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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Objašnjenje" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Objašnjenje"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Objašnjenje</span> </div> </a> <button aria-controls="toc-Objašnjenje-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>Sadržaj cjeline Objašnjenje</span> </button> <ul id="toc-Objašnjenje-sublist" class="vector-toc-list"> <li id="toc-Toplina" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Toplina"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Toplina</span> </div> </a> <ul id="toc-Toplina-sublist" class="vector-toc-list"> <li id="toc-Količina_topline" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Količina_topline"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Količina topline</span> </div> </a> <ul id="toc-Količina_topline-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Specifični_toplinski_kapacitet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specifični_toplinski_kapacitet"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Specifični toplinski kapacitet</span> </div> </a> <ul id="toc-Specifični_toplinski_kapacitet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specifični_toplinski_kapacitet_nekih_tvari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specifični_toplinski_kapacitet_nekih_tvari"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Specifični toplinski kapacitet nekih tvari</span> </div> </a> <ul id="toc-Specifični_toplinski_kapacitet_nekih_tvari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molarni_toplinski_kapacitet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Molarni_toplinski_kapacitet"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Molarni toplinski kapacitet</span> </div> </a> <ul id="toc-Molarni_toplinski_kapacitet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mayerova_relacija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mayerova_relacija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Mayerova relacija</span> </div> </a> <ul id="toc-Mayerova_relacija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Latentna_toplina" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Latentna_toplina"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Latentna toplina</span> </div> </a> <ul id="toc-Latentna_toplina-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Atomska_teorija_tlaka_i_specifične_topline" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Atomska_teorija_tlaka_i_specifične_topline"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Atomska teorija tlaka i specifične topline</span> </div> </a> <button aria-controls="toc-Atomska_teorija_tlaka_i_specifične_topline-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>Sadržaj cjeline Atomska teorija tlaka i specifične topline</span> </button> <ul id="toc-Atomska_teorija_tlaka_i_specifične_topline-sublist" class="vector-toc-list"> <li id="toc-Vrtnje_molekula_i_specifična_toplina" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vrtnje_molekula_i_specifična_toplina"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Vrtnje molekula i specifična toplina</span> </div> </a> <ul id="toc-Vrtnje_molekula_i_specifična_toplina-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specifični_toplinski_kapacitet_čvrstog_tijela" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specifični_toplinski_kapacitet_čvrstog_tijela"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Specifični toplinski kapacitet čvrstog tijela</span> </div> </a> <ul id="toc-Specifični_toplinski_kapacitet_čvrstog_tijela-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-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="Sadržaj" 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="Prikaz sadržaja stranice" > <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">Prikaz sadržaja stranice</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">Toplinski kapacitet</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="Idi na druge jezične varijante članka. Dostupan je na 71 jeziku" > <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-71" 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">71 jezik</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Warmtekapasiteit" title="Warmtekapasiteit – afrikaans" lang="af" hreflang="af" data-title="Warmtekapasiteit" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B3%D8%B9%D8%A9_%D8%AD%D8%B1%D8%A7%D8%B1%D9%8A%D8%A9" title="سعة حرارية – arapski" lang="ar" hreflang="ar" data-title="سعة حرارية" data-language-autonym="العربية" data-language-local-name="arapski" 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/Capacid%C3%A1_calor%C3%ADfica" title="Capacidá calorífica – asturijski" lang="ast" hreflang="ast" data-title="Capacidá calorífica" data-language-autonym="Asturianu" data-language-local-name="asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0stilik_tutumu" title="İstilik tutumu – azerbajdžanski" lang="az" hreflang="az" data-title="İstilik tutumu" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/W%C3%A4rmekapazit%C3%A4t" title="Wärmekapazität – Bavarian" lang="bar" hreflang="bar" data-title="Wärmekapazität" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BF%D0%BB%D0%B0%D1%91%D0%BC%D1%96%D1%81%D1%82%D0%B0%D1%81%D1%86%D1%8C" title="Цеплаёмістасць – bjeloruski" lang="be" hreflang="be" data-title="Цеплаёмістасць" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D0%B5%D0%BD_%D0%BA%D0%B0%D0%BF%D0%B0%D1%86%D0%B8%D1%82%D0%B5%D1%82" title="Топлинен капацитет – bugarski" lang="bg" hreflang="bg" data-title="Топлинен капацитет" data-language-autonym="Български" data-language-local-name="bugarski" 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%A4%E0%A6%BE%E0%A6%AA_%E0%A6%A7%E0%A6%BE%E0%A6%B0%E0%A6%95%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="তাপ ধারকত্ব – bangla" lang="bn" hreflang="bn" data-title="তাপ ধারকত্ব" data-language-autonym="বাংলা" data-language-local-name="bangla" 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/Toplotni_kapacitet" title="Toplotni kapacitet – bosanski" lang="bs" hreflang="bs" data-title="Toplotni kapacitet" data-language-autonym="Bosanski" data-language-local-name="bosanski" 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/Capacitat_t%C3%A8rmica" title="Capacitat tèrmica – katalonski" lang="ca" hreflang="ca" data-title="Capacitat tèrmica" data-language-autonym="Català" data-language-local-name="katalonski" 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%81%D8%B1%D8%A7%D9%88%D8%A7%D9%86%DB%8C_%DA%AF%DB%95%D8%B1%D9%85%DB%8C_%D8%AA%D8%A7%DB%8C%D8%A8%DB%95%D8%AA%DB%8C" title="فراوانی گەرمی تایبەتی – soranski kurdski" lang="ckb" hreflang="ckb" data-title="فراوانی گەرمی تایبەتی" data-language-autonym="کوردی" data-language-local-name="soranski kurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Tepeln%C3%A1_kapacita" title="Tepelná kapacita – češki" lang="cs" hreflang="cs" data-title="Tepelná kapacita" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C4%82%D1%88%C4%83_%D1%88%C4%83%D0%BD%C4%83%C3%A7%D1%82%D0%B0%D1%80%C4%83%D1%88" title="Ăшă шăнăçтарăш – čuvaški" lang="cv" hreflang="cv" data-title="Ăшă шăнăçтарăш" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" 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/Varmekapacitet" title="Varmekapacitet – danski" lang="da" hreflang="da" data-title="Varmekapacitet" data-language-autonym="Dansk" data-language-local-name="danski" 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/W%C3%A4rmekapazit%C3%A4t" title="Wärmekapazität – njemački" lang="de" hreflang="de" data-title="Wärmekapazität" data-language-autonym="Deutsch" data-language-local-name="njemački" 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%98%CE%B5%CF%81%CE%BC%CE%BF%CF%87%CF%89%CF%81%CE%B7%CF%84%CE%B9%CE%BA%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Θερμοχωρητικότητα – grčki" lang="el" hreflang="el" data-title="Θερμοχωρητικότητα" data-language-autonym="Ελληνικά" data-language-local-name="grčki" 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/Heat_capacity" title="Heat capacity – engleski" lang="en" hreflang="en" data-title="Heat capacity" data-language-autonym="English" data-language-local-name="engleski" 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/Varmokapacito" title="Varmokapacito – esperanto" lang="eo" hreflang="eo" data-title="Varmokapacito" 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/Capacidad_calor%C3%ADfica" title="Capacidad calorífica – španjolski" lang="es" hreflang="es" data-title="Capacidad calorífica" data-language-autonym="Español" data-language-local-name="španjolski" 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/Soojusmahtuvus" title="Soojusmahtuvus – estonski" lang="et" hreflang="et" data-title="Soojusmahtuvus" data-language-autonym="Eesti" data-language-local-name="estonski" 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/Bero-ahalmen" title="Bero-ahalmen – baskijski" lang="eu" hreflang="eu" data-title="Bero-ahalmen" data-language-autonym="Euskara" data-language-local-name="baskijski" 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%B8%D8%B1%D9%81%DB%8C%D8%AA_%DA%AF%D8%B1%D9%85%D8%A7%DB%8C%DB%8C" title="ظرفیت گرمایی – perzijski" lang="fa" hreflang="fa" data-title="ظرفیت گرمایی" data-language-autonym="فارسی" data-language-local-name="perzijski" 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/L%C3%A4mp%C3%B6kapasiteetti" title="Lämpökapasiteetti – finski" lang="fi" hreflang="fi" data-title="Lämpökapasiteetti" data-language-autonym="Suomi" data-language-local-name="finski" 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/Capacit%C3%A9_thermique" title="Capacité thermique – francuski" lang="fr" hreflang="fr" data-title="Capacité thermique" data-language-autonym="Français" data-language-local-name="francuski" 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/Toilleadh_teasa" title="Toilleadh teasa – irski" lang="ga" hreflang="ga" data-title="Toilleadh teasa" data-language-autonym="Gaeilge" data-language-local-name="irski" 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/Capacidade_calor%C3%ADfica" title="Capacidade calorífica – galicijski" lang="gl" hreflang="gl" data-title="Capacidade calorífica" data-language-autonym="Galego" data-language-local-name="galicijski" 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%A7%D7%99%D7%91%D7%95%D7%9C_%D7%97%D7%95%D7%9D" title="קיבול חום – hebrejski" lang="he" hreflang="he" data-title="קיבול חום" data-language-autonym="עברית" data-language-local-name="hebrejski" 