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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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_no.wikipedia.org&uselang=nb" class=""><span>Doner</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=Spesial:Opprett_konto&returnto=Kjemisk+potensial" title="Du oppfordres til å opprette en konto og logge inn, men det er ikke obligatorisk" class=""><span>Opprett konto</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=Spesial:Logg_inn&returnto=Kjemisk+potensial" title="Du oppfordres til å logge inn, men det er ikke 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data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Innhold</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skjul</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">(Til toppen)</div> </a> </li> <li id="toc-Formelle_egenskaper" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formelle_egenskaper"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Formelle egenskaper</span> </div> </a> <ul id="toc-Formelle_egenskaper-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideell_gass" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ideell_gass"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Ideell gass</span> </div> </a> <button aria-controls="toc-Ideell_gass-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>Vis/skjul underseksjonen Ideell gass</span> </button> <ul id="toc-Ideell_gass-sublist" class="vector-toc-list"> <li id="toc-Ideell_blanding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideell_blanding"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Ideell blanding</span> </div> </a> <ul id="toc-Ideell_blanding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kvantegasser" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kvantegasser"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kvantegasser</span> </div> </a> <ul id="toc-Kvantegasser-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Faselikevekt" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Faselikevekt"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Faselikevekt</span> </div> </a> <button aria-controls="toc-Faselikevekt-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>Vis/skjul underseksjonen Faselikevekt</span> </button> <ul id="toc-Faselikevekt-sublist" class="vector-toc-list"> <li id="toc-Raoults_lov" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Raoults_lov"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Raoults lov</span> </div> </a> <ul id="toc-Raoults_lov-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Litteratur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Litteratur"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Litteratur</span> </div> </a> <ul id="toc-Litteratur-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="Innhold" 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="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Kjemisk potensial</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="Gå til en artikkel på et annet språk. Tilgjengelig på 37 språk" > <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-37" 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">37 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kemisk_potential" title="Kemisk potential – dansk" lang="da" hreflang="da" data-title="Kemisk potential" data-language-autonym="Dansk" data-language-local-name="dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kemisk_potential" title="Kemisk potential – svensk" lang="sv" hreflang="sv" data-title="Kemisk potential" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Chemiese_potensiaal" title="Chemiese potensiaal – afrikaans" lang="af" hreflang="af" data-title="Chemiese potensiaal" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D9%87%D8%AF_%D9%83%D9%8A%D9%85%D9%8A%D8%A7%D8%A6%D9%8A" title="جهد كيميائي – arabisk" lang="ar" hreflang="ar" data-title="جهد كيميائي" data-language-autonym="العربية" data-language-local-name="arabisk" 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%A5%D1%96%D0%BC%D1%96%D1%87%D0%BD%D1%8B_%D0%BF%D0%B0%D1%82%D1%8D%D0%BD%D1%86%D1%8B%D1%8F%D0%BB" title="Хімічны патэнцыял – belarusisk" lang="be" hreflang="be" data-title="Хімічны