Ideal gas - Wikipedia

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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Types_of_ideal_gas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_ideal_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Types of ideal gas</span> </div> </a> <ul id="toc-Types_of_ideal_gas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_thermodynamic_ideal_gas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classical_thermodynamic_ideal_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Classical thermodynamic ideal gas</span> </div> </a> <button aria-controls="toc-Classical_thermodynamic_ideal_gas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Classical thermodynamic ideal gas subsection</span> </button> <ul id="toc-Classical_thermodynamic_ideal_gas-sublist" class="vector-toc-list"> <li id="toc-Ideal_gas_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal_gas_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Ideal gas law</span> </div> </a> <ul id="toc-Ideal_gas_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internal_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Internal energy</span> </div> </a> <ul id="toc-Internal_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Microscopic_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Microscopic_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Microscopic model</span> </div> </a> <ul id="toc-Microscopic_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Heat_capacity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Heat_capacity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Heat capacity</span> </div> </a> <ul id="toc-Heat_capacity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Entropy</span> </div> </a> <ul id="toc-Entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermodynamic_potentials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Thermodynamic_potentials"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Thermodynamic potentials</span> </div> </a> <ul id="toc-Thermodynamic_potentials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Speed_of_sound" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Speed_of_sound"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Speed of sound</span> </div> </a> <ul id="toc-Speed_of_sound-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Table_of_ideal_gas_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Table_of_ideal_gas_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Table of ideal gas equations</span> </div> </a> <ul id="toc-Table_of_ideal_gas_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideal_quantum_gases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ideal_quantum_gases"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ideal quantum gases</span> </div> </a> <button aria-controls="toc-Ideal_quantum_gases-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 quantum gases subsection</span> </button> <ul id="toc-Ideal_quantum_gases-sublist" class="vector-toc-list"> <li id="toc-Ideal_Boltzmann_gas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal_Boltzmann_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Ideal Boltzmann gas</span> </div> </a> <ul id="toc-Ideal_Boltzmann_gas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideal_Bose_and_Fermi_gases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal_Bose_and_Fermi_gases"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Ideal Bose and Fermi gases</span> </div> </a> <ul id="toc-Ideal_Bose_and_Fermi_gases-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </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">Ideal gas</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 68 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-68" 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">68 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%BA%D8%A7%D8%B2_%D9%85%D8%AB%D8%A7%D9%84%D9%8A" title="غاز مثالي – Arabic" lang="ar" hreflang="ar" data-title="غاز مثالي" 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/Gas_ideal" title="Gas ideal – Asturian" lang="ast" hreflang="ast" data-title="Gas ideal" 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/%C4%B0deal_qaz" title="İdeal qaz – Azerbaijani" lang="az" hreflang="az" data-title="İdeal qaz" 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%86%E0%A6%A6%E0%A6%B0%E0%A7%8D%E0%A6%B6_%E0%A6%97%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B8" title="আদর্শ গ্যাস – Bangla" lang="bn" hreflang="bn" data-title="আদর্শ গ্যাস" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/L%C3%AD-si%C3%B3ng_kh%C3%AC-th%C3%A9" title="Lí-sióng khì-thé – Minnan" lang="nan" hreflang="nan" data-title="Lí-sióng khì-thé" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%B4%D1%8D%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%B3%D0%B0%D0%B7" title="Ідэальны газ – Belarusian" lang="be" hreflang="be" data-title="Ідэальны газ" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%86%D0%B4%D1%8D%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%B3%D0%B0%D0%B7" title="Ідэальны газ – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Ідэальны газ" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B3%D0%B0%D0%B7" title="Идеален газ – Bulgarian" lang="bg" hreflang="bg" data-title="Идеален газ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Idealni_plin" title="Idealni plin – Bosnian" lang="bs" hreflang="bs" data-title="Idealni plin" 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/Gas_ideal" title="Gas ideal – Catalan" lang="ca" hreflang="ca" data-title="Gas ideal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%BB%D0%B0_%D0%B3%D0%B0%D0%B7" title="Идеалла газ – Chuvash" lang="cv" hreflang="cv" data-title="Идеалла газ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ide%C3%A1ln%C3%AD_plyn" title="Ideální plyn – Czech" lang="cs" hreflang="cs" data-title="Ideální plyn" 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/Idealgas" title="Idealgas – Danish" lang="da" hreflang="da" data-title="Idealgas" 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/Ideales_Gas" title="Ideales Gas – German" lang="de" hreflang="de" data-title="Ideales Gas" 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/Ideaalne_gaas" title="Ideaalne gaas – Estonian" lang="et" hreflang="et" data-title="Ideaalne gaas" 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%99%CE%B4%CE%B1%CE%BD%CE%B9%CE%BA%CF%8C_%CE%B1%CE%AD%CF%81%CE%B9%CE%BF" title="Ιδανικό αέριο – Greek" lang="el" hreflang="el" data-title="Ιδανικό αέριο" 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/Gas_ideal" title="Gas ideal – Spanish" lang="es" hreflang="es" data-title="Gas ideal" 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/Ideala_gaso" title="Ideala gaso – Esperanto" lang="eo" hreflang="eo" data-title="Ideala gaso" 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/Gas_ideal" title="Gas ideal – Basque" lang="eu" hreflang="eu" data-title="Gas ideal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%A7%D8%B2_%D8%A7%DB%8C%D8%AF%D9%87%E2%80%8C%D8%A2%D9%84" title="گاز ایده‌آل – Persian" lang="fa" hreflang="fa" data-title="گاز ایده‌آل" 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/Gaz_parfait" title="Gaz parfait – French" lang="fr" hreflang="fr" data-title="Gaz parfait" 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/G%C3%A1s_id%C3%A9alach" title="Gás idéalach – Irish" lang="ga" hreflang="ga" data-title="Gás idéalach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%86%E0%AA%A6%E0%AA%B0%E0%AB%8D%E0%AA%B6_%E0%AA%B5%E0%AA%BE%E0%AA%AF%E0%AB%81" title="આદર્શ વાયુ – Gujarati" lang="gu" hreflang="gu" data-title="આદર્શ વાયુ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%83%81%EA%B8%B0%EC%B2%B4" title="이상기체 – Korean" lang="ko" hreflang="ko" data-title="이상기체" 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%BB%D5%A4%D5%A5%D5%A1%D5%AC%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A3%D5%A1%D5%A6" title="Իդեալական գազ – Armenian" lang="hy" hreflang="hy" data-title="Իդեալական գազ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%B6_%E0%A4%97%E0%A5%88%E0%A4%B8" title="आदर्श गैस – Hindi" lang="hi" hreflang="hi" data-title="आदर्श गैस" 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/Idealni_plin" title="Idealni plin – Croatian" lang="hr" hreflang="hr" data-title="Idealni plin" 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/Gas_ideal" title="Gas ideal – Indonesian" lang="id" hreflang="id" data-title="Gas ideal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Kj%C3%B6rgas" title="Kjörgas – Icelandic" lang="is" hreflang="is" data-title="Kjörgas" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gas_ideale" title="Gas ideale – Italian" lang="it" hreflang="it" data-title="Gas ideale" 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%92%D7%96_%D7%90%D7%99%D7%93%D7%99%D7%90%D7%9C%D7%99" title="גז אידיאלי – Hebrew" lang="he" hreflang="he" data-title="גז אידיאלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%93%E1%83%94%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%98%E1%83%A0%E1%83%98" title="იდეალური აირი – Georgian" lang="ka" hreflang="ka" data-title="იდეალური აირი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_%D0%B3%D0%B0%D0%B7" title="Идеал газ – Kazakh" lang="kk" hreflang="kk" data-title="Идеал газ" 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%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%B4%D1%83%D1%83_%D0%B3%D0%B0%D0%B7" title="Идеалдуу газ – Kyrgyz" lang="ky" hreflang="ky" data-title="Идеалдуу газ" 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/Ide%C4%81la_g%C4%81ze" title="Ideāla gāze – Latvian" lang="lv" hreflang="lv" data-title="Ideāla gāze" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Idealiosios_dujos" title="Idealiosios dujos – Lithuanian" lang="lt" hreflang="lt" data-title="Idealiosios dujos" 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/Ide%C3%A1lis_g%C3%A1z" title="Ideális gáz – Hungarian" lang="hu" hreflang="hu" data-title="Ideális gáz" 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%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B3%D0%B0%D1%81" title="Идеален гас – Macedonian" lang="mk" hreflang="mk" data-title="Идеален