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%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE_%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%BF%E0%A4%A4%E0%A4%BE" title="ऊष्मा धारिता – hindski" lang="hi" hreflang="hi" data-title="ऊष्मा धारिता" data-language-autonym="हिन्दी" data-language-local-name="hindski" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Chal%C3%A8_espesifik" title="Chalè espesifik – haićanski kreolski" lang="ht" hreflang="ht" data-title="Chalè espesifik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haićanski kreolski" 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/H%C5%91kapacit%C3%A1s" title="Hőkapacitás – mađarski" lang="hu" hreflang="hu" data-title="Hőkapacitás" data-language-autonym="Magyar" data-language-local-name="mađarski" 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%8B%D5%A5%D6%80%D5%B4%D5%B8%D6%82%D5%B6%D5%A1%D5%AF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ջերմունակություն – armenski" lang="hy" hreflang="hy" data-title="Ջերմունակություն" data-language-autonym="Հայերեն" data-language-local-name="armenski" 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/Kapasitas_kalor" title="Kapasitas kalor – indonezijski" lang="id" hreflang="id" data-title="Kapasitas kalor" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Varmar%C3%BDmd" title="Varmarýmd – islandski" lang="is" hreflang="is" data-title="Varmarýmd" data-language-autonym="Íslenska" data-language-local-name="islandski" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Capacit%C3%A0_termica" title="Capacità termica – talijanski" lang="it" hreflang="it" data-title="Capacità termica" data-language-autonym="Italiano" data-language-local-name="talijanski" 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/%E7%86%B1%E5%AE%B9%E9%87%8F" title="熱容量 – japanski" lang="ja" hreflang="ja" data-title="熱容量" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%97%E1%83%91%E1%83%9D%E1%83%A2%E1%83%94%E1%83%95%E1%83%90%E1%83%93%E1%83%9D%E1%83%91%E1%83%90" title="თბოტევადობა – gruzijski" lang="ka" hreflang="ka" data-title="თბოტევადობა" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D1%8B%D0%BB%D1%83_%D1%81%D1%8B%D0%B9%D1%8B%D0%BC%D0%B4%D1%8B%D0%BB%D1%8B%D2%93%D1%8B" title="Жылу сыйымдылығы – kazaški" lang="kk" hreflang="kk" data-title="Жылу сыйымдылығы" data-language-autonym="Қазақша" data-language-local-name="kazaški" 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/%EC%97%B4%EC%9A%A9%EB%9F%89" title="열용량 – korejski" lang="ko" hreflang="ko" data-title="열용량" data-language-autonym="한국어" data-language-local-name="korejski" 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%96%D1%8B%D0%BB%D1%83%D1%83%D0%BB%D1%83%D0%BA_%D1%81%D1%8B%D0%B9%D1%8B%D0%BC%D0%B4%D1%83%D1%83%D0%BB%D1%83%D0%B3%D1%83" title="Жылуулук сыйымдуулугу – kirgiski" lang="ky" hreflang="ky" data-title="Жылуулук сыйымдуулугу" data-language-autonym="Кыргызча" data-language-local-name="kirgiski" 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/%C5%A0ilumin%C4%97_talpa" title="Šiluminė talpa – litavski" lang="lt" hreflang="lt" data-title="Šiluminė talpa" data-language-autonym="Lietuvių" data-language-local-name="litavski" 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/Siltumietilp%C4%ABba" title="Siltumietilpība – latvijski" lang="lv" hreflang="lv" data-title="Siltumietilpība" data-language-autonym="Latviešu" data-language-local-name="latvijski" 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%A2%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D1%81%D0%BA%D0%B8_%D0%BA%D0%B0%D0%BF%D0%B0%D1%86%D0%B8%D1%82%D0%B5%D1%82" title="Топлински капацитет – makedonski" lang="mk" hreflang="mk" data-title="Топлински капацитет" data-language-autonym="Македонски" data-language-local-name="makedonski" 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%A4%E0%B4%BE%E0%B4%AA%E0%B4%A7%E0%B4%BE%E0%B4%B0%E0%B4%BF%E0%B4%A4" title="താപധാരിത – malajalamski" lang="ml" hreflang="ml" data-title="താപധാരിത" data-language-autonym="മലയാളം" data-language-local-name="malajalamski" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Muatan_haba" title="Muatan haba – malajski" lang="ms" hreflang="ms" data-title="Muatan haba" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" 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/Warmtecapaciteit" title="Warmtecapaciteit – nizozemski" lang="nl" hreflang="nl" data-title="Warmtecapaciteit" data-language-autonym="Nederlands" data-language-local-name="nizozemski" 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/Varmekapasitet" title="Varmekapasitet – norveški nynorsk" lang="nn" hreflang="nn" data-title="Varmekapasitet" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški 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/Varmekapasitet" title="Varmekapasitet – norveški bokmål" lang="nb" hreflang="nb" data-title="Varmekapasitet" data-language-autonym="Norsk bokmål" data-language-local-name="norveški 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/Capacitat_termica" title="Capacitat termica – okcitanski" lang="oc" hreflang="oc" data-title="Capacitat termica" data-language-autonym="Occitan" data-language-local-name="okcitanski" 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/Pojemno%C5%9B%C4%87_cieplna" title="Pojemność cieplna – poljski" lang="pl" hreflang="pl" data-title="Pojemność cieplna" data-language-autonym="Polski" data-language-local-name="poljski" 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/Capacidade_t%C3%A9rmica" title="Capacidade térmica – portugalski" lang="pt" hreflang="pt" data-title="Capacidade térmica" data-language-autonym="Português" data-language-local-name="portugalski" 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/Capacitate_termic%C4%83" title="Capacitate termică – rumunjski" lang="ro" hreflang="ro" data-title="Capacitate termică" data-language-autonym="Română" data-language-local-name="rumunjski" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BF%D0%BB%D0%BE%D1%91%D0%BC%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Теплоёмкость – ruski" lang="ru" hreflang="ru" data-title="Теплоёмкость" data-language-autonym="Русский" data-language-local-name="ruski" 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/Toplinski_kapacitet" title="Toplinski kapacitet – srpsko-hrvatski" lang="sh" hreflang="sh" data-title="Toplinski kapacitet" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpsko-hrvatski" 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/Heat_capacity" title="Heat capacity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Heat capacity" 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/Tepeln%C3%A1_kapacita" title="Tepelná kapacita – slovački" lang="sk" hreflang="sk" data-title="Tepelná kapacita" data-language-autonym="Slovenčina" data-language-local-name="slovački" 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/Toplotna_kapaciteta" title="Toplotna kapaciteta – slovenski" lang="sl" hreflang="sl" data-title="Toplotna kapaciteta" data-language-autonym="Slovenščina" data-language-local-name="slovenski" 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/Kapaciteti_p%C3%ABr_ngrohje" title="Kapaciteti për ngrohje – albanski" lang="sq" hreflang="sq" data-title="Kapaciteti për ngrohje" data-language-autonym="Shqip" data-language-local-name="albanski" 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/Toplotni_kapacitet" title="Toplotni kapacitet – srpski" lang="sr" hreflang="sr" data-title="Toplotni kapacitet" data-language-autonym="Српски / srpski" data-language-local-name="srpski" 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/V%C3%A4rmekapacitet" title="Värmekapacitet – švedski" lang="sv" hreflang="sv" data-title="Värmekapacitet" data-language-autonym="Svenska" data-language-local-name="švedski" 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%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8A%E0%AE%A3%E0%AF%8D%E0%AE%AE%E0%AF%88" title="வெப்பக் கொண்மை – tamilski" lang="ta" hreflang="ta" data-title="வெப்பக் கொண்மை" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%89%E0%B0%B7%E0%B1%8D%E0%B0%A3_%E0%B0%B8%E0%B0%BE%E0%B0%AE%E0%B0%B0%E0%B1%8D%E0%B0%A5%E0%B1%8D%E0%B0%AF%E0%B0%82" title="ఉష్ణ సామర్థ్యం – teluški" lang="te" hreflang="te" data-title="ఉష్ణ సామర్థ్యం" data-language-autonym="తెలుగు" data-language-local-name="teluški" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%B8%D0%B3%D1%83%D0%BD%D2%B7%D0%BE%D0%B8%D1%88" title="Гармигунҷоиш – tadžički" lang="tg" hreflang="tg" data-title="Гармигунҷоиш" data-language-autonym="Тоҷикӣ" data-language-local-name="tadžički" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A3%E0%B9%89%E0%B8%AD%E0%B8%99%E0%B8%88%E0%B8%B3%E0%B9%80%E0%B8%9E%E0%B8%B2%E0%B8%B0" title="ความร้อนจำเพาะ – tajlandski" lang="th" hreflang="th" data-title="ความร้อนจำเพาะ" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Is%C4%B1_s%C4%B1%C4%9Fas%C4%B1" title="Isı sığası – turski" lang="tr" hreflang="tr" data-title="Isı sığası" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D2%96%D1%8B%D0%BB%D1%8B%D1%81%D1%8B%D0%B5%D1%88%D0%BB%D1%8B%D0%BA" title="Җылысыешлык – tatarski" lang="tt" hreflang="tt" data-title="Җылысыешлык" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarski" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BF%D0%BB%D0%BE%D1%94%D0%BC%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Теплоємність – ukrajinski" lang="uk" hreflang="uk" data-title="Теплоємність" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA%DB%8C_%DA%AF%D9%86%D8%AC%D8%A7%D8%A6%D8%B4" title="حرارتی گنجائش – urdski" lang="ur" hreflang="ur" data-title="حرارتی گنجائش" data-language-autonym="اردو" data-language-local-name="urdski" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nhi%E1%BB%87t_dung" title="Nhiệt dung – vijetnamski" lang="vi" hreflang="vi" data-title="Nhiệt dung" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%83%AD%E5%AE%B9%E9%87%8F" title="热容量 – wu kineski" lang="wuu" hreflang="wuu" data-title="热容量" data-language-autonym="吴语" data-language-local-name="wu kineski" 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/%E7%86%B1%E5%AE%B9%E9%87%8F" title="熱容量 – kineski" lang="zh" hreflang="zh" data-title="熱容量" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%86%B1%E5%AE%B9%E9%87%8F" title="熱容量 – kantonski" lang="yue" hreflang="yue" data-title="熱容量" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q179388#sitelinks-wikipedia" title="Poveznice na druge jezike" class="wbc-editpage">Uredi poveznice</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Imenski prostori"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Toplinski_kapacitet" title="Pogledaj sadržaj [c]" accesskey="c"><span>Stranica</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Razgovor:Toplinski_kapacitet" rel="discussion" 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title="Specifični toplinski kapacitet">Specifični toplinski kapacitet</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hr" dir="ltr"><table class="box-Uklopi_iz messagebox plainlinks" style="width:min(50em, 96%); margin: 0 auto; border-width: 1px 1px 1px 10px; border-style: solid; border-color: #2371CB; padding: 0.5em 1em;"> <tbody><tr style="width: 100%;"> <td style="width: 50px;"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Datoteka:Mergefrom_2.svg" class="mw-file-description" title="Toplinski kapacitet"><img alt="Toplinski kapacitet" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Mergefrom_2.svg/50px-Mergefrom_2.svg.png" decoding="async" width="50" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Mergefrom_2.svg/75px-Mergefrom_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Mergefrom_2.svg/100px-Mergefrom_2.svg.png 2x" data-file-width="50" data-file-height="20" /></a><figcaption>Toplinski kapacitet</figcaption></figure> </td> <td style="width: calc(100% - 50px);">Predloženo je da se u ovaj tekst <b>uklopi <a href="/wiki/Specifi%C4%8Dna_toplina" title="Specifična toplina">Specifična toplina</a></b>. (<a href="/wiki/Razgovor:Toplinski_kapacitet" title="Razgovor:Toplinski kapacitet">Rasprava</a>) </td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Translational_motion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif" decoding="async" width="300" height="263" class="mw-file-element" data-file-width="300" data-file-height="263" /></a><figcaption><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> <a href="/wiki/Idealni_plin" title="Idealni plin">idealnog plina</a> je mjera prosječne <a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">kinetičke energije</a> <a href="/wiki/Molekula" title="Molekula">molekula</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:171879main_LimbFlareJan12_lg.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/171879main_LimbFlareJan12_lg.jpg/300px-171879main_LimbFlareJan12_lg.jpg" decoding="async" width="300" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/171879main_LimbFlareJan12_lg.jpg/450px-171879main_LimbFlareJan12_lg.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/171879main_LimbFlareJan12_lg.jpg/600px-171879main_LimbFlareJan12_lg.jpg 2x" data-file-width="1014" data-file-height="712" /></a><figcaption><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> <a href="/wiki/Sunce" title="Sunce">Sunčeve</a> površine (<a href="/wiki/Fotosfera" title="Fotosfera">fotosfera</a>) je 5 778 K (5 505 °C). Toplina se od Sunca do <a href="/wiki/Zemlja" title="Zemlja">Zemlje</a> prenosi <a href="/wiki/Toplinsko_zra%C4%8Denje" title="Toplinsko zračenje">toplinskim zračenjem</a> (<a href="/wiki/Sun%C4%8Deva_svjetlost" title="Sunčeva svjetlost">Sunčeva svjetlost</a>).</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Sunshine_at_Dunstanburgh.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Sunshine_at_Dunstanburgh.JPG/300px-Sunshine_at_Dunstanburgh.JPG" decoding="async" width="300" height="403" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Sunshine_at_Dunstanburgh.JPG/450px-Sunshine_at_Dunstanburgh.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Sunshine_at_Dunstanburgh.JPG/600px-Sunshine_at_Dunstanburgh.JPG 2x" data-file-width="1920" data-file-height="2576" /></a><figcaption><a href="/wiki/Sun%C4%8Deva_svjetlost" title="Sunčeva svjetlost">Sunčeva svjetlost</a> svijetli kroz <a href="/wiki/Oblak" class="mw-redirect" title="Oblak">oblake</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Adijabatski_proces_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/thumb/b/be/Adijabatski_proces_1.png/300px-Adijabatski_proces_1.png" decoding="async" width="300" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/hr/thumb/b/be/Adijabatski_proces_1.png/450px-Adijabatski_proces_1.png 1.5x, //upload.wikimedia.org/wikipedia/hr/thumb/b/be/Adijabatski_proces_1.png/600px-Adijabatski_proces_1.png 2x" data-file-width="1000" data-file-height="860" /></a><figcaption>Za <a href="/wiki/Idealni_plin" title="Idealni plin">idealni plin</a> <a href="/wiki/Adijabatski_proces" title="Adijabatski proces">adijabatsko</a> povećanje obujma (<a href="/wiki/Volumen" class="mw-redirect" title="Volumen">volumena</a>) rezultira <a href="/wiki/Hla%C4%91enje" title="Hlađenje">hlađenjem</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Thermally_Agitated_Molecule.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/23/Thermally_Agitated_Molecule.gif" decoding="async" width="280" height="280" class="mw-file-element" data-file-width="280" data-file-height="280" /></a><figcaption>Toplinske <a href="/wiki/Vibracije" title="Vibracije">vibracije</a> dijelova <a href="/wiki/Bjelan%C4%8Devine" title="Bjelančevine">bjelančevine</a>: <a href="/wiki/Amplituda" title="Amplituda">amplituda</a> vibracija raste s <a href="/wiki/Temperatura" title="Temperatura">temperaturom</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Kochendes_wasser02.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kochendes_wasser02.jpg/300px-Kochendes_wasser02.jpg" decoding="async" width="300" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kochendes_wasser02.jpg/450px-Kochendes_wasser02.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kochendes_wasser02.jpg/600px-Kochendes_wasser02.jpg 2x" data-file-width="674" data-file-height="644" /></a><figcaption>Prilikom <a href="/wiki/Vrenje_(fizika)" title="Vrenje (fizika)">vrenja</a> vode potrebna je <a href="/wiki/Toplina_isparavanja" title="Toplina isparavanja">toplina isparivanja</a> da bi se <a href="/wiki/Voda" title="Voda">voda</a> pretvorila u <a href="/wiki/Vodena_para" title="Vodena para">vodenu paru</a>. Za cijelo vrijeme dok traje vrenje, <a href="/wiki/Temperatura" title="Temperatura">temperatura</a> se ne mijenja i ona je u normalnim uvjetima 100 <a href="/wiki/Celzij" title="Celzij">°C</a>. </figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Hot_metalwork.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Hot_metalwork.jpg/300px-Hot_metalwork.jpg" decoding="async" width="300" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Hot_metalwork.jpg/450px-Hot_metalwork.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Hot_metalwork.jpg/600px-Hot_metalwork.jpg 2x" data-file-width="1600" data-file-height="1163" /></a><figcaption>Iz vrućeg tijela rasprostire se toplina na sve strane nevidljivim <a href="/wiki/Toplinsko_zra%C4%8Denje" title="Toplinsko zračenje">toplinskim zrakama</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Joule%27s_heat_apparatus.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Joule%27s_heat_apparatus.JPG/300px-Joule%27s_heat_apparatus.JPG" decoding="async" width="300" height="454" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Joule%27s_heat_apparatus.JPG/450px-Joule%27s_heat_apparatus.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Joule%27s_heat_apparatus.JPG/600px-Joule%27s_heat_apparatus.JPG 2x" data-file-width="632" data-file-height="956" /></a><figcaption><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Jouleov</a> uređaj iz 1845.</figcaption></figure> <p><b>Toplinski kapacitet</b> (oznaka <i>C</i>) je <a href="/wiki/Fizikalna_veli%C4%8Dina" class="mw-redirect" title="Fizikalna veličina">fizikalna veličina</a> jednaka količniku <a href="/wiki/Toplina" title="Toplina">količine topline</a> <i>Q</i> dovedene nekome <a href="/wiki/Tijelo_(fizika)" title="Tijelo (fizika)">tijelu</a> i promjene <a href="/wiki/Temperatura" title="Temperatura">temperature</a> <i>ΔT</i> koju je dovedena <a href="/wiki/Toplina" title="Toplina">toplina</a> prouzročila: </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={\frac {Q}{\Delta T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\frac {Q}{\Delta T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a523403f82903eb711f75c7aa04990bb99fde61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.273ex; height:5.509ex;" alt="{\displaystyle C={\frac {Q}{\Delta T}}}"></span></dd></dl> <p><a href="/wiki/Mjerna_jedinica" title="Mjerna jedinica">Mjerna jedinica</a> je <a href="/wiki/D%C5%BEul" title="Džul">džul</a> po <a href="/wiki/Kelvin" title="Kelvin">kelvinu</a> (J/K).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Objašnjenje"><span id="Obja.C5.A1njenje"></span>Objašnjenje</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=1" title="Uredi odlomak: Objašnjenje" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=1" title="Uredi kôd odjeljka Objašnjenje"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Toplinski kapacitet nam pokazuje koliko topline treba dovesti kilogramu nekog materijala da mu se temperatura povisi za 1 K (1°C). Za plinove se razlikuju <b>specifični toplinski kapacitet</b> kod stalnoga <a href="/wiki/Tlak" title="Tlak">tlaka</a> C<sub>p</sub> i kod stalnoga volumena <i>C<sub>V</sub></i>. </p> <div class="mw-heading mw-heading3"><h3 id="Toplina">Toplina</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=2" title="Uredi odlomak: Toplina" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=2" title="Uredi kôd odjeljka Toplina"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="margin-left:0.3rem"><i><style data-mw-deduplicate="TemplateStyles:r7016135">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Datoteka:Vista-xmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/18px-Vista-xmag.