патэнцыял" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D1%87%D0%B5%D0%BD_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Химичен потенциал – bulgarsk" lang="bg" hreflang="bg" data-title="Химичен потенциал" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Potencial_qu%C3%ADmic" title="Potencial químic – katalansk" lang="ca" hreflang="ca" data-title="Potencial químic" data-language-autonym="Català" data-language-local-name="katalansk" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Химилле потенциал – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Химилле потенциал" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Chemick%C3%BD_potenci%C3%A1l" title="Chemický potenciál – tsjekkisk" lang="cs" hreflang="cs" data-title="Chemický potenciál" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Chemisches_Potential" title="Chemisches Potential – tysk" lang="de" hreflang="de" data-title="Chemisches Potential" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Keemiline_potentsiaal" title="Keemiline potentsiaal – estisk" lang="et" hreflang="et" data-title="Keemiline potentsiaal" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Chemical_potential" title="Chemical potential – engelsk" lang="en" hreflang="en" data-title="Chemical potential" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="anbefalt artikkel-merke"><a href="https://es.wikipedia.org/wiki/Potencial_qu%C3%ADmico" title="Potencial químico – spansk" lang="es" hreflang="es" data-title="Potencial químico" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kemia_potencialo" title="Kemia potencialo – esperanto" lang="eo" hreflang="eo" data-title="Kemia potencialo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%AA%D8%A7%D9%86%D8%B3%DB%8C%D9%84_%D8%B4%DB%8C%D9%85%DB%8C%D8%A7%DB%8C%DB%8C" title="پتانسیل شیمیایی – persisk" lang="fa" hreflang="fa" data-title="پتانسیل شیمیایی" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Potentiel_chimique" title="Potentiel chimique – fransk" lang="fr" hreflang="fr" data-title="Potentiel chimique" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%94%ED%95%99_%ED%8D%BC%ED%85%90%EC%85%9C" title="화학 퍼텐셜 – koreansk" lang="ko" hreflang="ko" data-title="화학 퍼텐셜" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%AB%D5%B4%D5%AB%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BA%D5%B8%D5%BF%D5%A5%D5%B6%D6%81%D5%AB%D5%A1%D5%AC" title="Քիմիական պոտենցիալ – armensk" lang="hy" hreflang="hy" data-title="Քիմիական պոտենցիալ" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Potensial_kimia" title="Potensial kimia – indonesisk" lang="id" hreflang="id" data-title="Potensial kimia" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Potenziale_chimico" title="Potenziale chimico – italiensk" lang="it" hreflang="it" data-title="Potenziale chimico" data-language-autonym="Italiano" data-language-local-name="italiensk" 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%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C_%D7%9B%D7%99%D7%9E%D7%99" title="פוטנציאל כימי – hebraisk" lang="he" hreflang="he" data-title="פוטנציאל כימי" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%A9miai_potenci%C3%A1l" title="Kémiai potenciál – ungarsk" lang="hu" hreflang="hu" data-title="Kémiai potenciál" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BC%D0%B8%D0%B9%D0%BD_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Химийн потенциал – mongolsk" lang="mn" hreflang="mn" data-title="Химийн потенциал" data-language-autonym="Монгол" data-language-local-name="mongolsk" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Chemische_potentiaal" title="Chemische potentiaal – nederlandsk" lang="nl" hreflang="nl" data-title="Chemische potentiaal" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8C%96%E5%AD%A6%E3%83%9D%E3%83%86%E3%83%B3%E3%82%B7%E3%83%A3%E3%83%AB" title="化学ポテンシャル – japansk" lang="ja" hreflang="ja" data-title="化学ポテンシャル" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Potencja%C5%82_chemiczny" title="Potencjał chemiczny – polsk" lang="pl" hreflang="pl" data-title="Potencjał chemiczny" data-language-autonym="Polski" data-language-local-name="polsk" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Potencial_qu%C3%ADmico" title="Potencial químico – portugisisk" lang="pt" hreflang="pt" data-title="Potencial químico" data-language-autonym="Português" data-language-local-name="portugisisk" 