гас" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B4%A6%E0%B5%BC%E0%B4%B6%E0%B4%B5%E0%B4%BE%E0%B4%A4%E0%B4%95%E0%B4%82" title="ആദർശവാതകം – Malayalam" lang="ml" hreflang="ml" data-title="ആദർശവാതകം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Gas_unggul" title="Gas unggul – Malay" lang="ms" hreflang="ms" data-title="Gas unggul" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ideaal_gas" title="Ideaal gas – Dutch" lang="nl" hreflang="nl" data-title="Ideaal gas" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%90%86%E6%83%B3%E6%B0%97%E4%BD%93" title="理想気体 – Japanese" lang="ja" hreflang="ja" data-title="理想気体" 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/Ideell_gass" title="Ideell gass – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ideell gass" 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/Ideell_gass" title="Ideell gass – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ideell gass" 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/Gas_perfiech" title="Gas perfiech – Occitan" lang="oc" hreflang="oc" data-title="Gas perfiech" 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/Ideal_gaz" title="Ideal gaz – Uzbek" lang="uz" hreflang="uz" data-title="Ideal gaz" 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-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%A7%D9%8A%DA%89%D9%8A%D8%A7%D9%84_%D8%BA%D8%A7%D8%B2%D8%A7%D8%AA" title="ايډيال غازات – Pashto" lang="ps" hreflang="ps" data-title="ايډيال غازات" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Gas_ideal" title="Gas ideal – Piedmontese" lang="pms" hreflang="pms" data-title="Gas ideal" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Gaz_doskona%C5%82y" title="Gaz doskonały – Polish" lang="pl" hreflang="pl" data-title="Gaz doskonały" 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/G%C3%A1s_ideal" title="Gás ideal – Portuguese" lang="pt" hreflang="pt" data-title="Gás ideal" 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/Gaz_ideal" title="Gaz ideal – Romanian" lang="ro" hreflang="ro" data-title="Gaz ideal" 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-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B3%D0%B0%D0%B7" title="Идеальный газ – Russian" lang="ru" hreflang="ru" data-title="Идеальный газ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gazi_ideal" title="Gazi ideal – Albanian" lang="sq" hreflang="sq" data-title="Gazi ideal" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ideal_gas" title="Ideal gas – Simple English" lang="en-simple" hreflang="en-simple" data-title="Ideal gas" 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/Ide%C3%A1lny_plyn" title="Ideálny plyn – Slovak" lang="sk" hreflang="sk" data-title="Ideálny plyn" 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/Idealni_plin" title="Idealni plin – Slovenian" lang="sl" hreflang="sl" data-title="Idealni plin" 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%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%BD%D0%B8_%D0%B3%D0%B0%D1%81" title="Идеални гас – Serbian" lang="sr" hreflang="sr" data-title="Идеални гас" 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/Idealni_gas" title="Idealni gas – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Idealni gas" 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/Ideaalikaasu" title="Ideaalikaasu – Finnish" lang="fi" hreflang="fi" data-title="Ideaalikaasu" 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/Ideal_gas" title="Ideal gas – Swedish" lang="sv" hreflang="sv" data-title="Ideal gas" 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%95%E0%AE%B0%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%B5%E0%AE%B3%E0%AE%BF%E0%AE%AE%E0%AE%AE%E0%AF%8D" title="கருத்தியல் வளிமம் – Tamil" lang="ta" hreflang="ta" data-title="கருத்தியல் வளிமம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D0%B0%D0%B7%D0%B8_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB%D3%A3" title="Гази идеалӣ – Tajik" lang="tg" hreflang="tg" data-title="Гази идеалӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0deal_gaz" title="İdeal gaz – Turkish" lang="tr" hreflang="tr" data-title="İdeal gaz" 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%86%D0%B4%D0%B5%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%B3%D0%B0%D0%B7" title="Ідеальний газ – Ukrainian" lang="uk" hreflang="uk" data-title="Ідеальний газ" 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/Kh%C3%AD_l%C3%BD_t%C6%B0%E1%BB%9Fng" title="Khí lý tưởng – Vietnamese" lang="vi" hreflang="vi" data-title="Khí lý tưởng" 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%90%86%E6%83%B3%E6%B0%94%E4%BD%93" title="理想气体 – Wu" lang="wuu" hreflang="wuu" data-title="理想气体" 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%90%86%E6%83%B3%E6%B0%A3%E9%AB%94" title="理想氣體 – Cantonese" lang="yue" hreflang="yue" data-title="理想氣體" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%90%86%E6%83%B3%E6%B0%A3%E9%AB%94" title="理想氣體 – Chinese" lang="zh" hreflang="zh" data-title="理想氣體" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q172280#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical model which approximates the behavior of real gases</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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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 thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a class="mw-selflink selflink">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Other</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>An <b>ideal gas</b> is a theoretical <a href="/wiki/Gas" title="Gas">gas</a> composed of many randomly moving <a href="/wiki/Point_particle" title="Point particle">point particles</a> that are not subject to <a href="/wiki/Intermolecular_force" title="Intermolecular force">interparticle interactions</a>.<sup id="cite_ref-tuckerman_1-0" class="reference"><a href="#cite_note-tuckerman-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The ideal gas concept is useful because it obeys the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a>, a simplified <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a>, and is amenable to analysis under <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. The requirement of zero interaction can often be relaxed if, for example, the interaction is <a href="/wiki/Elastic_collision" title="Elastic collision">perfectly elastic</a> or regarded as point-like collisions. </p><p>Under various conditions of temperature and pressure, many <a href="/wiki/Real_gas" title="Real gas">real gases</a> behave qualitatively like an ideal gas where the gas molecules (or atoms for <a href="/wiki/Monatomic_gas" title="Monatomic gas">monatomic gas</a>) play the role of the ideal particles. Many gases such as <a href="/wiki/Nitrogen" title="Nitrogen">nitrogen</a>, <a href="/wiki/Oxygen" title="Oxygen">oxygen</a>, <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a>, <a href="/wiki/Noble_gas" title="Noble gas">noble gases</a>, some heavier gases like <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a> and mixtures such as air, can be treated as ideal gases within reasonable tolerances<sup id="cite_ref-boles_2-0" class="reference"><a href="#cite_note-boles-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> over a considerable parameter range around <a href="/wiki/Standard_temperature_and_pressure" title="Standard temperature and pressure">standard temperature and pressure</a>. Generally, a gas behaves more like an ideal gas at higher <a href="/wiki/Temperature" title="Temperature">temperature</a> and lower <a href="/wiki/Pressure" title="Pressure">pressure</a>,<sup id="cite_ref-boles_2-1" class="reference"><a href="#cite_note-boles-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> as the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> due to intermolecular forces becomes less significant compared with the particles' <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>, and the size of the molecules becomes less significant compared to the empty space between them. One <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> of an ideal gas has a volume of <span class="nowrap"><span data-sort-value="6998227109546400000♠"></span>22.710<span style="margin-left:.25em;">954</span><span style="margin-left:.25em;">64</span>... L</span> (exact value based on <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision of the SI</a>)<sup id="cite_ref-mvol_3-0" class="reference"><a href="#cite_note-mvol-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> at standard temperature and pressure (a <a href="/wiki/Temperature" title="Temperature">temperature</a> of 273.15 <a href="/wiki/Kelvin" title="Kelvin">K</a> and an <a href="/wiki/Absolute_pressure" class="mw-redirect" title="Absolute pressure">absolute pressure</a> of exactly 10<sup>5</sup> <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">Pa</a>).