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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</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">(úvod)</div> </a> </li> <li id="toc-Značení" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Značení"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Značení</span> </div> </a> <ul id="toc-Značení-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Výpočet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Výpočet"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Výpočet</span> </div> </a> <ul id="toc-Výpočet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vlastnosti</span> </div> </a> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vnitřní_energie_ideálního_plynu" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vnitřní_energie_ideálního_plynu"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vnitřní energie ideálního plynu</span> </div> </a> <button aria-controls="toc-Vnitřní_energie_ideálního_plynu-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>Přepnout podsekci Vnitřní energie ideálního plynu</span> </button> <ul id="toc-Vnitřní_energie_ideálního_plynu-sublist" class="vector-toc-list"> <li id="toc-Ekvipartiční_teorém" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ekvipartiční_teorém"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Ekvipartiční teorém</span> </div> </a> <ul id="toc-Ekvipartiční_teorém-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-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="Obsah" 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="Přepnout obsah" > <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">Přepnout obsah</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">Vnitřní energie</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="Přejděte k článku v jiném jazyce. Je dostupný v 66 jazycích" > <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-66" 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">66 jazyků</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D8%A9_%D8%AF%D8%A7%D8%AE%D9%84%D9%8A%D8%A9" title="طاقة داخلية – arabština" lang="ar" hreflang="ar" data-title="طاقة داخلية" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Enerx%C3%ADa_interno" title="Enerxía interno – asturština" lang="ast" hreflang="ast" data-title="Enerxía interno" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D0%BD%D1%83%D1%82%D1%80%D0%B0%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Унутраная энергія – běloruština" lang="be" hreflang="be" data-title="Унутраная энергія" data-language-autonym="Беларуская" data-language-local-name="běloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9D%D1%83%D1%82%D1%80%D0%B0%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%8D%D1%80%D0%B3%D1%96%D1%8F" title="Нутраная энэргія – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Нутраная энэргія" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%8A%D1%82%D1%80%D0%B5%D1%88%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Вътрешна енергия – bulharština" lang="bg" hreflang="bg" data-title="Вътрешна енергия" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AD%E0%A7%8D%E0%A6%AF%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0%E0%A7%80%E0%A6%A3_%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="অভ্যন্তরীণ শক্তি – bengálština" lang="bn" hreflang="bn" data-title="অভ্যন্তরীণ শক্তি" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Unutra%C5%A1nja_energija" title="Unutrašnja energija – bosenština" lang="bs" hreflang="bs" data-title="Unutrašnja energija" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Energia_interna" title="Energia interna – katalánština" lang="ca" hreflang="ca" data-title="Energia interna" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%88%D8%B2%DB%95%DB%8C_%D9%86%D8%A7%D9%88%DB%95%DA%A9%DB%8C" title="وزەی ناوەکی – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="وزەی ناوەکی" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%D0%B0%D0%BB%D1%82%D0%B8_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Шалти энерги – čuvaština" lang="cv" hreflang="cv" data-title="Шалти энерги" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Indre_energi" title="Indre energi – dánština" lang="da" hreflang="da" data-title="Indre energi" data-language-autonym="Dansk" data-language-local-name="dánština" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Innere_Energie" title="Innere Energie – němčina" lang="de" hreflang="de" data-title="Innere Energie" data-language-autonym="Deutsch" data-language-local-name="němčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BD%CE%AD%CF%81%CE%B3%CE%B5%CE%B9%CE%B1" title="Εσωτερική ενέργεια – řečtina" lang="el" hreflang="el" data-title="Εσωτερική ενέργεια" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Internal_energy" title="Internal energy – angličtina" lang="en" hreflang="en" data-title="Internal energy" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Interna_energio" title="Interna energio – esperanto" lang="eo" hreflang="eo" data-title="Interna energio" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Energ%C3%ADa_interna" title="Energía interna – španělština" lang="es" hreflang="es" data-title="Energía interna" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Siseenergia" title="Siseenergia – estonština" lang="et" hreflang="et" data-title="Siseenergia" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Barne_energia" title="Barne energia – baskičtina" lang="eu" hreflang="eu" data-title="Barne energia" data-language-autonym="Euskara" data-language-local-name="baskičtina" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%B1%DA%98%DB%8C_%D8%AF%D8%B1%D9%88%D9%86%DB%8C" title="انرژی