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/Poten%C8%9Bial_chimic" title="Potențial chimic – rumensk" lang="ro" hreflang="ro" data-title="Potențial chimic" data-language-autonym="Română" data-language-local-name="rumensk" 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%A5%D0%B8%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Химический потенциал – russisk" lang="ru" hreflang="ru" data-title="Химический потенциал" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Chemical_potential" title="Chemical potential – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Chemical potential" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Chemick%C3%BD_potenci%C3%A1l" title="Chemický potenciál – slovakisk" lang="sk" hreflang="sk" data-title="Chemický potenciál" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kemi%C4%8Dni_potencial" title="Kemični potencial – slovensk" lang="sl" hreflang="sl" data-title="Kemični potencial" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kemiallinen_potentiaali" title="Kemiallinen potentiaali – finsk" lang="fi" hreflang="fi" data-title="Kemiallinen potentiaali" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%87%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%A3%E0%AF%8D%E0%AE%AA%E0%AF%81" title="வேதிப்பண்பு – tamil" lang="ta" hreflang="ta" data-title="வேதிப்பண்பு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A5%D1%96%D0%BC%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB" title="Хімічний потенціал – ukrainsk" lang="uk" hreflang="uk" data-title="Хімічний потенціал" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8C%96%E5%AD%B8%E5%8B%A2" title="化學勢 – kantonesisk" lang="yue" hreflang="yue" data-title="化學勢" data-language-autonym="粵語" data-language-local-name="kantonesisk" 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%8C%96%E5%AD%A6%E5%8A%BF" title="化学势 – kinesisk" lang="zh" hreflang="zh" data-title="化学势" data-language-autonym="中文" data-language-local-name="kinesisk" 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/Q737004#sitelinks-wikipedia" title="Rediger lenker til artikkelen på andre språk" class="wbc-editpage">Rediger lenker</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="Navnerom"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div 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class="mw-content-ltr mw-parser-output" lang="nb" dir="ltr"><table class="navbox" style="width: 18em; margin-left: 1em; float: right;"> <tbody><tr> <th colspan="2">Konjugerte variabler<br />i termodynamikk </th></tr> <tr> <td><a href="/wiki/Trykk" title="Trykk">Trykk</a></td> <td><a href="/wiki/Volum" title="Volum">Volum</a> </td></tr> <tr> <td>(<a href="/wiki/Spenning_(mekanikk)" title="Spenning (mekanikk)">Spenning </a>)</td> <td>(<a href="/wiki/Deformasjon#Tøyning" title="Deformasjon">Tøyning</a>) </td></tr> <tr> <td><a href="/wiki/Temperatur" title="Temperatur">Temperatur</a></td> <td><a href="/wiki/Entropi" title="Entropi">Entropi</a> </td></tr> <tr> <td><a class="mw-selflink selflink">Kjemisk potensial</a></td> <td><a href="/wiki/Partikkelnummer" title="Partikkelnummer">Partikkelnummer</a> </td></tr></tbody></table> <p><b>Kjemisk potensial</b> er et begrep innen <a href="/wiki/Termodynamikk" title="Termodynamikk">termodynamikken</a> som har stor betydning innen <a href="/wiki/Fysikalsk_kjemi" title="Fysikalsk kjemi">fysikalsk kjemi</a>. Det beskriver hvor mye energien til et system av partikler forandres når en ekstra partikkel legges til. Det ble først klarlagt av den amerikanske fysiker <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a> i <a href="/wiki/1873" title="1873">1873</a>. Det er direkte forbundet med <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs fri energi</a> for systemet. </p><p>Dets nytte tilsvarer bruken av <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektrisk potensial</a> for å beskrive bevegelsen til elektrisk ladete partikler eller <a href="/wiki/Gravitasjonspotensial" title="Gravitasjonspotensial">gravitasjonspotensial</a> for å beregne hvordan en massiv partikkel beveger seg på grunn av <a href="/wiki/Tyngdekraft" title="Tyngdekraft">tyngdekraften</a>. Partiklene vil spontant bevege seg slik at deres <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensielle energi</a> blir mindre. På