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p><p>The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size becomes important. It also fails for most heavy gases, such as many <a href="/wiki/Refrigerant" title="Refrigerant">refrigerants</a>,<sup id="cite_ref-boles_2-2" class="reference"><a href="#cite_note-boles-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and for gases with strong intermolecular forces, notably <a href="/wiki/Water_vapor" title="Water vapor">water vapor</a>. At high pressures, the volume of a real gas is often considerably larger than that of an ideal gas. At low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point of low temperature and high pressure, real gases undergo a <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a>, such as to a <a href="/wiki/Liquid" title="Liquid">liquid</a> or a <a href="/wiki/Solid" title="Solid">solid</a>. The model of an ideal gas, however, does not describe or allow phase transitions. These must be modeled by more complex equations of state. The deviation from the ideal gas behavior can be described by a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless quantity</a>, the <a href="/wiki/Compressibility_factor" title="Compressibility factor">compressibility factor</a>, <span class="texhtml mvar" style="font-style:italic;">Z</span>. </p><p>The ideal gas model has been explored in both the <a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a> (as in "<a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory</a>") and in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (as a "<a href="/wiki/Gas_in_a_box" title="Gas in a box">gas in a box</a>"). The ideal gas model has also been used to model the behavior of electrons in a metal (in the <a href="/wiki/Drude_model" title="Drude model">Drude model</a> and the <a href="/wiki/Free_electron_model" title="Free electron model">free electron model</a>), and it is one of the most important models in statistical mechanics. </p><p>If the pressure of an ideal gas is reduced in a <a href="/wiki/Throttling_process" class="mw-redirect" title="Throttling process">throttling process</a> the temperature of the gas does not change. (If the pressure of a real gas is reduced in a throttling process, its temperature either falls or rises, depending on whether its <a href="/wiki/Joule%E2%80%93Thomson_(Kelvin)_coefficient" class="mw-redirect" title="Joule–Thomson (Kelvin) coefficient">Joule–Thomson coefficient</a> is positive or negative.) </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Types_of_ideal_gas">Types of ideal gas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=1" title="Edit section: Types of ideal gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are three basic classes of ideal gas:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="to verify this whole section (June 2017)">citation needed</span></a></i>]</sup> </p> <ul><li>the classical or <a href="/wiki/Maxwell%E2%80%93Boltzmann_statistics" title="Maxwell–Boltzmann statistics">Maxwell–Boltzmann</a> ideal gas,</li> <li>the ideal quantum <a href="/wiki/Bose_gas" title="Bose gas">Bose gas</a>, composed of <a href="/wiki/Boson" title="Boson">bosons</a>, and</li> <li>the ideal quantum <a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a>, composed of <a href="/wiki/Fermion" title="Fermion">fermions</a>.</li></ul> <p>The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas. Both are essentially the same, except that the classical thermodynamic ideal gas is based on classical <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, and certain thermodynamic parameters such as the <a href="/wiki/Entropy" title="Entropy">entropy</a> are only specified to within an undetermined additive constant. The ideal quantum Boltzmann gas overcomes this limitation by taking the limit of the quantum Bose gas and quantum Fermi gas in the limit of high temperature to specify these additive constants. The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the <a href="/wiki/Sackur%E2%80%93Tetrode_equation" title="Sackur–Tetrode equation">Sackur–Tetrode equation</a> for the entropy of an ideal gas and the <a href="/wiki/Saha_ionization_equation" title="Saha ionization equation">Saha ionization equation</a> for a weakly ionized <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Classical_thermodynamic_ideal_gas">Classical thermodynamic ideal gas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=2" title="Edit section: Classical thermodynamic ideal gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classical thermodynamic properties of an ideal gas can be described by two <a href="/wiki/Equation_of_state" title="Equation of state">equations of state</a>:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ideal_gas_law">Ideal gas law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=3" title="Edit section: Ideal gas law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Ideal_gas_law_relationships.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Ideal_gas_law_relationships.svg/200px-Ideal_gas_law_relationships.svg.png" decoding="async" width="200" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Ideal_gas_law_relationships.svg/300px-Ideal_gas_law_relationships.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Ideal_gas_law_relationships.svg/400px-Ideal_gas_law_relationships.svg.png 2x" data-file-width="512" data-file-height="341" /></a><figcaption>Relationships between <a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's</a>, <a href="/wiki/Charles%27s_law" title="Charles's law">Charles's</a>, <a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's</a>, <a href="/wiki/Avogadro%27s_law" title="Avogadro's law">Avogadro's</a>, <a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">combined</a> and <span class="nowrap"><a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas laws</a></span>, with the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> <span class="texhtml"><var style="padding-right: 1px;">k</var><sub><var style="padding-right: 1px;"></var></sub> = <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"><var style="padding-right: 1px;">R</var></span><span class="sr-only">/</span><span class="den"><var style="padding-right: 1px;">N</var><sub><var style="padding-right: 1px;">A</var></sub></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">n</var> <var style="padding-right: 1px;">R</var></span><span class="sr-only">/</span><span class="den"><var style="padding-right: 1px;">N</var></span></span>⁠</span></span> (in each law, <a href="/wiki/Physical_property" title="Physical property">properties</a> circled are variable and properties not circled are held constant)</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></div> <p>The <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> is the equation of state for an ideal gas, given by: <span class="mwe-math-element" data-qid="Q11432"><a href="/w/index.php?title=Special:MathWikibase&qid=Q11432" style="color:inherit;"><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=nRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV=nRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934032db2ac1f12624f85a90eeba651dcf4af377" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.426ex; height:2.176ex;" alt="{\displaystyle PV=nRT}"></a></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">P</span> is the <a href="/wiki/Pressure" title="Pressure">pressure</a></li> <li><span class="texhtml mvar" style="font-style:italic;">V</span> is the <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">volume</a></li> <li><span class="texhtml mvar" style="font-style:italic;">n</span> is the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> of the gas (in <a href="/wiki/Mole_(unit)" title="Mole (unit)">moles</a>)</li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a></li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>, which must be expressed in units consistent with those chosen for pressure, volume and temperature. For example, in <a href="/wiki/SI_Units" class="mw-redirect" title="SI Units">SI units</a> <span class="texhtml mvar" style="font-style:italic;">R</span> = 8.3145 <a href="/wiki/Joule" title="Joule">J</a>⋅<a href="/wiki/Kelvin" title="Kelvin">K</a><sup>−1</sup>⋅<a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a><sup>−1</sup> when pressure is expressed in <a href="/wiki/Pascals" class="mw-redirect" title="Pascals">pascals</a>, volume in cubic <a href="/wiki/Meters" class="mw-redirect" title="Meters">meters</a>, and absolute temperature in <a href="/wiki/Kelvin" title="Kelvin">kelvin</a>.</li></ul> <p>The ideal gas law is an extension of experimentally discovered <a href="/wiki/Gas_laws" title="Gas laws">gas laws</a>. It can also be derived from microscopic considerations. </p><p>Real <a href="/wiki/Fluid" title="Fluid">fluids</a> at low <a href="/wiki/Density" title="Density">density</a> and high <a href="/wiki/Temperature" title="Temperature">temperature</a> approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it <a href="/wiki/Condensation" title="Condensation">condenses</a> from a gas into a liquid or as it <a href="/wiki/Deposition_(phase_transition)" title="Deposition (phase transition)">deposits</a> from a gas into a solid. This deviation is expressed as a <a href="/wiki/Compressibility_factor" title="Compressibility factor">compressibility factor</a>. </p><p>This equation is derived from </p> <ul><li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\propto {\frac {1}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\propto {\frac {1}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af5d27fb746798dd4fcbd49c72bd41c496b7a32b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.467ex; height:5.176ex;" alt="{\displaystyle V\propto {\frac {1}{P}}}"></span>;</li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\propto T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∝</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\propto T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd10ef227c1e2f9bb313f1c9eeaefc6886c6376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.522ex; height:2.176ex;" alt="{\displaystyle V\propto T}"></span>;</li> <li><a href="/wiki/Avogadro%27s_law" title="Avogadro's law">Avogadro's law</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\propto n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∝</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\propto n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af26e9124bcaa5c95326a66187ace220c93545a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.28ex; height:2.176ex;" alt="{\displaystyle V\propto n}"></span>.