درونی – perština" lang="fa" hreflang="fa" data-title="انرژی درونی" data-language-autonym="فارسی" data-language-local-name="perština" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Sis%C3%A4energia" title="Sisäenergia – finština" lang="fi" hreflang="fi" data-title="Sisäenergia" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89nergie_interne" title="Énergie interne – francouzština" lang="fr" hreflang="fr" data-title="Énergie interne" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fuinneamh_inmhe%C3%A1nach" title="Fuinneamh inmheánach – irština" lang="ga" hreflang="ga" data-title="Fuinneamh inmheánach" data-language-autonym="Gaeilge" data-language-local-name="irština" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Enerx%C3%ADa_interna" title="Enerxía interna – galicijština" lang="gl" hreflang="gl" data-title="Enerxía interna" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A4%D7%A0%D7%99%D7%9E%D7%99%D7%AA" title="אנרגיה פנימית – hebrejština" lang="he" hreflang="he" data-title="אנרגיה פנימית" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="आन्तरिक ऊर्जा – hindština" lang="hi" hreflang="hi" data-title="आन्तरिक ऊर्जा" data-language-autonym="हिन्दी" data-language-local-name="hindština" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Unutarnja_energija" title="Unutarnja energija – chorvatština" lang="hr" hreflang="hr" data-title="Unutarnja energija" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/En%C3%A8ji_t%C3%A8mik" title="Enèji tèmik – haitština" lang="ht" hreflang="ht" data-title="Enèji tèmik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitština" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bels%C5%91_energia" title="Belső energia – maďarština" lang="hu" hreflang="hu" data-title="Belső energia" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%A5%D6%80%D6%84%D5%AB%D5%B6_%D5%A7%D5%B6%D5%A5%D6%80%D5%A3%D5%AB%D5%A1" title="Ներքին էներգիա – arménština" lang="hy" hreflang="hy" data-title="Ներքին էներգիա" data-language-autonym="Հայերեն" data-language-local-name="arménština" 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/Energi_dalam" title="Energi dalam – indonéština" lang="id" hreflang="id" data-title="Energi dalam" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" 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/Energia_interna" title="Energia interna – italština" lang="it" hreflang="it" data-title="Energia interna" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%85%E9%83%A8%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC" title="内部エネルギー – japonština" lang="ja" hreflang="ja" data-title="内部エネルギー" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%86%D1%88%D0%BA%D1%96_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Ішкі энергия – kazaština" lang="kk" hreflang="kk" data-title="Ішкі энергия" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%B4%EB%B6%80_%EC%97%90%EB%84%88%EC%A7%80" title="내부 에너지 – korejština" lang="ko" hreflang="ko" data-title="내부 에너지" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D1%87%D0%BA%D0%B8_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Ички энергия – kyrgyzština" lang="ky" hreflang="ky" data-title="Ички энергия" data-language-autonym="Кыргызча" data-language-local-name="kyrgyzština" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vidin%C4%97_energija" title="Vidinė energija – litevština" lang="lt" hreflang="lt" data-title="Vidinė energija" data-language-autonym="Lietuvių" data-language-local-name="litevština" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Iek%C5%A1%C4%93j%C4%81_ener%C4%A3ija" title="Iekšējā enerģija – lotyština" lang="lv" hreflang="lv" data-title="Iekšējā enerģija" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%BD%D0%B0%D1%82%D1%80%D0%B5%D1%88%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Внатрешна енергија – makedonština" lang="mk" hreflang="mk" data-title="Внатрешна енергија" data-language-autonym="Македонски" data-language-local-name="makedonština" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%B0%E0%B4%BF%E0%B4%95_%E0%B4%8A%E0%B5%BC%E0%B4%9C%E0%B5%8D%E0%B4%9C%E0%B4%82" title="ആന്തരിക ഊർജ്ജം – malajálamština" lang="ml" hreflang="ml" data-title="ആന്തരിക ഊർജ്ജം" data-language-autonym="മലയാളം" data-language-local-name="malajálamština" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D0%BE%D1%82%D0%BE%D0%BE%D0%B4_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Дотоод энерги – mongolština" lang="mn" hreflang="mn" data-title="Дотоод энерги" data-language-autonym="Монгол" data-language-local-name="mongolština" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tenaga_dalaman" title="Tenaga dalaman – malajština" lang="ms" hreflang="ms" data-title="Tenaga dalaman" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inwendige_energie" title="Inwendige energie – nizozemština" lang="nl" hreflang="nl" data-title="Inwendige energie" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Indre_energi" title="Indre energi – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Indre energi" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Indre_energi" title="Indre energi – norština (bokmål)" lang="nb" hreflang="nb" data-title="Indre energi" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Energia_int%C3%A8rna" title="Energia intèrna – okcitánština" lang="oc" hreflang="oc" data-title="Energia intèrna" data-language-autonym="Occitan" data-language-local-name="okcitánština" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Energia_wewn%C4%99trzna" title="Energia wewnętrzna – polština" lang="pl" hreflang="pl" data-title="Energia wewnętrzna" data-language-autonym="Polski" data-language-local-name="polština" 