CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Carnot's theorem (thermodynamics) - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"13a5c482-780e-4aef-b7a9-a3ede0e7d5d3","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Carnot's_theorem_(thermodynamics)","wgTitle":"Carnot's theorem (thermodynamics)","wgCurRevisionId":1240277712,"wgRevisionId":1240277712,"wgArticleId":308803,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description is different from Wikidata","Eponymous theorems of physics","Laws of thermodynamics","Thought experiments in physics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Carnot's_theorem_(thermodynamics)","wgRelevantArticleId":308803,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true, "wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q2347139","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups", "ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/1200px-Carnot_heat_engine_2.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="529"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/800px-Carnot_heat_engine_2.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="352"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/640px-Carnot_heat_engine_2.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="282"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Carnot's theorem (thermodynamics) - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Carnot%27s_theorem_(thermodynamics)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Carnot%27s_theorem_(thermodynamics)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Carnot_s_theorem_thermodynamics rootpage-Carnot_s_theorem_thermodynamics skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Carnot%27s+theorem+%28thermodynamics%29" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Carnot%27s+theorem+%28thermodynamics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Carnot%27s+theorem+%28thermodynamics%29" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Carnot%27s+theorem+%28thermodynamics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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-Proof" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Proof</span> </div> </a> <button aria-controls="toc-Proof-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 Proof subsection</span> </button> <ul id="toc-Proof-sublist" class="vector-toc-list"> <li id="toc-Reversible_engines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reversible_engines"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Reversible engines</span> </div> </a> <ul id="toc-Reversible_engines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irreversible_engines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irreversible_engines"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Irreversible engines</span> </div> </a> <ul id="toc-Irreversible_engines-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition_of_thermodynamic_temperature" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_of_thermodynamic_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition of thermodynamic temperature</span> </div> </a> <ul id="toc-Definition_of_thermodynamic_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applicability_to_fuel_cells" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applicability_to_fuel_cells"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Applicability to fuel cells</span> </div> </a> <ul id="toc-Applicability_to_fuel_cells-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</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">5</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" > <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">Carnot's theorem (thermodynamics)</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 20 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-20" 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">20 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/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D9%83%D8%A7%D8%B1%D9%86%D9%88_(%D8%AF%D9%8A%D9%86%D8%A7%D9%85%D9%8A%D9%83%D8%A7_%D8%AD%D8%B1%D8%A7%D8%B1%D9%8A%D8%A9)" title="مبرهنة كارنو (ديناميكا حرارية) – Arabic" lang="ar" hreflang="ar" data-title="مبرهنة كارنو (ديناميكا حرارية)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0_%D0%9A%D0%B0%D1%80%D0%BD%D0%BE_(%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D0%BA%D0%B0)" title="Тэарэма Карно (тэрмадынаміка) – Belarusian" lang="be" hreflang="be" data-title="Тэарэма Карно (тэрмадынаміка)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Carnot_(termodin%C3%A0mica)" title="Teorema de Carnot (termodinàmica) – Catalan" lang="ca" hreflang="ca" data-title="Teorema de Carnot (termodinàmica)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Carnots_s%C3%A6tning_(termodynamik)" title="Carnots sætning (termodynamik) – Danish" lang="da" hreflang="da" data-title="Carnots sætning (termodynamik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%DA%A9%D8%A7%D8%B1%D9%86%D9%88" title="قضیه کارنو – Persian" lang="fa" hreflang="fa" data-title="قضیه کارنو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A5%8B_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="कार्नो प्रमेय – Hindi" lang="hi" hreflang="hi" data-title="कार्नो प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Carnot_(termodinamica)" title="Teorema di Carnot (termodinamica) – Italian" lang="it" hreflang="it" data-title="Teorema di Carnot (termodinamica)" 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%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%A8%D7%A0%D7%95_(%D7%AA%D7%A8%D7%9E%D7%95%D7%93%D7%99%D7%A0%D7%9E%D7%99%D7%A7%D7%94)" title="משפט קרנו (תרמודינמיקה) – Hebrew" lang="he" hreflang="he" data-title="משפט קרנו (תרמודינמיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%83%AB%E3%83%8E%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86_(%E7%86%B1%E5%8A%9B%E5%AD%A6)" title="カルノーの定理 (熱力学) – Japanese" lang="ja" hreflang="ja" data-title="カルノーの定理 (熱力学)" 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/Carnots_teorem" title="Carnots teorem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Carnots teorem" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Carnot" title="Teorema de Carnot – Portuguese" lang="pt" hreflang="pt" data-title="Teorema de Carnot" 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/Teorema_lui_Carnot_(termodinamic%C4%83)" title="Teorema lui Carnot (termodinamică) – Romanian" lang="ro" hreflang="ro" data-title="Teorema lui Carnot (termodinamică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9A%D0%B0%D1%80%D0%BD%D0%BE_(%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0)" title="Теорема Карно (термодинамика) – Russian" lang="ru" hreflang="ru" data-title="Теорема Карно (термодинамика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Carnotov_izrek_(termodinamika)" title="Carnotov izrek (termodinamika) – Slovenian" lang="sl" hreflang="sl" data-title="Carnotov