</li></ul> <p>After combining three laws we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\propto {\frac {nT}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>T</mi> </mrow> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\propto {\frac {nT}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c5e3218cbef6aac72ae0d98ef683515c3e7c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.753ex; height:5.176ex;" alt="{\displaystyle V\propto {\frac {nT}{P}}}"></span></dd></dl> <p>That is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=R\left({\frac {nT}{P}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>T</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=R\left({\frac {nT}{P}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b297e7b6ede9d8e68add90d0d2f36e878a5462bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.325ex; height:6.176ex;" alt="{\displaystyle V=R\left({\frac {nT}{P}}\right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV=nRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV=nRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934032db2ac1f12624f85a90eeba651dcf4af377" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.426ex; height:2.176ex;" alt="{\displaystyle PV=nRT}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Internal_energy">Internal energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=4" title="Edit section: Internal energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The other equation of state of an ideal gas must express <a href="/wiki/Joule%E2%80%93Thomson_effect#Joule's_second_law" title="Joule–Thomson effect">Joule's second law</a>, that the internal energy of a fixed mass of ideal gas is a function only of its temperature, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=U(n,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=U(n,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff4ba3771f82ae89bec9c7252f098cdcf085d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.538ex; height:2.843ex;" alt="{\displaystyle U=U(n,T)}"></span>. For the present purposes it is convenient to postulate an exemplary version of this law by writing: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\hat {c}}_{V}nRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\hat {c}}_{V}nRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8694bd77c197dcf14e9b4a3ece8decde872b9797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.464ex; height:2.509ex;" alt="{\displaystyle U={\hat {c}}_{V}nRT}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">U</span> is the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a></li> <li><span class="texhtml mvar" style="font-style:italic;">ĉ<sub>V</sub></span> is the dimensionless specific <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> at constant volume, approximately <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> for a <a href="/wiki/Monatomic_gas" title="Monatomic gas">monatomic gas</a>, <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">2</span></span>⁠</span> for <a href="/wiki/Diatomic" class="mw-redirect" title="Diatomic">diatomic</a> gas, and 3 for non-linear molecules if we treat translations and rotations classically and ignore quantum vibrational contribution and electronic excitation. These formulas arise from application of the classical <a href="/wiki/Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a> to the translational and rotational degrees of freedom. <sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <p>That <span class="texhtml mvar" style="font-style:italic;">U</span> for an ideal gas depends only on temperature is a consequence of the ideal gas law, although in the general case <span class="texhtml mvar" style="font-style:italic;">ĉ<sub>V</sub></span> depends on temperature and an integral is needed to compute <span class="texhtml mvar" style="font-style:italic;">U</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Microscopic_model">Microscopic model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=5" title="Edit section: Microscopic model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to switch from macroscopic quantities (left hand side of the following equation) to microscopic ones (right hand side), we use </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nR=Nk_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>R</mi> <mo>=</mo> <mi>N</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nR=Nk_{\mathrm {B} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb7b7acce8935e19c3ada9842df97b60a80469a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.928ex; height:2.509ex;" alt="{\displaystyle nR=Nk_{\mathrm {B} }}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><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> is the number of gas particles</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}"></span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> (<span class="nowrap"><span data-sort-value="6977138099999999999♠"></span>1.381<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−23</sup> J·K<sup>−1</sup></span>).</li></ul> <p>The probability distribution of particles by velocity or energy is given by the <a href="/wiki/Maxwell_speed_distribution" class="mw-redirect" title="Maxwell speed distribution">Maxwell speed distribution</a>. </p><p>The ideal gas model depends on the following assumptions: </p> <ul><li>The molecules of the gas are indistinguishable, small, hard spheres</li> <li>All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)</li> <li>Newton's laws apply</li> <li>The average distance between molecules is much larger than the size of the molecules</li> <li>The molecules are constantly moving in random directions with a distribution of speeds</li> <li>There are no attractive or repulsive forces between the molecules apart from those that determine their point-like collisions</li> <li>The only forces between the gas molecules and the surroundings are those that determine the point-like collisions of the molecules with the walls</li> <li>In the simplest case, there are no long-range forces between the molecules of the gas and the surroundings.</li></ul> <p>The assumption of spherical particles is necessary so that there are no rotational modes allowed, unlike in a diatomic gas. The following three assumptions are very related: molecules are hard, collisions are elastic, and there are no inter-molecular forces. The assumption that the space between particles is much larger than the particles themselves is of paramount importance, and explains why the ideal gas approximation fails at high pressures. </p> <div class="mw-heading mw-heading2"><h2 id="Heat_capacity">Heat capacity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=6" title="Edit section: Heat capacity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dimensionless <a href="/wiki/Heat_capacity#Heat_capacity_of_compressible_bodies" title="Heat capacity">heat capacity</a><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/MOS:BROKENSECTIONLINKS" class="mw-redirect" title="MOS:BROKENSECTIONLINKS"><span title="The anchor (Heat capacity of compressible bodies) has been deleted. (2025-01-14)">broken anchor</span></a></i>]</sup> at constant volume is generally defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}_{V}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{V}={\frac {1}{nR}}\left({\frac {\partial U}{\partial T}}\right)_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {c}}_{V}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{V}={\frac {1}{nR}}\left({\frac {\partial U}{\partial T}}\right)_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89907bd1edbf4b7ff017443ad8f406ca7e93dcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.172ex; height:6.176ex;" alt="{\displaystyle {\hat {c}}_{V}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{V}={\frac {1}{nR}}\left({\frac {\partial U}{\partial T}}\right)_{V}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">S</span> is the <a href="/wiki/Entropy" title="Entropy">entropy</a>. This quantity is generally a function of temperature due to intermolecular and intramolecular forces, but for moderate temperatures it is approximately constant. Specifically, the <a href="/wiki/Equipartition_Theorem" class="mw-redirect" title="Equipartition Theorem">Equipartition Theorem</a> predicts that the constant for a monatomic gas is <span class="texhtml mvar" style="font-style:italic;"><i>ĉ<sub>V</sub></i></span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> while for a diatomic gas it is <span class="texhtml mvar" style="font-style:italic;"><i>ĉ<sub>V</sub></i></span> = <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">2</span></span>⁠</span> if vibrations are neglected (which is often an excellent approximation). Since the heat capacity depends on the atomic or molecular nature of the gas, macroscopic measurements on heat capacity provide useful information on the microscopic structure of the molecules. </p><p>The dimensionless heat capacity at constant pressure of an ideal gas is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}_{P}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{P}={\frac {1}{nR}}\left({\frac {\partial H}{\partial T}}\right)_{P}={\hat {c}}_{V}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mi>T</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {c}}_{P}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{P}={\frac {1}{nR}}\left({\frac {\partial H}{\partial T}}\right)_{P}={\hat {c}}_{V}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90f8213291ab3b5dabff40adc020acbf823c0fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.253ex; height:6.176ex;" alt="{\displaystyle {\hat {c}}_{P}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{P}={\frac {1}{nR}}\left({\frac {\partial H}{\partial T}}\right)_{P}={\hat {c}}_{V}+1}"></span></dd></dl> <p>where <span class="texhtml"><i>H</i> = <i>U</i> + <i>PV</i></span> is the <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> of the gas. </p><p>Sometimes, a distinction is made between an ideal gas, where <span class="texhtml mvar" style="font-style:italic;"><i>ĉ<sub>V</sub></i></span> and <span class="texhtml mvar" style="font-style:italic;"><i>ĉ<sub>P</sub></i></span> could vary with temperature, and a <a href="/wiki/Perfect_gas" title="Perfect gas">perfect gas</a>, for which this is not the case. </p><p>The ratio of the constant volume and constant pressure heat capacity is the <a href="/wiki/Adiabatic_index" class="mw-redirect" title="Adiabatic index">adiabatic index</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cc143f85a25580e3eb9a5aec1840d4226c7362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.7ex; height:5.009ex;" alt="{\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}"></span></dd></dl> <p>For air, which is a mixture of gases that are mainly diatomic (nitrogen and oxygen), this ratio is often assumed to be 7/5, the value predicted by the classical <a href="/wiki/Equipartition_theorem" title="Equipartition theorem">Equipartition Theorem</a> for diatomic gases. </p> <div class="mw-heading