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/Energia_interna" title="Energia interna – portugalština" lang="pt" hreflang="pt" data-title="Energia interna" data-language-autonym="Português" data-language-local-name="portugalština" 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/Energie_intern%C4%83" title="Energie internă – rumunština" lang="ro" hreflang="ro" data-title="Energie internă" data-language-autonym="Română" data-language-local-name="rumunština" 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%92%D0%BD%D1%83%D1%82%D1%80%D0%B5%D0%BD%D0%BD%D1%8F%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Внутренняя энергия – ruština" lang="ru" hreflang="ru" data-title="Внутренняя энергия" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Unutarnja_energija" title="Unutarnja energija – srbochorvatština" lang="sh" hreflang="sh" data-title="Unutarnja energija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Internal_energy" title="Internal energy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Internal energy" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vn%C3%BAtorn%C3%A1_energia" title="Vnútorná energia – slovenština" lang="sk" hreflang="sk" data-title="Vnútorná energia" data-language-autonym="Slovenčina" data-language-local-name="slovenština" 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/Notranja_energija" title="Notranja energija – slovinština" lang="sl" hreflang="sl" data-title="Notranja energija" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Energjia_e_brendshme" title="Energjia e brendshme – albánština" lang="sq" hreflang="sq" data-title="Energjia e brendshme" data-language-autonym="Shqip" data-language-local-name="albánština" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Unutra%C5%A1nja_energija" title="Unutrašnja energija – srbština" lang="sr" hreflang="sr" data-title="Unutrašnja energija" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Inre_energi" title="Inre energi – švédština" lang="sv" hreflang="sv" data-title="Inre energi" data-language-autonym="Svenska" data-language-local-name="švédština" 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%85%E0%AE%95_%E0%AE%86%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%B2%E0%AF%8D" title="அக ஆற்றல் – tamilština" lang="ta" hreflang="ta" data-title="அக ஆற்றல்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" 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/%C4%B0%C3%A7_enerji" title="İç enerji – turečtina" lang="tr" hreflang="tr" data-title="İç enerji" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%BD%D1%83%D1%82%D1%80%D1%96%D1%88%D0%BD%D1%8F_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Внутрішня енергія – ukrajinština" lang="uk" hreflang="uk" data-title="Внутрішня енергія" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ichki_energiya" title="Ichki energiya – uzbečtina" lang="uz" hreflang="uz" data-title="Ichki energiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/N%E1%BB%99i_n%C4%83ng" title="Nội năng – vietnamština" lang="vi" hreflang="vi" data-title="Nội năng" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%86%85%E8%83%BD" title="内能 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="内能" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%86%85%E8%83%BD" title="内能 – čínština" lang="zh" hreflang="zh" data-title="内能" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%85%A7%E8%83%BD" title="內能 – čínština (klasická)" lang="lzh" hreflang="lzh" data-title="內能" data-language-autonym="文言" data-language-local-name="čínština (klasická)" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/L%C5%8De-l%C3%AAng" title="Lōe-lêng – čínština (dialekty Minnan)" lang="nan" hreflang="nan" data-title="Lōe-lêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="čínština (dialekty Minnan)" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedie, otevřené encyklopedie</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="cs" dir="ltr"><div class="noprint ambox labelced labelced-page labelced-page-type-content ambox-content plainlinks"> <div class="labelced_message"><div class="labelced_image ambox-image mbox-image"><span typeof="mw:File"><span><img alt="ikona" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/48px-Question_book-new.svg.png" decoding="async" width="48" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/72px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/96px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></span></span></div> <div class="labelced_message_inner"> <div class="labelced_message_headline ambox-text">Tento článek není dostatečně <a href="/wiki/Wikipedie:Uv%C3%A1d%C4%9Bn%C3%AD_zdroj%C5%AF" title="Wikipedie:Uvádění zdrojů">ozdrojován</a>, a může tedy obsahovat informace, které je třeba <a href="/wiki/Wikipedie:Ov%C4%9B%C5%99itelnost" title="Wikipedie:Ověřitelnost">ověřit</a>.</div> <div class="labelced_message_text hide-when-compact ambox-text">Jste-li s popisovaným předmětem seznámeni, pomozte doložit uvedená tvrzení doplněním <a href="/wiki/N%C3%A1pov%C4%9Bda:Reference_ve_Vizu%C3%A1ln%C3%ADm_editoru" title="Nápověda:Reference ve Vizuálním editoru">referencí</a> na <a href="/wiki/Wikipedie:V%C4%9Brohodn%C3%A9_zdroje" title="Wikipedie:Věrohodné zdroje">věrohodné zdroje</a>.