td>380</td> <td>2</td> <td>6.85</td> <td>530 </td></tr> <tr> <td><span class="texhtml"><b>c</b></span></td> <td>300</td> <td>200</td> <td>5.16</td> <td>430 </td></tr> <tr> <td><span class="texhtml"><b>d</b></span></td> <td>270</td> <td>1</td> <td>6.79</td> <td>430 </td></tr> <tr> <td><span class="texhtml"><b>e</b></span></td> <td>108</td> <td>13</td> <td>3.55</td> <td>100 </td></tr> <tr> <td><span class="texhtml"><b>f</b></span></td> <td>77.2</td> <td>1</td> <td>3.75</td> <td>100 </td></tr> <tr> <td><span class="texhtml"><b>g</b></span></td> <td>77.2</td> <td>1</td> <td>2.83</td> <td>28 </td></tr> <tr> <td><span class="texhtml"><b>h</b></span></td> <td>77.2</td> <td>1</td> <td>5.41</td> <td>230 </td></tr></tbody></table></dd></dl> <p>Points <span class="texhtml"><b>e</b></span> and <span class="texhtml"><b>g</b></span> are saturated liquids, and point <span class="texhtml"><b>h</b></span> is a saturated gas. </p> <div class="mw-heading mw-heading3"><h3 id="Throttling">Throttling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=15" title="Edit section: Throttling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Joule%E2%80%93Thomson_effect" title="Joule–Thomson effect">Joule–Thomson effect</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Schematic_of_throttling.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Schematic_of_throttling.png/220px-Schematic_of_throttling.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Schematic_of_throttling.png/330px-Schematic_of_throttling.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Schematic_of_throttling.png/440px-Schematic_of_throttling.png 2x" data-file-width="588" data-file-height="260" /></a><figcaption>Schematic diagram of a throttling in the steady state. Fluid enters the system (dotted rectangle) at point 1 and leaves it at point 2. The mass flow is <span class="texhtml mvar" style="font-style:italic;">ṁ</span> .</figcaption></figure> <p>One of the simple applications of the concept of enthalpy is the so-called throttling process, also known as <a href="/wiki/Joule%E2%80%93Thomson_effect" title="Joule–Thomson effect">Joule–Thomson expansion</a>. It concerns a steady adiabatic flow of a fluid through a flow resistance (valve, porous plug, or any other type of flow resistance) as shown in the figure. This process is very important, since it is at the heart of domestic <a href="/wiki/Refrigerator" title="Refrigerator">refrigerators</a>, where it is responsible for the temperature drop between ambient temperature and the interior of the refrigerator. It is also the final stage in many types of <a href="/wiki/Liquefier" class="mw-redirect" title="Liquefier">liquefiers</a>. </p><p>For a steady state flow regime, the enthalpy of the system (dotted rectangle) has to be constant. Hence </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\dot {m}}h_{1}-{\dot {m}}h_{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\dot {m}}h_{1}-{\dot {m}}h_{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2262eeeae8bf33aa827598a85ec4c2dbb8281b37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.196ex; height:2.509ex;" alt="{\displaystyle 0={\dot {m}}h_{1}-{\dot {m}}h_{2}~.}"></span> </p><p>Since the mass flow is constant, the specific enthalpies at the two sides of the flow resistance are the same: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}=h_{2}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}=h_{2}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be6e50789e0f751c3dc1d76ae054a27c84df5e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.177ex; height:2.509ex;" alt="{\displaystyle h_{1}=h_{2}\;,}"></span> </p><p>that is, the enthalpy per unit mass does not change during the throttling. The consequences of this relation can be demonstrated using the <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">T − s</span> </span> diagram above. </p> <div class="mw-heading mw-heading4"><h4 id="Example_1">Example 1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=16" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Point <b>c</b> is at 200 bar and room temperature (300 K). A Joule–Thomson expansion from 200 bar to 1 bar follows a curve of constant enthalpy of roughly 425 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den">kg</span></span>⁠</span> (not shown in the diagram) lying between the 400 and 450 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den">kg</span></span>⁠</span> isenthalps and ends in point <b>d</b>, which is at a temperature of about 270 K . Hence the expansion from 200 bar to 1 bar cools nitrogen from 300 K to 270 K . In the valve, there is a lot of friction, and a lot of entropy is produced, but still the final temperature is below the starting value. </p> <div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=17" title="Edit section: Example 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Point <b>e</b> is chosen so that it is on the saturated liquid line with <span class="nowrap"> <span class="texhtml"><i>h</i> = 100 </span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den">kg</span></span>⁠</span> .</span> It corresponds roughly with <span class="nowrap"> <span class="texhtml"><i>p</i> = 13</span> bar </span> and <span class="nowrap"> <span class="texhtml"><i>T</i> = 108</span> K .</span> Throttling from this point to a pressure of 1 bar ends in the two-phase region (point <b>f</b>). This means that a mixture of gas and liquid leaves the throttling valve. Since the enthalpy is an extensive parameter, the enthalpy in <span class="nowrap"><b>f</b> <span class="texhtml">( <i>h</i><sub><b>f</b></sub> )</span> </span> is equal to the enthalpy in <span class="nowrap"> <b>g</b> <span class="texhtml">( <i>h</i><sub><b>g</b></sub> )</span> </span> multiplied by the liquid fraction in <span class="nowrap"> <b>f</b> <span class="texhtml">( <i>x</i><sub><b>f</b></sub> )</span> </span> plus the enthalpy in <span class="nowrap"> <b>h</b> <span class="texhtml">( <i>h</i><sub><b>h</b></sub> )</span> </span> multiplied by the gas fraction in <span class="nowrap"><b>f</b> <span class="texhtml"> (1 − <i>x</i><sub><b>f</b></sub> )</span> .</span> So </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\mathbf {\mathsf {f}} }=x_{\mathbf {\mathsf {f}} }h_{\mathbf {\mathsf {g}} }+(1-x_{\mathbf {\mathsf {f}} })h_{\mathsf {\mathbf {h} }}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">f</mi> </mrow> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">h</mi> </mrow> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{\mathbf {\mathsf {f}} }=x_{\mathbf {\mathsf {f}} }h_{\mathbf {\mathsf {g}} }+(1-x_{\mathbf {\mathsf {f}} })h_{\mathsf {\mathbf {h} }}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe5ea90560d9157f031ca4d0de5607b4e7e3720" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.4ex; height:3.009ex;" alt="{\displaystyle h_{\mathbf {\mathsf {f}} }=x_{\mathbf {\mathsf {f}} }h_{\mathbf {\mathsf {g}} }+(1-x_{\mathbf {\mathsf {f}} })h_{\mathsf {\mathbf {h} }}~.}"></span> </p><p>With numbers: </p> <dl><dd><span class="nowrap"><span class="texhtml"> 100 = <i>x</i><sub><b>f</b></sub> × 28 + <span style="font-size:120%">(</span>1 − <i>x</i><sub><b>f</b></sub><span style="font-size:120%">)</span> × 230 </span> ,</span> so <span class="nowrap"><span class="texhtml"><i>x</i><sub><b>f</b></sub> = 0.64</span> .</span></dd></dl> <p>This means that the mass fraction of the liquid in the liquid–gas mixture that leaves the throttling valve is 64%. </p> <div class="mw-heading mw-heading3"><h3 id="Compressors">Compressors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=18" title="Edit section: Compressors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Compressor" title="Compressor">Compressor</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Schematic_of_compressor.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Schematic_of_compressor.png/220px-Schematic_of_compressor.png" decoding="async" width="220" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Schematic_of_compressor.png/330px-Schematic_of_compressor.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Schematic_of_compressor.png/440px-Schematic_of_compressor.png 2x" data-file-width="566" data-file-height="280" /></a><figcaption>Schematic diagram of a compressor in the steady state. Fluid enters the system (dotted rectangle) at point 1 and leaves it at point 2. The mass flow is <span class="texhtml mvar" style="font-style:italic;">ṁ</span>. A power <span class="texhtml mvar" style="font-style:italic;">P</span> is applied and a heat flow <span class="texhtml mvar" style="font-style:italic;">Q̇</span> is released to the surroundings at ambient temperature <span class="texhtml"><i>T</i><sub>a</sub></span> .