mw-heading2"><h2 id="Entropy">Entropy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=7" title="Edit section: Entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the results of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> only, we can go a long way in determining the expression for the <a href="/wiki/Entropy" title="Entropy">entropy</a> of an ideal gas. This is an important step since, according to the theory of <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potentials</a>, if we can express the entropy as a function of <span class="texhtml mvar" style="font-style:italic;">U</span> (<span class="texhtml mvar" style="font-style:italic;">U</span> is a thermodynamic potential), volume <span class="texhtml mvar" style="font-style:italic;">V</span> and the number of particles <span class="texhtml mvar" style="font-style:italic;">N</span>, then we will have a complete statement of the thermodynamic behavior of the ideal gas. We will be able to derive both the ideal gas law and the expression for internal energy from it. </p><p>Since the entropy is an <a href="/wiki/Exact_differential" title="Exact differential">exact differential</a>, using the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>, the change in entropy when going from a reference state 0 to some other state with entropy <span class="texhtml mvar" style="font-style:italic;">S</span> may be written as <span class="texhtml">Δ<i>S</i></span> where: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S=\int _{S_{0}}^{S}dS=\int _{T_{0}}^{T}\left({\frac {\partial S}{\partial T}}\right)_{V}\!dT+\int _{V_{0}}^{V}\left({\frac {\partial S}{\partial V}}\right)_{T}\!dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msubsup> <mi>d</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\int _{S_{0}}^{S}dS=\int _{T_{0}}^{T}\left({\frac {\partial S}{\partial T}}\right)_{V}\!dT+\int _{V_{0}}^{V}\left({\frac {\partial S}{\partial V}}\right)_{T}\!dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00335177c45a3d22c98fac58f9d05d83a614f903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.46ex; height:6.676ex;" alt="{\displaystyle \Delta S=\int _{S_{0}}^{S}dS=\int _{T_{0}}^{T}\left({\frac {\partial S}{\partial T}}\right)_{V}\!dT+\int _{V_{0}}^{V}\left({\frac {\partial S}{\partial V}}\right)_{T}\!dV}"></span></dd></dl> <p>where the reference variables may be functions of the <a href="/wiki/Number_of_particles" class="mw-redirect" title="Number of particles">number of particles</a> <span class="texhtml mvar" style="font-style:italic;">N</span>. Using the definition of the <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> at constant volume for the first differential and the appropriate <a href="/wiki/Maxwell_relation" class="mw-redirect" title="Maxwell relation">Maxwell relation</a> for the second we have: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S=\int _{T_{0}}^{T}{\frac {C_{V}}{T}}\,dT+\int _{V_{0}}^{V}\left({\frac {\partial P}{\partial T}}\right)_{V}dV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>d</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\int _{T_{0}}^{T}{\frac {C_{V}}{T}}\,dT+\int _{V_{0}}^{V}\left({\frac {\partial P}{\partial T}}\right)_{V}dV.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d15988e736c670f51c0c74437d501d10395dbd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.662ex; height:6.509ex;" alt="{\displaystyle \Delta S=\int _{T_{0}}^{T}{\frac {C_{V}}{T}}\,dT+\int _{V_{0}}^{V}\left({\frac {\partial P}{\partial T}}\right)_{V}dV.}"></span></dd></dl> <p>Expressing <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> in terms of <span class="texhtml mvar" style="font-style:italic;"><i>ĉ<sub>V</sub></i></span> as developed in the above section, differentiating the ideal gas equation of state, and integrating yields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S={\hat {c}}_{V}Nk\ln \left({\frac {T}{T_{0}}}\right)+Nk\ln \left({\frac {V}{V_{0}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>N</mi> <mi>k</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S={\hat {c}}_{V}Nk\ln \left({\frac {T}{T_{0}}}\right)+Nk\ln \left({\frac {V}{V_{0}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04124fadd982998c9f7dd5ee16aa5eedf024b2fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.7ex; height:6.176ex;" alt="{\displaystyle \Delta S={\hat {c}}_{V}Nk\ln \left({\frac {T}{T_{0}}}\right)+Nk\ln \left({\frac {V}{V_{0}}}\right)}"></span></dd></dl> <p>which implies that the entropy may be expressed as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=Nk\ln \left({\frac {VT^{{\hat {c}}_{V}}}{f(N)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>N</mi> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=Nk\ln \left({\frac {VT^{{\hat {c}}_{V}}}{f(N)}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0211ffae76a92ad3e7c5c597c9fd98584745b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.597ex; height:7.509ex;" alt="{\displaystyle S=Nk\ln \left({\frac {VT^{{\hat {c}}_{V}}}{f(N)}}\right)}"></span></dd></dl> <p>where all constants have been incorporated into the logarithm as <span class="texhtml"><i>f</i>(<i>N</i>)</span> which is some function of the particle number <span class="texhtml mvar" style="font-style:italic;">N</span> having the same dimensions as <span class="texhtml mvar" style="font-style:italic;">VT<sup>ĉ<sub>V</sub></sup></span> in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy be extensive. This will mean that when the extensive parameters (<span class="texhtml mvar" style="font-style:italic;">V</span> and <span class="texhtml mvar" style="font-style:italic;">N</span>) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(T,aV,aN)=aS(T,V,N).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>a</mi> <mi>V</mi> <mo>,</mo> <mi>a</mi> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(T,aV,aN)=aS(T,V,N).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11fcbd3e230c8b7aa3ccb760c25fee721712088d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.162ex; height:2.843ex;" alt="{\displaystyle S(T,aV,aN)=aS(T,V,N).}"></span></dd></dl> <p>From this we find an equation for the function <span class="texhtml"><i>f</i>(<i>N</i>)</span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle af(N)=f(aN).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>N</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle af(N)=f(aN).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b95c1ec3a824247fdb10264aac93652146a03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.508ex; height:2.843ex;" alt="{\displaystyle af(N)=f(aN).}"></span></dd></dl> <p>Differentiating this with respect to <span class="texhtml mvar" style="font-style:italic;">a</span>, setting <span class="texhtml mvar" style="font-style:italic;">a</span> equal to 1, and then solving the differential equation yields <span class="texhtml"><i>f</i>(<i>N</i>)</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(N)=\Phi N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(N)=\Phi N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec474447a6d5a4098a7cb015f6cabd655526c6e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.992ex; height:2.843ex;" alt="{\displaystyle f(N)=\Phi N}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">Φ</span> may vary for different gases, but will be independent of the thermodynamic state of the gas. It will have the dimensions of <span class="texhtml"><i>VT<sup>ĉ<sub>V</sub></sup></i>/<i>N</i></span>. Substituting into the equation for the entropy: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S}{Nk}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mrow> <mi>N</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi>N</mi> <mi mathvariant="normal">Φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S}{Nk}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc42f1cae2b044dab651e4f6ab964cb0a9fb010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.194ex; height:7.509ex;" alt="{\displaystyle {\frac {S}{Nk}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right).}"></span></dd></dl> <p>and using the expression for the internal energy of an ideal gas, the entropy may be written: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S}{Nk}}=\ln \left[{\frac {V}{N}}\,\left({\frac {U}{{\hat {c}}_{V}kN}}\right)^{{\hat {c}}_{V}}\,{\frac {1}{\Phi }}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mrow> <mi>N</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>N</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>k</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Φ</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S}{Nk}}=\ln \left[{\frac {V}{N}}\,\left({\frac {U}{{\hat {c}}_{V}kN}}\right)^{{\hat {c}}_{V}}\,{\frac {1}{\Phi }}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d5055ff166a2ab540fb032ec1d0ace534b5966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.741ex; height:7.676ex;" alt="{\displaystyle {\frac {S}{Nk}}=\ln \left[{\frac {V}{N}}\,\left({\frac {U}{{\hat {c}}_{V}kN}}\right)^{{\hat {c}}_{V}}\,{\frac {1}{\Phi }}\right]}"></span></dd></dl> <p>Since this is an expression for entropy in terms of <span class="texhtml mvar" style="font-style:italic;">U</span>, <span class="texhtml mvar" style="font-style:italic;">V</span>, and <span class="texhtml mvar" style="font-style:italic;">N</span>, it is a fundamental equation from which all other properties of the ideal gas may be derived. </p><p>This is about as far as we can go using thermodynamics alone. Note that the above equation is flawed – as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the <a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">third law of thermodynamics</a>. In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero. This is unphysical. The above equation is a good approximation only when the argument of the logarithm is much larger than unity – the concept of an ideal gas breaks down at low values of <span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">V</span><span class="sr-only">/</span><span class="den">N</span></span>⁠</span></span>. Nevertheless, there will be a "best" value of the constant in the sense that the predicted entropy is as close as possible to the actual entropy, given the flawed assumption of ideality. A quantum-mechanical derivation of this constant is developed in the derivation of the <a href="/wiki/Sackur%E2%80%93Tetrode_equation" title="Sackur–Tetrode equation">Sackur–Tetrode equation</a> which expresses the entropy of a monatomic (<span class="texhtml"><i>ĉ<sub>V</sub></i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>) ideal gas. In the Sackur–Tetrode theory the constant depends only upon the mass of the gas particle. The Sackur–Tetrode equation also suffers from a divergent entropy at absolute zero, but is a good approximation for the entropy of a monatomic ideal gas for high enough temperatures. </p><p>An alternative way of expressing the change in entropy: <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 {\Delta S}{Nk{\hat {c}}_{V}}}=\ln \left({\frac {P}{P_{0}}}\right)+\gamma \ln \left({\frac {V}{V_{0}}}\right)=\ln \left({\frac {PV^{\gamma }}{P_{0}V_{0}^{\gamma }}}\right)\implies PV^{\gamma }=\mathrm {const.