</div> </div> </div> </div> <p><b>Vnitřní energie</b> (též <b>termodynamická energie</b><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>) <a href="/wiki/T%C4%9Bleso" title="Těleso">tělesa</a> (<a href="/wiki/Termodynamick%C3%BD_syst%C3%A9m" title="Termodynamický systém">termodynamického systému</a>) je <a href="/wiki/Fyzik%C3%A1ln%C3%AD_veli%C4%8Dina#Veličiny_extenzivní,_intenzivní_a_protenzivní" title="Fyzikální veličina">extenzivní</a> <a href="/wiki/Fyzik%C3%A1ln%C3%AD_veli%C4%8Dina" title="Fyzikální veličina">veličina</a> představující v makroskopickém popisu souhrn <a href="/wiki/Energie" title="Energie">energií</a> všech <a href="/wiki/%C4%8C%C3%A1stice" title="Částice">částic</a>, z nichž se těleso skládá. Jde především o jejich <a href="/wiki/Kinetick%C3%A1_energie" title="Kinetická energie">kinetickou</a> a <a href="/wiki/Potenci%C3%A1ln%C3%AD_energie" title="Potenciální energie">potenciální energii</a>, ale může jít také o <a href="/wiki/Elektrick%C3%A1_energie" title="Elektrická energie">elektrickou</a> či <a href="/wiki/Chemick%C3%A1_energie" title="Chemická energie">chemickou energii</a>, apod. Kinetická a potenciální energie, kterou má těleso (soustava) jako celek, se do vnitřní energie nezahrnuje. </p><p>Jedná se o jeden ze základních <a href="/wiki/Termodynamick%C3%BD_potenci%C3%A1l" title="Termodynamický potenciál">termodynamických potenciálů</a>, z ní vycházejí <a href="/wiki/Termodynamick%C3%BD_potenci%C3%A1l#Definiční_vztahy" title="Termodynamický potenciál">definice</a> všech ostatních. </p><p>Vnitřní energie ovlivňuje vlastnosti a <a href="/wiki/Stav_t%C4%9Blesa" title="Stav tělesa">stav</a> <a href="/wiki/L%C3%A1tka" title="Látka">látky</a>. Např. kinetická energie částic se projevuje jako <a href="/wiki/Teplota" title="Teplota">teplota</a> tělesa, tzn. čím rychlejší pohyb částic, tím vyšší je teplota tělesa. Polohová energie částic se projevuje ve vlastnostech tělesa jako <a href="/wiki/Skupenstv%C3%AD" title="Skupenství">skupenství</a>, <a href="/wiki/Stla%C4%8Ditelnost" title="Stlačitelnost">stlačitelnost</a>/<a href="/wiki/Pru%C5%BEnost" title="Pružnost">pružnost</a> či <a href="/wiki/Pevnost_(fyzika)" title="Pevnost (fyzika)">pevnost</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Značení"><span id="Zna.C4.8Den.C3.AD"></span>Značení</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=1" title="Editace sekce: Značení" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=1" title="Editovat zdrojový kód sekce Značení"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Symbol <a href="/wiki/Fyzik%C3%A1ln%C3%AD_veli%C4%8Dina" title="Fyzikální veličina">veličiny</a>: <i>U</i></li> <li>Hlavní <a href="/wiki/Fyzik%C3%A1ln%C3%AD_jednotka" title="Fyzikální jednotka">jednotka</a> v <a href="/wiki/Soustava_SI" title="Soustava SI">SI</a> (řazená mezi <a href="/wiki/Odvozen%C3%A1_jednotka_SI" title="Odvozená jednotka SI">odvozené jednotky</a>): <a href="/wiki/Joule" title="Joule">joule</a>, značka jednotky: J;</li> <li>Rozměr: [J]=[m]<sup>2</sup>·[kg]·[s]<sup>−2</sup></li> <li>Další jednotky: viz <a href="/wiki/Energie" title="Energie">energie</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Výpočet"><span id="V.C3.BDpo.C4.8Det"></span>Výpočet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=2" title="Editace sekce: Výpočet" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=2" title="Editovat zdrojový kód sekce Výpočet"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Tepeln%C3%BD_pohyb" class="mw-redirect" title="Tepelný pohyb">Tepelnému pohybu</a> částic přísluší určitá kinetická energie. Pokud látka obsahuje <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> částic o stejné <a href="/wiki/Hmotnost" title="Hmotnost">hmotnosti</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 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/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> a velikost <a href="/wiki/Rychlost" title="Rychlost">rychlosti</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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-té částice je <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span>, pak celková kinetická energie takovéto soustavy pak bude </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_{k}=\sum _{i=1}^{n}{\frac {1}{2}}mv_{i}^{2}={\frac {1}{2}}m\sum _{i=1}^{n}v_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}=\sum _{i=1}^{n}{\frac {1}{2}}mv_{i}^{2}={\frac {1}{2}}m\sum _{i=1}^{n}v_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c89b45fb76135c146ead6e92f151794b4d10089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.314ex; height:6.843ex;" alt="{\displaystyle E_{k}=\sum _{i=1}^{n}{\frac {1}{2}}mv_{i}^{2}={\frac {1}{2}}m\sum _{i=1}^{n}v_{i}^{2}}"></span></dd></dl> <p>Je třeba si uvědomit, že těleso jako celek zůstává v <a href="/wiki/Mechanick%C3%BD_pohyb" title="Mechanický pohyb">klidu</a>. Rychlosti <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 \mathbf {v} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"></span> jednotlivých částic odpovídají jejich mikroskopickému pohybu. Pokud by se těleso pohybovalo, např. rychlostí <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 \mathbf {v} _{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/168079e60e95c0bbd564581db6c796b650de63ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.37ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{M}}"></span>, pak pro určení vnitřní energie <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-té částice se uvažuje s rychlostí <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 \mathbf {v} _{i}-\mathbf {v} _{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{i}-\mathbf {v} _{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48660888ef9d6707aade218d0a166b582c6b90c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.421ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} _{i}-\mathbf {v} _{M}}"></span>. </p><p>Kdyby měly všechny částice stejnou rychlost <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_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span>, byla by celková energie