Adiabatic process - Wikipedia

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class="vector-toc-list"> <li id="toc-Various_applications_of_the_adiabatic_assumption" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Various_applications_of_the_adiabatic_assumption"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Various applications of the adiabatic assumption</span> </div> </a> <ul id="toc-Various_applications_of_the_adiabatic_assumption-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Adiabatic_compression_and_expansion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Adiabatic_compression_and_expansion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Adiabatic compression and expansion</span> </div> </a> <ul id="toc-Adiabatic_compression_and_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideal_gas_(reversible_process)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ideal_gas_(reversible_process)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ideal gas (reversible process)</span> </div> </a> <button aria-controls="toc-Ideal_gas_(reversible_process)-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 Ideal gas (reversible process) subsection</span> </button> <ul id="toc-Ideal_gas_(reversible_process)-sublist" class="vector-toc-list"> <li id="toc-Example_of_adiabatic_compression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_of_adiabatic_compression"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example of adiabatic compression</span> </div> </a> <ul id="toc-Example_of_adiabatic_compression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adiabatic_free_expansion_of_a_gas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adiabatic_free_expansion_of_a_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Adiabatic free expansion of a gas</span> </div> </a> <ul id="toc-Adiabatic_free_expansion_of_a_gas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_P–V_relation_for_adiabatic_compression_and_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_P–V_relation_for_adiabatic_compression_and_expansion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Derivation of <i>P</i>–<i>V</i> relation for adiabatic compression and expansion</span> </div> </a> <ul id="toc-Derivation_of_P–V_relation_for_adiabatic_compression_and_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_discrete_formula_and_work_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_discrete_formula_and_work_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Derivation of discrete formula and work expression</span> </div> </a> <ul id="toc-Derivation_of_discrete_formula_and_work_expression-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graphing_adiabats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graphing_adiabats"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Graphing adiabats</span> </div> </a> <ul id="toc-Graphing_adiabats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Etymology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Etymology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Etymology</span> </div> </a> <ul id="toc-Etymology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conceptual_significance_in_thermodynamic_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conceptual_significance_in_thermodynamic_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Conceptual significance in thermodynamic theory</span> </div> </a> <ul id="toc-Conceptual_significance_in_thermodynamic_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Divergent_usages_of_the_word_adiabatic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Divergent_usages_of_the_word_adiabatic"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Divergent usages of the word <i>adiabatic</i></span> </div> </a> <ul id="toc-Divergent_usages_of_the_word_adiabatic-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">8</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">9</span> <span>References</span> </div> </a> <ul id="toc-References-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">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Adiabatic process</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 63 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-63" 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">63 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/Adiabaties" title="Adiabaties – Afrikaans" lang="af" hreflang="af" data-title="Adiabaties" 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/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_%D9%83%D8%B8%D9%88%D9%85%D8%A9" title="عملية كظومة – Arabic" lang="ar" hreflang="ar" data-title="عملية كظومة" 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/Procesu_adiab%C3%A1ticu" title="Procesu adiabáticu – Asturian" lang="ast" hreflang="ast" data-title="Procesu adiabáticu" 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/Adiabatik_proses" title="Adiabatik proses – Azerbaijani" lang="az" hreflang="az" data-title="Adiabatik proses" 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%B0%E0%A7%81%E0%A6%A6%E0%A7%8D%E0%A6%A7%E0%A6%A4%E0%A6%BE%E0%A6%AA%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE" title="রুদ্ধতাপীয় প্রক্রিয়া – Bangla" lang="bn" hreflang="bn" data-title="রুদ্ধতাপীয় প্রক্রিয়া" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D1%8B%D1%8F%D0%B1%D0%B0%D1%82%D0%BD%D1%8B_%D0%BF%D1%80%D0%B0%D1%86%D1%8D%D1%81" title="Адыябатны працэс – Belarusian" lang="be" hreflang="be" data-title="Адыябатны працэс" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B8%D0%B0%D0%B1%D0%B0%D1%82%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81" title="Адиабатен процес – Bulgarian" lang="bg" hreflang="bg" data-title="Адиабатен процес" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%91%E0%BE%B2%E0%BD%BC%E0%BD%91%E0%BC%8B%E0%BD%82%E0%BD%85%E0%BD%BC%E0%BD%91%E0%BC%8B%E0%BD%A2%E0%BE%92%E0%BE%B1%E0%BD%B4%E0%BD%91%E0%BC%8B%E0%BD%A2%E0%BD%B2%E0%BD%98%E0%BC%8D" title="དྲོད་གཅོད་རྒྱུད་རིམ། – Tibetan" lang="bo" hreflang="bo" data-title="དྲོད་གཅོད་རྒྱུད་རིམ།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Adijabatski_proces" title="Adijabatski proces – Bosnian" lang="bs" hreflang="bs" data-title="Adijabatski proces" 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/Proc%C3%A9s_adiab%C3%A0tic" title="Procés adiabàtic – Catalan" lang="ca" hreflang="ca" data-title="Procés adiabàtic" 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/Adiabatick%C3%BD_d%C4%9Bj" title="Adiabatický děj – Czech" lang="cs" hreflang="cs" data-title="Adiabatický děj" 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/Adiabatisk_proces" title="Adiabatisk proces – Danish" lang="da" hreflang="da" data-title="Adiabatisk proces" 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/Adiabatische_Zustands%C3%A4nderung" title="Adiabatische Zustandsänderung – German" lang="de" hreflang="de" data-title="Adiabatische Zustandsänderung" 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/Adiabaatiline_protsess" title="Adiabaatiline protsess – Estonian" lang="et" hreflang="et" data-title="Adiabaatiline protsess" 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%91%CE%B4%CE%B9%CE%B1%CE%B2%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%BC%CE%B5%CF%84%CE%B1%CE%B2%CE%BF%CE%BB%CE%AE" title="Αδιαβατική μεταβολή – Greek" lang="el" hreflang="el" data-title="Αδιαβατική μεταβολή" 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/Proceso_adiab%C3%A1tico" title="Proceso adiabático – Spanish" lang="es" hreflang="es" data-title="Proceso adiabático" 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/Adiabata_procezo" title="Adiabata procezo – Esperanto" lang="eo" hreflang="eo" data-title="Adiabata procezo" 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/Prozesu_adiabatiko" title="Prozesu adiabatiko – Basque" lang="eu" hreflang="eu" data-title="Prozesu adiabatiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%D8%A7%DB%8C%D9%86%D8%AF_%D8%A8%DB%8C%E2%80%8C%D8%AF%D8%B1%D8%B1%D9%88" title="فرایند بی‌دررو – Persian" lang="fa" hreflang="fa" data-title="فرایند بی‌دررو" 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/Processus_adiabatique" title="Processus adiabatique – French" lang="fr" hreflang="fr" data-title="Processus adiabatique" 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/Pr%C3%B3iseas_aidiabatach" title="Próiseas aidiabatach – Irish" lang="ga" hreflang="ga" data-title="Próiseas aidiabatach" 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/Proceso_adiab%C3%A1tico" title="Proceso adiabático – Galician" lang="gl" hreflang="gl" data-title="Proceso adiabático" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A8%EC%97%B4%EA%B3%BC%EC%A0%95" title="단열과정 – Korean" lang="ko" hreflang="ko" data-title="단열과정" 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%B1%D5%A4%D5%AB%D5%A1%D5%A2%D5%A1%D5%BF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A3%D5%B8%D6%80%D5%AE%D5%A8%D5%B6%D5%A9%D5%A1%D6%81" title="Ադիաբատական գործընթաց – Armenian" lang="hy" hreflang="hy" data-title="Ադիաբատական գործընթաց" 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%B0%E0%A5%81%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A5%8B%E0%A4%B7%E0%A5%8D%E0%A4%AE_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="रुद्धोष्म प्रक्रम – Hindi" lang="hi" hreflang="hi" data-title="रुद्धोष्म प्रक्रम" 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/Adijabatski_proces" title="Adijabatski proces – Croatian" lang="hr" hreflang="hr" data-title="Adijabatski proces" 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/Proses_adiabatik" title="Proses adiabatik – Indonesian" lang="id" hreflang="id" data-title="Proses adiabatik" 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/Trasformazione_adiabatica" title="Trasformazione adiabatica – Italian" lang="it" hreflang="it" data-title="Trasformazione adiabatica" 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%AA%D7%94%D7%9C%D7%99%D7%9A_%D7%90%D7%93%D7%99%D7%90%D7%91%D7%98%D7%99" title="תהליך אדיאבטי – Hebrew" lang="he" hreflang="he" data-title="תהליך אדיאבטי" 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%90%D0%B4%D0%B8%D0%B0%D0%B1%D0%B0%D1%82%D1%82%D1%8B%D2%9B_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%82%D0%B5%D1%80" title="Адиабаттық процестер – Kazakh" lang="kk" hreflang="kk" data-title="Адиабаттық процестер" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B4%D0%B8%D0%B0%D0%B1%D0%B0%D1%82%D0%B0_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81%D0%B8" title="Адиабата процесси – Kyrgyz" lang="ky" hreflang="ky" data-title="Адиабата процесси" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Adiab%C4%81tisks_process" title="Adiabātisks process – Latvian" lang="lv" hreflang="lv" data-title="Adiabātisks process" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Adiabatesche_Prozess" title="Adiabatesche Prozess – Luxembourgish" lang="lb" hreflang="lb" data-title="Adiabatesche Prozess" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Adiabatinis_procesas" title="Adiabatinis procesas – Lithuanian" lang="lt" hreflang="lt" data-title="Adiabatinis procesas" 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/Adiabatikus_%C3%A1llapotv%C3%A1ltoz%C3%A1s" title="Adiabatikus állapotváltozás – Hungarian" lang="hu" hreflang="hu" data-title="Adiabatikus állapotváltozás" 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%90%D0%B4%D0%B8%D1%98%D0%B0%D0%B1%D0%B0%D1%82%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81" title="Адијабатски процес – Macedonian" lang="mk" hreflang="mk" data-title="Адијабатски процес" 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%85%E0%B4%A1%E0%B4%AF%E0%B4%AC%E0%B4%BE%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B4%BF%E0%B4%95%E0%B5%8D_%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%AF" title="അഡയബാറ്റിക് പ്രക്രിയ – Malayalam" lang="ml" hreflang="ml" data-title="അഡയബാറ്റിക് പ്രക്രിയ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%B4%D0%B8%D0%B0%D0%B1%D0%B0%D1%82_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81" title="Адиабат процесс – Mongolian" lang="mn" hreflang="mn" data-title="Адиабат процесс" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Adiabatisch_proces" title="Adiabatisch proces – Dutch" lang="nl" hreflang="nl" data-title="Adiabatisch proces" 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/%E6%96%AD%E7%86%B1%E9%81%8E%E7%A8%8B" title="断熱過程 – Japanese" lang="ja" hreflang="ja" data-title="断熱過程" 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/Adiabatisk_prosess" title="Adiabatisk prosess – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Adiabatisk prosess" 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/Adiabatisk_prosess" title="Adiabatisk prosess – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Adiabatisk prosess" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Proc%C3%A8s_adiabatic" title="Procès adiabatic – Occitan" lang="oc" hreflang="oc" data-title="Procès adiabatic" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Adiabatik_jarayon" title="Adiabatik jarayon – Uzbek" lang="uz" hreflang="uz" data-title="Adiabatik jarayon" 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/Przemiana_adiabatyczna" title="Przemiana adiabatyczna – Polish" lang="pl" hreflang="pl" data-title="Przemiana adiabatyczna" 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/Sistema_adiab%C3%A1tico" title="Sistema adiabático – Portuguese" lang="pt" hreflang="pt" data-title="Sistema adiabático" 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/Proces_adiabatic" title="Proces adiabatic – Romanian" lang="ro" hreflang="ro" data-title="Proces adiabatic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%B4%D0%B8%D0%B0%D0%B1%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81" title="Адиабатический процесс – Russian" lang="ru" hreflang="ru" data-title="Адиабатический процесс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Adiabatic_process" title="Adiabatic process – Simple English" lang="en-simple" hreflang="en-simple" data-title="Adiabatic process" 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/Adiabatick%C3%BD_dej" title="Adiabatický dej – Slovak" lang="sk" hreflang="sk" data-title="Adiabatický dej" 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/Adiabatna_sprememba" title="Adiabatna sprememba – Slovenian" lang="sl" hreflang="sl" data-title="Adiabatna sprememba" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%B4%D0%B8%D1%98%D0%B0%D0%B1%D0%B0%D1%82%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81" title="Адијабатски процес – Serbian" lang="sr" hreflang="sr" data-title="Адијабатски процес" 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/Adijabatski_proces" title="Adijabatski proces – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Adijabatski proces" 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/Adiabaattinen_prosessi" title="Adiabaattinen prosessi – Finnish" lang="fi" hreflang="fi" data-title="Adiabaattinen prosessi" 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/Adiabatisk_process" title="Adiabatisk process – Swedish" lang="sv" hreflang="sv" data-title="Adiabatisk process" 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%AE%E0%AE%BE%E0%AE%B1%E0%AE%BE%E0%AE%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AF%86%E0%AE%AF%E0%AE%B2%E0%AF%8D%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88" title="மாறாவெப்பச் செயல்முறை – Tamil" lang="ta" hreflang="ta" data-title="மாறாவெப்பச் செயல்முறை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%A7%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AD%E0%B8%B0%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2%E0%B9%81%E0%B8%9A%E0%B8%95%E0%B8%B4%E0%B8%81" title="กระบวนการอะเดียแบติก – Thai" lang="th" hreflang="th" data-title="กระบวนการอะเดียแบติก" 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/Adyabatik" title="Adyabatik – Turkish" lang="tr" hreflang="tr" data-title="Adyabatik" 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%90%D0%B4%D1%96%D0%B0%D0%B1%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81" title="Адіабатичний процес – Ukrainian" lang="uk" hreflang="uk" data-title="Адіабатичний процес" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Qu%C3%A1_tr%C3%ACnh_%C4%91o%E1%BA%A1n_nhi%E1%BB%87t" title="Quá trình đoạn nhiệt – Vietnamese" lang="vi" hreflang="vi" data-title="Quá trình đoạn nhiệt" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BB%9D%E7%83%AD%E8%BF%87%E7%A8%8B" title="绝热过程 – Wu" lang="wuu" hreflang="wuu" data-title="绝热过程" 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%B5%95%E7%86%B1%E8%B3%AA%E9%81%8E%E7%A8%8B" title="絕熱質過程 – Cantonese" lang="yue" hreflang="yue" data-title="絕熱質過程" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li 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<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">Thermodynamic process in which no mass or heat is exchanged with surroundings</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">This article is about adiabatic processes in thermodynamics. 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class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium 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class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a class="mw-selflink selflink">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Other</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>An <b>adiabatic process</b> (<i>adiabatic</i> from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BC%80%CE%B4%CE%B9%CE%AC%CE%B2%CE%B1%CF%84%CE%BF%CF%82#Ancient_Greek" class="extiw" title="wikt:ἀδιάβατος">ἀδιάβατος</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>adiábatos</i></span>)</i> 'impassable') is a type of <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">thermodynamic process</a> that occurs without transferring <a href="/wiki/Heat" title="Heat">heat</a> between the <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a> and its <a href="/wiki/Environment_(systems)" class="mw-redirect" title="Environment (systems)">environment</a>. Unlike an <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal process</a>, an adiabatic process transfers energy to the surroundings only as <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> and/or mass flow.<sup id="cite_ref-Carathéodory_1-0" class="reference"><a href="#cite_note-Carathéodory-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bailyn_21_2-0" class="reference"><a href="#cite_note-Bailyn_21-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> As a key concept in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, the adiabatic process supports the theory that explains the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a>. The opposite term to "adiabatic" is <i>diabatic</i>. </p><p>Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation".<sup id="cite_ref-Bailyn_53_3-0" class="reference"><a href="#cite_note-Bailyn_53-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For example, the <a href="/wiki/Adiabatic_flame_temperature" title="Adiabatic flame temperature">adiabatic flame temperature</a> uses this approximation to calculate the upper limit of <a href="/wiki/Fire" title="Fire">flame</a> temperature by assuming combustion loses no heat to its surroundings. </p><p>In <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, adiabatic expansion and cooling of moist air, which can be triggered by winds flowing up and over a mountain for example, can cause the <a href="/wiki/Water_vapor" title="Water vapor">water vapor</a> pressure to exceed the <a href="/wiki/Vapor_pressure" title="Vapor pressure">saturation vapor pressure</a>. Expansion and cooling beyond the saturation vapor pressure is often idealized as a <i>pseudo-adiabatic process</i> whereby excess vapor instantly <a href="/wiki/Precipitation" title="Precipitation">precipitates</a> into water droplets. The change in temperature of an air undergoing pseudo-adiabatic expansion differs from air undergoing adiabatic expansion because <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a> is released by precipitation.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A process without transfer of heat to or from a system, so that <span class="texhtml"><i>Q</i> = 0</span>, is called adiabatic, and such a system is said to be adiabatically isolated.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The simplifying assumption frequently made is that a process is adiabatic. For example, the compression of a gas within a cylinder of an engine is assumed to occur so rapidly that on the time scale of the compression process, little of the system's energy can be transferred out as heat to the surroundings. Even though the cylinders are not insulated and are quite conductive, that process is idealized to be adiabatic. The same can be said to be true for the expansion process of such a system. </p><p>The assumption of adiabatic isolation is useful and often combined with other such idealizations to calculate a good first approximation of a system's behaviour. For example, according to <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a>, when sound travels in a gas, there is no time for heat conduction in the medium, and so the propagation of sound is adiabatic. For such an adiabatic process, the <a href="/wiki/Elastic_modulus" title="Elastic modulus">modulus of elasticity</a> (<a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>) can be expressed as <span class="texhtml"><i>E</i> = <i>γP</i></span>, where <span class="texhtml"><i>γ</i></span> is the <a href="/wiki/Heat_capacity_ratio" title="Heat capacity ratio">ratio of specific heats</a> at constant pressure and at <a href="/wiki/Constant_volume" class="mw-redirect" title="Constant volume">constant volume</a> (<span class="texhtml"><i>γ</i> = <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"><i>C<sub>p</sub></i></span><span class="sr-only">/</span><span class="den"><i>C<sub>v</sub></i></span></span>⁠</span></span>) and <span class="texhtml"><i>P</i></span> is the pressure of the gas. </p> <div class="mw-heading mw-heading3"><h3 id="Various_applications_of_the_adiabatic_assumption">Various applications of the adiabatic assumption</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=2" title="Edit section: Various applications of the adiabatic assumption"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a closed system, one may write the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> as <span class="texhtml">Δ<i>U</i> = <i>Q</i> − <i>W</i></span>, where <span class="texhtml">Δ<i>U</i></span> denotes the change of the system's internal energy, <span class="texhtml"><i>Q</i></span> the quantity of energy added to it as heat, and <span class="texhtml"><i>W</i></span> the work done by the system on its surroundings. </p> <ul><li>If the system has such rigid walls that work cannot be transferred in or out (<span class="texhtml"><i>W</i> = 0</span>), and the walls are not adiabatic and energy is added in the form of heat (<span class="texhtml"><i>Q</i> > 0</span>), and there is no phase change, then the temperature of the system will rise.</li> <li>If the system has such rigid walls that pressure–volume work cannot be done, but the walls are adiabatic (<span class="texhtml"><i>Q</i> = 0</span>), and energy is added as <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric</a> (constant volume) work in the form of friction or the stirring of a <a href="/wiki/Viscous" class="mw-redirect" title="Viscous">viscous</a> fluid within the system (<span class="texhtml"><i>W</i> < 0</span>), and there is no phase change, then the temperature of the system will rise.</li> <li>If the system walls are adiabatic (<span class="texhtml"><i>Q</i> = 0</span>) but not rigid (<span class="texhtml"><i>W</i> ≠ 0</span>), and, in a fictive idealized process, energy is added to the system in the form of frictionless, non-viscous pressure–volume work (<span class="texhtml"><i>W</i> < 0</span>), and there is no phase change, then the temperature of the system will rise. Such a process is called an <a href="/wiki/Isentropic_process" title="Isentropic process">isentropic process</a> and is said to be "reversible". Ideally, if the process were reversed the energy could be recovered entirely as work done by the system. If the system contains a compressible gas and is reduced in volume, the uncertainty of the position of the gas is reduced, and seemingly would reduce the entropy of the system, but the temperature of the system will rise as the process is isentropic (<span class="texhtml">Δ<i>S</i> = 0</span>). Should the work be added in such a way that friction or viscous forces are operating within the system, then the process is not isentropic, and if there is no phase change, then the temperature of the system will rise, the process is said to be "irreversible", and the work added to the system is not entirely recoverable in the form of work.</li> <li>If the walls of a system are not adiabatic, and energy is transferred in as heat, entropy is transferred into the system with the heat. Such a process is neither adiabatic nor isentropic, having <span class="texhtml"><i>Q</i> > 0</span>, and <span class="texhtml">Δ<i>S</i> > 0</span> according to the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>.</li></ul> <p>Naturally occurring adiabatic processes are irreversible (entropy is produced). </p><p>The transfer of energy as work into an adiabatically isolated system can be imagined as being of two idealized extreme kinds. In one such kind, no entropy is produced within the system (no friction, viscous dissipation, etc.), and the work is only pressure-volume work (denoted by <span class="texhtml"><i>P</i> d<i>V</i></span>). In nature, this ideal kind occurs only approximately because it demands an infinitely slow process and no sources of dissipation. </p><p>The other extreme kind of work is <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric</a> work (<span class="texhtml">d<i>V</i> = 0</span>), for which energy is added as work solely through friction or viscous dissipation within the system. A stirrer that transfers energy to a viscous fluid of an adiabatically isolated system with rigid walls, without phase change, will cause a rise in temperature of the fluid, but that work is not recoverable. Isochoric work is irreversible.