} \;{\text{for isentropic process}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>S</mi> </mrow> <mrow> <mi>N</mi> <mi>k</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <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> <mi>γ</mi> <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> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <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> <mo>.</mo> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for isentropic process</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta S}{Nk{\hat {c}}_{V}}}=\ln \left({\frac {P}{P_{0}}}\right)+\gamma \ln \left({\frac {V}{V_{0}}}\right)=\ln \left({\frac {PV^{\gamma }}{P_{0}V_{0}^{\gamma }}}\right)\implies PV^{\gamma }=\mathrm {const.} \;{\text{for isentropic process}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cead0227c301ac00e6bf96442d7a4ad8044a428" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.902ex; height:7.509ex;" alt="{\displaystyle {\frac {\Delta S}{Nk{\hat {c}}_{V}}}=\ln \left({\frac {P}{P_{0}}}\right)+\gamma \ln \left({\frac {V}{V_{0}}}\right)=\ln \left({\frac {PV^{\gamma }}{P_{0}V_{0}^{\gamma }}}\right)\implies PV^{\gamma }=\mathrm {const.} \;{\text{for isentropic process}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Thermodynamic_potentials">Thermodynamic potentials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=8" title="Edit section: Thermodynamic potentials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Thermodynamic potential</a></div> <p>Expressing the entropy as a function of <span class="texhtml mvar" style="font-style:italic;">T</span>, <span class="texhtml mvar" style="font-style:italic;">V</span>, and <span class="texhtml mvar" style="font-style:italic;">N</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S}{kN}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mrow> <mi>k</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi>N</mi> <mi mathvariant="normal">Φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S}{kN}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bde8bc2c00188179aa28c080d185906fb4635fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.547ex; height:7.509ex;" alt="{\displaystyle {\frac {S}{kN}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)}"></span></dd></dl> <p>The <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> of the ideal gas is calculated from the corresponding equation of state (see <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\left({\frac {\partial G}{\partial N}}\right)_{T,P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>,</mo> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\left({\frac {\partial G}{\partial N}}\right)_{T,P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfc9e00203eee9816d9e4834cb3dff4d7492b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.22ex; height:6.509ex;" alt="{\displaystyle \mu =\left({\frac {\partial G}{\partial N}}\right)_{T,P}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> and is equal to <span class="texhtml"><i>U</i> + <i>PV</i> − <i>TS</i></span> so that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (T,P)=kT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi>P</mi> <mi mathvariant="normal">Φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (T,P)=kT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ce612e4aec239c9262d365f1dedd3a13a06da1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.977ex; height:7.509ex;" alt="{\displaystyle \mu (T,P)=kT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}"></span></dd></dl> <p>The chemical potential is usually referenced to the potential at some standard pressure <i>P<sup>o</sup></i> so that, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{o}(T)=\mu (T,P^{o})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{o}(T)=\mu (T,P^{o})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f7d5b8ad72c6780c3fd8cbfdc678d0992e2faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.708ex; height:2.843ex;" alt="{\displaystyle \mu ^{o}(T)=\mu (T,P^{o})}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (T,P)=\mu ^{o}(T)+kT\ln \left({\frac {P}{Po}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>k</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mi>P</mi> <mi>o</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (T,P)=\mu ^{o}(T)+kT\ln \left({\frac {P}{Po}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aebd9dd0984986678a911c4672fa6f2476213ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.747ex; height:6.176ex;" alt="{\displaystyle \mu (T,P)=\mu ^{o}(T)+kT\ln \left({\frac {P}{Po}}\right)}"></span></dd></dl> <p>For a mixture (<i>j</i>=1,2,...) of ideal gases, each at partial pressure <i>P<sub>j</sub></i>, it can be shown that the chemical potential <i>μ<sub>j</sub></i> will be given by the above expression with the pressure <i>P</i> replaced by <i>P<sub>j</sub></i>. </p><p>The thermodynamic potentials for an ideal gas can now be written as functions of <span class="texhtml mvar" style="font-style:italic;">T</span>, <span class="texhtml mvar" style="font-style:italic;">V</span>, and <span class="texhtml mvar" style="font-style:italic;">N</span> as: </p> <dl><dd><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 U\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/435be01e77ffd4af9dd1ea8152a657958eecdf1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.176ex;" alt="{\displaystyle U\,}"></span> </td> <td> </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 ={\hat {c}}_{V}NkT\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\hat {c}}_{V}NkT\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301746543211030b3f49f33ee708641463870744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.539ex; height:2.509ex;" alt="{\displaystyle ={\hat {c}}_{V}NkT\,}"></span> </td></tr> <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 A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}"></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 =U-TS\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =U-TS\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461f1fdb844602487f4968510ba44b3968117a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.599ex; height:2.343ex;" alt="{\displaystyle =U-TS\,}"></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 =\mu N-NkT\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>μ</mi> <mi>N</mi> <mo>−</mo> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\mu N-NkT\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b55d53dcf124c540ac6afcd8175359e568931d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.057ex; height:2.676ex;" alt="{\displaystyle =\mu N-NkT\,}"></span> </td></tr> <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 H\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33810a0db43048d22f06d52f257b8e49e4e977a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.451ex; height:2.176ex;" alt="{\displaystyle H\,}"></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 =U+PV\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>P</mi> <mi>V</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =U+PV\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a007d40f065c354ed58cb5c41534336cb67c29f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.996ex; height:2.343ex;" alt="{\displaystyle =U+PV\,}"></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 ={\hat {c}}_{P}NkT\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\hat {c}}_{P}NkT\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b646a944001cb52e9849f7524f34572df963cc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.51ex; height:2.509ex;" alt="{\displaystyle ={\hat {c}}_{P}NkT\,}"></span> </td></tr> <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 G\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f6c4d01b8be460cde4156d1fde6af720f8ff12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.214ex; height:2.176ex;" alt="{\displaystyle G\,}"></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 =U+PV-TS\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>P</mi> <mi>V</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =U+PV-TS\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d498465460a1c6b1bec678777b23c76e94a3ccb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.972ex; height:2.343ex;" alt="{\displaystyle =U+PV-TS\,}"></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 =\mu N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>μ</mi> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\mu N\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391af324f730c383f86a26fcd47047d36864bfd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.306ex; height:2.676ex;" alt="{\displaystyle =\mu N\,}"></span> </td></tr></tbody></table></dd></dl> <p>where, as before, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}_{P}={\hat {c}}_{V}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {c}}_{P}={\hat {c}}_{V}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93bf596c3992088da69ae8044f5a33f9bb1ed600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.647ex; height:2.509ex;" alt="{\displaystyle {\hat {c}}_{P}={\hat {c}}_{V}+1}"></span>.