izrek (termodinamika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Carnots_sats_(termodynamik)" title="Carnots sats (termodynamik) – Swedish" lang="sv" hreflang="sv" data-title="Carnots sats (termodynamik)" 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%BE%E0%AE%B0%E0%AF%8D%E0%AE%A9%E0%AF%8B%E0%AE%B5%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D_(%E0%AE%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="கார்னோவின் தேற்றம் (வெப்பவியக்கவியல்) – Tamil" lang="ta" hreflang="ta" data-title="கார்னோவின் தேற்றம் (வெப்பவியக்கவியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Carnot_teoremi_(termodinamik)" title="Carnot teoremi (termodinamik) – Turkish" lang="tr" hreflang="tr" data-title="Carnot teoremi (termodinamik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1%D9%86%D9%88_%DA%A9%D8%A7_%D9%86%D8%B8%D8%B1%DB%8C%DB%81_(%D8%AD%D8%B1%D8%AD%D8%B1%DA%A9%DB%8C%D8%A7%D8%AA)" title="کارنو کا نظریہ (حرحرکیات) – Urdu" lang="ur" hreflang="ur" data-title="کارنو کا نظریہ (حرحرکیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8D%A1%E8%AB%BE%E5%AE%9A%E7%90%86_(%E7%86%B1%E5%8A%9B%E5%AD%B8)" title="卡諾定理 (熱力學) – Cantonese" lang="yue" hreflang="yue" data-title="卡諾定理 (熱力學)" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%A1%E8%AB%BE%E5%AE%9A%E7%90%86_(%E7%86%B1%E5%8A%9B%E5%AD%B8)" title="卡諾定理 (熱力學) – Chinese" lang="zh" hreflang="zh" data-title="卡諾定理 (熱力學)" 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/Q2347139#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Carnot%27s_theorem_(thermodynamics)" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Carnot%27s_theorem_(thermodynamics)"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Carnot%27s_theorem_(thermodynamics)"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Carnot%27s_theorem_(thermodynamics)" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Carnot%27s_theorem_(thermodynamics)" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&oldid=1240277712" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Carnot%27s_theorem_%28thermodynamics%29&id=1240277712&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCarnot%2527s_theorem_%28thermodynamics%29"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCarnot%2527s_theorem_%28thermodynamics%29"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Carnot%27s_theorem_%28thermodynamics%29&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2347139" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Maximum attainable efficiency of any heat engine</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 dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></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)"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a class="mw-selflink selflink">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a 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="hlist"><ul><li>History</li><li>Culture</li></ul></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Other</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Carnot's theorem</b>, also called <b>Carnot's rule</b>, is a principle of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> developed by <a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Nicolas Léonard Sadi Carnot</a> in 1824 that specifies limits on the maximum <a href="/wiki/Efficiency" title="Efficiency">efficiency</a> that any <a href="/wiki/Heat_engine" title="Heat engine">heat engine</a> can obtain. </p><p>Carnot's theorem states that all heat engines operating between the same two <a href="/wiki/Thermal_reservoir" title="Thermal reservoir">thermal or heat reservoirs</a> cannot have <a href="/wiki/Heat_engine#Efficiency" title="Heat engine">efficiencies</a> greater than a <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible</a> heat engine operating between the same reservoirs. A <a href="/wiki/Corollary" title="Corollary">corollary</a> of this theorem is that every reversible heat engine operating between a pair of heat reservoirs is equally efficient, regardless of the <a href="/wiki/Working_substance" class="mw-redirect" title="Working substance">working substance</a> employed or the operation details. Since a <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a> is also a reversible engine, the efficiency of all the reversible heat engines is determined as the efficiency of the Carnot heat engine that depends solely on the temperatures of its hot and cold reservoirs. </p><p>The maximum efficiency (i.e., the Carnot heat engine efficiency) of a heat engine operating between hot and cold reservoirs, denoted as <span class="texhtml">H</span> and <span class="texhtml">C</span> respectively, is the ratio of the temperature difference between the reservoirs to the hot reservoir temperature, expressed in the equation </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 \eta _{\text{max}}={\frac {T_{\mathrm {H} }-T_{\mathrm {C} }}{T_{\mathrm {H} }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{max}}={\frac {T_{\mathrm {H} }-T_{\mathrm {C} }}{T_{\mathrm {H} }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9239ded4457cb96bd9d664f62165970afb48728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.467ex; height:5.676ex;" alt="{\displaystyle \eta _{\text{max}}={\frac {T_{\mathrm {H} }-T_{\mathrm {C} }}{T_{\mathrm {H} }}},}"></span></dd></dl> <p>where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {H} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {H} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88428267cebc08c14abc957757f42b198811e302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {H} }}"></span>⁠</span> and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {C} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {C} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38820828ca2ea1ffd8de70606ae3dd263fc3b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {C} }}"></span>⁠</span> are the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperatures</a> of the hot and cold reservoirs, respectively, and the efficiency <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span>⁠</span> is the ratio of the <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> done by the engine (to the <a href="/wiki/Environment_(systems)" title="Environment (systems)">surroundings</a>) to the heat drawn out of the hot reservoir (to the engine). </p><p><span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\text{max}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{max}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83cdbbcebef40bc477cc553db83ddbf0397e4620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.447ex; height:2.176ex;" alt="{\displaystyle \eta _{\text{max}}}"></span>⁠</span> is greater than zero if and only if there is a temperature difference between the two thermal reservoirs. Since <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\text{max}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{max}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83cdbbcebef40bc477cc553db83ddbf0397e4620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.447ex; height:2.176ex;" alt="{\displaystyle \eta _{\text{max}}}"></span>⁠</span> is the upper limit of all reversible and irreversible heat engine efficiencies, it is concluded that work from a heat engine can be produced if and only if there is a temperature difference between two thermal reservoirs connecting to the engine. </p><p>Carnot's theorem is a consequence of the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>. Historically, it was based on contemporary <a href="/wiki/Caloric_theory" title="Caloric theory">caloric theory</a>, and preceded the establishment of the second law.