samme måte vil det kjemiske potensialet være høyere for en partikkel som befinner seg i et område hvor konsentrasjonen av partikler er høyere. Den vil derfor forsøke å bevege seg mot lavere kjemisk potensial. Dette vil tilsvare en bevegelse mot et område hvor konsentrasjonen er mindre. Denne prosessen er kjent som <a href="/wiki/Diffusjon" title="Diffusjon">diffusjon</a>. </p><p>I en <a href="/wiki/Kjemisk_reaksjon" title="Kjemisk reaksjon">kjemisk reaksjon</a> inngår flere komponenter av forskjellige partikler eller stoffer. Hver komponent vil ha sitt eget kjemiske potensial. Ved konstant trykk og temperatur vil likevekt oppstå når <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs fri energi</a> for systemet er minst mulig. Dette tilsvarer at de kjemiske potensialene for begynnelsestilstanden er like store som for sluttilstanden. Denne <a href="/wiki/Kjemisk_likevekt" title="Kjemisk likevekt">kjemiske likevekten</a> vil da være beskrevet ved konsentrasjoner som oppfyller <a href="/wiki/Massevirkningsloven" title="Massevirkningsloven">massevirkningsloven</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formelle_egenskaper">Formelle egenskaper</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=1" title="Rediger avsnitt: Formelle egenskaper" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=1" title="Rediger kildekoden til seksjonen Formelle egenskaper"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fra <a href="/wiki/Termodynamikkens_andre_hovedsetning" title="Termodynamikkens andre hovedsetning">termodynamikkens andre hovedsetning</a> følger at den har en <a href="/wiki/Indre_energi" title="Indre energi">indre energi</a> <i>U</i> for et system ved temperatur <i>T</i> og trykk <i>P</i> vll forandre seg med den infinitesemal størrelsen <span class="nowrap"><i>dU = TdS - PdV + μdN </i></span> når dets <a href="/wiki/Entropi" title="Entropi">entropi</a> forandres med <i>dS</i>, volumet med <i>dV</i> og partikkeltallet med <i>dN </i> hvor nå <i>μ</i> er dets kjemiske potensial. Herav følger at dette er git som økningen av den indre partikkel når én partikkel tiløres systemet på en slik måte at entropien og volumet forblir uforandret. Matematisk tilsvarer det at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\left({\partial U \over \partial N}\right)_{SV}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\left({\partial U \over \partial N}\right)_{SV}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27814ad0cc683e97abfbf910227b0bdc43ed3e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.695ex; height:6.176ex;" alt="{\displaystyle \mu =\left({\partial U \over \partial N}\right)_{SV}}"></span></dd></dl> <p>Under <a href="/wiki/Legendre-transformasjon" title="Legendre-transformasjon">Legendre-transformasjoner</a> til <a href="/wiki/Helmholtz_fri_energi" title="Helmholtz fri energi">Helmholtz fri energi</a> <i>F(T,V,N) </i> og <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs fri energi</a> <i>G(T,P,N) </i> vil denne termodynamiske deriverte bevare sin form da partikkeltallet <i>N </i> er en passiv variabel under disse transformasjonene. Det gir at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\left({\partial F \over \partial N}\right)_{TV}=\left({\partial G \over \partial N}\right)_{TP}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\left({\partial F \over \partial N}\right)_{TV}=\left({\partial G \over \partial N}\right)_{TP}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d739291e4db48317b022d717fd3d8d26f0a85323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.153ex; height:6.176ex;" alt="{\displaystyle \mu =\left({\partial F \over \partial N}\right)_{TV}=\left({\partial G \over \partial N}\right)_{TP}}"></span></dd></dl> <p>Denne siste deriverte har en enkel betydning. Da <i>G(T,P,N) </i> er <b>ekstensiv</b>, det vil si direkte proporsjonal med systemets størrelse, mens de to variable <i>T</i> og <i>P</i> er <b>intensive</b>, må Gibbs fri energi være direkte proporsjonal med partikkeltallet <i>N</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T,P,N)=N\mu (T,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,P,N)=N\mu (T,P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8de7b26d38c32ff6df6a8c526a09e176f5664c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.938ex; height:2.843ex;" alt="{\displaystyle G(T,P,N)=N\mu (T,P)}"></span></dd></dl> <p>Dette viser at det kjemiske potensialet er lik fri energi per partikkel, <i>μ = G/N</i>. Denne egenskapen fører også til at det må oppfylle <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs-Duhem-ligningen</a> som bestemmer dets avhengighet av trykk og temperatur.