soustavy </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_{k}={\frac {1}{2}}nmv_{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}={\frac {1}{2}}nmv_{k}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b07457bc0f73041b8f9959491303f3c0bed9f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.552ex; height:5.176ex;" alt="{\displaystyle E_{k}={\frac {1}{2}}nmv_{k}^{2}}"></span></dd></dl> <p>Má-li být celková energie v obou případech stejná, pak musí z předchozích vztahů platit </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}^{2}={\frac {1}{n}}\sum _{i=1}^{n}v_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}^{2}={\frac {1}{n}}\sum _{i=1}^{n}v_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173e51234d92ca2e3f9dc2645e24f9f690d638e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.857ex; height:6.843ex;" alt="{\displaystyle v_{k}^{2}={\frac {1}{n}}\sum _{i=1}^{n}v_{i}^{2}}"></span></dd></dl> <p>Rychlost <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_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> se nazývá <i><a href="/wiki/St%C5%99edn%C3%AD_kvadratick%C3%A1_rychlost" title="Střední kvadratická rychlost">střední kvadratickou rychlostí</a></i>. Kinetická energie určená pomocí střední kvadratické rychlosti se nazývá <b>střední kinetická energie</b>. </p><p>Vzájemná soudržnost částic, kterou lze pozorovat u <a href="/wiki/Pevn%C3%A1_l%C3%A1tka" title="Pevná látka">pevných látek</a> a <a href="/wiki/Kapalina" title="Kapalina">kapalin</a>, vede k závěru, že mezi molekulami musí působit určité síly. Tyto síly se projevují tím, že při <a href="/wiki/Vzd%C3%A1lenost" title="Vzdálenost">vzdálenostech</a> menších než je rozměr <a href="/wiki/Atom" title="Atom">atomu</a> se projevují jako <a href="/wiki/Odpudiv%C3%A1_s%C3%ADla" class="mw-redirect" title="Odpudivá síla">odpudivé</a>, a při vzdálenostech větších jako <a href="/wiki/P%C5%99ita%C5%BEliv%C3%A1_s%C3%ADla" class="mw-redirect" title="Přitažlivá síla">přitažlivé</a>. Tyto síly se nazývají <a href="/w/index.php?title=Molekul%C3%A1rn%C3%AD_s%C3%ADla&action=edit&redlink=1" class="new" title="Molekulární síla (stránka neexistuje)">molekulárními (popř. kohezními) silami</a>. Původ těchto sil lze částečně vysvětlit <a href="/wiki/Elektrostatika" title="Elektrostatika">elektrostatickým</a> přitahováním a odpuzováním molekul, avšak plný výklad podává až <a href="/wiki/Kvantov%C3%A1_fyzika" title="Kvantová fyzika">kvantová fyzika</a>. </p><p>Kohezní síly způsobují, že dvě blízké molekuly mají určitou <a href="/wiki/Potenci%C3%A1ln%C3%AD_energie" title="Potenciální energie">potenciální energii</a>. Odpudivým silám přísluší kladná potenciální energie <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"></span> a přitažlivým silám přísluší záporná potenciální energie <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6ee346e54f38302f47b5cf3016d8718f2040c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{2}}"></span>, přičemž hladinu <a href="/wiki/Nula" title="Nula">nulové</a> potenciální energie klademe do <a href="/wiki/Nekone%C4%8Dno" title="Nekonečno">nekonečna</a>. Celková potenciální energie je pak součtem těchto složek <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_{p}=E_{1}+E_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{p}=E_{1}+E_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e469051256f5dfb3bd55282bf1526319ceaee2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.252ex; height:2.843ex;" alt="{\displaystyle E_{p}=E_{1}+E_{2}}"></span>. </p><p>Celková vnitřní energie soustavy se určí jako suma kinetických a potenciálních energií jednotlivých částic, tzn. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=E_{k}+E_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=E_{k}+E_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e62ea63f79aef014a7328a2b21f3d40c8548f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.3ex; height:2.843ex;" alt="{\displaystyle U=E_{k}+E_{p}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=3" title="Editace sekce: Vlastnosti" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=3" title="Editovat zdrojový kód sekce Vlastnosti"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vnitřní energie určuje tepelný stav a <a href="/wiki/Skupenstv%C3%AD" title="Skupenství">skupenství</a> látky. </p><p>Vnitřní energii lze měnit </p> <ul><li><i>konáním práce</i> – Při konání <a href="/wiki/Pr%C3%A1ce_(fyzika)" title="Práce (fyzika)">práce</a> dochází působením <a href="/wiki/Vn%C4%9Bj%C5%A1%C3%AD_s%C3%ADla" class="mw-redirect" title="Vnější síla">vnějších sil</a> ke změně <a href="/wiki/Objem" title="Objem">objemu</a> nebo <a href="/wiki/Tlak" title="Tlak">tlaku</a> soustavy, což vede ke změně kinetické energie částic a tedy i celkové vnitřní energie soustavy.</li> <li><i>tepelnou výměnou</i> – Změnou <a href="/wiki/Teplota" title="Teplota">teploty</a> dochází ke změně kinetické energie částic, což má za následek změnu celkové vnitřní energie soustavy.