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit&section=1" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Carnot_theorem_paradox.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Carnot_theorem_paradox.svg/300px-Carnot_theorem_paradox.svg.png" decoding="async" width="300" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Carnot_theorem_paradox.svg/450px-Carnot_theorem_paradox.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Carnot_theorem_paradox.svg/600px-Carnot_theorem_paradox.svg.png 2x" data-file-width="632" data-file-height="394" /></a><figcaption>An impossible situation: A heat engine cannot drive a less-efficient reversible heat engine without violating the second law of thermodynamics. Quantities in this figure are the absolute values of energy transfers (heat and work). </figcaption></figure> <p>The proof of the Carnot theorem is a <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> or <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a> (a method to prove a statement by assuming its falsity and logically deriving a false or contradictory statement from this assumption), based on a situation like the right figure where two heat engines with different <a href="/wiki/Heat_engine#Efficiency" title="Heat engine">efficiencies</a> are operating between two <a href="/wiki/Thermal_reservoir" title="Thermal reservoir">thermal reservoirs</a> at different temperature. The relatively hotter reservoir is called the hot reservoir and the other reservoir is called the cold reservoir. A (not necessarily <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible</a>) heat engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> with a greater efficiency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{_{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{_{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f2b85ea0e5017e925359d50e8eb035d0844226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.954ex; height:2.343ex;" alt="{\displaystyle \eta _{_{M}}}"></span> is driving a reversible heat engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> with a less efficiency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{_{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{_{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14294036226803714c8f062f1eb8916a4bfca559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.461ex; height:2.343ex;" alt="{\displaystyle \eta _{_{L}}}"></span>, causing the latter to act as a <a href="/wiki/Heat_pump" title="Heat pump">heat pump</a>. The requirement for the engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> to be reversible is necessary to explain work <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> and heat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> associated with it by using its known efficiency. However, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{_{M}}>\eta _{_{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo>></mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{_{M}}>\eta _{_{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e070e4281105874e498026b3e425b7d7050c329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.513ex; height:2.509ex;" alt="{\displaystyle \eta _{_{M}}>\eta _{_{L}}}"></span>, the net heat flow would be backwards, i.e., into the hot reservoir: </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 Q_{\text{h}}^{\text{out}}=Q<{\frac {\eta _{_{M}}}{\eta _{_{L}}}}Q=Q_{\text{h}}^{\text{in}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> <mo>=</mo> <mi>Q</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </msub> </mfrac> </mrow> <mi>Q</mi> <mo>=</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{h}}^{\text{out}}=Q<{\frac {\eta _{_{M}}}{\eta _{_{L}}}}Q=Q_{\text{h}}^{\text{in}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aca49d841487df84e4251d5fe213a5c93a8fe466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.297ex; height:5.676ex;" alt="{\displaystyle Q_{\text{h}}^{\text{out}}=Q<{\frac {\eta _{_{M}}}{\eta _{_{L}}}}Q=Q_{\text{h}}^{\text{in}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> represents heat, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e52a9c666190fb8c7b34bafc6061acb2994f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle {\text{in}}}"></span> denotes input to an object, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc53ad8f465a0e7fda9737a3b6c8cef2954dfddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle {\text{out}}}"></span> for output from an object, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </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/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> for the hot thermal reservoir. If heat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\text{h}}^{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{h}}^{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072b8e99dd4b96a84bce804db6d6ec45a4c27011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.446ex; height:3.009ex;" alt="{\displaystyle Q_{\text{h}}^{\text{out}}}"></span> flows from the hot reservoir then it has the sign of + while if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\text{h}}^{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{h}}^{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f2e396074752df5c24bbdf457351cec1035e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.442ex; height:3.176ex;" alt="{\displaystyle Q_{\text{h}}^{\text{in}}}"></span> flows from the hot reservoir then it has the sign of -. This expression can be easily derived by using the definition of the <a href="/wiki/Heat_engine#Efficiency" title="Heat engine">efficiency</a> of a heat engine, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =W/Q_{\text{h}}^{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =W/Q_{\text{h}}^{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf5a2b74da0a76b63d574ebb2b7dfdf642396d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.308ex; height:3.176ex;" alt="{\displaystyle \eta =W/Q_{\text{h}}^{\text{in}}}"></span>, where work and heat in this expression are net quantities per engine cycle, and the conservation of energy for each engine as shown below. The sign convention of work <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>, with which the sign of + for work done by an engine to its surroundings, is employed. </p><p>The above expression means that heat into the hot reservoir from the engine pair (can be considered as a single engine) is greater than heat into the engine pair from the hot reservoir (i.e., the hot reservoir continuously gets energy). A reversible heat engine with a low efficiency delivers more heat (energy) to the hot reservoir for a given amount of work (energy) to this engine when it is being driven as a heat pump. All these mean that heat can transfer from cold to hot places without external work, and such a heat transfer is impossible by the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>. </p> <ul><li>It may seem odd that a hypothetical reversible heat pump with a low efficiency is used to violate the second law of thermodynamics, but the <a href="/wiki/Figure_of_merit" title="Figure of merit">figure of merit</a> for refrigerator units is not the efficiency, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W/Q_{\text{h}}^{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W/Q_{\text{h}}^{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad47478b29f762ad5db2e577386ed66f3f5f92e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.044ex; height:3.009ex;" alt="{\displaystyle W/Q_{\text{h}}^{\text{out}}}"></span>, but the <a href="/wiki/Coefficient_of_performance" title="Coefficient of performance">coefficient of performance</a> (COP),<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\text{c}}^{\text{out}}/W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{c}}^{\text{out}}/W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c491c2dc10424ff5c870bfc7aa6fbcc50a7418a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.044ex; height:2.843ex;" alt="{\displaystyle Q_{\text{c}}^{\text{out}}/W}"></span> where this <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> has the sign opposite to the above (+ for work done to the engine).</li></ul> <p>Let's find the values of work <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>and heat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> depicted in the right figure in which a reversible heat engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> with a less efficiency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{_{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{_{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14294036226803714c8f062f1eb8916a4bfca559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.461ex; height:2.343ex;" alt="{\displaystyle \eta _{_{L}}}"></span> is driven as a heat pump by a heat engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> with a more efficiency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{_{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{_{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f2b85ea0e5017e925359d50e8eb035d0844226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.954ex; height:2.343ex;" alt="{\displaystyle \eta _{_{M}}}"></span>. </p><p>The definition of the <a href="/wiki/Heat_engine#Efficiency" title="Heat engine">efficiency</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =W/Q_{\text{h}}^{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =W/Q_{\text{h}}^{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/506d07f14ae08c08c7df97498acb424d87e6b308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.312ex; height:3.009ex;" alt="{\displaystyle \eta =W/Q_{\text{h}}^{\text{out}}}"></span> for each engine and the following expressions can be made: </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 \eta _{M}={\frac {W_{M}}{Q_{\text{h}}^{{\text{out}},M}}}={\frac {\eta _{M}Q}{Q}}=\eta _{M},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mo>,</mo> <mi>M</mi> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>Q</mi> </mrow> <mi>Q</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{M}={\frac {W_{M}}{Q_{\text{h}}^{{\text{out}},M}}}={\frac {\eta _{M}Q}{Q}}=\eta _{M},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d087598b0d3c1d49f04c66077192edeb69211abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.427ex; height:7.009ex;" alt="{\displaystyle \eta _{M}={\frac {W_{M}}{Q_{\text{h}}^{{\text{out}},M}}}={\frac {\eta _{M}Q}{Q}}=\eta _{M},}"></span></dd></dl> <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 \eta _{L}={\frac {W_{L}}{Q_{\text{h}}^{{\text{out}},L}}}={\frac {-\eta _{M}Q}{-{\frac {\eta _{M}}{\eta _{L}}}Q}}=\eta _{L}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mo>,</mo> <mi>L</mi> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>Q</mi> </mrow> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mfrac> </mrow> <mi>Q</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{L}={\frac {W_{L}}{Q_{\text{h}}^{{\text{out}},L}}}={\frac {-\eta _{M}Q}{-{\frac {\eta _{M}}{\eta _{L}}}Q}}=\eta _{L}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c7346884de0faecc36ac6b88d213b4c218aabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:29.517ex; height:7.343ex;" alt="{\displaystyle \eta _{L}={\frac {W_{L}}{Q_{\text{h}}^{{\text{out}},L}}}={\frac {-\eta _{M}Q}{-{\frac {\eta _{M}}{\eta _{L}}}Q}}=\eta _{L}.}"></span></dd></dl> <p>The denominator of the second expression, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\text{h}}^{{\text{out}},L}=-{\frac {\eta _{M}}{\eta _{L}}}Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mo>,</mo> <mi>L</mi> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mfrac> </mrow> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\text{h}}^{{\text{out}},L}=-{\frac {\eta _{M}}{\eta _{L}}}Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2f58b01f9cd90fd293edfd496e61e93eebab1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.719ex; height:5.343ex;" alt="{\displaystyle Q_{\text{h}}^{{\text{out}},L}=-{\frac {\eta _{M}}{\eta _{L}}}Q}"></span>, is made to make the expression to be consistent, and it helps to fill the values of work and heat for the engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>. </p><p>For each engine, the absolute value of the energy entering the engine, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{abs}}^{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>abs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{abs}}^{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e6662eb56b9502a4fc9462f419c1060587b845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.331ex; height:3.176ex;" alt="{\displaystyle E_{\text{abs}}^{\text{in}}}"></span>, must be equal to the absolute value of the energy leaving from the engine, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{abs}}^{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>abs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{abs}}^{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f62d764055f3c3b041d829d626e617cc24c682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.402ex; height:3.009ex;" alt="{\displaystyle