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<span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solubility_prediction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solubility_prediction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Solubility prediction</span> </div> </a> <button aria-controls="toc-Solubility_prediction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Solubility prediction subsection</span> </button> <ul id="toc-Solubility_prediction-sublist" class="vector-toc-list"> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Enthalpy of fusion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 54 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-54" 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">54 languages</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/Smeltingswarmte" title="Smeltingswarmte – Afrikaans" lang="af" hreflang="af" data-title="Smeltingswarmte" 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%AD%D8%B1%D8%A7%D8%B1%D8%A9_%D8%A7%D9%84%D8%A7%D9%86%D8%B5%D9%87%D8%A7%D8%B1" title="حرارة الانصهار – Arabic" lang="ar" hreflang="ar" data-title="حرارة الانصهار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Entalp%C3%ADa_de_fusi%C3%B3n" title="Entalpía de fusión – Aragonese" lang="an" hreflang="an" data-title="Entalpía de fusión" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Entalp%C3%ADa_de_fusi%C3%B3n" title="Entalpía de fusión – Asturian" lang="ast" hreflang="ast" data-title="Entalpía de fusión" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%B2%E0%A6%A8%E0%A7%87%E0%A6%B0_%E0%A6%85%E0%A6%AD%E0%A7%8D%E0%A6%AF%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0%E0%A7%80%E0%A6%A3_%E0%A6%A4%E0%A6%BE%E0%A6%AA%E0%A6%97%E0%A6%A4%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%AC%E0%A6%BF%E0%A6%AD%E0%A6%AC" title="গলনের অভ্যন্তরীণ তাপগতীয় বিভব – Bangla" lang="bn" hreflang="bn" data-title="গলনের অভ্যন্তরীণ তাপগতীয় বিভব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%84%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D0%B0_%D0%BD%D0%B0_%D1%82%D0%BE%D0%BF%D0%B5%D0%BD%D0%B5" title="Специфична топлина на топене – Bulgarian" lang="bg" hreflang="bg" data-title="Специфична топлина на топене" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Toplota_topljenja" title="Toplota topljenja – Bosnian" lang="bs" hreflang="bs" data-title="Toplota topljenja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Calor_de_fusi%C3%B3" title="Calor de fusió – Catalan" lang="ca" hreflang="ca" data-title="Calor de fusió" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%C4%83%D1%80%D0%B0%D0%BD%D0%BD%D0%B8%D0%BD_%D0%BF%D0%B0%D0%B9%D0%BB%D0%B0%D0%B2%D0%BB%D0%B0_%C4%83%D1%88%D1%88%D0%B8" title="Шăраннин пайлавла ăшши – Chuvash" lang="cv" hreflang="cv" data-title="Шăраннин пайлавла ăшши" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/M%C4%9Brn%C3%A9_skupensk%C3%A9_teplo_t%C3%A1n%C3%AD" title="Měrné skupenské teplo tání – Czech" lang="cs" hreflang="cs" data-title="Měrné skupenské teplo tání" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Smeltevarme" title="Smeltevarme – Danish" lang="da" hreflang="da" data-title="Smeltevarme" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schmelzenthalpie" title="Schmelzenthalpie – German" lang="de" hreflang="de" data-title="Schmelzenthalpie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Sulamissoojus" title="Sulamissoojus – Estonian" lang="et" hreflang="et" data-title="Sulamissoojus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Entalp%C3%ADa_de_fusi%C3%B3n" title="Entalpía de fusión – Spanish" lang="es" hreflang="es" data-title="Entalpía de fusión" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Urtze-entalpia" title="Urtze-entalpia – Basque" lang="eu" hreflang="eu" data-title="Urtze-entalpia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D9%86%D8%AA%D8%A7%D9%84%D9%BE%DB%8C_%D8%B0%D9%88%D8%A8" title="آنتالپی ذوب – Persian" lang="fa" hreflang="fa" data-title="آنتالپی ذوب" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Enthalpie_de_fusion" title="Enthalpie de fusion – French" lang="fr" hreflang="fr" data-title="Enthalpie de fusion" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Entalp%C3%ADa_de_fusi%C3%B3n" title="Entalpía de fusión – Galician" lang="gl" hreflang="gl" data-title="Entalpía de fusión" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%B5%ED%95%B4%EC%97%B4" title="융해열 – Korean" lang="ko" hreflang="ko" data-title="융해열" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A5%D5%BD%D5%A1%D5%AF%D5%A1%D6%80%D5%A1%D6%80_%D5%B0%D5%A1%D5%AC%D5%B4%D5%A1%D5%B6_%D5%BB%D5%A5%D6%80%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Տեսակարար հալման ջերմություն – Armenian" lang="hy" hreflang="hy" data-title="Տեսակարար հալման ջերմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Toplina_taljenja" title="Toplina taljenja – Croatian" lang="hr" hreflang="hr" data-title="Toplina taljenja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kalor_peleburan" title="Kalor peleburan – Indonesian" lang="id" hreflang="id" data-title="Kalor peleburan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Enthalpia_de_fusion" title="Enthalpia de fusion – Interlingua" lang="ia" hreflang="ia" data-title="Enthalpia de fusion" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Entalpia_di_fusione" title="Entalpia di fusione – Italian" lang="it" hreflang="it" data-title="Entalpia