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Vnitřní_energie_ideálního_plynu"><span id="Vnit.C5.99n.C3.AD_energie_ide.C3.A1ln.C3.ADho_plynu"></span>Vnitřní energie ideálního plynu</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=4" title="Editace sekce: Vnitřní energie ideálního plynu" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=4" title="Editovat zdrojový kód sekce Vnitřní energie ideálního plynu"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U <a href="/wiki/Ide%C3%A1ln%C3%AD_plyn" title="Ideální plyn">ideálního plynu</a> je vnitřní energie dána součtem <a href="/wiki/Kinetick%C3%A1_energie" title="Kinetická energie">kinetických energií</a> jednotlivých <a href="/wiki/%C4%8C%C3%A1stice" title="Částice">částic</a> <a href="/wiki/Plyn" title="Plyn">plynu</a>. </p><p>Střední kinetická energie <a href="/wiki/Molekulov%C3%BD_pohyb" class="mw-redirect" title="Molekulový pohyb">molekulového pohybu</a>, tzn. <a href="/wiki/St%C5%99edn%C3%AD_hodnota" title="Střední hodnota">střední hodnota</a> kinetické energie plynu připadající na jednu částici, je </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {\sum _{i=1}^{n}{\frac {1}{2}}mv_{i}^{2}}{n}}={\frac {1}{2}}m{\frac {\sum _{i=1}^{n}v_{i}^{2}}{n}}={\frac {1}{2}}mv_{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {\sum _{i=1}^{n}{\frac {1}{2}}mv_{i}^{2}}{n}}={\frac {1}{2}}m{\frac {\sum _{i=1}^{n}v_{i}^{2}}{n}}={\frac {1}{2}}mv_{k}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14896ccf7188cef8787292a3ec5914bbb3e3700e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.136ex; height:6.343ex;" alt="{\displaystyle u={\frac {\sum _{i=1}^{n}{\frac {1}{2}}mv_{i}^{2}}{n}}={\frac {1}{2}}m{\frac {\sum _{i=1}^{n}v_{i}^{2}}{n}}={\frac {1}{2}}mv_{k}^{2}}"></span>,</dd></dl> <p>kde <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/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> je hmotnost libovolné částice plynu, <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> je rychlost <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-té částice, <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> je celkový počet částic a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d142b4083872eb72f81c1e20fd2c91d02b4a9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}"></span> je <a href="/wiki/St%C5%99edn%C3%AD_kvadratick%C3%A1_rychlost" title="Střední kvadratická rychlost">střední kvadratická rychlost</a>. Dosazením za střední kvadratickou rychlost dostaneme </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {1}{2}}mv_{k}^{2}={\frac {1}{2}}m{\left({\sqrt {\frac {3kT}{m}}}\right)}^{2}={\frac {3}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>3</mn> <mi>k</mi> <mi>T</mi> </mrow> <mi>m</mi> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {1}{2}}mv_{k}^{2}={\frac {1}{2}}m{\left({\sqrt {\frac {3kT}{m}}}\right)}^{2}={\frac {3}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a70e17ec460ee3bd90a6ad9eded5b6e1ac54ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.411ex; height:6.843ex;" alt="{\displaystyle u={\frac {1}{2}}mv_{k}^{2}={\frac {1}{2}}m{\left({\sqrt {\frac {3kT}{m}}}\right)}^{2}={\frac {3}{2}}kT}"></span>,</dd></dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> je <a href="/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta">Boltzmannova konstanta</a> 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 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> je <a href="/wiki/Termodynamick%C3%A1_teplota" title="Termodynamická teplota">termodynamická teplota</a>. </p><p>Střední kinetická energie částice je tedy přímo úměrná termodynamické teplotě plynu. </p><p>Celková vnitřní energie N částic neboli <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={\frac {N}{N_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {N}{N_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f0b1bb0c88fc7f1fb5c19eb758f8db2d19108e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.66ex; height:5.676ex;" alt="{\displaystyle n={\frac {N}{N_{A}}}}"></span> <a href="/wiki/Mol_(jednotka)" title="Mol (jednotka)">molů</a> je podle tohoto vztahu rovna: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=Nu={\frac {3}{2}}NkT={\frac {3}{2}}nRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>N</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>N</mi> <mi>k</mi> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=Nu={\frac {3}{2}}NkT={\frac {3}{2}}nRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4738eaacef133ca84e28410a24625e44f8c2795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.175ex; height:5.176ex;" alt="{\displaystyle U=Nu={\frac {3}{2}}NkT={\frac {3}{2}}nRT}"></span></dd></dl> <p>kde <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_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9233ef6a6a2c0576dbf65490ddb8307fde494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle N_{A}}"></span> je <a href="/wiki/Avogadrova_konstanta" title="Avogadrova konstanta">Avogadrova konstanta</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> je <a href="/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta">Boltzmannova konstanta</a> 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 R=N_{A}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=N_{A}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a9ff30c3f7204382aeeab2c1891039fa6f7492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.405ex; height:2.509ex;" alt="{\displaystyle R=N_{A}k}"></span> je <a href="/wiki/Mol%C3%A1rn%C3%AD_plynov%C3%A1_konstanta" title="Molární plynová konstanta">molární plynová konstanta</a>. </p><p>Z výsledků <a href="/wiki/Experiment" title="Experiment">experimentů</a> vyplývá, že vztahy pro vnitřní energii odpovídají jednoatomovým plynům, nevyhovují však dostatečně pro víceatomové plyny. U těchto plynů totiž k vnitřní energii přispívá také <a href="/wiki/Rota%C4%8Dn%C3%AD_pohyb" class="mw-redirect" title="Rotační pohyb">rotační pohyb</a> částic. Celková kinetická energie je pak součtem kinetické energie <a href="/wiki/Posuvn%C3%BD_pohyb" title="Posuvný pohyb">posuvného</a> a rotačního pohybu. </p><p><a href="/wiki/Symetrie" title="Symetrie">Symetrie</a> částice jednoatomového plynu neumožňuje zahrnout <a href="/wiki/Rotace" class="mw-disambig" title="Rotace">rotaci</a>, neboť při pootočení částice nedojde k žádné změně systému. Takovouto částici lze považovat za <a href="/wiki/Hmotn%C3%BD_bod" title="Hmotný bod">hmotný bod</a>. Částice jednoatomového plynu se tedy může pohybovat pouze posuvným pohybem podél tří prostorových os, tzn. má <i>tři <a href="/wiki/Stupe%C5%88_volnosti" title="Stupeň volnosti">stupně volnosti</a></i>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Molekula_2atomy_stupne_volnosti.