E_{\text{abs}}^{\text{out}}}"></span>. Otherwise, energy is continuously accumulated in an engine or the conservation of energy is violated by taking more energy from an engine than input energy to the engine: </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 E_{\text{M,abs}}^{\text{in}}=Q=(1-\eta _{M})Q+\eta _{M}Q=E_{\text{M,abs}}^{\text{out}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>M,abs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msubsup> <mo>=</mo> <mi>Q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>Q</mi> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>Q</mi> <mo>=</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>M,abs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{M,abs}}^{\text{in}}=Q=(1-\eta _{M})Q+\eta _{M}Q=E_{\text{M,abs}}^{\text{out}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cd4a72d797f3d48085a74289b1cc11cbd55d4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:42.93ex; height:3.509ex;" alt="{\displaystyle E_{\text{M,abs}}^{\text{in}}=Q=(1-\eta _{M})Q+\eta _{M}Q=E_{\text{M,abs}}^{\text{out}},}"></span></dd></dl> <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 E_{\text{L,abs}}^{\text{in}}=\eta _{M}Q+\eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)={\frac {\eta _{M}}{\eta _{L}}}Q=E_{\text{L,abs}}^{\text{out}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L,abs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msubsup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>Q</mi> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>Q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mfrac> </mrow> <mi>Q</mi> <mo>=</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L,abs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{L,abs}}^{\text{in}}=\eta _{M}Q+\eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)={\frac {\eta _{M}}{\eta _{L}}}Q=E_{\text{L,abs}}^{\text{out}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a6ee33286ff3dd92f51f1f02e0806aaa8edb45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.264ex; height:6.176ex;" alt="{\displaystyle E_{\text{L,abs}}^{\text{in}}=\eta _{M}Q+\eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)={\frac {\eta _{M}}{\eta _{L}}}Q=E_{\text{L,abs}}^{\text{out}}.}"></span></dd></dl> <p>In the second expression, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|Q_{\text{h}}^{{\text{out}},L}\right|=\left|-{\frac {\eta _{M}}{\eta _{L}}}Q\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>h</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mo>,</mo> <mi>L</mi> </mrow> </msubsup> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mfrac> </mrow> <mi>Q</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|Q_{\text{h}}^{{\text{out}},L}\right|=\left|-{\frac {\eta _{M}}{\eta _{L}}}Q\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f7ffee36b2042eb2ae9f618547721256744d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.575ex; height:4.176ex;" alt="{\textstyle \left|Q_{\text{h}}^{{\text{out}},L}\right|=\left|-{\frac {\eta _{M}}{\eta _{L}}}Q\right|}"></span> is used to find the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>Q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48223967943602693ee56e747d68b1a89b78072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.845ex; height:4.843ex;" alt="{\textstyle \eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)}"></span> describing the amount of heat taken from the cold reservoir, completing the absolute value expressions of work and heat in the right figure. </p><p>Having established that the right figure values are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines as shown below.<sup id="cite_ref-CarnotThoerem_3-0" class="reference"><a href="#cite_note-CarnotThoerem-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Reversible_engines">Reversible engines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit&section=2" title="Edit section: Reversible engines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To see that every <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible engine</a> operating between reservoirs at temperatures <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> must have the same efficiency, assume that two reversible heat engines have different efficiencies, and let the relatively more efficient engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> drive the relatively less efficient engine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> as a heat pump. As the right figure shows, this will cause heat to flow from the cold to the hot reservoir without external work, which violates the second law of thermodynamics. Therefore, both (reversible) heat engines have the same efficiency, and we conclude that: </p> <dl><dd><i>All reversible heat engines that operate between the same two thermal</i> (<i>heat</i>) <i>reservoirs have the same efficiency.</i></dd></dl> <p>The reversible heat engine efficiency can be determined by analyzing a Carnot heat engine as one of reversible heat engine. </p><p>This conclusion is an important result because it helps establish the <a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a>, which implies that the change in <a href="/wiki/Entropy" title="Entropy">entropy</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is unique for all reversible processes:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <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 _{a}^{b}{\frac {dQ_{\text{rev}}}{T}}}"> <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"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rev</mtext> </mrow> </msub> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S=\int _{a}^{b}{\frac {dQ_{\text{rev}}}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b478ed3c4e4e4b05a6081ac9b573e794db26645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.688ex; height:6.343ex;" alt="{\displaystyle \Delta S=\int _{a}^{b}{\frac {dQ_{\text{rev}}}{T}}}"></span></dd></dl> <p>as the entropy change, that is made during a transition from a <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> to a state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> in a <i>V-T</i> (Volume-Temperature) space, is the same over all reversible process paths between these two states. If this integral were not path independent, then entropy would not be a <a href="/wiki/State_function" title="State function">state variable</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Irreversible_engines">Irreversible engines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit&section=3" title="Edit section: Irreversible engines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider two engines, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>, which are irreversible and reversible respectively. We construct the machine shown in the right figure, 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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> driving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> as a heat pump. Then if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is more efficient than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>, the machine will violate the second law of thermodynamics. Since a Carnot heat engine is a reversible heat engine, and all reversible heat engines operate with the same efficiency between the same reservoirs, we have the first part of Carnot's theorem: </p> <dl><dd><i>No irreversible heat engine is more efficient than a Carnot heat engine operating between the same two thermal reservoirs.