</figcaption></figure> <p>A power <span class="texhtml mvar" style="font-style:italic;">P</span> is applied e.g. as electrical power. If the compression is <a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a>, the gas temperature goes up. In the reversible case it would be at constant entropy, which corresponds with a vertical line in the <span class="texhtml"><i>T</i> – <i>s</i></span> diagram. For example, compressing nitrogen from 1 bar (point <b>a</b>) to 2 bar (point <b>b</b>) would result in a temperature increase from 300 K to 380 K. In order to let the compressed gas exit at ambient temperature <span class="texhtml"><i>T</i><sub>a</sub></span>, heat exchange, e.g. by cooling water, is necessary. In the ideal case the compression is isothermal. The average heat flow to the surroundings is <span class="texhtml mvar" style="font-style:italic;">Q̇</span>. Since the system is in the steady state the first law gives </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=-{\dot {Q}}+{\dot {m}}\,h_{1}-{\dot {m}}\,h_{2}+P~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>P</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=-{\dot {Q}}+{\dot {m}}\,h_{1}-{\dot {m}}\,h_{2}+P~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4d7077ea6072801fcc93dcd9bd7c0a7d2d13b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.043ex; height:3.176ex;" alt="{\displaystyle 0=-{\dot {Q}}+{\dot {m}}\,h_{1}-{\dot {m}}\,h_{2}+P~.}"></span> </p><p>The minimal power needed for the compression is realized if the compression is reversible. In that case the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> for open systems gives </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=-{\frac {\,{\dot {Q}}\,}{T_{\mathsf {a}}}}+{\dot {m}}\,s_{1}-{\dot {m}}\,s_{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=-{\frac {\,{\dot {Q}}\,}{T_{\mathsf {a}}}}+{\dot {m}}\,s_{1}-{\dot {m}}\,s_{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a18099e9fd27e645eee7ffd63776f5823bfd041" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.57ex; height:6.176ex;" alt="{\displaystyle 0=-{\frac {\,{\dot {Q}}\,}{T_{\mathsf {a}}}}+{\dot {m}}\,s_{1}-{\dot {m}}\,s_{2}~.}"></span> </p><p>Eliminating <span class="texhtml mvar" style="font-style:italic;">Q̇</span> gives for the minimal power </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=h_{2}-h_{1}-T_{\mathsf {a}}\left(s_{2}-s_{1}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=h_{2}-h_{1}-T_{\mathsf {a}}\left(s_{2}-s_{1}\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3221e20e525e8b89aac68386cde675e0e3003ee1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.77ex; height:5.176ex;" alt="{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=h_{2}-h_{1}-T_{\mathsf {a}}\left(s_{2}-s_{1}\right)~.}"></span> </p><p>For example, compressing 1 kg of nitrogen from 1 bar to 200 bar costs at least :<span class="nowrap"> <span class="texhtml"> <span style="font-size:120%">(</span> <i>h</i><sub><b>c</b></sub> − <i>h</i><sub><b>a</b></sub> <span style="font-size:120%">)</span> − <i>T</i><sub>a</sub>( <i>s</i><sub><b>c</b></sub> − <i>s</i><sub><b>a</b></sub> )</span> .</span> With the data, obtained with the <span class="nowrap"> <span class="texhtml"><i>T</i> – <i>s</i></span> </span> diagram, we find a value of <span class="nowrap"> <span class="texhtml">(430 − 461) − 300 × (5.16 − 6.85) = 476 </span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> kJ </span><span class="sr-only">/</span><span class="den">kg</span></span>⁠</span> .</span> </p><p>The relation for the power can be further simplified by writing it as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=\int _{1}^{2}\left(\mathrm {d} h-T_{\mathsf {a}}\,\mathrm {d} s\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>h</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">a</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=\int _{1}^{2}\left(\mathrm {d} h-T_{\mathsf {a}}\,\mathrm {d} s\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f02f5f3a2683debb9749d52e0477793d7f9383" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.933ex; height:6.176ex;" alt="{\displaystyle {\frac {\,P_{\mathsf {min}}\,}{\dot {m}}}=\int _{1}^{2}\left(\mathrm {d} h-T_{\mathsf {a}}\,\mathrm {d} s\right)~.}"></span> </p><p>With </p> <dl><dd><span class="nowrap"> <span class="texhtml"> d<i>h</i> = <i>T</i> d<i>s</i> + <i>v</i> d<i>p</i> </span> ,</span></dd></dl> <p>this results in the final relation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\,P_{\mathsf {min}}}{\dot {m}}}=\int _{1}^{2}v\,\mathrm {d} p~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>v</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\,P_{\mathsf {min}}}{\dot {m}}}=\int _{1}^{2}v\,\mathrm {d} p~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5536af0219328d81f96df9b38d071fc1c406fcaa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.704ex; height:6.176ex;" alt="{\displaystyle {\frac {\,P_{\mathsf {min}}}{\dot {m}}}=\int _{1}^{2}v\,\mathrm {d} p~.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="History_and_etymology">History and etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=19" title="Edit section: History and etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <i>enthalpy</i> was coined relatively late in the history of thermodynamics, in the early 20th century. <a href="/wiki/Energy" title="Energy">Energy</a> was introduced in a modern sense by <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> in 1802, while <a href="/wiki/Entropy" title="Entropy">entropy</a> by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> in 1865. <i>Energy</i> uses the root of the <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Greek</a> word <span title="Ancient Greek (to 1453)-language text"><span lang="grc"><span class="texhtml">ἔργον</span></span></span> (<i>ergon</i>), meaning "work",<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> to express the idea of capacity to perform work. <i>Entropy</i> uses the Greek word <span title="Ancient Greek (to 1453)-language text"><span lang="grc">τροπή</span></span> (<i>tropē</i>) meaning <i>transformation</i> or <i>turning</i>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> <i>Enthalpy</i> uses the root of the Greek word <span title="Ancient Greek (to 1453)-language text"><span lang="grc"><span class="texhtml">θάλπος</span></span></span> (<i>thalpos</i>) "warmth, heat".<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>The term expresses the obsolete concept of <i>heat content</i>,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> as <span class="texhtml">d<i>H</i></span> refers to the amount of heat gained in a process at constant pressure only,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> but not in the general case when pressure is variable.<sup id="cite_ref-Laidler-Meiser-1982_19-1" class="reference"><a href="#cite_note-Laidler-Meiser-1982-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">J. W. Gibbs</a> used the term "a heat function for constant pressure" for clarity.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </p><p>Introduction of the concept of "heat content" <span class="texhtml mvar" style="font-style:italic;">H</span> is associated with <a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Benoît Paul Émile Clapeyron</a> and <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> (<a href="/wiki/Clausius%E2%80%93Clapeyron_relation" title="Clausius–Clapeyron relation">Clausius–Clapeyron relation</a>, 1850). </p><p>The term <i>enthalpy</i> first appeared in print in 1909.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> It is attributed to <a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Heike Kamerlingh Onnes</a>, who most likely introduced it orally the year before, at the first meeting of the Institute of Refrigeration in Paris.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> It gained currency only in the 1920s, notably with the <i><a href="/wiki/Enthalpy%E2%80%93entropy_chart" title="Enthalpy–entropy chart">Mollier Steam Tables and Diagrams</a></i>, published in 1927. </p><p>Until the 1920s, the symbol <span class="texhtml mvar" style="font-style:italic;">H</span> was used, somewhat inconsistently, for "heat" in general. The definition of <span class="texhtml mvar" style="font-style:italic;">H</span> as strictly limited to enthalpy or "heat content at constant pressure" was formally proposed by A. W. Porter in 1922.