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/27px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/36px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></a></span> Podrobniji članak o temi: <a href="/wiki/Toplina" title="Toplina">Toplina</a></div></i></div> <p><b>Toplina</b>, <b>toplinska energija</b> ili <b>količina topline</b> (oznaka <i>Q</i>) je <a href="/wiki/Fizikalna_veli%C4%8Dina" class="mw-redirect" title="Fizikalna veličina">fizikalna veličina</a> kojom se opisuje <a href="/wiki/Energija" title="Energija">energija</a> koja <a href="/wiki/Prijenos_topline" title="Prijenos topline">prelazi s toplijega tijela na hladnije</a>. <a href="/wiki/Mjerna_jedinica" title="Mjerna jedinica">Mjerna jedinica</a> topline jest <a href="/wiki/D%C5%BEul" title="Džul">džul</a> (J); stara jedinica bila je <a href="/wiki/Kalorija" title="Kalorija">kalorija</a> (1 cal = 4,186 J). Toplina koja se izmjenjuje pri dodiru dvaju tijela različitih <a href="/wiki/Temperatura" title="Temperatura">temperatura</a> ovisi o <a href="/wiki/Masa" title="Masa">masi</a> <i>m</i> <a href="/wiki/Tijelo_(fizika)" title="Tijelo (fizika)">tijela</a>, <a class="mw-selflink selflink">specifičnom toplinskom kapacitetu</a> <i>c</i> <a href="/wiki/Tvar" title="Tvar">tvari</a> od koje se tijelo sastoji, te o <a href="/wiki/Temperatura" title="Temperatura">temperaturnoj</a> razlici <i>Δt</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta Q=c\cdot m\cdot \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> <mo>=</mo> <mi>c</mi> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta Q=c\cdot m\cdot \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28df4476a28fc66919365b0a9407746a651d56eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.054ex; height:2.509ex;" alt="{\displaystyle \Delta Q=c\cdot m\cdot \Delta t}"></span></dd></dl> <p>Subjektivni osjećaj topline dobiva se dodirom s tijelima kojima je temperatura viša (toplo) ili niža (hladno) od temperature ljudskoga <a href="/wiki/Tijelo_(biologija)" title="Tijelo (biologija)">tijela</a>. Toplina se objektivno mjeri promatranjem djelovanja ugrijanih tijela na druga tijela (<a href="/wiki/Kalorimetrija" title="Kalorimetrija">kalorimetrija</a>). </p> <div class="mw-heading mw-heading4"><h4 id="Količina_topline"><span id="Koli.C4.8Dina_topline"></span>Količina topline</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=3" title="Uredi odlomak: Količina topline" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=3" title="Uredi kôd odjeljka Količina topline"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Želimo li masu od 1 <a href="/wiki/Kilogram" title="Kilogram">kilogram</a> <a href="/wiki/Voda" title="Voda">vode</a> ugrijati na <a href="/wiki/Plamenik" title="Plamenik">plameniku</a> od 10 °C na 20 °C, trebat će nam određeno <a href="/wiki/Vrijeme_(fizika)" title="Vrijeme (fizika)">vrijeme</a>. Za <a href="/wiki/Grijanje" title="Grijanje">grijanje</a> 1 kilograma vode na tom plameniku od 10 °C na 30 °C bit će potrebno duže vremena. Za grijanje 2 kilograma vode trebat će dvostruko duže vremena nego za grijanje 1 kilograma vode. Dakle potrebna količina topline za zagrijavanje vode je to veća što je veća masa vode i što je zagrijavamo na višu temperaturu. Prema tome 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 Q=m\cdot (t_{2}-t_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=m\cdot (t_{2}-t_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5eec4b3d44ef60cf4e8cb1655e4b57f0da953f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.094ex; height:2.843ex;" alt="{\displaystyle Q=m\cdot (t_{2}-t_{1})}"></span></dd></dl> <p>gdje je: <i>Q</i> - količina topline izražena u <a href="/wiki/D%C5%BEul" title="Džul">džulima</a> (J), <i>m</i> - <a href="/wiki/Masa" title="Masa">masa</a> vode u <a href="/wiki/Kilogram" title="Kilogram">kilogramima</a> (kg) i <i>t<sub>2</sub> - t<sub>1</sub></i> - razlika temperature u <a href="/wiki/Celzij" title="Celzij">celzijima</a> (°C). </p><p>Količinu topline mjerimo kao i svaku energiju u džulima. Međutim, još se upotrebljava kao jedinica topline kilo<a href="/wiki/Kalorija" title="Kalorija">kalorija</a> (kcal). 1 kcal je ona količina topline koja je potrebna da se 1 kilogram vode, kod normalnog <a href="/wiki/Tlak_zraka" class="mw-redirect" title="Tlak zraka">tlaka zraka</a> od 760 torra (1 <a href="/wiki/Standardna_atmosfera" title="Standardna atmosfera">atm</a>), zagrije od 14,5 °C na 15,5 °C. To je zbog toga što količine topline za zagrijavanje 1 kilograma vode, na primjer od 12 na 13 °C ili od 20 na 21 °C, nisu jednake. Međutim su razlike tako malene da se u praksi uzima da je za ugrijavanje 1 kilograma vode za 1 °C potrebna 1 kcal, bez obzira kod koje se to temperature vrši. Kod grijanja mora se toplina dovoditi, a kod ohlađivanja odvoditi. Kilokalorija (kcal), određena pri 15 °C, približno je jednaka 4,1855 kilodžula (kJ). </p><p>Za zagrijavanje 1 kilograma <a href="/wiki/%C5%BDeljezo" title="Željezo">željeza</a> trebat će manje vremena nego za zagrijavanje 1 kilograma <a href="/wiki/Opeka" title="Opeka">opeke</a>. Znači da za različite tvari treba različita količina topline da bi se 1 kilogram te tvari ugrijao za 1 °C. Količina topline u J ili kcal koja je potrebna da se 1 kg neke tvari ugrije za 1 °C zove se specifična toplina ili specifični toplinski kapacitet, a označuje se malim slovom <i>c</i>. </p><p>Prema tome, ako je za ugrijavanje 1 kg neke tvari potrebna specifična toplina <i>c</i>, to je za ugrijavanje <i>m</i> kg tvari potrebna <i>c ∙ m</i>. Za ugrijavanje <i>m</i> kg tvari od temperature <i>t<sub>1</sub></i> na temperaturu <i>t<sub>2</sub></i> potrebna je toplina: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=c\cdot m\cdot (t_{2}-t_{1})=c\cdot m\cdot \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>c</mi> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=c\cdot m\cdot (t_{2}-t_{1})=c\cdot m\cdot \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329dde5d6debaa386f7c8e042117286bae564604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.059ex; height:2.843ex;" alt="{\displaystyle Q=c\cdot m\cdot (t_{2}-t_{1})=c\cdot m\cdot \Delta t}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Specifični_toplinski_kapacitet"><span id="Specifi.C4.8Dni_toplinski_kapacitet"></span>Specifični toplinski kapacitet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=4" title="Uredi odlomak: Specifični toplinski kapacitet" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=4" title="Uredi kôd odjeljka Specifični toplinski kapacitet"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Specifični toplinski kapacitet</b> (oznaka <i>c</i>) je <a href="/wiki/Fizikalna_veli%C4%8Dina" class="mw-redirect" title="Fizikalna veličina">fizikalna veličina</a> jednaka toplinskom kapacitetu čvrstoga tijela ili tekućine <i>C</i> podijeljenom <a href="/wiki/Masa" title="Masa">masom</a> <i>m</i> <a href="/wiki/Krutine" title="Krutine">čvrstog tijela</a> ili <a href="/wiki/Teku%C4%87ine" title="Tekućine">tekućine</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {C}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {C}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a374d990e09b87f992368ffa5c4c1039a816efe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.982ex; height:5.343ex;" alt="{\displaystyle c={\frac {C}{m}}}"></span></dd></dl> <p><a href="/wiki/Mjerna_jedinica" title="Mjerna jedinica">Mjerna jedinica</a> je <a href="/wiki/D%C5%BEul" title="Džul">džul</a> po <a href="/wiki/Kilogram" title="Kilogram">kilogramu</a> i <a href="/wiki/Kelvin" title="Kelvin">kelvinu</a> (J · kg<sup>–1</sup> · K<sup>–1</sup>). Za plinove se razlikuju specifični toplinski kapacitet kod stalnoga <a href="/wiki/Tlak" title="Tlak">tlaka</a> <i>c<sub>p</sub></i> i kod stalnoga volumena <i>c<sub>V</sub></i>. Omjer tih dvaju kapaciteta: </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 X={\frac {C_{p}}{C_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\frac {C_{p}}{C_{V}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c93d4790ea33d5e3707721401b238f4dac4d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.072ex; height:6.176ex;" alt="{\displaystyle X={\frac {C_{p}}{C_{V}}}}"></span></dd></dl> <p>važna je veličina u opisivanju kružnih procesa u <a href="/wiki/Termodinamika" title="Termodinamika">termodinamici</a>, na primjer jednadžba je <a href="/wiki/Adijabatski_proces" title="Adijabatski proces">adijabate</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot V_{x}={\mbox{konstanta}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>konstanta</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V_{x}={\mbox{konstanta}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64340c8e1c23c03bb66824f6c7accd3e06ea7826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:18.59ex; height:2.509ex;" alt="{\displaystyle p\cdot V_{x}={\mbox{konstanta}}}"></span></dd></dl> <p>gdje je: <i>p</i> - <a href="/wiki/Tlak" title="Tlak">tlak</a>, a <i>V</i> - <a href="/wiki/Volumen" class="mw-redirect" title="Volumen">volumen</a> <a href="/wiki/Idealni_plin" title="Idealni plin">idealnoga plina</a>. </p><p>Toplinski kapacitet koji se odnosi na masu neke tvari naziva se specifični toplinski kapacitet. </p><p>Razlikujemo toplinske kapacitete pri stalnom <a href="/wiki/Tlak" title="Tlak">tlaku</a> <i>(p)</i> i pri stalnom <a href="/wiki/Volumen" class="mw-redirect" title="Volumen">volumenu</a> <i>(V)</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}=\left({\frac {Q}{T_{2}-T_{1}}}\right)_{p}=\left({\frac {\Delta Q}{\Delta T}}\right)_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}=\left({\frac {Q}{T_{2}-T_{1}}}\right)_{p}=\left({\frac {\Delta Q}{\Delta T}}\right)_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1264aa164385604c4594a6d9977412b7e41bd986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.989ex; height:6.509ex;" alt="{\displaystyle C_{p}=\left({\frac {Q}{T_{2}-T_{1}}}\right)_{p}=\left({\frac {\Delta Q}{\Delta T}}\right)_{p}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{V}=\left({\frac {Q}{T_{2}-T_{1}}}\right)_{V}=\left({\frac {\Delta Q}{\Delta T}}\right)_{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> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}=\left({\frac {Q}{T_{2}-T_{1}}}\right)_{V}=\left({\frac {\Delta Q}{\Delta T}}\right)_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648b9f7cae579f5acaae1380c3b7a45ad27c18d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.3ex; height:6.176ex;" alt="{\displaystyle C_{V}=\left({\frac {Q}{T_{2}-T_{1}}}\right)_{V}=\left({\frac {\Delta Q}{\Delta T}}\right)_{V}}"></span></dd></dl> <p><a href="/wiki/Mjerna_jedinica" title="Mjerna jedinica">Mjerna jedinica</a> za toplinski kapacitet u sustavu <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> je <a href="/wiki/D%C5%BEul" title="Džul">džul</a> po <a href="/wiki/Kelvin" title="Kelvin">kelvinu</a> (J/K).