<sup id="cite_ref-Castellan_1-0" class="reference"><a href="#cite_note-Castellan-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ideell_gass">Ideell gass</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=2" title="Rediger avsnitt: Ideell gass" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=2" title="Rediger kildekoden til seksjonen Ideell gass"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Tilstandsligning" title="Tilstandsligning">Tilstandsligningen</a> for en <a href="/wiki/Ideell_gass" title="Ideell gass">ideell gass</a> er <i>PV = Nk<sub>B</sub>T</i> hvor <i>k<sub>B</sub> </i> er <a href="/wiki/Boltzmanns_konstant" title="Boltzmanns konstant">Boltzmanns konstant</a>. Ved bruk av ren <a href="/wiki/Termodynamikk" title="Termodynamikk">termodynamikk</a> kan man da enkelt vise at det kjemiske potensialet må variere med trykket som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (T,P)=\mu _{0}(T,P_{0})+k_{B}T\ln(P/P_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (T,P)=\mu _{0}(T,P_{0})+k_{B}T\ln(P/P_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be29ba0bfc6c037b394e5369eb3b74d8f361d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.965ex; height:2.843ex;" alt="{\displaystyle \mu (T,P)=\mu _{0}(T,P_{0})+k_{B}T\ln(P/P_{0})}"></span></dd></dl> <p>Her er <i>P<sub>0</sub> </i> er et referansetrykk og <span class="nowrap"><i>μ<sub>0</sub>(T,P<sub>0</sub>) </i></span> er en ukjent funksjon av temperaturen. Denne kan bare finnes ved mer nøyaktige beregninger basert på <a href="/wiki/Statistisk_fysikk" title="Statistisk fysikk">statistisk fysikk</a>. Består gassen av ikke-relativistiske partikler med masse <i>m</i>, finner man da at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =k_{B}T\ln \left({P\Lambda ^{3} \over k_{B}T}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =k_{B}T\ln \left({P\Lambda ^{3} \over k_{B}T}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfa69e2418582a182267b7b22e68f56c09fb225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.824ex; height:6.343ex;" alt="{\displaystyle \mu =k_{B}T\ln \left({P\Lambda ^{3} \over k_{B}T}\right)}"></span></dd></dl> <p>når resultatatet uttrykkes ved <i>den termodynamiske bølgelengden</i> <span class="nowrap"><i>Λ = h/√(2πmk<sub>B</sub>T)</i></span> hvor <i>h </i> er <a href="/wiki/Plancks_konstant" title="Plancks konstant">Plancks konstant</a>. Det kjemiske potensialet for klassiske gasser tar i alminnelighet negative verdier, men i de fleste tilfeller er det bare potensialforskjeller som vil ha praktisk betydning. Og de kan være både positive og negative.<sup id="cite_ref-Rock_2-0" class="reference"><a href="#cite_note-Rock-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>I en reell gass er det vekselvirkninger mellom partiklene, og det kjemiske potensialet er mer vanskelig å beregne. Ofte skrives resulatatet da som for en ideell gass med den forskjell at trykket erstattes med <b>fugasiteten</b> som da er en funksjon av trykket. Er dette tilstrekkelig lavt slik at gassen er tilnærmet reell, blir fugasiteten lik trykket. </p> <div class="mw-heading mw-heading3"><h3 id="Ideell_blanding">Ideell blanding</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=3" title="Rediger avsnitt: Ideell blanding" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=3" title="Rediger kildekoden til seksjonen Ideell blanding"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Blander man sammen ideelle gasser, har man en <i>ideell gassblanding</i>. Ifølge <a href="/wiki/Daltons_lov" title="Daltons lov">Daltons lov</a> er det totale trykket <i>P </i> i gassen summen av partialtrykkene <i>P<sub>i</sub> </i> til hver av komponentene <i>i</i>. Hver slik komponent vil da ha et kjemisk potensial på formen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}(T,P_{i})=\mu _{0i}(T,P_{0})+k_{B}T\ln(P_{i}/P_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}(T,P_{i})=\mu _{0i}(T,P_{0})+k_{B}T\ln(P_{i}/P_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a15e0b0bbfef58329df095714d07f9c028e59ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.425ex; height:2.843ex;" alt="{\displaystyle \mu _{i}(T,P_{i})=\mu _{0i}(T,P_{0})+k_{B}T\ln(P_{i}/P_{0})}"></span></dd></dl> <p>Her er partialtrykket <i>P<sub>i</sub>   = x<sub>i</sub> P</i> hvor <i>x<sub>i</sub> = N<sub>i</sub> /N</i> er <a href="/wiki/Molfraksjon" class="mw-redirect" title="Molfraksjon">molfraksjonen</a> for partikler av type <i>i</i>. Uttrykt ved denne, kan da det kjemiske potensialet for denne komponenten skrives som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}(T,P_{i})=\mu _{0i}(T,P)+k_{B}T\ln x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}(T,P_{i})=\mu _{0i}(T,P)+k_{B}T\ln x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2c4a14677a88a18a424efa32e803118252b73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.33ex; height:2.843ex;" alt="{\displaystyle \mu _{i}(T,P_{i})=\mu _{0i}(T,P)+k_{B}T\ln x_{i}}"></span></dd></dl> <p>Her er nå det første leddet lik det kjemiske potensialet for en ren gass av denne komponenten alene og med trykk <i>P</i>. Da ln <i>x<sub>i</sub> ≤ 0</i>, er derfor det kjemiske potensialet i en gassblanding alltid mindre eller lik potensialet for den rene gassen ved samme totaltrykket. I den totale Gibbs fri energi <span class="nowrap"><i>G = ∑N<sub>i</sub>μ<sub>i</sub> </i></span> til blandingen gir bidragene fra partialtrykkene opphav til <a href="/wiki/Entropi" title="Entropi">blandingsentropien</a>.<sup id="cite_ref-Castellan_1-1" class="reference"><a href="#cite_note-Castellan-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Dette resultatet kan tas over til å definere en ideell blanding av forskjellige væsker. Da vil kreftene mellom partiklene være viktige. Men er disse tilnærmet de samme mellom alle partiklene og uavhengig av hvilken komponent de tilhører, vil væsken være en ideell blanding. Derfor er det kjemiske potensialet for hver komponent av samme form som om den skulle være i en ideell gassblanding. Derfor kan definere en væskeblanding som ideell når det kjemiske potensialet for hver komponent kan skrives på formen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}(T,x_{i})=\mu _{0i}(T)+k_{B}T\ln x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}(T,x_{i})=\mu _{0i}(T)+k_{B}T\ln x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24314da7b213c9fcfc1c17441d196a9d01d722c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.388ex; height:2.843ex;" alt="{\displaystyle \mu _{i}(T,x_{i})=\mu _{0i}(T)+k_{B}T\ln x_{i}}"></span></dd></dl> <p>hvis man ser bort fra en mulig trykkavhengighet i væsken. I virkeligheten viser de fleste reelle væsker en slik enkel oppførsel bare i svært begrenset omfang. </p> <div class="mw-heading mw-heading3"><h3 id="Kvantegasser">Kvantegasser</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=4" title="Rediger avsnitt: Kvantegasser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=4" title="Rediger kildekoden til seksjonen Kvantegasser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For relativistiske systemer er ikke partikkeltallet <i>N </i> veldefinert da partikler kan skapes ved pardannelse og forsvinne ved annihilasjon. Slike systemer vil derimot være spesifiserte ved de termodynamiske variable <i>T</i>, <i>P</i> og <i>μ</i>. Man benytter da at partikkeltallet <i>N</i> er den konjugerte variable til det kjemiske potensialet. Dermed kan <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs fri energi</a> <i>G(T,P,N)</i> <a href="/wiki/Legendre-transformasjon" title="Legendre-transformasjon">Legendre-transformeres</a> til den tilsvarende <a href="/wiki/Landau_fri_energi" title="Landau fri energi">Landau fri energi</a> <span class="nowrap"><i>Ω(T,P,μ)</i></span>. Denne benyttes spesielt til å beskrive termodynamiske systemer hvor <a href="/wiki/Kvantemekanikk" title="Kvantemekanikk">kvantemekanikken</a> dominerer. Dette kan være en gass av <a href="/wiki/Boson" title="Boson">bosoner</a> hvor det vil oppstå en <a href="/wiki/Bose-Einstein-kondensasjon" title="Bose-Einstein-kondensasjon">Bose-Einstein-kondensasjon</a> i grensen hvor det kjemiske potensialet nærmer seg null. Derimot for <a href="/wiki/Fermigass" class="mw-redirect" title="Fermigass">fermigasser</a> ved tilstrekkelig lave temperaturer, tar det kjemiske potensialet positive verdier og definerer det som kalles <a href="/w/index.php?title=Fermi-energien&action=edit&redlink=1" class="new" title="Fermi-energien (ikke skrevet ennå)">Fermi-energien</a> for gassen.