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Molekula_2atomy_stupne_volnosti.svg/220px-Molekula_2atomy_stupne_volnosti.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Molekula_2atomy_stupne_volnosti.svg/330px-Molekula_2atomy_stupne_volnosti.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Molekula_2atomy_stupne_volnosti.svg/440px-Molekula_2atomy_stupne_volnosti.svg.png 2x" data-file-width="603" data-file-height="402" /></a><figcaption>Pohyby dvouatomové molekuly.</figcaption></figure> <p>Dvouatomovou <a href="/wiki/Molekula" title="Molekula">molekulu</a> se lze představit jako dva pevně spojené hmotné body. Taková částice může kromě posuvného pohybu konat také pohyb rotační kolem dvou vzájemně <a href="/wiki/Ortogonalita" title="Ortogonalita">kolmých</a> os, které jsou kolmé k <a href="/wiki/Osa_rotace" class="mw-redirect" title="Osa rotace">ose rotace</a> částice. Kromě tří stupňů volnosti připadajících na posuvný pohyb má tedy tato částice další dva stupně volnosti, které připadají na rotační pohyb, tzn. celkem <i>pět stupňů volnosti</i>. </p><p>Podobně u tříatomových a víceatomových molekul připadají na posuvný pohyb tři stupně volnosti a na rotační pohyb další tři stupně volnosti, neboť tato částice může rotovat kolem všech prostorových os. Celkem je to tedy <i>šest stupňů volnosti</i>. </p> <table class="wikitable"> <tbody><tr> <td><b>Plyn</b> </td> <td><b>Počet <a href="/wiki/Stupe%C5%88_volnosti" title="Stupeň volnosti">stupňů volnosti</a></b> </td></tr> <tr> <td>jednoatomový </td> <td>3 </td></tr> <tr> <td>dvouatomový </td> <td>5 </td></tr> <tr> <td>tříatomový a víceatomový </td> <td>6 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ekvipartiční_teorém"><span id="Ekviparti.C4.8Dn.C3.AD_teor.C3.A9m"></span>Ekvipartiční teorém</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=5" title="Editace sekce: Ekvipartiční teorém" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=5" title="Editovat zdrojový kód sekce Ekvipartiční teorém"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="uvodni-upozorneni hatnote"> Podrobnější informace naleznete v článku <a href="/wiki/Ekviparti%C4%8Dn%C3%AD_teor%C3%A9m" title="Ekvipartiční teorém">Ekvipartiční teorém</a>.</div> <p>Pokud na tři stupně volnosti částice jednoatomového plynu připadá <a href="/wiki/Energie" title="Energie">energie</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 u={\frac {3}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {3}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6d25835f9fd9afbd4703334c7d1e8108b62f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.274ex; height:5.176ex;" alt="{\displaystyle u={\frac {3}{2}}kT}"></span>, pak lze předpokládat, že na jeden <a href="/wiki/Stupe%C5%88_volnosti" title="Stupeň volnosti">stupeň volnosti</a> připadá energie </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}={\frac {1}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}={\frac {1}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279b77c0648d6a5943cd392854b47142f835882a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.328ex; height:5.176ex;" alt="{\displaystyle u_{1}={\frac {1}{2}}kT}"></span></dd></dl> <p>Tento závěr je důsledkem předpokladu o <a href="/wiki/Rovnom%C4%9Brn%C3%A9_rozd%C4%9Blen%C3%AD" title="Rovnoměrné rozdělení">rovnoměrném rozdělení</a> energie, který je znám jako <b>ekvipartiční teorém</b>. <a href="/wiki/Ekviparti%C4%8Dn%C3%AD_teor%C3%A9m" title="Ekvipartiční teorém">Ekvipartiční teorém</a> lze formulovat následovně: </p> <dl><dd><b>Energie soustavy je rovnoměrně rozdělena na všechny platné stupně volnosti.</b></dd></dl> <p>Pro <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> stupňů volnosti je možné tuto skutečnost zapsat v <a href="/wiki/Matematika" title="Matematika">matematické</a> formě </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}={\frac {i}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}={\frac {i}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94afe1ab78bf639d4aa9bb29bb4516c816dee784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.074ex; height:5.176ex;" alt="{\displaystyle u_{i}={\frac {i}{2}}kT}"></span></dd></dl> <p>Ekvipartiční teorém lze využít např. pro určení <a href="/wiki/Mol%C3%A1rn%C3%AD_tepeln%C3%A1_kapacita" title="Molární tepelná kapacita">molární tepelné kapacity</a> pří stálém tlaku <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_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c18988639f2937d3fe4dc00dfa0b0bd7e0bec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.158ex; height:2.509ex;" alt="{\displaystyle C_{V}}"></span>, kterou lze získat prostřednictvím vztahu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{V}=U_{T+1}-U_{T}={\frac {i}{2}}R(T+1)-{\frac {i}{2}}RT={\frac {i}{2}}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}=U_{T+1}-U_{T}={\frac {i}{2}}R(T+1)-{\frac {i}{2}}RT={\frac {i}{2}}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ad9fddf1dc819b1deba69cb89d021bd74d2839" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.561ex; height:5.176ex;" alt="{\displaystyle C_{V}=U_{T+1}-U_{T}={\frac {i}{2}}R(T+1)-{\frac {i}{2}}RT={\frac {i}{2}}R}"></span></dd></dl> <p>Pro jednotlivé typy plynů platí následující vztahy. </p> <table class="wikitable"> <tbody><tr> <td><b>Plyn</b> </td> <td><b>vnitřní energie na jednu