</i></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Definition_of_thermodynamic_temperature">Definition of thermodynamic temperature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit&section=4" title="Edit section: Definition of thermodynamic temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Thermodynamic_temperature#Definition_of_thermodynamic_temperature" title="Thermodynamic temperature">Definition of thermodynamic temperature</a></div> <p>The efficiency of a heat engine is the work done by the engine divided by the heat introduced to the engine per engine cycle or </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ={\frac {w_{\text{cy}}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cy</mtext> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta ={\frac {w_{\text{cy}}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d70b698ee7713eb238f197d7911a74790210a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.286ex; height:5.676ex;" alt="{\displaystyle \eta ={\frac {w_{\text{cy}}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{\text{cy}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cy</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{\text{cy}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed5394caa3afcb00d063eaee51daaba02937251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.494ex; height:2.343ex;" alt="{\displaystyle w_{\text{cy}}}"></span> is the work done by the engine, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e816a560387c62d11111a6be463c8422242392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.518ex; height:2.009ex;" alt="{\displaystyle q_{C}}"></span> is the heat to the cold reservoir from the engine, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a92c586758805ac6d0945c192f6728edb25d705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.728ex; height:2.009ex;" alt="{\displaystyle q_{H}}"></span> is the heat to the engine from the hot reservoir, per cycle. Thus, the efficiency depends only on <span class="mwe-math-element"><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 {q_{C}}{q_{H}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{C}}{q_{H}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2705bbdf1a417f07cc9eb856416dc315211e4eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.565ex; height:5.343ex;" alt="{\displaystyle {\frac {q_{C}}{q_{H}}}}"></span>.<sup id="cite_ref-SignProblem_6-0" class="reference"><a href="#cite_note-SignProblem-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Because all reversible heat engines operating between temperatures <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> must have the same efficiency, the efficiency of a reversible heat engine is a function of only the two reservoir temperatures: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ee1c3c07382b80d2c62b1454d94afe21a7f160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.673ex; height:5.343ex;" alt="{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}"></span>.</td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>In addition, a reversible heat engine operating between temperatures <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0936b6ba76bf0d900bb7315c99f64c5376f5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{3}}"></span> must have the same efficiency as one consisting of two cycles, one between <span class="mwe-math-element"><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> and another (intermediate) temperature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>, and the second between <span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0936b6ba76bf0d900bb7315c99f64c5376f5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{3}}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1}<T_{2}<T_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}<T_{2}<T_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71604fbb1db4ccef629cf0be69506a40b1dfb90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.432ex; height:2.509ex;" alt="{\displaystyle T_{1}<T_{2}<T_{3}}"></span>). This can only be the case if </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78775ea1d08bb51618ea147bbcd85f821c5e5b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.078ex; height:5.343ex;" alt="{\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3})}"></span>.</td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>Specializing to the case that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> is a fixed reference temperature: the temperature of the triple point of water as 273.16. (Of course any reference temperature and any positive numerical value could be used — the choice here corresponds to the <a href="/wiki/Kelvin" title="Kelvin">Kelvin</a> scale.) Then for any <span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0936b6ba76bf0d900bb7315c99f64c5376f5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{3}}"></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(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>273.16</mn> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>273.16</mn> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a158621a989a22f4eafc16dc0513f2017702de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.49ex; height:6.509ex;" alt="{\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}.}"></span></dd></dl> <p>Therefore, if thermodynamic temperature is 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 T'=273.16\cdot f(T_{1},T),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>273.16</mn> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'=273.16\cdot f(T_{1},T),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e7b9528a1946d7a93b7b45bad7e41aeb7ad1d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.458ex; height:3.009ex;" alt="{\displaystyle T'=273.16\cdot f(T_{1},T),}"></span></dd></dl> <p>then the function viewed as a function of thermodynamic temperature, 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 f(T_{2},T_{3})={\frac {T_{3}'}{T_{2}'}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{2},T_{3})={\frac {T_{3}'}{T_{2}'}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a912e7ab29c4260befb4811fbd3fa9e6fa97407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.939ex; height:6.509ex;" alt="{\displaystyle f(T_{2},T_{3})={\frac {T_{3}'}{T_{2}'}}.