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Howard_34-0" class="reference"><a href="#cite_note-Howard-34"><span class="cite-bracket">[</span>31<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=Enthalpy&action=edit&section=20" 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-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><span class="mwe-math-element"><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 T={\frac {T}{\,V\,}}\left({\frac {\partial ({\frac {\,nRT\,}{P}})}{\partial T}}\right)_{\mathsf {p}}={\frac {nRT}{PV}}=1~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>α</mi> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mrow> <mspace width="thinmathspace" /> <mi>V</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>n</mi> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> </mrow> <mi>P</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mi>P</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\alpha T={\frac {T}{\,V\,}}\left({\frac {\partial ({\frac {\,nRT\,}{P}})}{\partial T}}\right)_{\mathsf {p}}={\frac {nRT}{PV}}=1~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275ac6c4d35625b41ffdf1e5075a5dc4182d67ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:38.211ex; height:8.176ex;" alt="{\displaystyle \;\alpha T={\frac {T}{\,V\,}}\left({\frac {\partial ({\frac {\,nRT\,}{P}})}{\partial T}}\right)_{\mathsf {p}}={\frac {nRT}{PV}}=1~.}"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"> Howard (2002) quotes <a href="/wiki/J._R._Partington" title="J. R. Partington">J. R. Partington</a> in <i>An Advanced Treatise on Physical Chemistry</i> (1949) as saying that the function <i>H</i> was "usually called the heat content."</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"> Volume I of Gibbs' <i>Collected Works</i><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> does not contain the word <i>enthalpy</i>, but uses the phrase <i>"heat function for constant pressure"</i> instead, for the same quantity.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></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=Enthalpy&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 14em;"> <ul><li><a href="/wiki/Calorimetry" title="Calorimetry">Calorimetry</a></li> <li><a href="/wiki/Calorimeter" title="Calorimeter">Calorimeter</a></li> <li><a href="/wiki/Departure_function" title="Departure function">Departure function</a></li> <li><a href="/wiki/Hess%27s_law" title="Hess's law">Hess's law</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic process</a></li> <li><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws of thermodynamics</a></li> <li><a href="/wiki/Stagnation_enthalpy" title="Stagnation enthalpy">Stagnation enthalpy</a></li> <li><a href="/wiki/Standard_enthalpy_of_formation" title="Standard enthalpy of formation">Standard enthalpy of formation</a></li> <li><a href="/wiki/Thermodynamic_databases" class="mw-redirect" title="Thermodynamic databases">Thermodynamic databases</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=22" 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 reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-GoldBk-enth-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-GoldBk-enth_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-GoldBk-enth_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">IUPAC</a>, <i><a href="/wiki/IUPAC_books#Gold_Book" class="mw-redirect" title="IUPAC books">Compendium of Chemical Terminology</a></i>, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "<a rel="nofollow" class="external text" href="https://goldbook.iupac.org/terms/view/E02141.html">enthalpy</a>". <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fgoldbook.E02141">10.1351/goldbook.E02141</a></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZemansky1968" class="citation book cs1">Zemansky, Mark W. (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/heatthermodynami0000zema"><i>Heat and Thermodynamics</i></a></span> (5th ed.). New York, NY: McGraw-Hill. chapter 11 <a rel="nofollow" class="external text" href="https://archive.org/details/heatthermodynami0000zema/page/275">p 275</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Heat+and+Thermodynamics&rft.place=New+York%2C+NY&rft.pages=chapter-11+p-275&rft.edition=5th&rft.pub=McGraw-Hill&rft.date=1968&rft.aulast=Zemansky&rft.aufirst=Mark+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fheatthermodynami0000zema&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_WylenSonntag1985" class="citation book cs1">van Wylen, G.J.; Sonntag, R.E. (1985). <i>Fundamentals of Classical Thermodynamics</i> (3rd ed.). New York, NY: John Wiley & Sons. section 5.5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-82933-1" title="Special:BookSources/978-0-471-82933-1"><bdi>978-0-471-82933-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Classical+Thermodynamics&rft.place=New+York%2C+NY&rft.pages=section-5.5&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.isbn=978-0-471-82933-1&rft.aulast=van+Wylen&rft.aufirst=G.J.&rft.au=Sonntag%2C+R.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkinsde_Paula2006" class="citation book cs1">Atkins, Peter; de Paula, Julio (2006). <i>Atkins' Physical Chemistry</i> (8th ed.). W.H.Freeman. p. 51. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-8759-8" title="Special:BookSources/0-7167-8759-8"><bdi>0-7167-8759-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atkins%27+Physical+Chemistry&rft.pages=51&rft.edition=8th&rft.pub=W.H.Freeman&rft.date=2006&rft.isbn=0-7167-8759-8&rft.aulast=Atkins&rft.aufirst=Peter&rft.au=de+Paula%2C+Julio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaidlerMeiser1999" class="citation book cs1">Laidler, Keith J.; Meiser, John H. (1999). <i>Physical Chemistry</i> (3rd ed.). Boston, MA: Houghton Mifflin. p. 66. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-395-91848-0" title="Special:BookSources/0-395-91848-0"><bdi>0-395-91848-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry&rft.place=Boston%2C+MA&rft.pages=66&rft.edition=3rd&rft.pub=Houghton+Mifflin&rft.date=1999&rft.isbn=0-395-91848-0&rft.aulast=Laidler&rft.aufirst=Keith+J.&rft.au=Meiser%2C+John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFÇengelBolesKanoglu2019" class="citation book cs1">Çengel, Yunus A.; Boles, Michael A.; Kanoglu, Mehmet (2019). <i>Thermodynamics: an engineering approach</i> (Ninth ed.). New York, NY: McGraw-Hill Education. p. 123. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-259-82267-4" title="Special:BookSources/978-1-259-82267-4"><bdi>978-1-259-82267-4</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%3A+an+engineering+approach&rft.place=New+York%2C+NY&rft.pages=123&rft.edition=Ninth&rft.pub=McGraw-Hill+Education&rft.date=2019&rft.isbn=978-1-259-82267-4&rft.aulast=%C3%87engel&rft.aufirst=Yunus+A.&rft.au=Boles%2C+Michael+A.&rft.au=Kanoglu%2C+Mehmet&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=e)nqa/lpw"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἐνθάλπειν</span></span></a>, <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=e)n"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἐν</span></span></a>, <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=qa/lpw"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">θάλπειν</span></span></a>. <a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell, Henry George</a>; <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott, Robert</a>; <i><a href="/wiki/A_Greek%E2%80%93English_Lexicon" title="A Greek–English Lexicon">A Greek–English Lexicon</a></i> at the <a href="/wiki/Perseus_Project" class="mw-redirect" title="Perseus Project">Perseus Project</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Thermodynamics/The_Four_Laws_of_Thermodynamics/First_Law_of_Thermodynamics">"1st Law of Thermodynamics"</a>. <i>Chemistry LibreTexts</i>. 2013-10-02<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Chemistry+LibreTexts&rft.atitle=1st+Law+of+Thermodynamics&rft.date=2013-10-02&rft_id=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FSupplemental_Modules_%28Physical_and_Theoretical_Chemistry%29%2FThermodynamics%2FThe_Four_Laws_of_Thermodynamics%2FFirst_Law_of_Thermodynamics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-Guggenheim-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Guggenheim_9-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuggenheim1959" class="citation book cs1">Guggenheim, E.A. (1959). <i>Thermodynamics</i>. Amsterdam, NL: North-Holland Publishing Company.