<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> The second law of thermodynamics observes that a natural process, of transfer of energy as work, always consists at least of isochoric work and often both of these extreme kinds of work. Every natural process, adiabatic or not, is irreversible, with <span class="texhtml">Δ<i>S</i> > 0</span>, as friction or viscosity are always present to some extent. </p> <div class="mw-heading mw-heading2"><h2 id="Adiabatic_compression_and_expansion">Adiabatic compression and expansion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=3" title="Edit section: Adiabatic compression and expansion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The adiabatic compression of a gas causes a rise in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. In contrast, <a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">free expansion</a> is an <a href="/wiki/Isothermal" class="mw-redirect" title="Isothermal">isothermal</a> process for an ideal gas. </p><p><b>Adiabatic compression</b> occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a <a href="/wiki/Piston" title="Piston">piston</a> compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. This finds practical application in <a href="/wiki/Diesel_engines" class="mw-redirect" title="Diesel engines">diesel engines</a> which rely on the lack of heat dissipation during the compression stroke to elevate the fuel vapor temperature sufficiently to ignite it. </p><p>Adiabatic compression occurs in the <a href="/wiki/Earth%27s_atmosphere" class="mw-redirect" title="Earth's atmosphere">Earth's atmosphere</a> when an <a href="/wiki/Air_mass" title="Air mass">air mass</a> descends, for example, in a <a href="/wiki/Katabatic_wind" title="Katabatic wind">Katabatic wind</a>, <a href="/wiki/Foehn_wind" title="Foehn wind">Foehn wind</a>, or <a href="/wiki/Chinook_wind" title="Chinook wind">Chinook wind</a> flowing downhill over a mountain range. When a parcel of air descends, the pressure on the parcel increases. Because of this increase in pressure, the parcel's volume decreases and its temperature increases as work is done on the parcel of air, thus increasing its internal energy, which manifests itself by a rise in the temperature of that mass of air. The parcel of air can only slowly dissipate the energy by conduction or radiation (heat), and to a first approximation it can be considered adiabatically isolated and the process an adiabatic process. </p><p><b>Adiabatic expansion</b> occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand in size, thus causing it to do work on its surroundings. When the pressure applied on a parcel of gas is reduced, the gas in the parcel is allowed to expand; as the volume increases, the temperature falls as its internal energy decreases. Adiabatic expansion occurs in the Earth's atmosphere with <a href="/wiki/Orographic_lifting" class="mw-redirect" title="Orographic lifting">orographic lifting</a> and <a href="/wiki/Lee_waves" class="mw-redirect" title="Lee waves">lee waves</a>, and this can form <a href="/wiki/Pileus_(meteorology)" title="Pileus (meteorology)">pilei</a> or <a href="/wiki/Lenticular_cloud" title="Lenticular cloud">lenticular clouds</a>. </p><p>Due in part to adiabatic expansion in mountainous areas, snowfall infrequently occurs in some parts of the <a href="/wiki/Sahara_desert" class="mw-redirect" title="Sahara desert">Sahara desert</a>.<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>Adiabatic expansion does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is via <a href="/wiki/Adiabatic_demagnetization" class="mw-redirect" title="Adiabatic demagnetization">adiabatic demagnetisation</a>, where the change in <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> on a magnetic material is used to provide adiabatic expansion. Also, the contents of an <a href="/wiki/Expanding_universe" class="mw-redirect" title="Expanding universe">expanding universe</a> can be described (to first order) as an adiabatically expanding fluid. (See <a href="/wiki/Heat_death_of_the_universe" title="Heat death of the universe">heat death of the universe</a>.) </p><p>Rising magma also undergoes adiabatic expansion before eruption, particularly significant in the case of magmas that rise quickly from great depths such as <a href="/wiki/Kimberlite" title="Kimberlite">kimberlites</a>.<sup id="cite_ref-Kavanagh_9-0" class="reference"><a href="#cite_note-Kavanagh-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In the Earth's convecting mantle (the asthenosphere) beneath the <a href="/wiki/Lithosphere" title="Lithosphere">lithosphere</a>, the mantle temperature is approximately an adiabat. The slight decrease in temperature with shallowing depth is due to the decrease in pressure the shallower the material is in the Earth.<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>Such temperature changes can be quantified using the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a>, or the <a href="/wiki/Hydrostatic_equation" class="mw-redirect" title="Hydrostatic equation">hydrostatic equation</a> for atmospheric processes. </p><p>In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist. </p> <div class="mw-heading mw-heading2"><h2 id="Ideal_gas_(reversible_process)"><span id="Ideal_gas_.28reversible_process.29"></span>Ideal gas (reversible process)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=4" title="Edit section: Ideal gas (reversible process)"><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/Reversible_adiabatic_process" class="mw-redirect" title="Reversible adiabatic process">Reversible adiabatic process</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Adiabatic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Adiabatic.svg/260px-Adiabatic.svg.png" decoding="async" width="260" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Adiabatic.svg/390px-Adiabatic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Adiabatic.svg/520px-Adiabatic.svg.png 2x" data-file-width="301" data-file-height="259" /></a><figcaption>For a simple substance, during an adiabatic process in which the volume increases, the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> of the working substance must decrease.</figcaption></figure> <p>The mathematical equation for an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the <a href="/wiki/Polytropic_process" title="Polytropic process">polytropic process</a> equation<sup id="cite_ref-Bailyn_53_3-1" class="reference"><a href="#cite_note-Bailyn_53-3"><span class="cite-bracket">[</span>3<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 PV^{\gamma }={\text{constant}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV^{\gamma }={\text{constant}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7689e87b98497792f98a0a628611afb997e158a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.2ex; height:2.676ex;" alt="{\displaystyle PV^{\gamma }={\text{constant}},}"></span> </p><p>where <span class="texhtml"><i>P</i></span> is pressure, <span class="texhtml"><i>V</i></span> is volume, and <span class="texhtml"><i>γ</i></span> is the <a href="/wiki/Adiabatic_index" class="mw-redirect" title="Adiabatic index">adiabatic index</a> or heat capacity ratio defined 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 \gamma ={\frac {C_{P}}{C_{V}}}={\frac {f+2}{f}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>f</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {f+2}{f}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15ec38d29ac99b8a1c02f87cc58aeae512a1b75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.218ex; height:5.843ex;" alt="{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {f+2}{f}}.}"></span> </p><p>Here <span class="texhtml"><i>C<sub>P</sub></i></span> is the <a href="/wiki/Specific_heat" class="mw-redirect" title="Specific heat">specific heat</a> for constant pressure, <span class="texhtml"><i>C<sub>V</sub></i></span> is the specific heat for constant volume, and <span class="texhtml"><i>f</i></span> is the number of <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> (3 for a monatomic gas, 5 for a diatomic gas or a gas of linear molecules such as carbon dioxide). </p><p>For a monatomic ideal gas, <span class="texhtml"><i>γ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span></span>, and for a diatomic gas (such as <a href="/wiki/Nitrogen" title="Nitrogen">nitrogen</a> and <a href="/wiki/Oxygen" title="Oxygen">oxygen</a>, the main components of air), <span class="texhtml"><i>γ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Note that the above formula is only applicable to classical ideal gases (that is, gases far above absolute zero temperature) and not <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein</a> or <a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermi gases</a>. </p><p>One can also use the ideal gas law to rewrite the above relationship between <span class="texhtml"><i>P</i></span> and <span class="texhtml"><i>V</i></span> as <sup id="cite_ref-Bailyn_53_3-2" class="reference"><a href="#cite_note-Bailyn_53-3"><span class="cite-bracket">[</span>3<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 {\begin{aligned}P^{1-\gamma }T^{\gamma }&={\text{constant}},\\TV^{\gamma -1}&={\text{constant}}\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> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P^{1-\gamma }T^{\gamma }&={\text{constant}},\\TV^{\gamma -1}&={\text{constant}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c35648038a1b4bb2fe37fa8c7e8e8835e19b1fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.056ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}P^{1-\gamma }T^{\gamma }&={\text{constant}},\\TV^{\gamma -1}&={\text{constant}}\end{aligned}}}"></span> </p><p>where <i>T</i> is the absolute or <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Example_of_adiabatic_compression">Example of adiabatic compression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=5" title="Edit section: Example of adiabatic compression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The compression stroke in a <a href="/wiki/Gasoline_engine" class="mw-redirect" title="Gasoline engine">gasoline engine</a> can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm<sup>3</sup> = 0.001 m<sup>3</sup>); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so <span class="texhtml"><i>γ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span></span>); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure). </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P_{1}V_{1}^{\gamma }&=\mathrm {constant} _{1}\\&=100\,000~{\text{Pa}}\times (0.001~{\text{m}}^{3})^{\frac {7}{5}}\\&=10^{5}\times 6.31\times 10^{-5}~{\text{Pa}}\,{\text{m}}^{21/5}\\&=6.31~{\text{Pa}}\,{\text{m}}^{21/5},\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> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>100</mn> <mspace width="thinmathspace" /> <mn>000</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0.001</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>×</mo> <mn>6.31</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>6.31</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{1}V_{1}^{\gamma }&=\mathrm {constant} _{1}\\&=100\,000~{\text{Pa}}\times (0.001~{\text{m}}^{3})^{\frac {7}{5}}\\&=10^{5}\times 6.31\times 10^{-5}~{\text{Pa}}\,{\text{m}}^{21/5}\\&=6.31~{\text{Pa}}\,{\text{m}}^{21/5},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c520d222e78191d65493bdcf1f596904b5cdb3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:36.459ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}P_{1}V_{1}^{\gamma }&=\mathrm {constant} _{1}\\&=100\,000~{\text{Pa}}\times (0.001~{\text{m}}^{3})^{\frac {7}{5}}\\&=10^{5}\times 6.31\times 10^{-5}~{\text{Pa}}\,{\text{m}}^{21/5}\\&=6.31~{\text{Pa}}\,{\text{m}}^{21/5},\end{aligned}}}"></span> </p><p>so the adiabatic constant for this example is about 6.31 Pa m<sup>4.2</sup>. </p><p>The gas is now compressed to a 0.1 L (0.0001 m<sup>3</sup>) volume, which we assume happens quickly enough that no heat enters or leaves the gas through the walls. The adiabatic constant remains the same, but with the resulting pressure unknown </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P_{2}V_{2}^{\gamma }&=\mathrm {constant} _{1}\\&=6.31~{\text{Pa}}\,{\text{m}}^{21/5}\\&=P\times (0.0001~{\text{m}}^{3})^{\frac {7}{5}},\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> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>6.31</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0.0001</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{2}V_{2}^{\gamma }&=\mathrm {constant} _{1}\\&=6.31~{\text{Pa}}\,{\text{m}}^{21/5}\\&=P\times (0.0001~{\text{m}}^{3})^{\frac {7}{5}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3085bbcb9d7e20a058461ece2eac3acd382295e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:28.246ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}P_{2}V_{2}^{\gamma }&=\mathrm {constant} _{1}\\&=6.31~{\text{Pa}}\,{\text{m}}^{21/5}\\&=P\times (0.0001~{\text{m}}^{3})^{\frac {7}{5}},\end{aligned}}}"></span> </p><p>We can now solve for the final pressure<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P_{2}&=P_{1}\left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }\\&=100\,000~{\text{Pa}}\times {\text{10}}^{7/5}\\&=2.51\times 10^{6}~{\text{Pa}}\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> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>100</mn> <mspace width="thinmathspace" /> <mn>000</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>10</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2.51</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{2}&=P_{1}\left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }\\&=100\,000~{\text{Pa}}\times {\text{10}}^{7/5}\\&=2.51\times 10^{6}~{\text{Pa}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5858b2c648338bb77f48fd77ac214ab781be3af8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:24.948ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}P_{2}&=P_{1}\left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }\\&=100\,000~{\text{Pa}}\times {\text{10}}^{7/5}\\&=2.51\times 10^{6}~{\text{Pa}}\end{aligned}}}"></span> </p><p>or 25.1 bar. This pressure increase is more than a simple 10:1 compression ratio would indicate; this is because the gas is not only compressed, but the work done to compress the gas also increases its internal energy, which manifests itself by a rise in the gas temperature and an additional rise in pressure above what would result from a simplistic calculation of 10 times the original pressure. </p><p>We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law, <i>PV</i> = <i>nRT</i> (<i>n</i> is amount of gas in moles and <i>R</i> the gas constant for that gas). Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant (<i>nR</i>) is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {PV}{T}}&=\mathrm {constant} _{2}\\&={\frac {10^{5}~{\text{Pa}}\times 10^{-3}~{\text{m}}^{3}}{300~{\text{K}}}}\\&=0.333~{\text{Pa}}\,{\text{m}}^{3}{\text{K}}^{-1}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>V</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>300</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>0.333</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {PV}{T}}&=\mathrm {constant} _{2}\\&={\frac {10^{5}~{\text{Pa}}\times 10^{-3}~{\text{m}}^{3}}{300~{\text{K}}}}\\&=0.333~{\text{Pa}}\,{\text{m}}^{3}{\text{K}}^{-1}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1ef78467fc87d756d89bb20725532705d5a0f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:26.829ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {PV}{T}}&=\mathrm {constant} _{2}\\&={\frac {10^{5}~{\text{Pa}}\times 10^{-3}~{\text{m}}^{3}}{300~{\text{K}}}}\\&=0.333~{\text{Pa}}\,{\text{m}}^{3}{\text{K}}^{-1}.\end{aligned}}}"></span> </p><p>We know the compressed gas has <span class="texhtml mvar" style="font-style:italic;">V</span> = 0.1 L and <span class="texhtml mvar" style="font-style:italic;">P</span> = <span class="nowrap"><span data-sort-value="7006251000000000000♠"></span>2.51<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>6</sup> Pa</span>, so we can solve for temperature: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T&={\frac {PV}{\mathrm {constant} _{2}}}\\&={\frac {2.51\times 10^{6}~{\text{Pa}}\times 10^{-4}~{\text{m}}^{3}}{0.333~{\text{Pa}}\,{\text{m}}^{3}{\text{K}}^{-1}}}\\&=753~{\text{K}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>T</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>V</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2.51</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>0.333</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Pa</mtext> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>753</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T&={\frac {PV}{\mathrm {constant} _{2}}}\\&={\frac {2.51\times 10^{6}~{\text{Pa}}\times 10^{-4}~{\text{m}}^{3}}{0.333~{\text{Pa}}\,{\text{m}}^{3}{\text{K}}^{-1}}}\\&=753~{\text{K}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/043e44539245ba28178b77c49987362f0af1c70d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:31.071ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}T&={\frac {PV}{\mathrm {constant} _{2}}}\\&={\frac {2.51\times 10^{6}~{\text{Pa}}\times 10^{-4}~{\text{m}}^{3}}{0.333~{\text{Pa}}\,{\text{m}}^{3}{\text{K}}^{-1}}}\\&=753~{\text{K}}.\end{aligned}}}"></span> </p><p>That is a final temperature of 753 K, or 479 °C, or 896 °F, well above the ignition point of many fuels. This is why a high-compression engine requires fuels specially formulated to not self-ignite (which would cause <a href="/wiki/Engine_knocking" title="Engine knocking">engine knocking</a> when operated under these conditions of temperature and pressure), or that a <a href="/wiki/Supercharger" title="Supercharger">supercharger</a> with an <a href="/wiki/Intercooler" title="Intercooler">intercooler</a> to provide a pressure boost but with a lower temperature rise would be advantageous. A diesel engine operates under even more extreme conditions, with compression ratios of 16:1 or more being typical, in order to provide a very high gas pressure, which ensures immediate ignition of the injected fuel. </p> <div class="mw-heading mw-heading3"><h3 id="Adiabatic_free_expansion_of_a_gas">Adiabatic free expansion of a gas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=6" title="Edit section: Adiabatic free expansion of a gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></div> <p>For an adiabatic free expansion of an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a>, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the first law of thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_P–V_relation_for_adiabatic_compression_and_expansion"><span id="Derivation_of_P.E2.80.93V_relation_for_adiabatic_compression_and_expansion"></span>Derivation of <i>P</i>–<i>V</i> relation for adiabatic compression and expansion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=7" title="Edit section: Derivation of P–V relation for adiabatic compression and expansion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definition of an adiabatic process is that heat transfer to the system is zero, <span class="texhtml"><i>δQ</i> = 0</span>. Then, according to the first law of thermodynamics, </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU+\delta W=\delta Q=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>+</mo> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU+\delta W=\delta Q=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa56200384a8e6a771a692fc72ab0ad506b32a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.216ex; height:2.676ex;" alt="{\displaystyle dU+\delta W=\delta Q=0,}"></span></td> <td></td> <td class="nowrap"><span id="math_a1" class="reference nourlexpansion" style="font-weight:bold;">a1</span></td></tr></tbody></table> <p>where <span class="texhtml"><i>dU</i></span> is the change in the internal energy of the system and <span class="texhtml"><i>δW</i></span> is work done <i>by</i> the system. Any work (<span class="texhtml"><i>δW</i></span>) done must be done at the expense of internal energy <span class="texhtml"><i>U</i></span>, since no heat <span class="texhtml"><i>δQ</i></span> is being supplied from the surroundings. Pressure–volume work <span class="texhtml"><i>δW</i></span> done <i>by</i> the system is defined as </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=P\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=P\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1472cad4087edb556f54854771e08921e9809b3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.365ex; height:2.343ex;" alt="{\displaystyle \delta W=P\,dV.}"></span></td> <td></td> <td class="nowrap"><span id="math_a2" class="reference nourlexpansion" style="font-weight:bold;">a2</span></td></tr></tbody></table> <p>However, <span class="texhtml"><i>P</i></span> does not remain constant during an adiabatic process but instead changes along with <span class="texhtml"><i>V</i></span>. </p><p>It is desired to know how the values of <span class="texhtml"><i>dP</i></span> and <span class="texhtml"><i>dV</i></span> relate to each other as the adiabatic process proceeds. For an ideal gas (recall ideal gas law <span class="texhtml"><i>PV</i> = <i>nRT</i></span>) the internal energy is given by </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\alpha nRT=\alpha PV,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <mi>T</mi> <mo>=</mo> <mi>α</mi> <mi>P</mi> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\alpha nRT=\alpha PV,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6362425d1763182a68ea58f95e65629584e5e001" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.929ex; height:2.509ex;" alt="{\displaystyle U=\alpha nRT=\alpha PV,}"></span></td> <td></td> <td class="nowrap"><span id="math_a3" class="reference nourlexpansion" style="font-weight:bold;">a3</span></td></tr></tbody></table> <p>where <span class="texhtml"><i>α</i></span> is the number of degrees of freedom divided by 2, <span class="texhtml"><i>R</i></span> is the <a href="/wiki/Universal_gas_constant" class="mw-redirect" title="Universal gas constant">universal gas constant</a> and <span class="texhtml"><i>n</i></span> is the number of moles in the system (a constant). </p><p>Differentiating equation (a3) yields </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}dU&=\alpha nR\,dT\\&=\alpha \,d(PV)\\&=\alpha (P\,dV+V\,dP).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>+</mo> <mi>V</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}dU&=\alpha nR\,dT\\&=\alpha \,d(PV)\\&=\alpha (P\,dV+V\,dP).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc7595168ab63ad9e31f37d86569cbf5e43da9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.904ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}dU&=\alpha nR\,dT\\&=\alpha \,d(PV)\\&=\alpha (P\,dV+V\,dP).\end{aligned}}}"></span></td> <td></td> <td class="nowrap"><span id="math_a4" class="reference nourlexpansion" style="font-weight:bold;">a4</span></td></tr></tbody></table> <p>Equation (a4) is often expressed as <span class="texhtml"><i>dU</i> = <i>nC<sub>V</sub> dT</i></span> because <span class="texhtml"><i>C<sub>V</sub></i> = <i>αR</i></span>. </p><p>Now substitute equations (a2) and (a4) into equation (a1) to obtain </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\,dV=\alpha P\,dV+\alpha V\,dP,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>=</mo> <mi>α</mi> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>+</mo> <mi>α</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -P\,dV=\alpha P\,dV+\alpha V\,dP,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cded6b019f199748b899e13ffe9cf52268fcab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.776ex; height:2.509ex;" alt="{\displaystyle -P\,dV=\alpha P\,dV+\alpha V\,dP,}"></span> </p><p>factorize <span class="texhtml">−<i>P</i> <i>dV</i></span>: </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 +1)P\,dV=\alpha V\,dP,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>=</mo> <mi>α</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(\alpha +1)P\,dV=\alpha V\,dP,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd0c28989fdc69f239e5091a32865df6f225645" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.612ex; height:2.843ex;" alt="{\displaystyle -(\alpha +1)P\,dV=\alpha V\,dP,}"></span> </p><p>and divide both sides by <span class="texhtml"><i>PV</i></span>: </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 +1){\frac {dV}{V}}=\alpha {\frac {dP}{P}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(\alpha +1){\frac {dV}{V}}=\alpha {\frac {dP}{P}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06dcb2b53728d0716a0fa7406cc5aff244b64a05" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.977ex; height:5.509ex;" alt="{\displaystyle -(\alpha +1){\frac {dV}{V}}=\alpha {\frac {dP}{P}}.}"></span> </p><p>After integrating the left and right sides from <span class="texhtml"><i>V</i><sub>0</sub></span> to <span class="texhtml"><i>V</i></span> and from <span class="texhtml"><i>P</i><sub>0</sub></span> to <span class="texhtml"><i>P</i></span> and changing the sides respectively, </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 \ln \left({\frac {P}{P_{0}}}\right)=-{\frac {\alpha +1}{\alpha }}\ln \left({\frac {V}{V_{0}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>α</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left({\frac {P}{P_{0}}}\right)=-{\frac {\alpha +1}{\alpha }}\ln \left({\frac {V}{V_{0}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba73f306f1b412ec092e48db9a05ce34d6a7ec3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.616ex; height:6.176ex;" alt="{\displaystyle \ln \left({\frac {P}{P_{0}}}\right)=-{\frac {\alpha +1}{\alpha }}\ln \left({\frac {V}{V_{0}}}\right).}"></span> </p><p>Exponentiate both sides, substitute <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>α</i> + 1</span><span class="sr-only">/</span><span class="den"><i>α</i></span></span>⁠</span></span> with <span class="texhtml"><i>γ</i></span>, the heat capacity ratio </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {P}{P_{0}}}\right)=\left({\frac {V}{V_{0}}}\right)^{-\gamma },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {P}{P_{0}}}\right)=\left({\frac {V}{V_{0}}}\right)^{-\gamma },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b04856485c6c51943bad66a39038ffa3f73385" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.619ex; height:6.509ex;" alt="{\displaystyle \left({\frac {P}{P_{0}}}\right)=\left({\frac {V}{V_{0}}}\right)^{-\gamma },}"></span> </p><p>and eliminate the negative sign to obtain </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {P}{P_{0}}}\right)=\left({\frac {V_{0}}{V}}\right)^{\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {P}{P_{0}}}\right)=\left({\frac {V_{0}}{V}}\right)^{\gamma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b48940733fe56e04a8b2512a427a2672af5c4ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.341ex; height:6.176ex;" alt="{\displaystyle \left({\frac {P}{P_{0}}}\right)=\left({\frac {V_{0}}{V}}\right)^{\gamma }.