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Specifični toplinski kapacitet ili kapacitet po jedinici <a href="/wiki/Masa" title="Masa">mase</a> se može izraziti i kao: </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_{p,V}={\frac {C_{p,V}}{m}}={\frac {1}{m}}\left({\frac {\Delta Q}{\Delta T}}\right)_{p,V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>V</mi> </mrow> </msub> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p,V}={\frac {C_{p,V}}{m}}={\frac {1}{m}}\left({\frac {\Delta Q}{\Delta T}}\right)_{p,V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c620536b6ebc1fd5adb218ba9078e015085956a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.951ex; height:6.676ex;" alt="{\displaystyle c_{p,V}={\frac {C_{p,V}}{m}}={\frac {1}{m}}\left({\frac {\Delta Q}{\Delta T}}\right)_{p,V}}"></span></dd></dl> <p>Mjerna jedinica SI je džul po <a href="/wiki/Kilogram" title="Kilogram">kilogram</a>-kelvinu (J/kgK). </p> <div class="mw-heading mw-heading3"><h3 id="Specifični_toplinski_kapacitet_nekih_tvari"><span id="Specifi.C4.8Dni_toplinski_kapacitet_nekih_tvari"></span>Specifični toplinski kapacitet nekih tvari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=5" title="Uredi odlomak: Specifični toplinski kapacitet nekih tvari" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=5" title="Uredi kôd odjeljka Specifični toplinski kapacitet nekih tvari"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Specifični toplinski kapacitet nekih tvari: </p> <table class="wikitable"> <tbody><tr> <th><a href="/wiki/Tvar" title="Tvar">Tvar</a></th> <th>c (J kg<sup>-1</sup> K<sup>-1</sup>) </th></tr> <tr> <td><a href="/wiki/Voda" title="Voda">voda</a></td> <td>4 181,3 </td></tr> <tr> <td><a href="/wiki/Ulja" title="Ulja">ulje</a></td> <td>3 800 </td></tr> <tr> <td><a href="/wiki/Alkoholi" title="Alkoholi">alkohol</a></td> <td>2 500 </td></tr> <tr> <td><a href="/wiki/Led" title="Led">led</a></td> <td>2 100 </td></tr> <tr> <td><a href="/wiki/Aluminij" title="Aluminij">aluminij</a></td> <td>900 </td></tr> <tr> <td><a href="/wiki/Staklo" title="Staklo">staklo</a></td> <td>800 </td></tr> <tr> <td><a href="/wiki/%C5%BDeljezo" title="Željezo">željezo</a></td> <td>460 </td></tr> <tr> <td><a href="/wiki/Cink" title="Cink">cink</a></td> <td>390 </td></tr> <tr> <td><a href="/wiki/Bakar_(element)" class="mw-redirect" title="Bakar (element)">bakar</a></td> <td>380 </td></tr> <tr> <td><a href="/wiki/%C5%BDiva" title="Živa">živa</a></td> <td>140 </td></tr></tbody></table> <p>Produkt <i>c ∙ m</i>, to jest količina topline koju treba tijelo mase <i>m</i> kg da se zagrije za 1 °C, zove se toplinski kapacitet <a href="/wiki/Fizikalno_tijelo" class="mw-redirect" title="Fizikalno tijelo">fizikalnog tijela</a>. </p><p>Kako voda ima veliku specifičnu toplinu, to ju je teško ugrijati. Zagrijana pak voda sadrži veliku količinu topline, što se mnogo iskorištava u <a href="/wiki/Tehnika" title="Tehnika">tehnici</a>. Voda se upotrebljava kao nosilac topline kod <a href="/wiki/Centralno_grijanje" title="Centralno grijanje">centralnog grijanja</a> i u <a href="/wiki/Parni_kotao" class="mw-redirect" title="Parni kotao">parnim kotlovima</a> (<a href="/wiki/Generator_pare" title="Generator pare">generatorima pare</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Molarni_toplinski_kapacitet">Molarni toplinski kapacitet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=6" title="Uredi odlomak: Molarni toplinski kapacitet" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=6" title="Uredi kôd odjeljka Molarni toplinski kapacitet"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Molarni toplinski kapacitet</b> - kapacitet po <a href="/wiki/Mol_(mjerna_jedinica)" title="Mol (mjerna jedinica)">molu</a> tvari: </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_{p,V,mol.}={\frac {C_{p,V}}{n}}={\frac {1}{n}}\left({\frac {\Delta Q}{\Delta T}}\right)_{p,V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>m</mi> <mi>o</mi> <mi>l</mi> <mo>.</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>V</mi> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p,V,mol.}={\frac {C_{p,V}}{n}}={\frac {1}{n}}\left({\frac {\Delta Q}{\Delta T}}\right)_{p,V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0dfe6728a0b50eacad8ad70cc8d65799af2b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.95ex; height:6.676ex;" alt="{\displaystyle c_{p,V,mol.}={\frac {C_{p,V}}{n}}={\frac {1}{n}}\left({\frac {\Delta Q}{\Delta T}}\right)_{p,V}}"></span></dd></dl> <p>Mjerna jedinica SI je džul po mol-kelvinu (J/molK). Molarni se toplinski kapaciteti elemenata na dovoljno visokoj temperaturi (~ 300 K) međusobno vrlo malo razlikuju i kreću se oko 26,12 J/mol K . </p> <div class="mw-heading mw-heading3"><h3 id="Mayerova_relacija">Mayerova relacija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=7" title="Uredi odlomak: Mayerova relacija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=7" title="Uredi kôd odjeljka Mayerova relacija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="margin-left:0.3rem"><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r7016135"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Datoteka:Vista-xmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/18px-Vista-xmag.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/27px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/36px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></a></span> Podrobniji članak o temi: <a href="/wiki/Robert_Mayer" title="Robert Mayer">Robert Mayer</a></div></i></div> <p><b>Mayerova relacija</b> daje vezu između <i>c</i><sub>p,mol.</sub> i <i>c</i><sub>V,mol.</sub> idealnog plina: </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_{p}-C_{V}=n\cdot R\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mi>R</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}-C_{V}=n\cdot R\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64974fff86c66e242d5300235c08d84c0c98730f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.043ex; height:2.843ex;" alt="{\displaystyle C_{p}-C_{V}=n\cdot R\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{p,mol.}-c_{V,mol.}=R\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>m</mi> <mi>o</mi> <mi>l</mi> <mo>.</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mo>,</mo> <mi>m</mi> <mi>o</mi> <mi>l</mi> <mo>.</mo> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p,mol.}-c_{V,mol.}=R\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf868f5e8c9125705ab05c6085a0ca4da402c72c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.949ex; height:2.843ex;" alt="{\displaystyle c_{p,mol.}-c_{V,mol.}=R\,}"></span></dd></dl> <p>gdje je <i>n</i> <a href="/wiki/Mno%C5%BEina_tvari" title="Množina tvari">množina tvari</a>, a <i>R</i> je <a href="/wiki/Plinska_konstanta" title="Plinska konstanta">plinska konstanta</a>, <i>R</i> = 8,314 J/mol K. </p> <div class="mw-heading mw-heading3"><h3 id="Latentna_toplina">Latentna toplina</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=8" title="Uredi odlomak: Latentna toplina" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=8" title="Uredi kôd odjeljka Latentna toplina"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="margin-left:0.3rem"><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r7016135"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Datoteka:Vista-xmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/18px-Vista-xmag.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/27px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/36px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></a></span> Podrobniji članak o temi: <a href="/wiki/Latentna_toplina" title="Latentna toplina">Latentna toplina</a></div></i></div> <p>Različite okolnosti mogu utjecati na promjenu specifične topline tvari, tipično promjena agregatnog stanja. Također, promjene tlaka ili volumena tvari tijekom grijanja (što se posebno izraženo vidi kod plinova) utječu na mjereǌe specifične topline, iako ne utječu nužno ili izraženo na sam iznos specifične topline. Na primjer, možemo promatrati <a href="/wiki/Voda" title="Voda">vodu</a> i iznos ǌene specifične topline u tri različite okolnosti: </p> <ul><li>pri temperaturi od 100 °C (para): 2,08 kJ/(kg·K)</li> <li>pri temperaturi od 25 °C (tekućina): 4,1813 kJ/(kg·K)</li> <li>pri temperaturi od -10 °C (led): 2,05 kJ/(kg·K)</li></ul> <p>Toplina koju je tvari potrebno dovesti da bi se promijenilo agregatno staǌe se ne ubraja u specifičnu toplinu, jer temperatura tijela za to vrijeme niti ne raste, već se naziva <a href="/wiki/Latentna_toplina" title="Latentna toplina">latentnom toplinom</a>. <a href="/wiki/Latentna_toplina" title="Latentna toplina">Latentna toplina</a> je jako male temperature i zato se ne koristi često. </p> <div class="mw-heading mw-heading2"><h2 id="Atomska_teorija_tlaka_i_specifične_topline"><span id="Atomska_teorija_tlaka_i_specifi.C4.8Dne_topline"></span>Atomska teorija tlaka i specifične topline</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=9" title="Uredi odlomak: Atomska teorija tlaka i specifične topline" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=9" title="Uredi kôd odjeljka Atomska teorija tlaka i specifične topline"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prema atomističkim predodžbama <a href="/wiki/Tlak" title="Tlak">tlak</a> plina uzrokovan je udarcima <a href="/wiki/Molekula" title="Molekula">molekula</a> o zid posude. Da pojednostavimo stvarne odnose, pretpostavit ćemo da se u kocki sve molekule kreću istom <a href="/wiki/Brzina" title="Brzina">brzinom</a>. Dalje možemo zamisliti da se jedna trećina molekula kreće gore-dolje, druga trećina lijevo-desno, a ostatak naprijed-nazad. Prema svakom zidu kocke kreće se dakle 1/6 svih molekula. Pri udaru o zid promijeni se <a href="/wiki/Smjer" class="mw-disambig" title="Smjer">smjer</a> brzine. Ima li molekula prije <a href="/wiki/Sraz" title="Sraz">sraza</a> <a href="/wiki/Impuls_sile" title="Impuls sile">impuls sile</a> <i>p</i> + <i>m∙v</i>, to poslije odbijanja (<a href="/wiki/Refleksija" title="Refleksija">refleksije</a>) ima impuls - <i>m∙v</i>. Odbijanjem od zida promijeni se impuls molekula za 2∙<i>m∙v</i>. Toliki impuls preuzme zid kocke. Mnogobrojni udari molekula očituju se kao neprekinuti (kontinuirani) tlak plina. Tlak plina je <a href="/wiki/Pritisak" title="Pritisak">pritisak</a> (<a href="/wiki/Sila" title="Sila">sila</a>) na jedinicu <a href="/wiki/Povr%C5%A1ina" title="Površina">površine</a>. Sila je po <a href="/wiki/Newtonovi_zakoni_gibanja" title="Newtonovi zakoni gibanja">Newtonovom zakonu gibanja</a> jednaka promjeni impulsa sile u jedinici vremena. Tlak plina dobivamo dakle, tako da proračunamo, koliki impulsi prenesu molekule u jednoj sekundi na 1 <a href="/wiki/%C4%8Cetvorni_metar" title="Četvorni metar">m<sup>2</sup></a> zida. </p><p>Očito je da će na 1 m<sup>2</sup> zida kocke u jednoj <a href="/wiki/Sekunda" title="Sekunda">sekundi</a> udariti one molekule koje se nalaze pred zidom u stupcu koji je dug upravo toliko, kolika je <a href="/wiki/Brzina" title="Brzina">brzina</a> molekula. Broj molekula u 1 m<sup>3</sup> označit ćemo s <i>n</i>. U stupcu dugom <i>v</i> s presjekom od 1 m<sup>2</sup> nalazi se <i>n∙v</i>. Jedna šestina tih molekula kreće se prema zidu i prenese u sekundi na zid impuls