}"></span></dd></dl><p> It follows immediately that</p><table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}'}{T_{H}'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mo>′</mo> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}'}{T_{H}'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5bb37627f35319bf11335a19b297e08fbfe43a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.656ex; height:6.676ex;" alt="{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}'}{T_{H}'}}}"></span>.</td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span>)</b></td></tr></tbody></table> <p>Substituting this equation back into the above equation <span class="mwe-math-element"><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 {q_{C}}{q_{H}}}=f(T_{H},T_{C})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ee1c3c07382b80d2c62b1454d94afe21a7f160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.673ex; height:5.343ex;" alt="{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}"></span> gives a relationship for the efficiency in terms of thermodynamic temperatures: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}'}{T_{H}'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mo>′</mo> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}'}{T_{H}'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10089ffbee46be3e4b17a320feb4f88d1688680b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.822ex; height:6.676ex;" alt="{\displaystyle \eta =1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}'}{T_{H}'}}}"></span>.</td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Applicability_to_fuel_cells">Applicability to fuel cells</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit&section=5" title="Edit section: Applicability to fuel cells"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <a href="/wiki/Fuel_cell" title="Fuel cell">fuel cells</a> can generate useful power when all components of the system are at the same temperature (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=T_{H}=T_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=T_{H}=T_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233890e553656f326f6eda99cd3062dcf3a8d354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.721ex; height:2.509ex;" alt="{\displaystyle T=T_{H}=T_{C}}"></span>), they are clearly not limited by Carnot's theorem, which states that no power can be generated when <span class="mwe-math-element"><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_{H}=T_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{H}=T_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7151385fd41ac1c8a940b4b767efe36c8c3b2d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.986ex; height:2.509ex;" alt="{\displaystyle T_{H}=T_{C}}"></span>. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells instead convert <a href="/wiki/Chemical_energy" title="Chemical energy">chemical energy</a> to work.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Nevertheless, the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> still provides restrictions on fuel cell energy conversion.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Carnot_battery" title="Carnot battery">Carnot battery</a> is a type of energy storage system that stores electricity in <a href="/wiki/Thermal_energy_storage" title="Thermal energy storage">thermal energy storage</a> and converts the stored heat back to electricity through thermodynamic cycles.<sup id="cite_ref-DumontFrate2020_9-0" class="reference"><a href="#cite_note-DumontFrate2020-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </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=Carnot%27s_theorem_(thermodynamics)&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Endoreversible_thermodynamics#Novikov_engine" title="Endoreversible thermodynamics">Chambadal–Novikov efficiency</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle#Coefficient_of_performance" title="Heat pump and refrigeration cycle">Heating and cooling efficiency bounds</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJohn_Murrell2009" class="citation web cs1">John Murrell (2009). <a rel="nofollow" class="external text" href="http://www.slashdocs.com/myvnxz/thermodynamics.html">"A Very Brief History of Thermodynamics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">May 2,</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Very+Brief+History+of+Thermodynamics&rft.date=2009&rft.au=John+Murrell&rft_id=http%3A%2F%2Fwww.slashdocs.com%2Fmyvnxz%2Fthermodynamics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091122191251/http://www.sussex.ac.uk/chemistry/documents/a_thermodynamics_history.pdf">Archive copy</a> at the <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a><span style="font-size:smaller;"> PDF (142 <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091122191251/http://www.sussex.ac.uk/chemistry/documents/a_thermodynamics_history.pdf">Archived</a> November 22, 2009, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> KB)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTiplerMosca,_G.2008" class="citation book cs1">Tipler, Paul; Mosca, G. (2008). "19.2, 19.7". <i>Physics for Scientists and Engineers</i> (6th ed.). Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781429201322" title="Special:BookSources/9781429201322"><bdi>9781429201322</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=19.2%2C+19.7&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=6th&rft.pub=Freeman&rft.date=2008&rft.isbn=9781429201322&rft.aulast=Tipler&rft.aufirst=Paul&rft.au=Mosca%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" class="Z3988"></span></span> </li> <li id="cite_note-CarnotThoerem-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-CarnotThoerem_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://seit.unsw.adfa.edu.au/staff/sites/hrp/Literature/articles/CarnotTheorem.pdf">"Lecture 10: Carnot theorem"</a> <span class="cs1-format">(PDF)</span>. Feb 7, 2005<span class="reference-accessdate">. Retrieved <span class="nowrap">October 5,</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lecture+10%3A+Carnot+theorem&rft.date=2005-02-07&rft_id=http%3A%2F%2Fseit.unsw.adfa.edu.au%2Fstaff%2Fsites%2Fhrp%2FLiterature%2Farticles%2FCarnotTheorem.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOhanian1994" class="citation book cs1">Ohanian, Hans (1994). <i>Principles of Physics</i>. W.W. Norton and Co. p. 438. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/039395773X" title="Special:BookSources/039395773X"><bdi>039395773X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Physics&rft.pages=438&rft.pub=W.W.+Norton+and+Co.