}"></span> </p><p>Therefore, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {P}{P_{0}}}\right)\left({\frac {V}{V_{0}}}\right)^{\gamma }=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {P}{P_{0}}}\right)\left({\frac {V}{V_{0}}}\right)^{\gamma }=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96f46338e52e2740fe572b9bf27ee2a906ded2ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.89ex; height:6.176ex;" alt="{\displaystyle \left({\frac {P}{P_{0}}}\right)\left({\frac {V}{V_{0}}}\right)^{\gamma }=1,}"></span> </p><p>and </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_{0}V_{0}^{\gamma }=PV^{\gamma }=\mathrm {constant} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mo>=</mo> <mi>P</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}V_{0}^{\gamma }=PV^{\gamma }=\mathrm {constant} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffc9f770ce6d9f3deea83bddfb42d390fa5b43a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.887ex; height:3.176ex;" alt="{\displaystyle P_{0}V_{0}^{\gamma }=PV^{\gamma }=\mathrm {constant} .}"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U=\alpha RnT_{2}-\alpha RnT_{1}=\alpha Rn\Delta T.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <mi>α</mi> <mi>R</mi> <mi>n</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>α</mi> <mi>R</mi> <mi>n</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mi>R</mi> <mi>n</mi> <mi mathvariant="normal">Δ</mi> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=\alpha RnT_{2}-\alpha RnT_{1}=\alpha Rn\Delta T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5ec717083dc0b68bf929d53873f7a701461fd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.737ex; height:2.509ex;" alt="{\displaystyle \Delta U=\alpha RnT_{2}-\alpha RnT_{1}=\alpha Rn\Delta T.}"></span></td> <td></td> <td class="nowrap"><span id="math_b1" class="reference nourlexpansion" style="font-weight:bold;">b1</span></td></tr></tbody></table> <p>At the same time, the work done by the pressure–volume changes as a result from this process, is equal to </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{V_{1}}^{V_{2}}P\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{V_{1}}^{V_{2}}P\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d41e2837e40b41475f275dcc5f128708259dfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.19ex; height:6.509ex;" alt="{\displaystyle W=\int _{V_{1}}^{V_{2}}P\,dV.}"></span></td> <td></td> <td class="nowrap"><span id="math_b2" class="reference nourlexpansion" style="font-weight:bold;">b2</span></td></tr></tbody></table> <p>Since we require the process to be adiabatic, the following equation needs to be true </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U+W=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>+</mo> <mi>W</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U+W=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7126064e7942b7c5ceb3e5b6b19fc5ba975014c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.902ex; height:2.343ex;" alt="{\displaystyle \Delta U+W=0.}"></span></td> <td></td> <td class="nowrap"><span id="math_b3" class="reference nourlexpansion" style="font-weight:bold;">b3</span></td></tr></tbody></table> <p>By the previous derivation, </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV^{\gamma }={\text{constant}}=P_{1}V_{1}^{\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV^{\gamma }={\text{constant}}=P_{1}V_{1}^{\gamma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd3f246be1ba1fc74032ad7f8f12c576cad3fea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.887ex; height:3.176ex;" alt="{\displaystyle PV^{\gamma }={\text{constant}}=P_{1}V_{1}^{\gamma }.}"></span></td> <td></td> <td class="nowrap"><span id="math_b4" class="reference nourlexpansion" style="font-weight:bold;">b4</span></td></tr></tbody></table> <p>Rearranging (b4) 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 P=P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e341ab8c3ef415206b49231cf8e6c6f5aeaa5e94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.829ex; height:6.176ex;" alt="{\displaystyle P=P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }.}"></span> </p><p>Substituting this into (b2) 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 W=\int _{V_{1}}^{V_{2}}P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{V_{1}}^{V_{2}}P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41a2e3c232bdb5cebbc545ec94944f61f8cc2cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.782ex; height:6.509ex;" alt="{\displaystyle W=\int _{V_{1}}^{V_{2}}P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }\,dV.}"></span> </p><p>Integrating, we obtain the expression for work, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}W=P_{1}V_{1}^{\gamma }{\frac {V_{2}^{1-\gamma }-V_{1}^{1-\gamma }}{1-\gamma }}\\&={\frac {P_{2}V_{2}-P_{1}V_{1}}{1-\gamma }}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>W</mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W=P_{1}V_{1}^{\gamma }{\frac {V_{2}^{1-\gamma }-V_{1}^{1-\gamma }}{1-\gamma }}\\&={\frac {P_{2}V_{2}-P_{1}V_{1}}{1-\gamma }}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0ab7fb270465ffebef3a5a04bebc3c15758787" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:43.168ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}W=P_{1}V_{1}^{\gamma }{\frac {V_{2}^{1-\gamma }-V_{1}^{1-\gamma }}{1-\gamma }}\\&={\frac {P_{2}V_{2}-P_{1}V_{1}}{1-\gamma }}.\end{aligned}}}"></span> </p><p>Substituting <span class="texhtml"><i>γ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>α</i> + 1</span><span class="sr-only">/</span><span class="den"><i>α</i></span></span>⁠</span></span> in the second term, </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 W=-\alpha P_{1}V_{1}^{\gamma }\left(V_{2}^{1-\gamma }-V_{1}^{1-\gamma }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha P_{1}V_{1}^{\gamma }\left(V_{2}^{1-\gamma }-V_{1}^{1-\gamma }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d803b27962578a64e9be5065a7f0b9d24ec4d41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.739ex; height:4.843ex;" alt="{\displaystyle W=-\alpha P_{1}V_{1}^{\gamma }\left(V_{2}^{1-\gamma }-V_{1}^{1-\gamma }\right).}"></span> </p><p>Rearranging, </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 W=-\alpha P_{1}V_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha P_{1}V_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5322f95129c595374f6edbd3348e6168ecc0f4ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.783ex; height:7.509ex;" alt="{\displaystyle W=-\alpha P_{1}V_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"></span> </p><p>Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases), </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 W=-\alpha nRT_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba65b193d70df439a2a8eb9cebf85bc82d10259c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.397ex; height:7.509ex;" alt="{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"></span> </p><p>By the continuous formula, </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_{2}}{P_{1}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{-\gamma },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P_{2}}{P_{1}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{-\gamma },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0422e71ec644d607eb861b2172b5bfdb1074d3fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.198ex; height:6.509ex;" alt="{\displaystyle {\frac {P_{2}}{P_{1}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{-\gamma },}"></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 \left({\frac {P_{2}}{P_{1}}}\right)^{-{\frac {1}{\gamma }}}={\frac {V_{2}}{V_{1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{-{\frac {1}{\gamma }}}={\frac {V_{2}}{V_{1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2952bbecd3d1bfdfc532913b55e20879022de83f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.866ex; height:7.176ex;" alt="{\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{-{\frac {1}{\gamma }}}={\frac {V_{2}}{V_{1}}}.}"></span> </p><p>Substituting into the previous expression for <span class="texhtml"><i>W</i></span>, </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 W=-\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8679c25c3c68d194757b9585fcd5464e77ea49e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:34.193ex; height:8.843ex;" alt="{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"></span> </p><p>Substituting this expression and (b1) in (b3) 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 \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>n</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e361b75661eff982ba071306453abcebc985ffa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:44.069ex; height:8.843ex;" alt="{\displaystyle \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"></span> </p><p>Simplifying, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T_{2}-T_{1}&=T_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right),\\{\frac {T_{2}}{T_{1}}}-1&=\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1,\\T_{2}&=T_{1}\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}.\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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T_{2}-T_{1}&=T_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right),\\{\frac {T_{2}}{T_{1}}}-1&=\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1,\\T_{2}&=T_{1}\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6a58c8748311149a6955e0cd2e2ed407ad7e00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:33.719ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}T_{2}-T_{1}&=T_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right),\\{\frac {T_{2}}{T_{1}}}-1&=\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1,\\T_{2}&=T_{1}\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}.\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_discrete_formula_and_work_expression">Derivation of discrete formula and work expression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=8" title="Edit section: Derivation of discrete formula and work expression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The change in internal energy of a system, measured from state 1 to state 2, is equal to </p><p>At the same time, the work done by the pressure–volume changes as a result from this process, is equal to </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{V_{1}}^{V_{2}}P\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{V_{1}}^{V_{2}}P\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d41e2837e40b41475f275dcc5f128708259dfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.19ex; height:6.509ex;" alt="{\displaystyle W=\int _{V_{1}}^{V_{2}}P\,dV.}"></span></td> <td></td> <td class="nowrap"><span id="math_c2" class="reference nourlexpansion" style="font-weight:bold;">c2</span></td></tr></tbody></table> <p>Since we require the process to be adiabatic, the following equation needs to be true </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta U+W=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>+</mo> <mi>W</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U+W=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7126064e7942b7c5ceb3e5b6b19fc5ba975014c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.902ex; height:2.343ex;" alt="{\displaystyle \Delta U+W=0.}"></span></td> <td></td> <td class="nowrap"><span id="math_c3" class="reference nourlexpansion" style="font-weight:bold;">c3</span></td></tr></tbody></table> <p>By the previous derivation, </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV^{\gamma }={\text{constant}}=P_{1}V_{1}^{\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV^{\gamma }={\text{constant}}=P_{1}V_{1}^{\gamma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd3f246be1ba1fc74032ad7f8f12c576cad3fea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.887ex; height:3.176ex;" alt="{\displaystyle PV^{\gamma }={\text{constant}}=P_{1}V_{1}^{\gamma }.}"></span></td> <td></td> <td class="nowrap"><span id="math_c4" class="reference nourlexpansion" style="font-weight:bold;">c4</span></td></tr></tbody></table> <p>Rearranging (c4) 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 P=P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e341ab8c3ef415206b49231cf8e6c6f5aeaa5e94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.829ex; height:6.176ex;" alt="{\displaystyle P=P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }.}"></span> </p><p>Substituting this into (c2) 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 W=\int _{V_{1}}^{V_{2}}P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }\,dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{V_{1}}^{V_{2}}P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }\,dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41a2e3c232bdb5cebbc545ec94944f61f8cc2cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.782ex; height:6.509ex;" alt="{\displaystyle W=\int _{V_{1}}^{V_{2}}P_{1}\left({\frac {V_{1}}{V}}\right)^{\gamma }\,dV.}"></span> </p><p>Integrating we obtain the expression for work, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=P_{1}V_{1}^{\gamma }{\frac {V_{2}^{1-\gamma }-V_{1}^{1-\gamma }}{1-\gamma }}={\frac {P_{2}V_{2}-P_{1}V_{1}}{1-\gamma }}.