jednak: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {1}{6}}\cdot n\cdot v\cdot 2\cdot m\cdot v={\frac {1}{3}}\cdot n\cdot m\cdot v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>n</mi> <mo>⋅</mo> <mi>v</mi> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>n</mi> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {1}{6}}\cdot n\cdot v\cdot 2\cdot m\cdot v={\frac {1}{3}}\cdot n\cdot m\cdot v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9947af1fd9a9ea26fe9188c415f652bb33f456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:37.356ex; height:5.176ex;" alt="{\displaystyle p={\frac {1}{6}}\cdot n\cdot v\cdot 2\cdot m\cdot v={\frac {1}{3}}\cdot n\cdot m\cdot v^{2}}"></span></dd></dl> <p>To je tlak plina. Vidi se, da je tlak plina dan brojem molekula u 1 m<sup>3</sup> i njihovom <a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">kinetičkom energijom</a> 1/2∙m∙v<sup>2</sup>. To je razvidan rezultat. Gornju jednadžbu možemo još pisati u drugom obliku, da za <i>n</i> stavimo omjer između <i>L</i> - broja molekula u <a href="/wiki/Mol_(mjerna_jedinica)" title="Mol (mjerna jedinica)">molu</a> i <i>V</i> - <a href="/wiki/Volumen" class="mw-redirect" title="Volumen">volumena</a> mola: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot V={\frac {1}{3}}\cdot L\cdot m\cdot v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>L</mi> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V={\frac {1}{3}}\cdot L\cdot m\cdot v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39700e15226cdf4a12c418eaf76cbe823d02aca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:20.665ex; height:5.176ex;" alt="{\displaystyle p\cdot V={\frac {1}{3}}\cdot L\cdot m\cdot v^{2}}"></span></dd></dl> <p>Tlak na zid kocke opažamo kao stalan tlak, iako je on uistinu posljedica isprekidanih udaraca molekula. Ta kontinuiranost tlaka je znak da se plin sastoji od vrlo velikog broj molekula. Mnoštvo sitnih molekula proizvodi učinke, koji se u makroskopskim dimenzijama očituju kao kontinuirane osobine ili trajne sile. </p><p>U teorijskoj jednadžbi možemo po <a href="/wiki/Gay-Lussacov_zakon" title="Gay-Lussacov zakon">Gay-Lussacovu zakonu</a> umnožak <i>p∙V</i> zamijeniti s <i>R∙T</i>. Tada dobivamo jednadžbu: </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 R\cdot T={\frac {1}{3}}\cdot L\cdot m\cdot v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⋅</mo> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>L</mi> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cdot T={\frac {1}{3}}\cdot L\cdot m\cdot v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e00f057a4e9629897a8cf500cf0cc0e1ddd0f1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.019ex; height:5.176ex;" alt="{\displaystyle R\cdot T={\frac {1}{3}}\cdot L\cdot m\cdot v^{2}}"></span></dd></dl> <p>Odatle dobivamo za kvadrat brzine izraz: </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^{2}={\frac {3\cdot R\cdot T}{L\cdot m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>T</mi> </mrow> <mrow> <mi>L</mi> <mo>⋅</mo> <mi>m</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={\frac {3\cdot R\cdot T}{L\cdot m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9cab448403e56fd0f0ccc704c4776e50522682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.037ex; height:5.176ex;" alt="{\displaystyle v^{2}={\frac {3\cdot R\cdot T}{L\cdot m}}}"></span></dd></dl> <p>Broj i masa molekula zasad su nepoznate, ali je njihov umnožak iskustveno ustanovljena veličina. <i>L∙m</i> je masa jednog mola plina. Kako se razabire, brzina je dana samim poznatim veličinama. Po toj jednadžbi izračunate su brzine molekula kod temperature <a href="/wiki/Ledi%C5%A1te" title="Ledište">ledišta</a> <a href="/wiki/Voda" title="Voda">vode</a> za različite plinove: </p> <table class="wikitable"> <caption>Brzine molekula </caption> <tbody><tr> <th scope="col"><a href="/wiki/Plin" title="Plin">Plin</a> </th> <th scope="col"><a href="/wiki/Masa" title="Masa">Masa</a> <a href="/wiki/Mol_(mjerna_jedinica)" title="Mol (mjerna jedinica)">mola</a> (<a href="/wiki/Gram" class="mw-redirect" title="Gram">g</a>) </th> <th scope="col"><a href="/wiki/Brzina" title="Brzina">Brzina</a> (<a href="/wiki/Metar_u_sekundi" title="Metar u sekundi">m/s</a>) </th></tr> <tr> <td><a href="/wiki/Vodik" title="Vodik">Vodik</a> </td> <td>2 </td> <td>1 840 </td></tr> <tr> <td><a href="/wiki/Du%C5%A1ik" title="Dušik">Dušik</a> </td> <td>28 </td> <td>490 </td></tr> <tr> <td><a href="/wiki/Kisik" title="Kisik">Kisik</a> </td> <td>32 </td> <td>460 </td></tr> <tr> <td><a href="/wiki/Radon" title="Radon">Radon</a> </td> <td>222 </td> <td>170 </td></tr> </tbody></table> <p>Da bi se ispitala ispravnost atomske teorije, morale su se pokusima odrediti brzine molekula. Taj zadatak je pokusima riješio <a href="/wiki/Otto_Stern" title="Otto Stern">O. Stern</a> 1920. On je pomoću <a href="/wiki/Elektri%C4%8Dna_struja" title="Električna struja">električne struje</a> ugrijao nit <a href="/wiki/Srebro" title="Srebro">srebra</a> tako da je isparivala atome. Atome je pustio da jure u evakuiranoj staklenoj cijevi. </p><p>Otvorima na uzastopnim zastorima atomi su se u cijevi izdvojili u homogeni snop. Takvu se snopu vrlo točno mogla izmjeriti brzina. Čitava se aparatura stavila u brzu vrtnju. Atomi srebra zbog toga nisu na ploči pali točno iza otvora na zastoru, nego malo više prema gore. Taj pomak dan je vremenom što ga atom srebra treba da bi prošao razmak od zastora do ploče. Tako izmjerene brzine molekula potpuno su se slagale s teorijom. </p><p>Energija plina dana je zbrojem kinetičkih energija svih molekula. <a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">Kinetička energija</a> pojedine molekule jednaka je 1/2∙m∙v<sup>2</sup>, a cjelokupna energija mola umnošku 1/2∙L∙m∙v<sup>2</sup>. Energiju mola dakle dobivamo da gornju atomističku jednadžbu pomnožimo s 3/2. Učinivši to, dobivamo za <a href="/wiki/Energija" title="Energija">energiju</a> <i>U</i> mola plina: </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 {3}{2}}\cdot R\cdot T}"> <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> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {3}{2}}\cdot R\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713983c284dba9b80a6c9b47b845605b090a02a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.638ex; height:5.176ex;" alt="{\displaystyle U={\frac {3}{2}}\cdot R\cdot T}"></span></dd></dl> <p>Energija je u bitnom dana umnoškom plinske konstante i temperature. Taj zaključak teorije može se pokusima točno provjeriti. Energiju plina možemo povećati ili umanjiti tako, da plin grijemo ili hladimo. Prema gornjoj jednadžbi, <a href="/wiki/Specifi%C4%8Dna_toplina" title="Specifična toplina">specifična toplina</a> plina jednaka 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 C_{V}={\frac {3}{2}}\cdot 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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}={\frac {3}{2}}\cdot R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b352ab25a58c301c2aa49c5b7ffc8e9ca1f6275a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.698ex; height:5.176ex;" alt="{\displaystyle C_{V}={\frac {3}{2}}\cdot R}"></span></dd></dl> <p>Ova teorijska vrijednost vrlo se dobro slaže s mjerenjem na jednoatomnim plinovima, kao što su <a href="/wiki/Helij" title="Helij">helij</a>, <a href="/wiki/Neon" title="Neon">neon</a>, <a href="/wiki/Argon" title="Argon">argon</a> i tako dalje. Za druge plinove, gdje su molekule građene od dva ili više atoma, daju <a href="/wiki/Mjerenje" title="Mjerenje">mjerenja</a> veće vrijednosti. To je jasno ako pomislimo da energija višeatomnih molekula može biti sadržana i u vrtnji atoma oko unutarnjih osi, a ne samo u prostornom gibanju molekula kroz prostor. Specifične topline višeatomnih plinova proračunat će se kasnije. </p><p>Taj primjer lijepo osvjetljuje kako atomistička teorija povezuje različita svojstva materije. Na <a href="/wiki/Plinska_konstanta" title="Plinska konstanta">plinsku konstantu</a> <i>R</i> ispitivači su nadošli ispitujući tlak plinova. U početku se nije očekivalo da će postojati neka uska veza između plinske konstante i energije plina. Atomistika ujedinjuje energetsko ponašanje plinova s njihovim napetostima. Takvo povezivanje različitih osobina materije u izraz jedinstvene stvarnosti je glavna crta atomske teorije. </p><p>Što je niža temperatura plina, to su polaganija gibanja molekula. na apsolutnoj nuli nastupa potpuno mirovanje. Jasno je da se ispod te temperature materija ne može ohladiti. Hlađenje se sastoji u oduzimanju kinetičke energije molekula. No kad sve molekule miruju, ne može se više ništa energije oduzeti. Materija se nalazi u stanju najniže energije. To je puno fizičko značenje apsolutne nule temperature. </p><p>Dijeleći ukupnu energiju mola <a href="/wiki/Avogadrov_broj" title="Avogadrov broj">Avogadrovim brojem</a> <i>L</i>, dobivamo prosječnu energiju čestice u plinu: </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 {3}{2}}\cdot k\cdot T}"> <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> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {3}{2}}\cdot k\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d73dc8e9962a7c0388128ac1fd56f5c473ecedd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.085ex; height:5.176ex;" alt="{\displaystyle U={\frac {3}{2}}\cdot k\cdot T}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {R}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {R}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d613649b6bf13eb30ec9e41ca32d73aec6f1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.91ex; height:5.176ex;" alt="{\displaystyle k={\frac {R}{L}}}"></span></dd></dl> <p>gdje je: k - <a href="/wiki/Boltzmannova_konstanta" title="Boltzmannova konstanta">Boltzmannova konstanta</a>, koja se može izračunati, čim je Avogadrov broj određen (broj molekula u molu). </p> <div class="mw-heading mw-heading3"><h3 id="Vrtnje_molekula_i_specifična_toplina"><span id="Vrtnje_molekula_i_specifi.C4.8Dna_toplina"></span>Vrtnje molekula i specifična toplina</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=10" title="Uredi odlomak: Vrtnje molekula i specifična toplina" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=10" title="Uredi kôd odjeljka Vrtnje molekula i specifična toplina"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pored pravocrtnog (<a href="/wiki/Translacija" title="Translacija">translacijskog</a>) gibanja mogu se molekule okretati i oko sebe samih. <a href="/wiki/Energija" title="Energija">Energija</a> <a href="/wiki/Vrtnja" title="Vrtnja">vrtnje</a> ili rotacijska energija zavisi