</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=Amsterdam%2C+NL&rft.pub=North-Holland+Publishing+Company&rft.date=1959&rft.aulast=Guggenheim&rft.aufirst=E.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoranShapiro2006" class="citation book cs1">Moran, M.J.; Shapiro, H. N. (2006). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsengi00mora_077"><i>Fundamentals of Engineering Thermodynamics</i></a></span> (5th ed.). John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsengi00mora_077/page/n523">511</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470030370" title="Special:BookSources/9780470030370"><bdi>9780470030370</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Engineering+Thermodynamics&rft.pages=511&rft.edition=5th&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=9780470030370&rft.aulast=Moran&rft.aufirst=M.J.&rft.au=Shapiro%2C+H.+N.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffundamentalsengi00mora_077&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIribarneGodson1981" class="citation book cs1">Iribarne, J.V.; Godson, W.L. (1981). <i>Atmospheric Thermodynamics</i> (2nd ed.). Dordrecht, NL: Kluwer Academic Publishers. pp. 235–236. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-277-1297-2" title="Special:BookSources/90-277-1297-2"><bdi>90-277-1297-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atmospheric+Thermodynamics&rft.place=Dordrecht%2C+NL&rft.pages=235-236&rft.edition=2nd&rft.pub=Kluwer+Academic+Publishers&rft.date=1981&rft.isbn=90-277-1297-2&rft.aulast=Iribarne&rft.aufirst=J.V.&rft.au=Godson%2C+W.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTschoegl2000" class="citation book cs1">Tschoegl, N.W. (2000). <i>Fundamentals of Equilibrium and Steady-State Thermodynamics</i>. Amsterdam, NL: Elsevier. p. 17. <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"><bdi>0-444-50426-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Equilibrium+and+Steady-State+Thermodynamics&rft.place=Amsterdam%2C+NL&rft.pages=17&rft.pub=Elsevier&rft.date=2000&rft.isbn=0-444-50426-5&rft.aulast=Tschoegl&rft.aufirst=N.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCallen1985" class="citation book cs1">Callen, H.B. (1985) [1960]. <i>Thermodynamics and an Introduction to Thermostatistics</i> (1st (1960), 2nd (1985) ed.). New York, NY: John Wiley & Sons. Chapter 5. <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"><bdi>0-471-86256-8</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+and+an+Introduction+to+Thermostatistics&rft.place=New+York%2C+NY&rft.pages=Chapter-5&rft.edition=1st+%281960%29%2C+2nd+%281985%29&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.isbn=0-471-86256-8&rft.aulast=Callen&rft.aufirst=H.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMünster1970" class="citation book cs1">Münster, A. (1970). <i>Classical Thermodynamics</i>. Translated by Halberstadt, E.S. London, UK: Wiley–Interscience. p. 6. <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"><bdi>0-471-62430-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Thermodynamics&rft.place=London%2C+UK&rft.pages=6&rft.pub=Wiley%E2%80%93Interscience&rft.date=1970&rft.isbn=0-471-62430-6&rft.aulast=M%C3%BCnster&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></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="CITEREFReif1967" class="citation book cs1">Reif, F. (1967). <i>Statistical Physics</i>. London, UK: McGraw-Hill.</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=London%2C+UK&rft.pub=McGraw-Hill&rft.date=1967&rft.aulast=Reif&rft.aufirst=F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKittelKroemer1980" class="citation book cs1">Kittel, C.; Kroemer, H. (1980). <i>Thermal Physics</i>. London, UK: Freeman.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermal+Physics&rft.place=London%2C+UK&rft.pub=Freeman&rft.date=1980&rft.aulast=Kittel&rft.aufirst=C.&rft.au=Kroemer%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" 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="CITEREFRathakrishnan2015" class="citation book cs1">Rathakrishnan (2015). <i>High Enthalpy Gas Dynamics</i>. John Wiley and Sons Singapore Pte. Ltd. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1118821893" title="Special:BookSources/978-1118821893"><bdi>978-1118821893</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=High+Enthalpy+Gas+Dynamics&rft.pub=John+Wiley+and+Sons+Singapore+Pte.+Ltd.&rft.date=2015&rft.isbn=978-1118821893&rft.au=Rathakrishnan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-Laidler-Meiser-1982-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Laidler-Meiser-1982_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Laidler-Meiser-1982_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaidlerMeiser1982" class="citation book cs1"><a href="/wiki/Keith_J._Laidler" title="Keith J. Laidler">Laidler, K.J.</a>; Meiser, John H. (1982). <i>Physical Chemistry</i>. Benjamin / Cummings. p. 53. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-5682-3" title="Special:BookSources/978-0-8053-5682-3"><bdi>978-0-8053-5682-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry&rft.pages=53&rft.pub=Benjamin+%2F+Cummings&rft.date=1982&rft.isbn=978-0-8053-5682-3&rft.aulast=Laidler&rft.aufirst=K.J.&rft.au=Meiser%2C+John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></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="CITEREFPetrucciHarwoodHerring2002" class="citation book cs1">Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/generalchemistry00hill"><i>General Chemistry</i></a></span> (8th ed.). Prentice Hall. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/generalchemistry00hill/page/237">237–238</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-014329-7" title="Special:BookSources/978-0-13-014329-7"><bdi>978-0-13-014329-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Chemistry&rft.pages=237-238&rft.edition=8th&rft.pub=Prentice+Hall&rft.date=2002&rft.isbn=978-0-13-014329-7&rft.aulast=Petrucci&rft.aufirst=Ralph+H.&rft.au=Harwood%2C+William+S.&rft.au=Herring%2C+F.+Geoffrey&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeneralchemistry00hill&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoranShapiro2006" class="citation book cs1">Moran, M.J.; Shapiro, H.N. (2006). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsengi00mora_077"><i>Fundamentals of Engineering Thermodynamics</i></a></span> (5th ed.). John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsengi00mora_077/page/n141">129</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470030370" title="Special:BookSources/9780470030370"><bdi>9780470030370</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Engineering+Thermodynamics&rft.pages=129&rft.edition=5th&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=9780470030370&rft.aulast=Moran&rft.aufirst=M.J.&rft.au=Shapiro%2C+H.N.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffundamentalsengi00mora_077&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Figure composed with data obtained with <a href="/w/index.php?title=RefProp&action=edit&redlink=1" class="new" title="RefProp (page does not exist)">RefProp</a>, NIST Standard Reference Database 23.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=e)/rgon"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἔργον</span></span></a> in <a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell</a> and <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott</a>.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=troph/"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">τροπή</span></span></a> in <a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell</a> and <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott</a>.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=qa/lpos"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">θάλπος</span></span></a> in <a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell</a> and <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott</a>.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTinocoSauerWang1995" class="citation book cs1">Tinoco, Ignacio Jr.; Sauer, Kenneth; Wang, James C. (1995). <i>Physical Chemistry</i> (3rd ed.). Prentice-Hall. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-186545-7" title="Special:BookSources/978-0-13-186545-7"><bdi>978-0-13-186545-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry&rft.pages=41&rft.edition=3rd&rft.pub=Prentice-Hall&rft.date=1995&rft.isbn=978-0-13-186545-7&rft.aulast=Tinoco&rft.aufirst=Ignacio+Jr.&rft.au=Sauer%2C+Kenneth&rft.au=Wang%2C+James+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><a href="#CITEREFGibbs1948">Gibbs (1948)</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHendersonEyringJost1967" class="citation book cs1">Henderson, Douglas; Eyring, Henry; Jost, Wilhelm (1967). <i>Physical Chemistry: An advanced treatise</i>. Academic Press. p. 29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry%3A+An+advanced+treatise&rft.pages=29&rft.pub=Academic+Press&rft.date=1967&rft.aulast=Henderson&rft.aufirst=Douglas&rft.au=Eyring%2C+Henry&rft.au=Jost%2C+Wilhelm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"> <a href="#CITEREFDalton1909">Dalton (1909)</a>, p. 864, footnote (1).</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><a href="#CITEREFLaidler1995">Laidler (1995)</a>, p. 110; <a href="#CITEREFvan_Ness2003">van Ness (2003)</a>, p. 486.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPorter1922" class="citation journal cs1">Porter, Alfred W. (1922). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=njp.32101076787439;view=1up;seq=157">"The Generation and Utilisation of Cold. A General Discussion"</a>. <i>Transactions of the Faraday Society</i>. <b>18</b>: 139–143 esp. p. 140. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2Ftf9221800139">10.1039/tf9221800139</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Faraday+Society&rft.atitle=The+Generation+and+Utilisation+of+Cold.+A+General+Discussion&rft.volume=18&rft.pages=139-143+esp.+p.+140&rft.date=1922&rft_id=info%3Adoi%2F10.1039%2Ftf9221800139&rft.aulast=Porter&rft.aufirst=Alfred+W.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dnjp.32101076787439%3Bview%3D1up%3Bseq%3D157&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></span> </li> <li id="cite_note-Howard-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Howard_34-0">^</a></b></span> <span class="reference-text"> <a href="#CITEREFHoward2002">Howard (2002)</a>, p. 697.</span> </li> </ol></div> <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=Enthalpy&action=edit&section=23" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 25em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDalton1909" class="citation journal cs1">Dalton, J.P. (1909). <a rel="nofollow" class="external text" href="http://www.dwc.knaw.nl/DL/publications/PU00013601.pdf">"Researches on the Joule–Kelvin effect, especially at low temperatures. I. Calculations for hydrogen"</a> <span class="cs1-format">(PDF)</span>. <i>Koninklijke Akademie van Wetenschappen te Amsterdam [Proceedings of the Royal Academy of Sciences at Amsterdam, Section of Sciences]</i>. <b>11</b>: 863–873. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1908KNAB...11..863D">1908KNAB...11..863D</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Koninklijke+Akademie+van+Wetenschappen+te+Amsterdam+%5BProceedings+of+the+Royal+Academy+of+Sciences+at+Amsterdam%2C+Section+of+Sciences%5D&rft.atitle=Researches+on+the+Joule%E2%80%93Kelvin+effect%2C+especially+at+low+temperatures.+I.+Calculations+for+hydrogen&rft.volume=11&rft.pages=863-873&rft.date=1909&rft_id=info%3Abibcode%2F1908KNAB...11..863D&rft.aulast=Dalton&rft.aufirst=J.P.&rft_id=http%3A%2F%2Fwww.dwc.knaw.nl%2FDL%2Fpublications%2FPU00013601.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbs1948" class="citation book cs1"><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs, J.W.</a> (1948). <i>The Collected Works of J. Willard Gibbs</i>. Vol. I. New Haven, CT: Yale University Press. p. 88.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Collected+Works+of+J.+Willard+Gibbs&rft.place=New+Haven%2C+CT&rft.pages=88&rft.pub=Yale+University+Press&rft.date=1948&rft.aulast=Gibbs&rft.aufirst=J.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaase1971" class="citation book cs1">Haase, R. (1971). Jost, W. (ed.). <i>Physical Chemistry: An advanced treatise</i>. New York, NY: Academic. p. 29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry%3A+An+advanced+treatise&rft.place=New+York%2C+NY&rft.pages=29&rft.pub=Academic&rft.date=1971&rft.aulast=Haase&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Hoff2006" class="citation book cs1">de Hoff, R. (2006). <i>Thermodynamics in Materials Science</i>. Boca Raton, FL: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780849340659" title="Special:BookSources/9780849340659"><bdi>9780849340659</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+in+Materials+Science&rft.place=Boca+Raton%2C+FL&rft.pub=CRC+Press&rft.date=2006&rft.isbn=9780849340659&rft.aulast=de+Hoff&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoward2002" class="citation journal cs1">Howard, Irmgard K. (2002). "<i>H</i> is for enthalpy, thanks to Heike Kamerlingh Onnes and Alfred W. Porter". <i><a href="/wiki/Journal_of_Chemical_Education" title="Journal of Chemical Education">Journal of Chemical Education</a></i>. <b>79</b> (6): 697–698. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JChEd..79..697H">2002JChEd..79..697H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fed079p697">10.1021/ed079p697</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Chemical+Education&rft.atitle=H+is+for+enthalpy%2C+thanks+to+Heike+Kamerlingh+Onnes+and+Alfred+W.+Porter&rft.volume=79&rft.issue=6&rft.pages=697-698&rft.date=2002&rft_id=info%3Adoi%2F10.1021%2Fed079p697&rft_id=info%3Abibcode%2F2002JChEd..79..697H&rft.aulast=Howard&rft.aufirst=Irmgard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKittelKroemer1980" class="citation book cs1">Kittel, C.; Kroemer, H. (1980). <i>Thermal Physics</i>. New York, NY: S. R. Furphy & Co. p. 246.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermal+Physics&rft.place=New+York%2C+NY&rft.pages=246&rft.pub=S.+R.+Furphy+%26+Co.&rft.date=1980&rft.aulast=Kittel&rft.aufirst=C.&rft.au=Kroemer%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaidler1995" class="citation book cs1"><a href="/wiki/Keith_J._Laidler" title="Keith J. Laidler">Laidler, K.J.</a> (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/worldofphysicalc0000laid"><i>The World of Physical Chemistry</i></a></span>. Oxford, UK: Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/worldofphysicalc0000laid/page/110">110</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-855597-1" title="Special:BookSources/978-0-19-855597-1"><bdi>978-0-19-855597-1</bdi></a> – via archive.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+World+of+Physical+Chemistry&rft.place=Oxford%2C+UK&rft.pages=110&rft.pub=Oxford+University+Press&rft.date=1995&rft.isbn=978-0-19-855597-1&rft.aulast=Laidler&rft.aufirst=K.J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fworldofphysicalc0000laid&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Ness2003" class="citation journal cs1">van Ness, Hendrick C. (2003). "<i>H</i> is for enthalpy". <i>Journal of Chemical Education</i>. <b>80</b> (6): 486. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003JChEd..80..486V">2003JChEd..80..486V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fed080p486.1">10.1021/ed080p486.1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Chemical+Education&rft.atitle=H+is+for+enthalpy&rft.volume=80&rft.issue=6&rft.pages=486&rft.date=2003&rft_id=info%3Adoi%2F10.1021%2Fed080p486.1&rft_id=info%3Abibcode%2F2003JChEd..80..486V&rft.aulast=van+Ness&rft.aufirst=Hendrick+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy&action=edit&section=24" title="Edit section: External links"><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="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric. <a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/Enthalpy.html">"Enthalpy"</a>. <i>Eric Weisstein's World of Physics</i> – via scienceworld.wolfram.com.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Eric+Weisstein%27s+World+of+Physics&rft.atitle=Enthalpy&rft.aulast=Weisstein&rft.aufirst=Eric&rft_id=http%3A%2F%2Fscienceworld.wolfram.com%2Fphysics%2FEnthalpy.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/firlaw.html">"<i>Enthalpy</i>"</a>. Thermodynamics hypertext. Physics and Astronomy Department. <a href="/wiki/Georgia_State_University" title="Georgia State University">Georgia State University</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Enthalpy&rft.series=Physics+and+Astronomy+Department&rft.pub=Georgia+State+University&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Fthermo%2Ffirlaw.