}"> <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"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=P_{1}V_{1}^{\gamma }{\frac {V_{2}^{1-\gamma }-V_{1}^{1-\gamma }}{1-\gamma }}={\frac {P_{2}V_{2}-P_{1}V_{1}}{1-\gamma }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5242dd4f6fe54fc1385740319f86eac05f573034" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.417ex; height:6.843ex;" alt="{\displaystyle W=P_{1}V_{1}^{\gamma }{\frac {V_{2}^{1-\gamma }-V_{1}^{1-\gamma }}{1-\gamma }}={\frac {P_{2}V_{2}-P_{1}V_{1}}{1-\gamma }}.}"></span> </p><p>Substituting <span class="texhtml"><i>γ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>α</i> + 1</span><span class="sr-only">/</span><span class="den"><i>α</i></span></span>⁠</span></span> in second term, </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 W=-\alpha P_{1}V_{1}^{\gamma }\left(V_{2}^{1-\gamma }-V_{1}^{1-\gamma }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha P_{1}V_{1}^{\gamma }\left(V_{2}^{1-\gamma }-V_{1}^{1-\gamma }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d803b27962578a64e9be5065a7f0b9d24ec4d41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.739ex; height:4.843ex;" alt="{\displaystyle W=-\alpha P_{1}V_{1}^{\gamma }\left(V_{2}^{1-\gamma }-V_{1}^{1-\gamma }\right).}"></span> </p><p>Rearranging, </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 W=-\alpha P_{1}V_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha P_{1}V_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5322f95129c595374f6edbd3348e6168ecc0f4ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.783ex; height:7.509ex;" alt="{\displaystyle W=-\alpha P_{1}V_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"></span> </p><p>Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases), </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 W=-\alpha nRT_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba65b193d70df439a2a8eb9cebf85bc82d10259c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.397ex; height:7.509ex;" alt="{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {V_{2}}{V_{1}}}\right)^{1-\gamma }-1\right).}"></span> </p><p>By the continuous formula, </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_{2}}{P_{1}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{-\gamma },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P_{2}}{P_{1}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{-\gamma },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0422e71ec644d607eb861b2172b5bfdb1074d3fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.198ex; height:6.509ex;" alt="{\displaystyle {\frac {P_{2}}{P_{1}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{-\gamma },}"></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 \left({\frac {P_{2}}{P_{1}}}\right)^{-{\frac {1}{\gamma }}}={\frac {V_{2}}{V_{1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{-{\frac {1}{\gamma }}}={\frac {V_{2}}{V_{1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2952bbecd3d1bfdfc532913b55e20879022de83f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.866ex; height:7.176ex;" alt="{\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{-{\frac {1}{\gamma }}}={\frac {V_{2}}{V_{1}}}.}"></span> </p><p>Substituting into the previous expression for <span class="texhtml"><i>W</i></span>, </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 W=-\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8679c25c3c68d194757b9585fcd5464e77ea49e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:34.193ex; height:8.843ex;" alt="{\displaystyle W=-\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"></span> </p><p>Substituting this expression and (c1) in (c3) 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 \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>n</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>n</mi> <mi>R</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e361b75661eff982ba071306453abcebc985ffa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:44.069ex; height:8.843ex;" alt="{\displaystyle \alpha nR(T_{2}-T_{1})=\alpha nRT_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right).}"></span> </p><p>Simplifying, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T_{2}-T_{1}&=T_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right),\\{\frac {T_{2}}{T_{1}}}-1&=\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1,\\T_{2}&=T_{1}\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}.\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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>γ</mi> </mfrac> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T_{2}-T_{1}&=T_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right),\\{\frac {T_{2}}{T_{1}}}-1&=\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1,\\T_{2}&=T_{1}\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6a58c8748311149a6955e0cd2e2ed407ad7e00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:33.719ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}T_{2}-T_{1}&=T_{1}\left(\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1\right),\\{\frac {T_{2}}{T_{1}}}-1&=\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}-1,\\T_{2}&=T_{1}\left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}.\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Graphing_adiabats">Graphing adiabats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=9" title="Edit section: Graphing adiabats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Entropyandtemp.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Entropyandtemp.PNG/350px-Entropyandtemp.PNG" decoding="async" width="350" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Entropyandtemp.PNG/525px-Entropyandtemp.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Entropyandtemp.PNG/700px-Entropyandtemp.PNG 2x" data-file-width="1770" data-file-height="1170" /></a><figcaption><i>P</i>–<i>V</i> diagram with a superposition of adiabats and isotherms: <div><ul><li style="margin-bottom: 0">The isotherms are the red curves and the adiabats are the black curves.</li><li style="margin-bottom: 0">The adiabats are isentropic.</li><li style="margin-bottom: 0">Volume is the horizontal axis and pressure is the vertical axis.</li></ul></div></figcaption></figure> <p>An adiabat is a curve of constant <a href="/wiki/Entropy" title="Entropy">entropy</a> in a diagram. Some properties of adiabats on a <i>P</i>–<i>V</i> diagram are indicated. These properties may be read from the classical behaviour of ideal gases, except in the region where <i>PV</i> becomes small (low temperature), where quantum effects become important. </p> <ol><li>Every adiabat <a href="/wiki/Asymptotically" class="mw-redirect" title="Asymptotically">asymptotically</a> approaches both the <i>V</i> axis and the <i>P</i> axis (just like <a href="/wiki/Isotherms" class="mw-redirect" title="Isotherms">isotherms</a>).</li> <li>Each adiabat intersects each isotherm exactly once.</li> <li>An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination (more vertical).</li> <li>If isotherms are concave towards the north-east direction (45° from V-axis), then adiabats are concave towards the east north-east (31° from V-axis).</li> <li>If adiabats and isotherms are graphed at regular intervals of entropy and temperature, respectively (like altitude on a contour map), then as the eye moves towards the axes (towards the south-west), it sees the density of isotherms stay constant, but it sees the density of adiabats grow. The exception is very near absolute zero, where the density of adiabats drops sharply and they become rare (see <a href="/wiki/Nernst%27s_theorem" class="mw-redirect" title="Nernst's theorem">Nernst's theorem</a>).<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="No quantitative formula is identified in that article as Nernst's theorem, only a qualitative "Impossibility Principle" (March 2015)">clarification needed</span></a></i>]</sup></li></ol> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=10" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <i>adiabatic</i> (<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="/ˌ/: secondary stress follows">ˌ</span><span title="/æ/: 'a' in 'bad'">æ</span><span title="'d' in 'dye'">d</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ə/: 'a' in 'about'">ə</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'b' in 'buy'">b</span><span title="/æ/: 'a' in 'bad'">æ</span><span title="'t' in 'tie'">t</span><span title="/ɪ/: 'i' in 'kit'">ɪ</span><span title="'k' in 'kind'">k</span></span>/</a></span></span>) is an anglicization of the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> term ἀδιάβατος "impassable" (used by <a href="/wiki/Xenophon" title="Xenophon">Xenophon</a> of rivers). It is used in the thermodynamic sense by <a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a> (1866),<sup id="cite_ref-Rankine_1866_13-0" class="reference"><a href="#cite_note-Rankine_1866-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Partington_122_14-0" class="reference"><a href="#cite_note-Partington_122-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and adopted by <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a> in 1871 (explicitly attributing the term to Rankine).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The etymological origin corresponds here to an impossibility of <a href="/wiki/Heat" title="Heat">transfer of energy as heat</a> and of transfer of matter across the wall. </p><p>The Greek word ἀδιάβατος is formed from <a href="/wiki/Privative_a" class="mw-redirect" title="Privative a">privative ἀ-</a> ("not") and διαβατός, "passable", in turn deriving from διά ("through"), and βαῖνειν ("to walk, go, come").<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Furthermore, in <a href="/wiki/Atmospheric_thermodynamics" title="Atmospheric thermodynamics">atmospheric thermodynamics</a>, a diabatic process is one in which heat is exchanged.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> An adiabatic process is the opposite – a process in which no heat is exchanged. </p> <div class="mw-heading mw-heading2"><h2 id="Conceptual_significance_in_thermodynamic_theory">Conceptual significance in thermodynamic theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=11" title="Edit section: Conceptual significance in thermodynamic theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The adiabatic process has been important for thermodynamics since its early days. It was important in the work of <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a> because it provided a way of nearly directly relating quantities of heat and work. </p><p>Energy can enter or leave a thermodynamic system enclosed by walls that prevent <a href="/wiki/Mass_transfer" title="Mass transfer">mass transfer</a> only as heat or work. Therefore, a quantity of work in such a system can be related almost directly to an equivalent quantity of heat in a cycle of two limbs. The first limb is an isochoric adiabatic work process increasing the system's <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>; the second, an isochoric and workless heat transfer returning the system to its original state. Accordingly, Rankine measured quantity of heat in units of work, rather than as a calorimetric quantity.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1854, Rankine used a quantity that he called "the thermodynamic function" that later was called entropy, and at that time he wrote also of the "curve of no transmission of heat",<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> which he later called an adiabatic curve.<sup id="cite_ref-Rankine_1866_13-1" class="reference"><a href="#cite_note-Rankine_1866-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Besides its two isothermal limbs, Carnot's cycle has two adiabatic limbs. </p><p>For the foundations of thermodynamics, the conceptual importance of this was emphasized by Bryan,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> by Carathéodory,<sup id="cite_ref-Carathéodory_1-1" class="reference"><a href="#cite_note-Carathéodory-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and by Born.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The reason is that calorimetry presupposes a type of temperature as already defined before the statement of the first law of thermodynamics, such as one based on empirical scales. Such a presupposition involves making the distinction between empirical temperature and absolute temperature. Rather, the definition of absolute thermodynamic temperature is best left till the second law is available as a conceptual basis.<sup id="cite_ref-Bailyn_Ch_3_22-0" class="reference"><a href="#cite_note-Bailyn_Ch_3-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In the eighteenth century, the law of conservation of energy was not yet fully formulated or established, and the nature of heat was debated. One approach to these problems was to regard heat, measured by calorimetry, as a primary substance that is conserved in quantity. By the middle of the nineteenth century, it was recognized as a form of energy, and the law of conservation of energy was thereby also recognized. The view that eventually established itself, and is currently regarded as right, is that the law of conservation of energy is a primary axiom, and that heat is to be analyzed as consequential. In this light, heat cannot be a component of the total energy of a single body because it is not a <a href="/wiki/State_function" title="State function">state variable</a> but, rather, a variable that describes a transfer between two bodies. The adiabatic process is important because it is a logical ingredient of this current view.