također od temperature te daje svoj doprinos specifičnoj toplini plina. Molekulu možemo shvatiti kao kruto tijelo, jer atomi imaju stalne međusobne razmake. S obzirom na to da nema vanjskih <a href="/wiki/Sila" title="Sila">sila</a>, <a href="/wiki/Te%C5%BEi%C5%A1te" title="Težište">težište</a> molekule kreće se s konstantnom brzinom. To gibanje zasad nas ne zanima i pretpostavit ćemo da težište molekule miruje. Molekula se vrti oko neke čvrste osi koja prolazi kroz težište. Općenito se rotacijska energija dobije iz pravocrtnog gibanja tako da se masa zamijeni <a href="/wiki/Moment_tromosti" class="mw-redirect" title="Moment tromosti">momentima tromosti</a> s obzirom na glavne osi, a komponente impulsa <i>p<sub>x</sub>, p<sub>y</sub> i p<sub>z</sub></i> komponentama momenta impulsa. Prema tome, za prosječnu rotacijsku energiju dobivamo iste izraze kao i za pravocrtno gibanje. Na svaki stupanj slobode otpada prosječna energija 1/2∙k∙T. Prosječna rotacijska energija molekule jednaka 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 E_{rot}={\frac {3}{2}}\cdot k\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}={\frac {3}{2}}\cdot k\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4083af87f2b19b83f7a33a3c6e1bea6e4c0a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.383ex; height:5.176ex;" alt="{\displaystyle E_{rot}={\frac {3}{2}}\cdot k\cdot T}"></span></dd></dl> <p>Energija molekule plina zbroj je kinetičke i rotacijske energije: </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=E_{kin}+E_{rot}={\frac {3}{2}}\cdot k\cdot T+{\frac {3}{2}}\cdot k\cdot T=3\cdot k\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{kin}+E_{rot}={\frac {3}{2}}\cdot k\cdot T+{\frac {3}{2}}\cdot k\cdot T=3\cdot k\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad2c7defe4d6b4d4562a26dda19fc72e8609e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.966ex; height:5.176ex;" alt="{\displaystyle E=E_{kin}+E_{rot}={\frac {3}{2}}\cdot k\cdot T+{\frac {3}{2}}\cdot k\cdot T=3\cdot k\cdot T}"></span></dd></dl> <p>Pomnoživši to Avogadrovim brojem <i>L</i>, dobivamo energiju jednog mola plina: </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=3\cdot L\cdot k\cdot T=3\cdot R\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>L</mi> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=3\cdot L\cdot k\cdot T=3\cdot R\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324f49693e917fed52de8934aab31cc8803babb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.531ex; height:2.176ex;" alt="{\displaystyle U=3\cdot L\cdot k\cdot T=3\cdot R\cdot T}"></span></dd></dl> <p>Iz ove jednadžbe proizlazi da je specifična toplina (kod konstantnog volumena) jednaka: </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}=3\cdot 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> <mn>3</mn> <mo>⋅</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}=3\cdot R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7aecb924c689e9ecfbcee71f06ce657ec63188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.862ex; height:2.509ex;" alt="{\displaystyle C_{V}=3\cdot R}"></span></dd></dl> <p>To se slaže s pokusima i mjerenjima. Odstupanje se opaža za plinove sastavljene od dvoatomnih molekula. Tamo, čini se, otpada vrtnja oko one osi koja prolazi kroz obje molekule. S obzirom na tu os moment tromosti molekule gotovo isčezava. Time imamo stupanj slobode manje. Prosječna energija dvoatomne molekule nije jednaka 6∙1/2∙<i>k∙T</i>, nego 5∙1/2∙<i>k∙T</i>. Za specifičnu toplinu dvoatomnih plinova dobivamo: </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}={\frac {5}{2}}\cdot 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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}={\frac {5}{2}}\cdot R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09051170239632d888ec78779d24247d37c184d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.698ex; height:5.176ex;" alt="{\displaystyle C_{V}={\frac {5}{2}}\cdot R}"></span></dd></dl> <p>prema starim mjernim jedinicama, <i>R</i> je približno 2 <a href="/wiki/Kalorija" title="Kalorija">cal</a>/°C. Specifična toplina višeatomnih plinova iznosi dakle po teoriji 6 cal/°C, a dvoatomnih plinova 5 cal/°C. Ove vrijednosti izvrsno se slažu s iskustvom. </p><p>Od kinetičke energije molekula zavisi u kakvom se <a href="/wiki/Agregatna_stanja" class="mw-redirect" title="Agregatna stanja">agregatnom stanju</a> nalazi <a href="/wiki/Materija" title="Materija">materija</a>. Dok se molekule kreću s velikim brzinama, bez važnosti su međusobne sile molekula. Svaka molekula juri kroz prostor nezavisno o stanju. Za velike kinetičke energije molekula materija se nalazi u plinskom stanju. Da plin pretvorimo u tekućinu moramo molekulama oduzeti mnogo energije. Plin postaje tekućina ako ga ohladimo na određenu nisku temperatura. Na toj niskoj kritičnoj temperaturi molekule ostaju sve duže jedna uz drugu pa se počinju gomilati u male hrpe. Postepenim oduzimanjem kinetičke energije plin postaje tekućina. U tekućini molekule kližu jedna uz drugu. Materija je zadobila stalnu <a href="/wiki/Gusto%C4%87a" title="Gustoća">gustoću</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Specifični_toplinski_kapacitet_čvrstog_tijela"><span id="Specifi.C4.8Dni_toplinski_kapacitet_.C4.8Dvrstog_tijela"></span>Specifični toplinski kapacitet čvrstog tijela</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=11" title="Uredi odlomak: Specifični toplinski kapacitet čvrstog tijela" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=11" title="Uredi kôd odjeljka Specifični toplinski kapacitet čvrstog tijela"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prema kinetičkoj teoriji materije, <a href="/wiki/Termodinamika" title="Termodinamika">termodinamička</a> svojstva <a href="/wiki/Krutine" title="Krutine">čvrstih tijela</a> određena su <a href="/wiki/Gibanje" title="Gibanje">gibanjima</a> <a href="/wiki/Atom" title="Atom">atoma</a> oko njihovih stalnih mjesta u <a href="/wiki/Kristalna_re%C5%A1etka" title="Kristalna rešetka">kristalnoj rešetki</a>. Premda <a href="/wiki/Kristal" title="Kristal">kristal</a> titra kao cjelina, to ipak pri prvom približenju (aproksimaciji) možemo uzeti da svaki atom titra za sebe oko svojeg položaja <a href="/wiki/Ravnote%C5%BEa_(mehanika)" title="Ravnoteža (mehanika)">ravnoteže</a>. Time smo čvrsto tijelo sveli na sustav <a href="/wiki/Harmonijsko_titranje" title="Harmonijsko titranje">harmoničkih oscilatora</a>, što olakšava proračun njegove <a href="/wiki/Energija" title="Energija">energije</a>. </p><p><a href="/wiki/Titranje" title="Titranje">Titranje</a> atoma oko položaja ravnoteže očituje se kao <a href="/wiki/Temperatura" title="Temperatura">temperatura</a> tijela. Razumije se da u tijelu neće svi atomi baš jednako snažno (intenzivno) titrati kao što ni brzine plinskih molekula nisu sve jednake. No ipak, kod svake temperature uspostavit će se određena raspodjela energije titranja. Nama se ovdje odmah nameće pitanje: Kolika je prosječna energija titranja pri zadanoj temperaturi? </p><p>Pogledajmo najprije kolika je kinetička i potencijalna energija atoma u rešetki. Titranje je dano funkcijom: </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 x=A\cdot \cos \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=A\cdot \cos \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1a20be00f3780f190985376e39a15e57d448b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.634ex; height:2.176ex;" alt="{\displaystyle x=A\cdot \cos \omega t}"></span></dd></dl> <p><a href="/wiki/Brzina" title="Brzina">Brzina</a> je jednaka derivaciji: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dt}}=-\,\omega \cdot A\cdot \sin \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mspace width="thinmathspace" /> <mi>ω</mi> <mo>⋅</mo> <mi>A</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}=-\,\omega \cdot A\cdot \sin \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e15a792f5e6b69fcc31d06e187224ee1d08db8c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.751ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}=-\,\omega \cdot A\cdot \sin \omega t}"></span></dd></dl> <p><a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">Kinetička energija</a>, prema tome, jednaka 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 E_{kin}={\frac {1}{2}}\cdot m\cdot {\frac {dx}{dt}}={\frac {1}{2}}\cdot m\cdot \omega ^{2}\cdot A^{2}\cdot \sin ^{2}\omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{kin}={\frac {1}{2}}\cdot m\cdot {\frac {dx}{dt}}={\frac {1}{2}}\cdot m\cdot \omega ^{2}\cdot A^{2}\cdot \sin ^{2}\omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099c0c68fa07412f7b24b02dea4e24be9fa5c645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.969ex; height:5.509ex;" alt="{\displaystyle E_{kin}={\frac {1}{2}}\cdot m\cdot {\frac {dx}{dt}}={\frac {1}{2}}\cdot m\cdot \omega ^{2}\cdot A^{2}\cdot \sin ^{2}\omega t}"></span></dd></dl> <p>Kad kinetička energija postigne maksimum, ona postaje jednaka cjelokupnoj energiji. Energija harmoničkog oscilatora iznosi dakle: </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={\frac {1}{2}}\cdot m\cdot \omega ^{2}\cdot A^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}\cdot m\cdot \omega ^{2}\cdot A^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661c0e2c9fefe3e5ce348fe80d4ec39d622a7eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.248ex; height:5.176ex;" alt="{\displaystyle E={\frac {1}{2}}\cdot m\cdot \omega ^{2}\cdot A^{2}}"></span></dd></dl> <p><a href="/wiki/Potencijalna_energija" title="Potencijalna energija">Potencijalna energija</a> jednaka je razlici između cjelokupne energije i kinetičke energije: </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_{pot}=E-E_{kin}=E-E\cdot \sin ^{2}\omega t=E\cdot \cos ^{2}\omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo>−</mo> <mi>E</mi> <mo>⋅</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>=</mo> <mi>E</mi> <mo>⋅</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{pot}=E-E_{kin}=E-E\cdot \sin ^{2}\omega t=E\cdot \cos ^{2}\omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa839e4fc52fa800d4ffe59bcf6f5dd6bc7f2a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.38ex; height:3.343ex;" alt="{\displaystyle E_{pot}=E-E_{kin}=E-E\cdot \sin ^{2}\omega t=E\cdot \cos ^{2}\omega t}"></span></dd></dl> <p>Kosinus i sinus su jednake funkcije, pomaknute samo u fazi. Odatle proizlazi da je za <a href="/wiki/Harmoni%C4%8Dko_titranje" class="mw-redirect" title="Harmoničko titranje">harmoničko titranje</a> kinetička energija u prosjeku jednaka potencijalnoj. Energija