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061010093705/http://www.chem.tamu.edu/class/majors/tutorialnotefiles/enthalpy.htm">"Enthalpy example calculations"</a> (tutorial notes). Chemistry Department. <a href="/wiki/Texas_A%26M_University" title="Texas A&M University">Texas A&M University</a>. Archived from <a rel="nofollow" class="external text" href="http://www.chem.tamu.edu/class/majors/tutorialnotefiles/enthalpy.htm">the original</a> on 10 October 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Enthalpy+example+calculations&rft.series=Chemistry+Department&rft.pub=Texas+A%26M+University&rft_id=http%3A%2F%2Fwww.chem.tamu.edu%2Fclass%2Fmajors%2Ftutorialnotefiles%2Fenthalpy.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy" class="Z3988"></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 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Heating,_ventilation,_and_air_conditioning" style="vertical-align: middle;;padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:HVAC" title="Template:HVAC"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:HVAC" title="Template talk:HVAC"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:HVAC" title="Special:EditPage/Template:HVAC"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Heating,_ventilation,_and_air_conditioning" style="font-size:114%;margin:0 4em"><a href="/wiki/Heating,_ventilation,_and_air_conditioning" title="Heating, ventilation, and air conditioning">Heating, ventilation, and air conditioning</a></div></th></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 hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Air_changes_per_hour" title="Air changes per hour">Air changes per hour</a></li> <li><a href="/wiki/Bake-out" title="Bake-out">Bake-out</a></li> <li><a href="/wiki/Building_envelope" title="Building envelope">Building envelope</a></li> <li><a href="/wiki/Convection" title="Convection">Convection</a></li> <li><a href="/wiki/Dilution_(equation)" title="Dilution (equation)">Dilution</a></li> <li><a href="/wiki/Domestic_energy_consumption" title="Domestic energy consumption">Domestic energy consumption</a></li> <li><a class="mw-selflink selflink">Enthalpy</a></li> <li><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></li> <li><a href="/wiki/Gas_compressor" class="mw-redirect" title="Gas compressor">Gas compressor</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pump and refrigeration cycle</a></li> <li><a href="/wiki/Heat_transfer" title="Heat transfer">Heat transfer</a></li> <li><a href="/wiki/Humidity" title="Humidity">Humidity</a></li> <li><a href="/wiki/Infiltration_(HVAC)" title="Infiltration (HVAC)">Infiltration</a></li> <li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li> <li><a href="/wiki/Noise_control" title="Noise control">Noise control</a></li> <li><a href="/wiki/Outgassing" title="Outgassing">Outgassing</a></li> <li><a href="/wiki/Particulates" title="Particulates">Particulates</a></li> <li><a href="/wiki/Psychrometrics" title="Psychrometrics">Psychrometrics</a></li> <li><a href="/wiki/Sensible_heat" title="Sensible heat">Sensible heat</a></li> <li><a href="/wiki/Stack_effect" title="Stack effect">Stack effect</a></li> <li><a href="/wiki/Thermal_comfort" title="Thermal comfort">Thermal comfort</a></li> <li><a href="/wiki/Thermal_destratification" title="Thermal destratification">Thermal destratification</a></li> <li><a href="/wiki/Thermal_mass" title="Thermal mass">Thermal mass</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Vapour_pressure_of_water" title="Vapour pressure of water">Vapour pressure of water</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Technology</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absorption-compression_heat_pump" title="Absorption-compression heat pump">Absorption-compression heat pump</a></li> <li><a href="/wiki/Absorption_refrigerator" title="Absorption refrigerator">Absorption refrigerator</a></li> <li><a href="/wiki/Air_barrier" title="Air barrier">Air barrier</a></li> <li><a href="/wiki/Air_conditioning" title="Air conditioning">Air conditioning</a></li> <li><a href="/wiki/Antifreeze" title="Antifreeze">Antifreeze</a></li> <li><a href="/wiki/Automobile_air_conditioning" class="mw-redirect" title="Automobile air conditioning">Automobile air conditioning</a></li> <li><a href="/wiki/Autonomous_building" title="Autonomous building">Autonomous building</a></li> <li><a href="/wiki/Building_insulation_material" title="Building insulation material">Building insulation materials</a></li> <li><a href="/wiki/Central_heating" title="Central heating">Central heating</a></li> <li><a href="/wiki/Central_solar_heating" title="Central solar heating">Central solar heating</a></li> <li><a href="/wiki/Chilled_beam" title="Chilled beam">Chilled beam</a></li> <li><a href="/wiki/Chilled_water" title="Chilled water">Chilled water</a></li> <li><a href="/wiki/Constant_air_volume" title="Constant air volume">Constant air volume</a> (CAV)</li> <li><a href="/wiki/Coolant" title="Coolant">Coolant</a></li> <li><a href="/wiki/Cross_ventilation" title="Cross ventilation">Cross ventilation</a></li> <li><a href="/wiki/Dedicated_outdoor_air_system" title="Dedicated outdoor air system">Dedicated outdoor air system</a> (DOAS)</li> <li><a href="/wiki/Deep_water_source_cooling" title="Deep water source cooling">Deep water source cooling</a></li> <li><a href="/wiki/Demand_controlled_ventilation" title="Demand controlled ventilation">Demand controlled ventilation</a> (DCV)</li> <li><a href="/wiki/Displacement_ventilation" title="Displacement ventilation">Displacement ventilation</a></li> <li><a href="/wiki/District_cooling" title="District cooling">District cooling</a></li> <li><a href="/wiki/District_heating" title="District heating">District heating</a></li> <li><a href="/wiki/Electric_heating" title="Electric heating">Electric heating</a></li> <li><a href="/wiki/Energy_recovery_ventilation" class="mw-redirect" title="Energy recovery ventilation">Energy recovery ventilation</a> (ERV)</li> <li><a href="/wiki/Firestop" title="Firestop">Firestop</a></li> <li><a href="/wiki/Forced-air" title="Forced-air">Forced-air</a></li> <li><a href="/wiki/Forced-air_gas" title="Forced-air gas">Forced-air gas</a></li> <li><a href="/wiki/Free_cooling" title="Free cooling">Free cooling</a></li> <li><a href="/wiki/Heat_recovery_ventilation" title="Heat recovery ventilation">Heat recovery ventilation</a> (HRV)</li> <li><a href="/wiki/Hybrid_heat" title="Hybrid heat">Hybrid heat</a></li> <li><a href="/wiki/Hydronics" title="Hydronics">Hydronics</a></li> <li><a href="/wiki/Ice_storage_air_conditioning" title="Ice storage air conditioning">Ice storage air conditioning</a></li> <li><a href="/wiki/Kitchen_ventilation" title="Kitchen ventilation">Kitchen ventilation</a></li> <li><a href="/wiki/Mixed-mode_ventilation" title="Mixed-mode ventilation">Mixed-mode ventilation</a></li> <li><a href="/wiki/Microgeneration" title="Microgeneration">Microgeneration</a></li> <li><a href="/wiki/Passive_cooling" title="Passive cooling">Passive cooling</a></li> <li><a href="/wiki/Passive_daytime_radiative_cooling" title="Passive daytime radiative cooling">Passive daytime radiative cooling</a></li> <li><a href="/wiki/Passive_house" title="Passive house">Passive house</a></li> <li><a href="/wiki/Passive_ventilation" title="Passive ventilation">Passive ventilation</a></li> <li><a href="/wiki/Radiant_heating_and_cooling" title="Radiant heating and cooling">Radiant heating and cooling</a></li> <li><a href="/wiki/Radiant_cooling" class="mw-redirect" title="Radiant cooling">Radiant cooling</a></li> <li><a href="/wiki/Radiant_heating" class="mw-redirect" title="Radiant heating">Radiant heating</a></li> <li><a href="/wiki/Radon_mitigation" title="Radon mitigation">Radon mitigation</a></li> <li><a href="/wiki/Refrigeration" title="Refrigeration">Refrigeration</a></li> <li><a href="/wiki/Renewable_heat" title="Renewable heat">Renewable heat</a></li> <li><a href="/wiki/Room_air_distribution" title="Room air distribution">Room air distribution</a></li> <li><a href="/wiki/Solar_air_heat" title="Solar air heat">Solar air heat</a></li> <li><a href="/wiki/Solar_combisystem" title="Solar combisystem">Solar combisystem</a></li> <li><a href="/wiki/Solar_cooling" class="mw-redirect" title="Solar cooling">Solar cooling</a></li> <li><a href="/wiki/Solar_heating" class="mw-redirect" title="Solar heating">Solar heating</a></li> <li><a