<sup id="cite_ref-Bailyn_Ch_3_22-1" class="reference"><a href="#cite_note-Bailyn_Ch_3-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Divergent_usages_of_the_word_adiabatic">Divergent usages of the word <i>adiabatic</i></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=12" title="Edit section: Divergent usages of the word adiabatic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This present article is written from the viewpoint of macroscopic thermodynamics, and the word <i>adiabatic</i> is used in this article in the traditional way of thermodynamics, introduced by Rankine. It is pointed out in the present article that, for example, if a compression of a gas is rapid, then there is little time for heat transfer to occur, even when the gas is not adiabatically isolated by a definite wall. In this sense, a rapid compression of a gas is sometimes approximately or loosely said to be <i>adiabatic</i>, though often far from isentropic, even when the gas is not adiabatically isolated by a definite wall. </p><p>Some authors, like <a href="/wiki/Brian_Pippard" title="Brian Pippard">Pippard</a>, recommend using "adiathermal" to refer to processes where no heat-exchange occurs (such as Joule expansion), and "adiabatic" to reversible quasi-static adiathermal processes (so that rapid compression of a gas is <i>not</i> "adiabatic").<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> And <a href="/wiki/Keith_J._Laidler" title="Keith J. Laidler">Laidler</a> has summarized the complicated etymology of "adiabatic".<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> and <a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">quantum statistical mechanics</a>, however, use the word <i>adiabatic</i> <a href="/wiki/Adiabatic_theorem#Diabatic_vs._adiabatic_processes" title="Adiabatic theorem">in a very different sense</a>, one that can at times seem almost opposite to the classical thermodynamic sense. In quantum theory, the word <i>adiabatic</i> can mean something perhaps near <a href="/wiki/Isentropic_process" title="Isentropic process">isentropic</a>, or perhaps near <a href="/wiki/Quasistatic_process" title="Quasistatic process">quasi-static</a>, but the usage of the word is very different between the two disciplines. </p><p>On the one hand, in quantum theory, if a perturbative element of compressive work is done almost infinitely slowly (that is to say quasi-statically), it is said to have been done <i>adiabatically</i>. The idea is that the shapes of the eigenfunctions change slowly and continuously, so that no quantum jump is triggered, and the change is virtually reversible. While the occupation numbers are unchanged, nevertheless there is change in the energy levels of one-to-one corresponding, pre- and post-compression, eigenstates. Thus a perturbative element of work has been done without heat transfer and without introduction of random change within the system. For example, <a href="/wiki/Max_Born" title="Max Born">Max Born</a> writes </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Actually, it is usually the 'adiabatic' case with which we have to do: i.e. the limiting case where the external force (or the reaction of the parts of the system on each other) acts very slowly. In this case, to a very high approximation </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 c_{1}^{2}=1,\,\,c_{2}^{2}=0,\,\,c_{3}^{2}=0,\,...\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1}^{2}=1,\,\,c_{2}^{2}=0,\,\,c_{3}^{2}=0,\,...\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e984cd85b09f52aa81274216e2923e153449f36d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.139ex; height:3.176ex;" alt="{\displaystyle c_{1}^{2}=1,\,\,c_{2}^{2}=0,\,\,c_{3}^{2}=0,\,...\,,}"></span> </p><p> that is, there is no probability for a transition, and the system is in the initial state after cessation of the perturbation. Such a slow perturbation is therefore reversible, as it is classically.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>On the other hand, in quantum theory, if a perturbative element of compressive work is done rapidly, it changes the occupation numbers and energies of the eigenstates in proportion to the <a href="/wiki/Selection_rule" title="Selection rule">transition moment integral</a> and in accordance with <a href="/wiki/Time-dependent_perturbation_theory" class="mw-redirect" title="Time-dependent perturbation theory">time-dependent perturbation theory</a>, as well as perturbing the functional form of the eigenstates themselves. In that theory, such a rapid change is said not to be <i>adiabatic</i>, and the contrary word <i>diabatic</i> is applied to it. </p><p>Recent research<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> suggests that the power absorbed from the perturbation corresponds to the rate of these non-adiabatic transitions. This corresponds to the classical process of energy transfer in the form of heat, but with the relative time scales reversed in the quantum case. Quantum adiabatic processes occur over relatively long time scales, while classical adiabatic processes occur over relatively short time scales. It should also be noted that the concept of 'heat' (in reference to the quantity of <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a> transferred) breaks down at the quantum level, and the specific form of energy (typically electromagnetic) must be considered instead. The small or negligible absorption of energy from the perturbation in a quantum adiabatic process provides a good justification for identifying it as the quantum analogue of adiabatic processes in classical thermodynamics, and for the reuse of the term. </p><p>In classical thermodynamics, such a rapid change would still be called adiabatic because the system is adiabatically isolated, and there is no transfer of energy as heat. The strong irreversibility of the change, due to viscosity or other <a href="/wiki/Entropy_production" title="Entropy production">entropy production</a>, does not impinge on this classical usage. </p><p>Thus for a mass of gas, in macroscopic thermodynamics, words are so used that a compression is sometimes loosely or approximately said to be adiabatic if it is rapid enough to avoid significant heat transfer, even if the system is not adiabatically isolated. But in quantum statistical theory, a compression is not called adiabatic if it is rapid, even if the system is adiabatically isolated in the classical thermodynamic sense of the term. The words are used differently in the two disciplines, as stated just above. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fire_piston" title="Fire piston">Fire piston</a></li> <li><a href="/wiki/Heat_burst" title="Heat burst">Heat burst</a></li></ul> <dl><dt>Related physics topics</dt></dl> <ul><li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First law of thermodynamics</a></li> <li><a href="/wiki/Entropy_(classical_thermodynamics)" title="Entropy (classical thermodynamics)">Entropy (classical thermodynamics)</a></li> <li><a href="/wiki/Adiabatic_conductivity" title="Adiabatic conductivity">Adiabatic conductivity</a></li> <li><a href="/wiki/Adiabatic_lapse_rate" class="mw-redirect" title="Adiabatic lapse rate">Adiabatic lapse rate</a></li> <li><a href="/wiki/Total_air_temperature" title="Total air temperature">Total air temperature</a></li> <li><a href="/wiki/Magnetic_refrigeration" title="Magnetic refrigeration">Magnetic refrigeration</a></li> <li><a href="/wiki/Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li></ul> <dl><dt>Related thermodynamic processes</dt></dl> <ul><li><a href="/wiki/Cyclic_process" class="mw-redirect" title="Cyclic process">Cyclic process</a></li> <li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric process</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic process</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic process</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric process</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal process</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic process</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic process</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Carathéodory-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Carathéodory_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Carathéodory_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCarathéodory1909" class="citation journal cs1"><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory, C.</a> (1909). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428268">"Untersuchungen über die Grundlagen der Thermodynamik"</a>. <i>Mathematische Annalen</i>. <b>67</b> (3): <span class="nowrap">355–</span>386. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01450409">10.1007/BF01450409</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118230148">118230148</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Untersuchungen+%C3%BCber+die+Grundlagen+der+Thermodynamik&rft.volume=67&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E355-%3C%2Fspan%3E386&rft.date=1909&rft_id=info%3Adoi%2F10.1007%2FBF01450409&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118230148%23id-name%3DS2CID&rft.aulast=Carath%C3%A9odory&rft.aufirst=C.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428268&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span>. A translation may be found <a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf">here</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191012152205/http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf">Archived</a> 2019-10-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Also a mostly reliable <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations">translation is to be found</a> in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKestin1976" class="citation book cs1">Kestin, J. (1976). <i>The Second Law of Thermodynamics</i>. Stroudsburg, Pennsylvania: Dowden, Hutchinson & Ross.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Second+Law+of+Thermodynamics&rft.place=Stroudsburg%2C+Pennsylvania&rft.pub=Dowden%2C+Hutchinson+%26+Ross&rft.date=1976&rft.aulast=Kestin&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-Bailyn_21-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bailyn_21_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBailyn1994" class="citation book cs1">Bailyn, M. (1994). <i>A Survey of Thermodynamics</i>. New York, New York: American Institute of Physics Press. p. 21. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88318-797-3" title="Special:BookSources/0-88318-797-3"><bdi>0-88318-797-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Thermodynamics&rft.place=New+York%2C+New+York&rft.pages=21&rft.pub=American+Institute+of+Physics+Press&rft.date=1994&rft.isbn=0-88318-797-3&rft.aulast=Bailyn&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-Bailyn_53-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bailyn_53_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bailyn_53_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Bailyn_53_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Bailyn, M. (1994), pp. 52–53.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://glossary.ametsoc.org/wiki/Pseudoadiabatic_process">"pseudoadiabatic process"</a>. <a href="/wiki/American_Meteorological_Society" title="American Meteorological Society">American Meteorological Society</a><span class="reference-accessdate">. Retrieved <span class="nowrap">November 3,</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=pseudoadiabatic+process&rft.pub=American+Meteorological+Society&rft_id=http%3A%2F%2Fglossary.ametsoc.org%2Fwiki%2FPseudoadiabatic_process&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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="CITEREFTisza1966" class="citation book cs1"><a href="/wiki/L%C3%A1szl%C3%B3_Tisza" title="László Tisza">Tisza, L.</a> (1966). <i>Generalized Thermodynamics</i>. Cambridge, Massachusetts: MIT Press. p. 48. <q>(adiabatic partitions inhibit the transfer of heat and mass)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Thermodynamics&rft.place=Cambridge%2C+Massachusetts&rft.pages=48&rft.pub=MIT+Press&rft.date=1966&rft.aulast=Tisza&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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">Münster, A. (1970), p. 48: "mass is an adiabatically inhibited variable."