<sup id="cite_ref-Huang_3-0" class="reference"><a href="#cite_note-Huang-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Faselikevekt">Faselikevekt</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=5" title="Rediger avsnitt: Faselikevekt" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=5" title="Rediger kildekoden til seksjonen Faselikevekt"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mange stoffer kan eksistere i forskjellige faser eller <a href="/wiki/Aggregattilstander" class="mw-redirect" title="Aggregattilstander">aggregattilstander</a>. For eksempel kan H<sub>2</sub>O eksistere være fast som is, flytende som vann eller som en gass i form av vanndamp. To faser av samme stoff er i <a href="/wiki/Termodynamisk_likevekt" title="Termodynamisk likevekt">termodynamisk likevekt</a> med hverandre når de har samme samme trykk og temperatur. I tillegg vil de også være i <b>kjemisk likevekt</b> når de kan utveksle partikler seg i mellom. Det betyr at de også må ha samme kjemiske potensial. Det følger direkte fra kravet om at i likevekt må <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs fri energi</a> for det koblede systemet være et minimum. </p><p>I alminnelighet er en infinitesemal forandring av <a href="/wiki/Gibbs_fri_energi" title="Gibbs fri energi">Gibbs fri energi</a> gitt ved <span class="nowrap"><i>dG = - SdT + VdP + μdN</i></span>. Ved konstant trykk og temperatur vil de to første bidragene være lik null. Den siste termen får bidrag fra begge fasene. Betegner man disse med de greske bokstavene α og β, vil overføring av et lite antall partikler <i>ΔN</i> mellom dem medføre en forandring av den fri energien gitt ved <span class="nowrap"><i>ΔG = (μ<sub>α</sub> - μ<sub>β</sub>)ΔN</i></span>. Dette skyldes at antall partikler som den ene fasen mister, bidrar til å øke like mye antallet av partikler i den andre fasen. Dermed har man den fundamentale ligningen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\alpha }(T,P)=\mu _{\beta }(T,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\alpha }(T,P)=\mu _{\beta }(T,P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fef102eda34905654d4b71f6af22ae32117650f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.81ex; height:3.009ex;" alt="{\displaystyle \mu _{\alpha }(T,P)=\mu _{\beta }(T,P)}"></span></dd></dl> <p>for faselikevekt definert ved <span class="nowrap"><i>ΔG = 0</i></span>. Kjenner man de to kjemiske potensialene, sier denne betingelsen hvordan trykket ved likevekt avhenger av temperaturen. Disse to variable er ikke lenger uavhengige av hverandre. For likevekt mellom væske og dens damp, gir dette opphav til <a href="/wiki/Clausius-Clapeyron-ligningen" class="mw-redirect" title="Clausius-Clapeyron-ligningen">Clausius-Clapeyron-ligningen</a>. Kommer systemet ut av likevekt, vil den fasen være stabil som har lavest kjemisk potensial. Trykk og temperatur kan da variere fritt.<sup id="cite_ref-Castellan_1-2" class="reference"><a href="#cite_note-Castellan-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Raoults_lov">Raoults lov</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=6" title="Rediger avsnitt: Raoults lov" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=6" title="Rediger kildekoden til seksjonen Raoults lov"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En væske som består av forskjellige stoffer, vil på samme måte også kunne være i likevekt med sin egen damp ved. Men dampen vil inneholde partikler fra de forskjellige stoffene i væsken. Hvis man antar at denne kan beskrives som en ideell gass, vil det kjemiske potensialet for komponent <i>i</i> med <a href="/wiki/Partialtrykk" title="Partialtrykk">partialtrykk</a> <i>P<sub>i</sub></i> være </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}(g)=\mu _{i0}(g)+k_{B}T\ln(P_{i}/P_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}(g)=\mu _{i0}(g)+k_{B}T\ln(P_{i}/P_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc76a8dd9412668830595f3f804456e73e09cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.478ex; height:2.843ex;" alt="{\displaystyle \mu _{i}(g)=\mu _{i0}(g)+k_{B}T\ln(P_{i}/P_{0})}"></span></dd></dl> <p>Antas også at væsken er en ideell blanding med en fraksjon <i>x<sub>i</sub> </i> med partikler av den samme komponenten, vil den i denne fasen ha et kjemisk potensial </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}(v)=\mu _{i0}(v)+k_{B}T\ln x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}(v)=\mu _{i0}(v)+k_{B}T\ln x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f398714e8a779571696151595727d104819b5141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.207ex; height:2.843ex;" alt="{\displaystyle \mu _{i}(v)=\mu _{i0}(v)+k_{B}T\ln x_{i}}"></span></dd></dl> <p>Det første leddet her er potensialet for en ren væske som består av denne komponenten alene. Den ville da ha et <a href="/wiki/Damptrykk" title="Damptrykk">damptrykk</a> <span class="nowrap"><i>P<sub>i0</sub> </i></span> gitt ved likevektsbetingelsen <span class="nowrap"><i>μ<sub>i0</sub>(v) = μ<sub>i0</sub>(g) + k<sub>B</sub>T </i>ln(<i>P<sub>i0</sub>/P<sub>0</sub> </i></span> i overensstemmelse med <a href="/wiki/Clausius-Clapeyron-ligningen" class="mw-redirect" title="Clausius-Clapeyron-ligningen">Clausius-Clapeyron-ligningen</a>. Brukes dette nå i likevektsbetingelsen <span class="nowrap"><i>μ<sub>i</sub>(v) = μ<sub>i</sub>(g)</i></span> for dampen over den blandete væsken, finner man at denne komponenten har et <a href="/wiki/Partialtrykk" title="Partialtrykk">partialtrykk</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}=x_{i}P_{i0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}=x_{i}P_{i0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d2f6aa855e098cb22f47914eb9b7375dbef9e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.634ex; height:2.509ex;" alt="{\displaystyle P_{i}=x_{i}P_{i0}}"></span></dd></dl> <p>Det er altså redusert i forhold til <a href="/wiki/Damptrykk" title="Damptrykk">damptrykket</a> i den rene væsken med en faktor som er <a href="/wiki/Molfraksjon" class="mw-redirect" title="Molfraksjon">molfraksjonen</a> til komponenten i væskeblandingen. Dette er <a href="/wiki/Raoults_lov" title="Raoults lov">Raoults lov</a>. </p><p>For blandinger av reelle væsker gjelder denne loven kun tilnærmet for den dominerende komponenten i en tynn blanding hvor en væske er tilsatt en liten mengde væsker av andre slag. Men likevel danner den et godt, teoretisk utgangspunkt for å beskrive slike virkelige blandinger.<sup id="cite_ref-Rock_2-1" class="reference"><a href="#cite_note-Rock-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=7" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=7" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Castellan-1"><b>^</b> <a href="#cite_ref-Castellan_1-0"><sup>a</sup></a> <a href="#cite_ref-Castellan_1-1"><sup>b</sup></a> <a href="#cite_ref-Castellan_1-2"><sup>c</sup></a> <span class="reference-text">G.W. Castellan, <i>Physical Chemistry</i>, Addison-Wesley Publishing Company, New York (1971). <a href="/wiki/Spesial:Bokkilder/0201103869" class="internal mw-magiclink-isbn">ISBN 0-20-110386-9</a>.</span> </li> <li id="cite_note-Rock-2"><b>^</b> <a href="#cite_ref-Rock_2-0"><sup>a</sup></a> <a href="#cite_ref-Rock_2-1"><sup>b</sup></a> <span class="reference-text">P.A. Rock, <i>Chemical Thermodynamics</i>, University Science Books, Oxford (1983). <a href="/wiki/Spesial:Bokkilder/0198557125" class="internal mw-magiclink-isbn">ISBN 0-19-855712-5</a>.</span> </li> <li id="cite_note-Huang-3"><b><a href="#cite_ref-Huang_3-0">^</a></b> <span class="reference-text">K. Huang, <i>Statistical Mechanics</i>, J. Wiley & Sons, New York (1987). <a href="/wiki/Spesial:Bokkilder/0471859133" class="internal mw-magiclink-isbn">ISBN 0-471-85913-3</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Litteratur">Litteratur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kjemisk_potensial&veaction=edit&section=8" title="Rediger avsnitt: Litteratur" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kjemisk_potensial&action=edit&section=8" title="Rediger kildekoden til seksjonen Litteratur"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>D. ter Haar and <a href="/wiki/Harald_Wergeland" title="Harald Wergeland">H. Wergeland</a>, <i>Elements of Thermodynamics</i>, Addison-Wesley Publishing Co. 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