částici</b> </td> <td><b>molární vnitřní energie</b> </td> <td><b>molární tepelná kapacita při stálém objemu</b> </td> <td><b>molární tepelná kapacita za stálého tlaku</b> </td></tr> <tr> <td>jednoatomový </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 {\frac {3}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d6b9d393b94f01f8943a0c0c3ff4c51a6b508fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.846ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}kT}"></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 {\frac {3}{2}}RT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}RT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35a01d24ccb37eb9c05aa1fd8f724cf12f60a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.399ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}RT}"></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 {\frac {3}{2}}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9892b1d26924b7db4892578f5b8392da125f9eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.763ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}R}"></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 {\frac {5}{2}}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{2}}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42682ba5128fd462c0fa01a40497a72e2ba5454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.763ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{2}}R}"></span> </td></tr> <tr> <td>dvouatomový </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 {\frac {5}{2}}kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{2}}kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e7bf7ea7006af516b61b4069a743721e77ad48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.846ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{2}}kT}"></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 {\frac {5}{2}}RT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{2}}RT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0353ac4fdcb24756dc8e8d6b04a929f23d840219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.399ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{2}}RT}"></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 {\frac {5}{2}}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{2}}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42682ba5128fd462c0fa01a40497a72e2ba5454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.763ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{2}}R}"></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 {\frac {7}{2}}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{2}}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912c6720b8535b27b1bd192875a974c0bee7ed85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.763ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{2}}R}"></span> </td></tr> <tr> <td>tříatomový a víceatomový </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 3kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53662bda208090eaf8ae010d3d7e92c8415f1944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.01ex; height:2.176ex;" alt="{\displaystyle 3kT}"></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 3RT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3RT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcbea54613015a47080dc3edcc9750a05cd73d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.563ex; height:2.176ex;" alt="{\displaystyle 3RT}"></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 3R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7a1e9e19106a30f8a46bacdba16f0ac9a5dc45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.926ex; height:2.176ex;" alt="{\displaystyle 3R}"></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 4R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2aad08a148667477e01529f097394cfe97ce523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.926ex; height:2.176ex;" alt="{\displaystyle 4R}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Reference">Reference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=6" title="Editace sekce: Reference" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=6" title="Editovat zdrojový kód sekce Reference"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">ČSN ISO 80000-5 Veličiny a jednotky - Část 5: Termodynamika</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=7" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=7" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kinetick%C3%A1_teorie_l%C3%A1tek" title="Kinetická teorie látek">Kinetická teorie látek</a></li> <li><a href="/wiki/Chemick%C3%A1_energie" title="Chemická energie">Chemická energie</a></li> <li><a href="/wiki/Energie" title="Energie">Energie</a></li> <li><a href="/wiki/Entalpie" title="Entalpie">Entalpie</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&veaction=edit&section=8" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vnit%C5%99n%C3%AD_energie&action=edit&section=8" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="sisterproject sisterproject-wiktionary"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/24px-Wiktionary-logo-cs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/32px-Wiktionary-logo-cs.svg.png 2x" data-file-width="411" data-file-height="411" /></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Slovníkové heslo </span><span class="sisterproject_text_target"><a href="https://cs.wiktionary.org/wiki/energie" class="extiw" title="wikt:energie">energie</a></span><span class="sisterproject_text_suffix"> ve Wikislovníku</span></span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r23078045">.mw-parser-output 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