</dd></dl> <p>The most informative way of writing the potentials is in terms of their natural variables, since each of these equations can be used to derive all of the other thermodynamic variables of the system. In terms of their natural variables, the thermodynamic potentials of a single-species ideal gas are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V,N)={\hat {c}}_{V}Nk\left({\frac {N\Phi }{V}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>N</mi> <mi>k</mi> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V,N)={\hat {c}}_{V}Nk\left({\frac {N\Phi }{V}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c8e90cb797e80ad649074e8de16275bae9d365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.088ex; height:7.009ex;" alt="{\displaystyle U(S,V,N)={\hat {c}}_{V}Nk\left({\frac {N\Phi }{V}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{V}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(T,V,N)=NkT\left({\hat {c}}_{V}-\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)\right)}"> <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>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi>N</mi> <mi mathvariant="normal">Φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V,N)=NkT\left({\hat {c}}_{V}-\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6972801195182be597dc3502e57b54005659de5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.151ex; height:7.509ex;" alt="{\displaystyle A(T,V,N)=NkT\left({\hat {c}}_{V}-\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)\right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S,P,N)={\hat {c}}_{P}Nk\left({\frac {P\Phi }{k}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{P}}}"> <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>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>N</mi> <mi>k</mi> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,P,N)={\hat {c}}_{P}Nk\left({\frac {P\Phi }{k}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2d5bca49c53299fc169b38ceb7d5e9019fd62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.955ex; height:7.009ex;" alt="{\displaystyle H(S,P,N)={\hat {c}}_{P}Nk\left({\frac {P\Phi }{k}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{P}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T,P,N)=NkT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}"> <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>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mi>P</mi> <mi mathvariant="normal">Φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,P,N)=NkT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce6e7b700c760b1ef826dd13df2a7fc1ef0f7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.563ex; height:7.509ex;" alt="{\displaystyle G(T,P,N)=NkT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}"></span></dd></dl> <p>In <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, the relationship between the <a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a> and the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> is fundamental, and is used to calculate the <a href="/wiki/Thermodynamic_properties" class="mw-redirect" title="Thermodynamic properties">thermodynamic properties</a> of matter; see <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29">configuration integral</a> for more details. </p> <div class="mw-heading mw-heading2"><h2 id="Speed_of_sound">Speed of sound</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=9" title="Edit section: Speed of sound"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Speed_of_sound#Equations" title="Speed of sound">speed of sound</a> in an ideal gas is given by the Newton-Laplace formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\text{sound}}={\sqrt {\frac {K_{s}}{\rho }}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sound</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\text{sound}}={\sqrt {\frac {K_{s}}{\rho }}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51223561696e7e83af1be01b0020d716cb81031d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.079ex; height:7.676ex;" alt="{\displaystyle c_{\text{sound}}={\sqrt {\frac {K_{s}}{\rho }}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}},}"></span></dd></dl> <p>where the isentropic <a href="/wiki/Bulk_modulus" title="Bulk modulus">Bulk modulus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db909b0b88fc7eb4455c152ca1d2723bb6e512f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.248ex; height:6.176ex;" alt="{\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}.}"></span> </p><p>For an isentropic process of an ideal gas, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV^{\gamma }=\mathrm {const} \Rightarrow P\propto \left({\frac {1}{V}}\right)^{\gamma }\propto \rho ^{\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"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">⇒</mo> <mi>P</mi> <mo>∝</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>V</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>∝</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV^{\gamma }=\mathrm {const} \Rightarrow P\propto \left({\frac {1}{V}}\right)^{\gamma }\propto \rho ^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b340120d47cf5a65bb3a59efde9ee896808192b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.247ex; height:6.176ex;" alt="{\displaystyle PV^{\gamma }=\mathrm {const} \Rightarrow P\propto \left({\frac {1}{V}}\right)^{\gamma }\propto \rho ^{\gamma }}"></span>, therefore </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\text{sound}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {\gamma P}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sound</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>γ</mi> <mi>P</mi> </mrow> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>γ</mi> <mi>R</mi> <mi>T</mi> </mrow> <mi>M</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\text{sound}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {\gamma P}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b564246e7d81e82aba6accbd8a930f440be8bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.384ex; height:7.843ex;" alt="{\displaystyle c_{\text{sound}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {\gamma P}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}}"></span></dd></dl> <p>Here, </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">γ</span> is the <a href="/wiki/Adiabatic_index" class="mw-redirect" title="Adiabatic index">adiabatic index</a> (<span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">ĉ<sub>P</sub></span><span class="sr-only">/</span><span class="den">ĉ<sub>V</sub></span></span>⁠</span></span>)</li> <li><span class="texhtml mvar" style="font-style:italic;">s</span> is the <a href="/wiki/Entropy" title="Entropy">entropy</a> per particle of the gas.</li> <li><span class="texhtml mvar" style="font-style:italic;">ρ</span> is the <a href="/wiki/Mass_density" class="mw-redirect" title="Mass density">mass density</a> of the gas.</li> <li><span class="texhtml mvar" style="font-style:italic;">P</span> is the <a href="/wiki/Pressure" title="Pressure">pressure</a> of the gas.</li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Universal_gas_constant" class="mw-redirect" title="Universal gas constant">universal gas constant</a></li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the <a href="/wiki/Temperature" title="Temperature">temperature</a></li> <li><span class="texhtml mvar" style="font-style:italic;">M</span> is the <a href="/wiki/Molar_mass" title="Molar mass">molar mass</a> of the gas.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Table_of_ideal_gas_equations">Table of ideal gas equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=10" title="Edit section: Table of ideal gas equations"><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/Table_of_thermodynamic_equations#Ideal_gas" title="Table of thermodynamic equations">Table of thermodynamic equations § Ideal gas</a></div> <div class="mw-heading mw-heading2"><h2 id="Ideal_quantum_gases">Ideal quantum gases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=11" title="Edit section: Ideal quantum gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the above-mentioned <a href="/wiki/Sackur%E2%80%93Tetrode_equation" title="Sackur–Tetrode equation">Sackur–Tetrode equation</a>, the best choice of the entropy constant was found to be proportional to the quantum <a href="/wiki/Thermal_wavelength" class="mw-redirect" title="Thermal wavelength">thermal wavelength</a> of a particle, and the point at which the argument of the logarithm becomes zero is roughly equal to the point at which the average distance between particles becomes equal to the thermal wavelength. In fact, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum</a> theory itself predicts the same thing. Any gas behaves as an ideal gas at high enough temperature and low enough density, but at the point where the Sackur–Tetrode equation begins to break down, the gas will begin to behave as a quantum gas, composed of either <a href="/wiki/Boson" title="Boson">bosons</a> or <a href="/wiki/Fermion" title="Fermion">fermions</a>. (See the <a href="/wiki/Gas_in_a_box" title="Gas in a box">gas in a box</a> article for a derivation of the ideal quantum gases, including the ideal Boltzmann gas.) </p><p>Gases tend to behave as an ideal gas over a wider range of pressures when the temperature reaches the <a href="/wiki/Boyle_temperature" title="Boyle temperature">Boyle temperature</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Ideal_Boltzmann_gas">Ideal Boltzmann gas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=12" title="Edit section: Ideal Boltzmann gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ideal Boltzmann gas yields the same results as the classical thermodynamic gas, but makes the following identification for the undetermined constant <span class="texhtml mvar" style="font-style:italic;">Φ</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ={\frac {T^{\frac {3}{2}}\Lambda ^{3}}{g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mi>g</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ={\frac {T^{\frac {3}{2}}\Lambda ^{3}}{g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763e961f985b9b54c3682ca854ef240db9deecc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.736ex; height:7.176ex;" alt="{\displaystyle \Phi ={\frac {T^{\frac {3}{2}}\Lambda ^{3}}{g}}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">Λ</span> is the <a href="/wiki/Thermal_de_Broglie_wavelength" title="Thermal de Broglie wavelength">thermal de Broglie wavelength</a> of the gas and <span class="texhtml mvar" style="font-style:italic;">g</span> is the <a href="/wiki/Degenerate_energy_level" class="mw-redirect" title="Degenerate energy level">degeneracy</a> of states. </p> <div class="mw-heading mw-heading3"><h3 id="Ideal_Bose_and_Fermi_gases">Ideal Bose and Fermi gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ideal_gas&action=edit&section=13" title="Edit section: Ideal Bose and Fermi gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An ideal <a href="/wiki/Bose_gas" title="Bose gas">gas of bosons</a> (e.g. a <a href="/wiki/Photon_gas" title="Photon gas">photon gas</a>) will be governed by <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a> and the distribution of energy will be in the form of a <a href="/wiki/Bose%E2%80%93Einstein_distribution" class="mw-redirect" title="Bose–Einstein distribution">Bose–Einstein distribution</a>. An ideal <a href="/wiki/Fermi_gas" title="Fermi gas">gas of fermions</a> will be governed by <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac statistics</a> and the distribution of energy will be in the form of a <a href="/wiki/Fermi%E2%80%93Dirac_distribution" class="mw-redirect" title="Fermi–Dirac distribution">Fermi–Dirac distribution</a>. </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=Ideal_gas&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Compressibility_factor" title="Compressibility factor">Compressibility factor</a> – Correction factor which describes the deviation of a real gas from ideal gas behavior</li> <li><a href="/wiki/Dynamical_billiards" title="Dynamical billiards">Dynamical billiards</a> – Dynamical system abstract an ideal game of billiards, with elastic collisions off boundaries – billiard balls as a model of an ideal gas</li> <li><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a> – Thermodynamics<span style="display:none" class="category-spaceless-annotation">Pages displaying short descriptions with no spaces</span></li> <li><a href="/wiki/Scale-free_ideal_gas" title="Scale-free ideal gas">Scale-free ideal gas</a> – Ideal gas with no physical scale</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=Ideal_gas&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Notes</dt> <dd></dd></dl> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Until 1982, STP was defined as a temperature of 273.15 <a href="/wiki/Kelvin" title="Kelvin">K</a> and an <a href="/wiki/Absolute_pressure" class="mw-redirect" title="Absolute pressure">absolute pressure</a> of exactly 1 <a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atm</a>. The volume of one mole of an ideal gas at this temperature and pressure is 22.413962(13) litres.<sup id="cite_ref-mvolstd_4-0" class="reference"><a href="#cite_note-mvolstd-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> IUPAC recommends that the former use of this definition should be discontinued;<sup id="cite_ref-IUPAC_5-0" class="reference"><a href="#cite_note-IUPAC-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> however, some textbooks still use these old values.</span> </li> </ol></div></div> <dl><dt>References</dt> <dd></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-tuckerman-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-tuckerman_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTuckerman2010" class="citation book cs1">Tuckerman, Mark E. (2010). <i>Statistical Mechanics: Theory and Molecular Simulation</i> (1st ed.). p. 87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852526-4" title="Special:BookSources/978-0-19-852526-4"><bdi>978-0-19-852526-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Mechanics%3A+Theory+and+Molecular+Simulation&rft.pages=87&rft.edition=1st&rft.date=2010&rft.isbn=978-0-19-852526-4&rft.aulast=Tuckerman&rft.aufirst=Mark+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> <li id="cite_note-boles-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-boles_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-boles_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-boles_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCengelBoles2001" class="citation book cs1">Cengel, Yunus A.; Boles, Michael A. (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/thermodynamicsen00ceng_0/page/89"><i>Thermodynamics: An Engineering Approach</i></a></span> (4th ed.). McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/thermodynamicsen00ceng_0/page/89">89</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-238332-1" title="Special:BookSources/0-07-238332-1"><bdi>0-07-238332-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics%3A+An+Engineering+Approach&rft.pages=89&rft.edition=4th&rft.pub=McGraw-Hill&rft.date=2001&rft.isbn=0-07-238332-1&rft.aulast=Cengel&rft.aufirst=Yunus+A.&rft.au=Boles%2C+Michael+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthermodynamicsen00ceng_0%2Fpage%2F89&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> <li id="cite_note-mvol-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-mvol_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://physics.nist.gov/cgi-bin/cuu/Value?mvol">"CODATA Value: molar volume of ideal gas (273.15 K, 100 kPa)"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=CODATA+Value%3A+molar+volume+of+ideal+gas+%28273.15+K%2C+100+kPa%29&rft_id=http%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fmvol&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> <li id="cite_note-mvolstd-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-mvolstd_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://physics.nist.gov/cgi-bin/cuu/Value?mvolstd">"CODATA Value: molar volume of ideal gas (273.15 K, 101.325 kPa)"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-02-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=CODATA+Value%3A+molar+volume+of+ideal+gas+%28273.15+K%2C+101.325+kPa%29&rft_id=http%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fmvolstd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> <li id="cite_note-IUPAC-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-IUPAC_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCalvert1990" class="citation journal cs1">Calvert, J. G. (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fpac199062112167">"Glossary of atmospheric chemistry terms (Recommendations 1990)"</a>. <i>Pure and Applied Chemistry</i>. <b>62</b> (11): <span class="nowrap">2167–</span>2219. <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/10.1351%2Fpac199062112167">10.1351/pac199062112167</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pure+and+Applied+Chemistry&rft.atitle=Glossary+of+atmospheric+chemistry+terms+%28Recommendations+1990%29&rft.volume=62&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2167-%3C%2Fspan%3E2219&rft.date=1990&rft_id=info%3Adoi%2F10.1351%2Fpac199062112167&rft.aulast=Calvert&rft.aufirst=J.+G.&rft_id=https%3A%2F%2Fdoi.org%2F10.1351%252Fpac199062112167&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdkins1983" class="citation book cs1">Adkins, C. J. (1983). <i>Equilibrium Thermodynamics</i> (3rd ed.). Cambridge, UK: Cambridge University Press. pp. <span class="nowrap">116–</span>120. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-25445-0" title="Special:BookSources/0-521-25445-0"><bdi>0-521-25445-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Equilibrium+Thermodynamics&rft.place=Cambridge%2C+UK&rft.pages=%3Cspan+class%3D%22nowrap%22%3E116-%3C%2Fspan%3E120&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=1983&rft.isbn=0-521-25445-0&rft.aulast=Adkins&rft.aufirst=C.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" 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="CITEREFTschoegl2000" class="citation book cs1">Tschoegl, N. W. (2000). <i>Fundamentals of Equilibrium and Steady-State Thermodynamics</i>. Amsterdam: Elsevier. p. 88. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-50426-5" title="Special:BookSources/0-444-50426-5"><bdi>0-444-50426-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Equilibrium+and+Steady-State+Thermodynamics&rft.place=Amsterdam&rft.pages=88&rft.pub=Elsevier&rft.date=2000&rft.isbn=0-444-50426-5&rft.aulast=Tschoegl&rft.aufirst=N.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAttard,_Phil2012" class="citation book cs1">Attard, Phil (2012). <i>Non-equilibrium thermodynamics and statistical mechanics : foundations and applications</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780191639760" title="Special:BookSources/9780191639760"><bdi>9780191639760</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/810281588">810281588</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-equilibrium+thermodynamics+and+statistical+mechanics+%3A+foundations+and+applications&rft.pub=Oxford+University+Press&rft.date=2012&rft_id=info%3Aoclcnum%2F810281588&rft.isbn=9780191639760&rft.au=Attard%2C+Phil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+gas" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Ideal_gas&oldid=1269393940">https://en.wikipedia.org/w/index.php?title=Ideal_gas&oldid=1269393940</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Ideal_gas" title="Category:Ideal gas">Ideal gas</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_June_2017" title="Category:Articles with unsourced statements from June 2017">Articles with unsourced statements from June 2017</a></li><li><a href="/wiki/Category:Pages_with_broken_anchors" title="Category:Pages with broken anchors">Pages with broken anchors</a></li><li><a href="/wiki/Category:Pages_displaying_short_descriptions_with_no_spaces_via_Module:Annotated_link" title="Category:Pages displaying short descriptions with no spaces via Module:Annotated link">Pages displaying short descriptions with no spaces via Module:Annotated link</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 January 2025, at 12:59<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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