href="/wiki/Thermal_insulation" title="Thermal insulation">Thermal insulation</a></li> <li><a href="/wiki/Thermosiphon" title="Thermosiphon">Thermosiphon</a></li> <li><a href="/wiki/Underfloor_air_distribution" title="Underfloor air distribution">Underfloor air distribution</a></li> <li><a href="/wiki/Underfloor_heating" title="Underfloor heating">Underfloor heating</a></li> <li><a href="/wiki/Vapor_barrier" title="Vapor barrier">Vapor barrier</a></li> <li><a href="/wiki/Vapor-compression_refrigeration" title="Vapor-compression refrigeration">Vapor-compression refrigeration</a> (VCRS)</li> <li><a href="/wiki/Variable_air_volume" title="Variable air volume">Variable air volume</a> (VAV)</li> <li><a href="/wiki/Variable_refrigerant_flow" title="Variable refrigerant flow">Variable refrigerant flow</a> (VRF)</li> <li><a href="/wiki/Ventilation_(architecture)" title="Ventilation (architecture)">Ventilation</a></li> <li><a href="/wiki/Water_heat_recycling" title="Water heat recycling">Water heat recycling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Components</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Air_conditioner_inverter" class="mw-redirect" title="Air conditioner inverter">Air conditioner inverter</a></li> <li><a href="/wiki/Air_door" title="Air door">Air door</a></li> <li><a href="/wiki/Air_filter" title="Air filter">Air filter</a></li> <li><a href="/wiki/Air_handler" title="Air handler">Air handler</a></li> <li><a href="/wiki/Air_ioniser" title="Air ioniser">Air ionizer</a></li> <li><a href="/wiki/Air-mixing_plenum" title="Air-mixing plenum">Air-mixing plenum</a></li> <li><a href="/wiki/Air_purifier" title="Air purifier">Air purifier</a></li> <li><a href="/wiki/Air_source_heat_pump" title="Air source heat pump">Air source heat pump</a></li> <li><a href="/wiki/Attic_fan" title="Attic fan">Attic fan</a></li> <li><a href="/wiki/Automatic_balancing_valve" title="Automatic balancing valve">Automatic balancing valve</a></li> <li><a href="/wiki/Back_boiler" title="Back boiler">Back boiler</a></li> <li><a href="/wiki/Barrier_pipe" title="Barrier pipe">Barrier pipe</a></li> <li><a href="/wiki/Blast_damper" title="Blast damper">Blast damper</a></li> <li><a href="/wiki/Boiler" title="Boiler">Boiler</a></li> <li><a href="/wiki/Centrifugal_fan" title="Centrifugal fan">Centrifugal fan</a></li> <li><a href="/wiki/Ceramic_heater" title="Ceramic heater">Ceramic heater</a></li> <li><a href="/wiki/Chiller" title="Chiller">Chiller</a></li> <li><a href="/wiki/Condensate_pump" title="Condensate pump">Condensate pump</a></li> <li><a href="/wiki/Condenser_(heat_transfer)" title="Condenser (heat transfer)">Condenser</a></li> <li><a href="/wiki/Condensing_boiler" title="Condensing boiler">Condensing boiler</a></li> <li><a href="/wiki/Convection_heater" title="Convection heater">Convection heater</a></li> <li><a href="/wiki/Compressor" title="Compressor">Compressor</a></li> <li><a href="/wiki/Cooling_tower" title="Cooling tower">Cooling tower</a></li> <li><a href="/wiki/Damper_(flow)" title="Damper (flow)">Damper</a></li> <li><a href="/wiki/Dehumidifier" title="Dehumidifier">Dehumidifier</a></li> <li><a href="/wiki/Duct_(flow)" title="Duct (flow)">Duct</a></li> <li><a href="/wiki/Economizer" title="Economizer">Economizer</a></li> <li><a href="/wiki/Electrostatic_precipitator" title="Electrostatic precipitator">Electrostatic precipitator</a></li> <li><a href="/wiki/Evaporative_cooler" title="Evaporative cooler">Evaporative cooler</a></li> <li><a href="/wiki/Evaporator" title="Evaporator">Evaporator</a></li> <li><a href="/wiki/Exhaust_hood" class="mw-redirect" title="Exhaust hood">Exhaust hood</a></li> <li><a href="/wiki/Expansion_tank" title="Expansion tank">Expansion tank</a></li> <li><a href="/wiki/Fan_(machine)" title="Fan (machine)">Fan</a></li> <li><a href="/wiki/Fan_coil_unit" title="Fan coil unit">Fan coil unit</a></li> <li><a href="/wiki/Fan_filter_unit" title="Fan filter unit">Fan filter unit</a></li> <li><a href="/wiki/Fan_heater" title="Fan heater">Fan heater</a></li> <li><a href="/wiki/Fire_damper" title="Fire damper">Fire damper</a></li> <li><a href="/wiki/Fireplace" title="Fireplace">Fireplace</a></li> <li><a href="/wiki/Fireplace_insert" title="Fireplace insert">Fireplace insert</a></li> <li><a href="/wiki/Freeze_stat" title="Freeze stat">Freeze stat</a></li> <li><a href="/wiki/Flue" title="Flue">Flue</a></li> <li><a href="/wiki/Freon" title="Freon">Freon</a></li> <li><a href="/wiki/Fume_hood" title="Fume hood">Fume hood</a></li> <li><a href="/wiki/Furnace_(house_heating)" class="mw-redirect" title="Furnace (house heating)">Furnace</a></li> <li><a href="/wiki/Gas_compressor" class="mw-redirect" title="Gas compressor">Gas compressor</a></li> <li><a href="/wiki/Gas_heater" title="Gas heater">Gas heater</a></li> <li><a href="/wiki/Gasoline_heater" title="Gasoline heater">Gasoline heater</a></li> <li><a href="/wiki/Grease_duct" title="Grease duct">Grease duct</a></li> <li><a href="/wiki/Grille_(architecture)" title="Grille (architecture)">Grille</a></li> <li><a href="/wiki/Ground-coupled_heat_exchanger" title="Ground-coupled heat exchanger">Ground-coupled heat exchanger</a></li> <li><a href="/wiki/Ground_source_heat_pump" title="Ground source heat pump">Ground source heat pump</a></li> <li><a href="/wiki/Heat_exchanger" title="Heat exchanger">Heat exchanger</a></li> <li><a href="/wiki/Heat_pipe" title="Heat pipe">Heat pipe</a></li> <li><a href="/wiki/Heat_pump" title="Heat pump">Heat pump</a></li> <li><a href="/wiki/Heating_film" title="Heating film">Heating film</a></li> <li><a href="/wiki/Heating_system" title="Heating system">Heating system</a></li> <li><a href="/wiki/HEPA" title="HEPA">HEPA</a></li> <li><a href="/wiki/High_efficiency_glandless_circulating_pump" title="High efficiency glandless circulating pump">High efficiency glandless circulating pump</a></li> <li><a href="/wiki/High-pressure_cut-off_switch" class="mw-redirect" title="High-pressure cut-off switch">High-pressure cut-off switch</a></li> <li><a href="/wiki/Humidifier" title="Humidifier">Humidifier</a></li> <li><a href="/wiki/Infrared_heater" title="Infrared heater">Infrared heater</a></li> <li><a href="/wiki/Inverter_compressor" title="Inverter compressor">Inverter compressor</a></li> <li><a href="/wiki/Kerosene_heater" title="Kerosene heater">Kerosene heater</a></li> <li><a href="/wiki/Louver" title="Louver">Louver</a></li> <li><a href="/wiki/Mechanical_room" title="Mechanical room">Mechanical room</a></li> <li><a href="/wiki/Oil_heater" title="Oil heater">Oil heater</a></li> <li><a href="/wiki/Packaged_terminal_air_conditioner" title="Packaged terminal air conditioner">Packaged terminal air conditioner</a></li> <li><a href="/wiki/Plenum_space" title="Plenum space">Plenum space</a></li> <li><a href="/wiki/Pressurisation_ductwork" title="Pressurisation ductwork">Pressurisation ductwork</a></li> <li><a href="/wiki/Process_duct_work" title="Process duct work">Process duct work</a></li> <li><a href="/wiki/Radiator_(heating)" title="Radiator (heating)">Radiator</a></li> <li><a href="/wiki/Radiator_reflector" title="Radiator reflector">Radiator reflector</a></li> <li><a href="/wiki/Recuperator" title="Recuperator">Recuperator</a></li> <li><a href="/wiki/Refrigerant" title="Refrigerant">Refrigerant</a></li> <li><a href="/wiki/Register_(air_and_heating)" title="Register (air and heating)">Register</a></li> <li><a href="/wiki/Reversing_valve" title="Reversing valve">Reversing valve</a></li> <li><a href="/wiki/Run-around_coil" title="Run-around coil">Run-around coil</a></li> <li><a href="/wiki/Sail_switch" title="Sail switch">Sail switch</a></li> <li><a href="/wiki/Scroll_compressor" title="Scroll compressor">Scroll compressor</a></li> <li><a href="/wiki/Solar_chimney" title="Solar chimney">Solar chimney</a></li> <li><a href="/wiki/Solar-assisted_heat_pump" title="Solar-assisted heat pump">Solar-assisted heat pump</a></li> <li><a href="/wiki/Space_heater" title="Space heater">Space heater</a></li> <li><a href="/wiki/Smoke_canopy" title="Smoke canopy">Smoke canopy</a></li> <li><a href="/wiki/Smoke_damper" title="Smoke damper">Smoke damper</a></li> <li><a href="/wiki/Smoke_exhaust_ductwork" title="Smoke exhaust ductwork">Smoke exhaust ductwork</a></li> <li><a href="/wiki/Thermal_expansion_valve" title="Thermal expansion valve">Thermal expansion valve</a></li> <li><a href="/wiki/Thermal_wheel" title="Thermal wheel">Thermal wheel</a></li> <li><a href="/wiki/Thermostatic_radiator_valve" title="Thermostatic