</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMünster1970" class="citation book cs1">Münster, A. (1970). <i>Classical Thermodynamics</i>. Translated by Halberstadt, E. S. London: Wiley–Interscience. p. 45. <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&rft.pages=45&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%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnight2022" class="citation web cs1">Knight, Jasper (31 January 2022). <a rel="nofollow" class="external text" href="https://theconversation.com/snowfall-in-the-sahara-desert-an-unusual-weather-phenomenon-176037">"Snowfall in the Sahara desert: an unusual weather phenomenon"</a>. <i>The Conversation</i><span class="reference-accessdate">. Retrieved <span class="nowrap">3 March</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Conversation&rft.atitle=Snowfall+in+the+Sahara+desert%3A+an+unusual+weather+phenomenon&rft.date=2022-01-31&rft.aulast=Knight&rft.aufirst=Jasper&rft_id=https%3A%2F%2Ftheconversation.com%2Fsnowfall-in-the-sahara-desert-an-unusual-weather-phenomenon-176037&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-Kavanagh-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kavanagh_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKavanaghSparks2009" class="citation journal cs1">Kavanagh, J. L.; Sparks, R. S. J. (2009). <a rel="nofollow" class="external text" href="https://monash.academia.edu/JanineKavanagh/Papers/114092/Temperature_changes_in_ascending_kimberlite_magma">"Temperature changes in ascending kimberlite magmas"</a>. <i>Earth and Planetary Science Letters</i>. <b>286</b> (<span class="nowrap">3–</span>4). <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>: <span class="nowrap">404–</span>413. <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/2009E&PSL.286..404K">2009E&PSL.286..404K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.epsl.2009.07.011">10.1016/j.epsl.2009.07.011</a><span class="reference-accessdate">. Retrieved <span class="nowrap">18 February</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Earth+and+Planetary+Science+Letters&rft.atitle=Temperature+changes+in+ascending+kimberlite+magmas&rft.volume=286&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E404-%3C%2Fspan%3E413&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.epsl.2009.07.011&rft_id=info%3Abibcode%2F2009E%26PSL.286..404K&rft.aulast=Kavanagh&rft.aufirst=J.+L.&rft.au=Sparks%2C+R.+S.+J.&rft_id=https%3A%2F%2Fmonash.academia.edu%2FJanineKavanagh%2FPapers%2F114092%2FTemperature_changes_in_ascending_kimberlite_magma&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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="CITEREFTurcotte_and_Schubert2002" class="citation book cs1">Turcotte and Schubert (2002). <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/geodynamics00dltu"><i>Geodynamics</i></a></span>. Cambridge: Cambridge University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/geodynamics00dltu/page/n199">185</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66624-4" title="Special:BookSources/0-521-66624-4"><bdi>0-521-66624-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geodynamics&rft.place=Cambridge&rft.pages=185&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=0-521-66624-4&rft.au=Turcotte+and+Schubert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeodynamics00dltu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><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/adiab.html">"Adiabatic Process"</a>. <i>HyperPhysics</i>. Georgia State University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=HyperPhysics&rft.atitle=Adiabatic+Process&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Fthermo%2Fadiab.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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="CITEREFAtkinsde_Paula2006" class="citation book cs1">Atkins, Peter; de Paula, Giulio (2006). <i>Atkins' Physical Chemistry</i> (8th ed.). W. H. Freeman. p. 48. <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=48&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+Giulio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-Rankine_1866-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rankine_1866_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rankine_1866_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine, William John MacQuorn</a> (1866). On the theory of explosive gas engines, <i>The Engineer</i>, July 27, 1866; at page 467 of the reprint in <i><a href="//archive.org/details/miscellaneoussci00rank" class="extiw" title="iarchive:miscellaneoussci00rank">Miscellaneous Scientific Papers</a></i>, edited by W. J. Millar, 1881, Charles Griffin, London.</span> </li> <li id="cite_note-Partington_122-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Partington_122_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartington1949" class="citation cs2"><a href="/wiki/J.R._Partington" class="mw-redirect" title="J.R. Partington">Partington, J. R.</a> (1949), <i>An Advanced Treatise on Physical Chemistry.</i>, vol. 1, Fundamental Principles. The Properties of Gases, London: <a href="/wiki/Longman" title="Longman">Longmans, Green and Co.</a>, p. 122</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Advanced+Treatise+on+Physical+Chemistry.&rft.place=Fundamental+Principles.+The+Properties+of+Gases%2C+London&rft.pages=122&rft.pub=Longmans%2C+Green+and+Co.&rft.date=1949&rft.aulast=Partington&rft.aufirst=J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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="CITEREFMaxwell1871" class="citation cs2"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J. C.</a> (1871), <a rel="nofollow" class="external text" href="https://archive.org/details/theoryheat04maxwgoog"><i>Theory of Heat</i></a> (first ed.), London: <a href="/wiki/Longman" title="Longman">Longmans, Green and Co.</a>, p. 129</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Heat&rft.place=London&rft.pages=129&rft.edition=first&rft.pub=Longmans%2C+Green+and+Co.&rft.date=1871&rft.aulast=Maxwell&rft.aufirst=J.+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryheat04maxwgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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"><a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell, H. G.</a>, <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott, R.</a> (1940). <i>A Greek-English Lexicon</i>, Clarendon Press, Oxford, UK.</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://glossary.ametsoc.org/wiki/Diabatic_process">"diabatic process"</a>. American Meteorological Society<span class="reference-accessdate">. Retrieved <span class="nowrap">24 November</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=diabatic+process&rft.pub=American+Meteorological+Society&rft_id=https%3A%2F%2Fglossary.ametsoc.org%2Fwiki%2FDiabatic_process&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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="CITEREFRankine1854" class="citation journal cs1">Rankine, W. J. MacQ. (1854). "On the geometrical representation of the expansive action of heat, and theory of thermodynamic engines". <i>Proceedings of the Royal Society</i>. <b>144</b>: <span class="nowrap">115–</span>175.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society&rft.atitle=On+the+geometrical+representation+of+the+expansive+action+of+heat%2C+and+theory+of+thermodynamic+engines&rft.volume=144&rft.pages=%3Cspan+class%3D%22nowrap%22%3E115-%3C%2Fspan%3E175&rft.date=1854&rft.aulast=Rankine&rft.aufirst=W.+J.+MacQ.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/stream/miscellaneoussci00rank#page/340/mode/1up">Miscellaneous Scientific Papers p. 339</a></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRankine1854" class="citation journal cs1">Rankine, W. J. MacQ. (1854). "On the geometrical representation of the expansive action of heat, and theory of thermodynamic engines". <i>Proceedings of the Royal Society</i>. <b>144</b>: <span class="nowrap">115–</span>175.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society&rft.atitle=On+the+geometrical+representation+of+the+expansive+action+of+heat%2C+and+theory+of+thermodynamic+engines&rft.volume=144&rft.pages=%3Cspan+class%3D%22nowrap%22%3E115-%3C%2Fspan%3E175&rft.date=1854&rft.aulast=Rankine&rft.aufirst=W.+J.+MacQ.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/stream/miscellaneoussci00rank#page/341/mode/1up/search/transmission">Miscellaneous Scientific Papers p. 341</a>.</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="CITEREFBryan1907" class="citation book cs1"><a href="/wiki/George_H._Bryan" title="George H. Bryan">Bryan, G. H.</a> (1907). <a rel="nofollow" class="external text" href="https://archive.org/details/ost-physics-thermodynamicsin00bryauoft"><i>Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications</i></a>. Leipzig: B. G. Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics.+An+Introductory+Treatise+dealing+mainly+with+First+Principles+and+their+Direct+Applications&rft.place=Leipzig&rft.pub=B.+G.+Teubner&rft.date=1907&rft.aulast=Bryan&rft.aufirst=G.+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fost-physics-thermodynamicsin00bryauoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" 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="CITEREFBorn1949" class="citation book cs1"><a href="/wiki/Max_Born" title="Max Born">Born, M.</a> (1949). <a rel="nofollow" class="external text" href="https://archive.org/details/naturalphilosoph032159mbp"><i>Natural Philosophy of Cause and Chance</i></a>. London: Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Natural+Philosophy+of+Cause+and+Chance&rft.place=London&rft.pub=Oxford+University+Press&rft.date=1949&rft.aulast=Born&rft.aufirst=M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnaturalphilosoph032159mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> <li id="cite_note-Bailyn_Ch_3-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bailyn_Ch_3_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bailyn_Ch_3_22-1"><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="CITEREFBailyn1994" class="citation book cs1">Bailyn, M. (1994). "Chapter 3". <i>A Survey of Thermodynamics</i>. 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Translated by <a href="/wiki/Robert_Oppenheimer" class="mw-redirect" title="Robert Oppenheimer">Oppenheimer, Robert</a>: <span class="nowrap">354–</span>357. <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/1927Natur.119..354B">1927Natur.119..354B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F119354a0">10.1038/119354a0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Physical+aspects+of+quantum+mechanics&rft.volume=119&rft.issue=2992&rft.pages=%3Cspan+class%3D%22nowrap%22%3E354-%3C%2Fspan%3E357&rft.date=1927&rft_id=info%3Adoi%2F10.1038%2F119354a0&rft_id=info%3Abibcode%2F1927Natur.119..354B&rft.aulast=Born&rft.aufirst=M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1038%252F119354a0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMandalHunt2020" class="citation journal cs1">Mandal, Anirban; Hunt, Katharine L. C. (2020-03-14). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5140009">"Variance of the energy of a quantum system in a time-dependent perturbation: Determination by nonadiabatic transition probabilities"</a>. <i>The Journal of Chemical Physics</i>. <b>152</b> (10): 104110. <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/2020JChPh.152j4110M">2020JChPh.152j4110M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5140009">10.1063/1.5140009</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9606">0021-9606</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32171229">32171229</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:212731108">212731108</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Chemical+Physics&rft.atitle=Variance+of+the+energy+of+a+quantum+system+in+a+time-dependent+perturbation%3A+Determination+by+nonadiabatic+transition+probabilities&rft.volume=152&rft.issue=10&rft.pages=104110&rft.date=2020-03-14&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A212731108%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2020JChPh.152j4110M&rft.issn=0021-9606&rft_id=info%3Adoi%2F10.1063%2F1.5140009&rft_id=info%3Apmid%2F32171229&rft.aulast=Mandal&rft.aufirst=Anirban&rft.au=Hunt%2C+Katharine+L.+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252F1.5140009&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></span> </li> </ol></div> <dl><dt>General</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilbey2004" class="citation book cs1">Silbey, Robert J.; et al. (2004). <i>Physical chemistry</i>. Hoboken, New Jersey: Wiley. p. 55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-21504-2" title="Special:BookSources/978-0-471-21504-2"><bdi>978-0-471-21504-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+chemistry&rft.place=Hoboken%2C+New+Jersey&rft.pages=55&rft.pub=Wiley&rft.date=2004&rft.isbn=978-0-471-21504-2&rft.aulast=Silbey&rft.aufirst=Robert+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdiabatic+process" class="Z3988"></span></li> <li>Nave, Carl Rod. "<a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html">Adiabatic Processes</a>". HyperPhysics.</li> <li>Thorngren, Dr. Jane R. "<a rel="nofollow" class="external text" href="http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm">Adiabatic Processes</a>". Daphne – A Palomar College Web Server, 21 July 1995. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110509121743/http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm">Archived</a> 2011-05-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adiabatic_process&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/adiabatic" class="extiw" title="wiktionary:adiabatic">adiabatic</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <p><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Adiabatic_processes" class="extiw" title="commons:Category:Adiabatic processes">Adiabatic processes</a> at Wikimedia Commons </p> <ul><li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html#c1:">Article in HyperPhysics Encyclopaedia</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Adiabatic process - Wikipedia