&rft.date=1994&rft.isbn=039395773X&rft.aulast=Ohanian&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://faculty.wwu.edu/vawter/PhysicsNet/Topics/ThermLaw2/ThermalProcesses.html">http://faculty.wwu.edu/vawter/PhysicsNet/Topics/ThermLaw2/ThermalProcesses.html</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131228111404/http://faculty.wwu.edu/vawter/PhysicsNet/Topics/ThermLaw2/ThermalProcesses.html">Archived</a> 2013-12-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, and <a rel="nofollow" class="external free" href="http://www.itp.phys.ethz.ch/education/hs10/stat/slides/Laws_TD.pdf">http://www.itp.phys.ethz.ch/education/hs10/stat/slides/Laws_TD.pdf</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131213183127/http://www.itp.phys.ethz.ch/education/hs10/stat/slides/Laws_TD.pdf">Archived</a> 2013-12-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Both retrieved 13 December 2013.</span> </li> <li id="cite_note-SignProblem-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-SignProblem_6-0">^</a></b></span> <span class="reference-text"> The sign of <i>q<sub>C</sub></i> > 0 for the waste heat lost by the system violates the sign convention of <a href="/wiki/Heat" title="Heat">heat</a>.</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.google.com/viewer?a=v&q=cache:KkaC5XqVzVgJ:filer.case.edu/org/hsfcproject/Unit%252036/36.1.3%2520Fuel%2520Cell%2520versus%2520Carnot%2520Efficiency.ppt+%22carnot+limit%22+%22fuel+cell%22&hl=en&gl=us&pid=bl&srcid=ADGEESj0jJDxV7N40e6SinnUfC9X_wqU6GHB4aPX6BlFRSYsfeixt1hGvyGLMAthvrl_Ws6k4r-Z8AdtYkjDqhVrIUwjO6rSA8y5JnGnqt73LWITIpc3KdeTUBe24GkaZcLs1BpHUxh5&sig=AHIEtbROSMlotD3mw6tegO0_v-uhvVwI2g">"Fuel Cell versus Carnot Efficiency"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">Feb 20,</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fuel+Cell+versus+Carnot+Efficiency&rft_id=https%3A%2F%2Fdocs.google.com%2Fviewer%3Fa%3Dv%26q%3Dcache%3AKkaC5XqVzVgJ%3Afiler.case.edu%2Forg%2Fhsfcproject%2FUnit%25252036%2F36.1.3%252520Fuel%252520Cell%252520versus%252520Carnot%252520Efficiency.ppt%2B%2522carnot%2Blimit%2522%2B%2522fuel%2Bcell%2522%26hl%3Den%26gl%3Dus%26pid%3Dbl%26srcid%3DADGEESj0jJDxV7N40e6SinnUfC9X_wqU6GHB4aPX6BlFRSYsfeixt1hGvyGLMAthvrl_Ws6k4r-Z8AdtYkjDqhVrIUwjO6rSA8y5JnGnqt73LWITIpc3KdeTUBe24GkaZcLs1BpHUxh5%26sig%3DAHIEtbROSMlotD3mw6tegO0_v-uhvVwI2g&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" 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="CITEREFJacobJain2005" class="citation conference cs1">Jacob, Kallarackel T; Jain, Saurabh (July 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304083422/http://eprints.iisc.ernet.in/42739/"><i>Fuel cell efficiency redefined : Carnot limit reassessed</i></a>. Q1 - Ninth International Symposium on Solid Oxide Fuel Cells (SOFC IX). USA. Archived from <a rel="nofollow" class="external text" href="http://eprints.iisc.ernet.in/42739/">the original</a> on 2016-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Fuel+cell+efficiency+redefined+%3A+Carnot+limit+reassessed&rft.place=USA&rft.date=2005-07&rft.aulast=Jacob&rft.aufirst=Kallarackel+T&rft.au=Jain%2C+Saurabh&rft_id=http%3A%2F%2Feprints.iisc.ernet.in%2F42739%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" class="Z3988"></span></span> </li> <li id="cite_note-DumontFrate2020-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-DumontFrate2020_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDumontFratePillaiLecompte2020" class="citation journal cs1">Dumont, Olivier; Frate, Guido Francesco; Pillai, Aditya; Lecompte, Steven; De paepe, Michel; Lemort, Vincent (2020). "Carnot battery technology: A state-of-the-art review". <i>Journal of Energy Storage</i>. <b>32</b>: 101756. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.est.2020.101756">10.1016/j.est.2020.101756</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2268%2F251473">2268/251473</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2352-152X">2352-152X</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:225019981">225019981</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Energy+Storage&rft.atitle=Carnot+battery+technology%3A+A+state-of-the-art+review&rft.volume=32&rft.pages=101756&rft.date=2020&rft_id=info%3Ahdl%2F2268%2F251473&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A225019981%23id-name%3DS2CID&rft.issn=2352-152X&rft_id=info%3Adoi%2F10.1016%2Fj.est.2020.101756&rft.aulast=Dumont&rft.aufirst=Olivier&rft.au=Frate%2C+Guido+Francesco&rft.au=Pillai%2C+Aditya&rft.au=Lecompte%2C+Steven&rft.au=De+paepe%2C+Michel&rft.au=Lemort%2C+Vincent&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACarnot%27s+theorem+%28thermodynamics%29" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Carnot%27s_theorem_(thermodynamics)&oldid=1240277712">https://en.wikipedia.org/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&oldid=1240277712</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">Categories</a>: <ul><li><a href="/wiki/Category:Eponymous_theorems_of_physics" title="Category:Eponymous theorems of physics">Eponymous theorems of physics</a></li><li><a href="/wiki/Category:Laws_of_thermodynamics" title="Category:Laws of thermodynamics">Laws of thermodynamics</a></li><li><a href="/wiki/Category:Thought_experiments_in_physics" title="Category:Thought experiments in physics">Thought experiments in physics</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:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><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></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 August 2024, at 14:43<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. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Carnot%27s_theorem_(thermodynamics)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5cd4cd96d5-vtnm8","wgBackendResponseTime":173,"wgPageParseReport":{"limitreport":{"cputime":"0.412","walltime":"0.608","ppvisitednodes":{"value":1917,"limit":1000000},"postexpandincludesize":{"value":69945,"limit":2097152},"templateargumentsize":{"value":2801,"limit":2097152},"expansiondepth":{"value":10,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":125711,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 366.127 1 -total"," 36.55% 133.805 1 Template:Thermodynamics"," 34.45% 126.135 1 Template:Reflist"," 20.90% 76.515 3 Template:Cite_web"," 18.28% 66.911 1 Template:Short_description"," 10.96% 40.121 2 Template:Pagetype"," 10.88% 39.841 3 Template:Sidebar"," 8.50% 31.137 10 Template:Flatlist"," 4.73% 17.312 5 Template:Main_other"," 4.17% 15.283 1 Template:Main"]},"scribunto":{"limitreport-timeusage":{"value":"0.173","limit":"10.000"},"limitreport-memusage":{"value":5240790,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-697b7966c5-djg8x","timestamp":"20241126175707","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Carnot's theorem (thermodynamics)","url":"https:\/\/en.wikipedia.org\/wiki\/Carnot%27s_theorem_(thermodynamics)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q2347139","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q2347139","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-08-31T22:04:27Z","dateModified":"2024-08-14T14:43:38Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/2\/22\/Carnot_heat_engine_2.svg","headline":"principle regarding the maximum efficiency of heat engines"}</script> </body> </html>