radiator valve">Thermostatic radiator valve</a></li> <li><a href="/wiki/Trickle_vent" title="Trickle vent">Trickle vent</a></li> <li><a href="/wiki/Trombe_wall" title="Trombe wall">Trombe wall</a></li> <li><a href="/wiki/TurboSwing" title="TurboSwing">TurboSwing</a></li> <li><a href="/wiki/Turning_vanes_(HVAC)" title="Turning vanes (HVAC)">Turning vanes</a></li> <li><a href="/wiki/Ultra-low_particulate_air" title="Ultra-low particulate air">Ultra-low particulate air</a> (ULPA)</li> <li><a href="/wiki/Whole-house_fan" title="Whole-house fan">Whole-house fan</a></li> <li><a href="/wiki/Windcatcher" title="Windcatcher">Windcatcher</a></li> <li><a href="/wiki/Wood-burning_stove" title="Wood-burning stove">Wood-burning stove</a></li> <li><a href="/wiki/Zone_valve" title="Zone valve">Zone valve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurement<br />and control</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Air_flow_meter" title="Air flow meter">Air flow meter</a></li> <li><a href="/wiki/Aquastat" title="Aquastat">Aquastat</a></li> <li><a href="/wiki/BACnet" title="BACnet">BACnet</a></li> <li><a href="/wiki/Blower_door" title="Blower door">Blower door</a></li> <li><a href="/wiki/Building_automation" title="Building automation">Building automation</a></li> <li><a href="/wiki/Carbon_dioxide_sensor" title="Carbon dioxide sensor">Carbon dioxide sensor</a></li> <li><a href="/wiki/Clean_air_delivery_rate" title="Clean air delivery rate">Clean air delivery rate</a> (CADR)</li> <li><a href="/wiki/Control_valve" title="Control valve">Control valve</a></li> <li><a href="/wiki/Gas_detector" title="Gas detector">Gas detector</a></li> <li><a href="/wiki/Home_energy_monitor" title="Home energy monitor">Home energy monitor</a></li> <li><a href="/wiki/Humidistat" title="Humidistat">Humidistat</a></li> <li><a href="/wiki/HVAC_control_system" title="HVAC control system">HVAC control system</a></li> <li><a href="/wiki/Infrared_thermometer" title="Infrared thermometer">Infrared thermometer</a></li> <li><a href="/wiki/Intelligent_buildings" class="mw-redirect" title="Intelligent buildings">Intelligent buildings</a></li> <li><a href="/wiki/LonWorks" title="LonWorks">LonWorks</a></li> <li><a href="/wiki/Minimum_efficiency_reporting_value" title="Minimum efficiency reporting value">Minimum efficiency reporting value</a> (MERV)</li> <li><a href="/wiki/Normal_temperature_and_pressure" class="mw-redirect" title="Normal temperature and pressure">Normal temperature and pressure</a> (NTP)</li> <li><a href="/wiki/OpenTherm" title="OpenTherm">OpenTherm</a></li> <li><a href="/wiki/Programmable_communicating_thermostat" title="Programmable communicating thermostat">Programmable communicating thermostat</a></li> <li><a href="/wiki/Programmable_thermostat" title="Programmable thermostat">Programmable thermostat</a></li> <li><a href="/wiki/Psychrometrics" title="Psychrometrics">Psychrometrics</a></li> <li><a href="/wiki/Room_temperature" title="Room temperature">Room temperature</a></li> <li><a href="/wiki/Smart_thermostat" title="Smart thermostat">Smart thermostat</a></li> <li><a href="/wiki/Standard_temperature_and_pressure" title="Standard temperature and pressure">Standard temperature and pressure</a> (STP)</li> <li><a href="/wiki/Thermographic_camera" class="mw-redirect" title="Thermographic camera">Thermographic camera</a></li> <li><a href="/wiki/Thermostat" title="Thermostat">Thermostat</a></li> <li><a href="/wiki/Thermostatic_radiator_valve" title="Thermostatic radiator valve">Thermostatic radiator valve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Professions,<br />trades,<br />and services</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Architectural_acoustics" title="Architectural acoustics">Architectural acoustics</a></li> <li><a href="/wiki/Architectural_engineering" title="Architectural engineering">Architectural engineering</a></li> <li><a href="/wiki/Architectural_technologist" title="Architectural technologist">Architectural technologist</a></li> <li><a href="/wiki/Building_services_engineering" title="Building services engineering">Building services engineering</a></li> <li><a href="/wiki/Building_information_modeling" title="Building information modeling">Building information modeling</a> (BIM)</li> <li><a href="/wiki/Deep_energy_retrofit" title="Deep energy retrofit">Deep energy retrofit</a></li> <li><a href="/wiki/Duct_cleaning" class="mw-redirect" title="Duct cleaning">Duct cleaning</a></li> <li><a href="/wiki/Duct_leakage_testing" title="Duct leakage testing">Duct leakage testing</a></li> <li><a href="/wiki/Environmental_engineering" title="Environmental engineering">Environmental engineering</a></li> <li><a href="/wiki/Hydronic_balancing" title="Hydronic balancing">Hydronic balancing</a></li> <li><a href="/wiki/Kitchen_exhaust_cleaning" title="Kitchen exhaust cleaning">Kitchen exhaust cleaning</a></li> <li><a href="/wiki/Mechanical_engineering" title="Mechanical engineering">Mechanical engineering</a></li> <li><a href="/wiki/Mechanical,_electrical,_and_plumbing" title="Mechanical, electrical, and plumbing">Mechanical, electrical, and plumbing</a></li> <li><a href="/wiki/Mold_growth,_assessment,_and_remediation" class="mw-redirect" title="Mold growth, assessment, and remediation">Mold growth, assessment, and remediation</a></li> <li><a href="/wiki/Refrigerant_reclamation" title="Refrigerant reclamation">Refrigerant reclamation</a></li> <li><a href="/wiki/Testing,_adjusting,_balancing" title="Testing, adjusting, balancing">Testing, adjusting, balancing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Industry <br />organizations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Air_Conditioning,_Heating_and_Refrigeration_Institute" title="Air Conditioning, Heating and Refrigeration Institute">AHRI</a></li> <li><a href="/wiki/Air_Movement_and_Control_Association" title="Air Movement and Control Association">AMCA</a></li> <li><a href="/wiki/ASHRAE" title="ASHRAE">ASHRAE</a></li> <li><a href="/wiki/ASTM_International" title="ASTM International">ASTM International</a></li> <li><a href="/wiki/Building_Research_Establishment" title="Building Research Establishment">BRE</a></li> <li><a href="/wiki/BSRIA" title="BSRIA">BSRIA</a></li> <li><a href="/wiki/Chartered_Institution_of_Building_Services_Engineers" title="Chartered Institution of Building Services Engineers">CIBSE</a></li> <li><a href="/wiki/Institute_of_Refrigeration" title="Institute of Refrigeration">Institute of Refrigeration</a></li> <li><a href="/wiki/International_Institute_of_Refrigeration" title="International Institute of Refrigeration">IIR</a></li> <li><a href="/wiki/Leadership_in_Energy_and_Environmental_Design" class="mw-redirect" title="Leadership in Energy and Environmental Design">LEED</a></li> <li><a href="/wiki/Sheet_Metal_and_Air_Conditioning_Contractors%27_National_Association" title="Sheet Metal and Air Conditioning Contractors' National Association">SMACNA</a></li> <li><a href="/wiki/Uniform_Mechanical_Code" title="Uniform Mechanical Code">UMC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Health and safety</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indoor_air_quality" title="Indoor air quality">Indoor air quality</a> (IAQ)</li> <li><a href="/wiki/Passive_smoking" title="Passive smoking">Passive smoking</a></li> <li><a href="/wiki/Sick_building_syndrome" title="Sick building syndrome">Sick building syndrome</a> (SBS)</li> <li><a href="/wiki/Volatile_organic_compound" title="Volatile organic compound">Volatile organic compound</a> (VOC)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;text-align: middle;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ASHRAE_Handbook" title="ASHRAE Handbook">ASHRAE Handbook</a></li> <li><a href="/wiki/Building_science" title="Building science">Building science</a></li> <li><a href="/wiki/Fireproofing" title="Fireproofing">Fireproofing</a></li> <li><a href="/wiki/Glossary_of_HVAC_terms" title="Glossary of HVAC terms">Glossary of HVAC terms</a></li> <li><a href="/wiki/Warm_Spaces" title="Warm Spaces">Warm Spaces</a></li> <li><a href="/wiki/World_Refrigeration_Day" title="World Refrigeration Day">World Refrigeration Day</a></li> <li><a href="/wiki/Template:Home_automation" title="Template:Home automation">Template:Home automation</a></li> <li><a href="/wiki/Template:Solar_energy" title="Template:Solar energy">Template:Solar energy</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px 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