titranja, gledana u dugom periodu, jednako se raspodjeljuje na kinetičku i potencijalnu energiju. </p><p>Prije smo već na jednostavan način proračunali kolika je prosječna kinetička energija u toplinski uravnoteženom sustavu. Naš izvod vrijedio je bez obzira na to da li dolazi i potencijalna energija. Na jednodimenzijalno gibanje otpada prosječno kinetička energija jednaka: </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_{kin}={\frac {1}{2}}\cdot k\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{kin}={\frac {1}{2}}\cdot k\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1c70a68cbf3f5342fdd12941647ef52cdf973f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.66ex; height:5.176ex;" alt="{\displaystyle E_{kin}={\frac {1}{2}}\cdot k\cdot T}"></span></dd></dl> <p>Tolikoj kinetičkoj energiji pridolazi jednako velika potencijalna energija. Prosječna energija linearnog harmoničkog oscilatora iznosi prema tome: </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\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=k\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6356998035ac38feef26fd03aca239c8e9498c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.401ex; height:2.176ex;" alt="{\displaystyle E=k\cdot T}"></span></dd></dl> <p>Taj rezultat je od temeljne važnosti. Kad je sustav harmoničkih oscilatora u termodinamičkoj ravnoteži, njihova prosječna energija iznosi <i>k∙T</i>. </p><p>Sistematski možemo čvrsto tijelo predočiti trodimenzionalnim nizom atoma koji titraju kao harmonički oscilatori. Razumije se, neki atomi titrat će jače, drugi slabije. No to se nas ne tiče. Njihovu prosječnu energiju znamo. Ovdje je još potrebno dodati da svaki atom može titrati u dubinu, širinu i visinu. Na svaki od ta tri smjera otpada u prosjeku energija <i>k∙T</i>. Prosječna energija <a href="/wiki/Atom" title="Atom">atoma</a> ili <a href="/wiki/Ion" title="Ion">iona</a> u <a href="/wiki/Kristal" title="Kristal">kristalu</a> jednaka je, dakle: </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=3\cdot k\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=3\cdot k\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac14c7073b5d76517bc9564efb1e9fdaa0effa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.242ex; height:2.176ex;" alt="{\displaystyle E=3\cdot k\cdot T}"></span></dd></dl> <p>Energiju <a href="/wiki/Mol_(mjerna_jedinica)" title="Mol (mjerna jedinica)">mola</a> čvrstog tijela dobijemo kad tu energiju pomnožimo <a href="/wiki/Avogadrov_broj" title="Avogadrov broj">Avogadrovim brojem</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=3\cdot L\cdot k\cdot T=3\cdot R\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>L</mi> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mi>T</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=3\cdot L\cdot k\cdot T=3\cdot R\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324f49693e917fed52de8934aab31cc8803babb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.531ex; height:2.176ex;" alt="{\displaystyle U=3\cdot L\cdot k\cdot T=3\cdot R\cdot T}"></span></dd></dl> <p>Za specifičnu toplinu s obzirom na mol proizlazi odatle izraz: </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}=3\cdot 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> <mn>3</mn> <mo>⋅</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}=3\cdot R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7aecb924c689e9ecfbcee71f06ce657ec63188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.862ex; height:2.509ex;" alt="{\displaystyle C_{V}=3\cdot R}"></span></dd></dl> <p>U praksi se obično računa sa specifičnim toplinskim kapacitetom obzirom na 1 <a href="/wiki/Gram" class="mw-redirect" title="Gram">gram</a>. Pomnožimo li tu specifičnu toplinu težinom mola ili <a href="/wiki/Relativna_atomska_masa" title="Relativna atomska masa">atomskom težinom</a>, dobivamo našu veličinu <i>C<sub>V</sub></i>. Ta veličina, a ne specifični toplinski kapacitet s obzirom na 1 g, <a href="/wiki/Konstanta" title="Konstanta">konstanta</a> je za čvrsta tijela. Pokusima su taj zakon dokazali 1818. istraživači <a href="/wiki/Pierre_Louis_Dulong" title="Pierre Louis Dulong">P. L. Dulong</a> i <a href="/w/index.php?title=Alexis_Th%C3%A9r%C3%A8se_Petit&action=edit&redlink=1" class="new" title="Alexis Thérèse Petit (stranica ne postoji)">A. T. Petit</a>. Oni su ustanovili da specifični toplinski kapaciteti čvrstih tijela pomnoženi atomskom težinom iznose 6 <a href="/wiki/Kalorija" title="Kalorija">cal</a>/°C. Kako je <i>R</i> jednako 2 cal/°C, to vidimo da klasična teorija daje ispravne specifične toplinske kapacitete. </p> <table class="wikitable"> <caption>Specifični toplinski kapaciteti kod sobne temperature </caption> <tbody><tr> <th scope="col"><a href="/wiki/Kemijski_element" title="Kemijski element">Kemijski element</a> </th> <th scope="col"><a href="/wiki/Atomska_te%C5%BEina" class="mw-redirect" title="Atomska težina">Atomska težina</a> </th> <th scope="col">Specifični toplinski kapaciteti na 1 gram </th> <th scope="col"><i>C<sub>V</sub></i> </th></tr> <tr> <td><a href="/wiki/Litij" title="Litij">Litij</a> </td> <td>7 </td> <td>0,8 </td> <td>5,6 </td></tr> <tr> <td><a href="/wiki/Magnezij" title="Magnezij">Magnezij</a> </td> <td>24 </td> <td>0,25 </td> <td>6 </td></tr> <tr> <td><a href="/wiki/%C5%BDeljezo" title="Željezo">Željezo</a> </td> <td>56 </td> <td>0,11 </td> <td>6,1 </td></tr> <tr> <td><a href="/wiki/Srebro" title="Srebro">Srebro</a> </td> <td>108 </td> <td>0,055 </td> <td>6 </td></tr> <tr> <td><a href="/wiki/Olovo_(element)" title="Olovo (element)">Olovo</a> </td> <td>207 </td> <td>0,031 </td> <td>6,4 </td></tr> <tr> <td><a href="/wiki/Torij" title="Torij">Torij</a> </td> <td>232 </td> <td>0,026 </td> <td>6 </td></tr> <tr> <td><a href="/wiki/Dijamant" title="Dijamant">Dijamant</a> </td> <td>12 </td> <td>0,12 </td> <td>1,4 </td></tr> </tbody></table> <p>Našla su se ipak odstupanja od klasične teorije. Tako specifični toplinski kapacitet dijamanta kod sobne temperature pomnožena s 12 iznosi 1,4. Dakle, manja je od teorijske vrijednosti. Tek kad se dijamant ugrije do 1 000 °C, postaje <i>C<sub>V</sub></i> jednak 6. Još gora je situacija kad se ispituju specifične temperature na niskim temperaturama. Pokusima je utvrđeno da specifične temperature svih čvrstih tijela isčezavaju kod apsolutne nule temperature. Za najniže temperature specifični toplinski kapacitet pada kao kub temperature: </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}={\mbox{konstanta}}\cdot T^{3}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>konstanta</mtext> </mstyle> </mrow> <mo>⋅</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}={\mbox{konstanta}}\cdot T^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b81ab279421d05dd82207de3e94fa9be88d4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.735ex; height:3.009ex;" alt="{\displaystyle C_{V}={\mbox{konstanta}}\cdot T^{3}}"></span></dd></dl> <p>Stavimo li <i>T</i> = 0, imamo <i>C<sub>V</sub></i> = 0. Ovaj pad specifičnog toplinskog kapaciteta ne možemo shvatiti u <a href="/wiki/Klasi%C4%8Dna_fizika" title="Klasična fizika">klasičnoj fizici</a>. Tu bi specifični toplinski kapaciteti bili strogo konstante. Međutim, pokusima se ta konstantnost opaža samo kod viših temperatura, normalno kod sobnih temperatura. Klasična teorija vrijedi samo za visoke temperature, no potpuno zatajuje kod niskih. </p><p>Moglo bi se pomisliti da teškoće proizlaze odatle što smo uzeli prejednostavan model. Čvrsto tijelo ne titra kao skup harmoničkih oscilatora, nego se zapravo kroza nj šire elestični valovi. Energija čvrstog tijela jednaka je sumi energija elastičnih ravnih valova. Taj model upotrijebit ćemo zaista u <a href="/wiki/Kvantna_fizika" title="Kvantna fizika">kvantnoj teoriji</a>. Međutim, u klasičnoj teoriji on ne daje različit rezultat. Energija ravnog vala formalno se podudara s energijom harmoničkog oscilatora, pa za prosječnu energiju ravnog vala izlazi u klasičnoj teoriji <i>k∙T</i>, a time konstantni specifični toplinski kapacitet. Klasična teorija ne može, ma kako profinjenim modelima, objasniti isčezavanje specifičnog toplinskog kapaciteta pri <a href="/wiki/Apsolutna_nula" class="mw-redirect" title="Apsolutna nula">apsolutnoj nuli</a>. </p><p>Te je pojave objasnila kvantna teorija. Došli smo do granice, gdje prestaju klasični zakoni, a počinje <a href="/wiki/Kvantna_fizika" title="Kvantna fizika">kvantna fizika</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Toplinski_kapacitet&veaction=edit&section=12" title="Uredi odlomak: Izvori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Toplinski_kapacitet&action=edit&section=12" title="Uredi kôd odjeljka Izvori"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r6541845">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"> <b>toplinski kapacitet</b>, <a rel="nofollow" class="external autonumber" href="http://www.enciklopedija.hr/Natuknica.aspx?ID=61792">[1]</a> "Hrvatska enciklopedija", Leksikografski zavod Miroslav Krleža, www.enciklopedija.hr, 2017.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"> <b>specifični toplinski kapacitet</b>, <a rel="nofollow" class="external autonumber" href="http://www.enciklopedija.hr/natuknica.aspx?ID=70165">[2]</a> "Hrvatska enciklopedija", Leksikografski zavod Miroslav Krleža, www.enciklopedija.hr, 2017.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"> <a href="/wiki/Ivan_Supek" title="Ivan Supek">Ivan Supek</a>: "Nova fizika", Školska knjiga Zagreb, 1966.</span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Dobavljeno iz "<a dir="ltr" href="https://hr.wikipedia.org/w/index.php?title=Toplinski_kapacitet&oldid=7022031">https://hr.wikipedia.org/w/index.php?title=Toplinski_kapacitet&oldid=7022031</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorije</a>: <ul><li><a href="/wiki/Kategorija:%C4%8Clanci_u_koje_treba_uklopiti_drugi_tekst" title="Kategorija:Članci u koje treba uklopiti drugi tekst">Članci u koje treba uklopiti drugi tekst</a></li><li><a href="/wiki/Kategorija:Fizikalne_veli%C4%8Dine" title="Kategorija:Fizikalne veličine">Fizikalne veličine</a></li><li><a href="/wiki/Kategorija:Termodinamika" title="Kategorija:Termodinamika">Termodinamika</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Ova stranica posljednji je put uređivana 16. rujna 2024. u 10:10.</li> <li id="footer-info-copyright">Tekst je dostupan pod licencijom <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.hr">Creative Commons: Imenuj autora/Dijeli pod istim uvjetima</a>; mogu se primjenjivati i dodatni uvjeti. 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