di fusione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%9C%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%99%E1%83%A3%E1%83%97%E1%83%A0%E1%83%98_%E1%83%A1%E1%83%98%E1%83%97%E1%83%91%E1%83%9D" title="დნობის კუთრი სითბო – Georgian" lang="ka" hreflang="ka" data-title="დნობის კუთრი სითბო" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B0%D0%BB%D2%9B%D1%8B%D1%82%D1%83_%D0%B6%D1%8B%D0%BB%D1%83%D0%BB%D1%8B%D2%93%D1%8B" title="Балқыту жылулығы – Kazakh" lang="kk" hreflang="kk" data-title="Балқыту жылулығы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Lydymosi_%C5%A1iluma" title="Lydymosi šiluma – Lithuanian" lang="lt" hreflang="lt" data-title="Lydymosi šiluma" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/manri_dujbi%27o_nejni" title="manri dujbi'o nejni – Lojban" lang="jbo" hreflang="jbo" data-title="manri dujbi'o nejni" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Olvad%C3%A1sh%C5%91" title="Olvadáshő – Hungarian" lang="hu" hreflang="hu" data-title="Olvadáshő" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A6%E0%B5%8D%E0%B4%B0%E0%B4%B5%E0%B5%80%E2%80%8C%E0%B4%95%E0%B4%B0%E0%B4%A3_%E0%B4%B2%E0%B5%80%E0%B4%A8%E0%B4%A4%E0%B4%BE%E0%B4%AA%E0%B4%82" title="ദ്രവീ‌കരണ ലീനതാപം – Malayalam" lang="ml" hreflang="ml" data-title="ദ്രവീ‌കരണ ലീനതാപം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Entalpi_pelakuran" title="Entalpi pelakuran – Malay" lang="ms" hreflang="ms" data-title="Entalpi pelakuran" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Smeltwarmte" title="Smeltwarmte – Dutch" lang="nl" hreflang="nl" data-title="Smeltwarmte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%9E%8D%E8%A7%A3%E7%86%B1" title="融解熱 – Japanese" lang="ja" hreflang="ja" data-title="融解熱" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Smeltevarme" title="Smeltevarme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Smeltevarme" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Smeltevarme" title="Smeltevarme – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Smeltevarme" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Solishtirma_erish_issiqligi" title="Solishtirma erish issiqligi – Uzbek" lang="uz" hreflang="uz" data-title="Solishtirma erish issiqligi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ciep%C5%82o_topnienia" title="Ciepło topnienia – Polish" lang="pl" hreflang="pl" data-title="Ciepło topnienia" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Calor_de_fus%C3%A3o" title="Calor de fusão – Portuguese" lang="pt" hreflang="pt" data-title="Calor de fusão" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Entalpie_de_topire" title="Entalpie de topire – Romanian" lang="ro" hreflang="ro" data-title="Entalpie de topire" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BF%D0%BB%D0%BE%D1%82%D0%B0_%D0%BF%D0%BB%D0%B0%D0%B2%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Удельная теплота плавления – Russian" lang="ru" hreflang="ru" data-title="Удельная теплота плавления" data-language-autonym="Русский" data-language-local-name="Russian" 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<figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg" class="mw-file-description"><img alt="A log-log plot of the enthalpies of melting and boiling versus the melting and boiling temperatures for the pure elements. The linear relationship between the enthalpy of melting the temperature is known as Richard's rule." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg/300px-Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg.png" decoding="async" width="300" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg/450px-Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg/600px-Enthalpies_of_melting_and_boiling_for_pure_elements_versus_temperatures_of_transition.svg.png 2x" data-file-width="499" data-file-height="386" /></a><figcaption>Enthalpies of melting and boiling for pure elements versus temperatures of transition, demonstrating <a href="/wiki/Trouton%27s_rule" title="Trouton's rule">Trouton's rule</a></figcaption></figure> <p>In <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, the <b>enthalpy of fusion</b> of a <a href="/wiki/Chemical_substance" title="Chemical substance">substance</a>, also known as (<b>latent</b>) <b>heat of fusion</b>, is the change in its <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> resulting from providing <a href="/wiki/Energy" title="Energy">energy</a>, typically <a href="/wiki/Heat" title="Heat">heat</a>, to a specific quantity of the substance to change its <a href="/wiki/State_of_matter" title="State of matter">state</a> from a <a href="/wiki/Solid" title="Solid">solid</a> to a <a href="/wiki/Liquid" title="Liquid">liquid</a>, at <a href="/wiki/Isobaric_process" title="Isobaric process">constant pressure</a>. </p><p>The enthalpy of fusion is the amount of energy required to convert one mole of solid into liquid. For example, when <a href="/wiki/Melting" title="Melting">melting</a> 1 kg of ice (at 0 °C under a <a href="https://commons.wikimedia.org/wiki/File:Phase_diagram_of_water.svg" class="extiw" title="c:File:Phase diagram of water.svg">wide range of pressures</a>), 333.55 kJ of energy is absorbed with no <a href="/wiki/Temperature" title="Temperature">temperature</a> change. The <b>heat of solidification</b> (when a substance <a href="/wiki/Freezing" title="Freezing">changes from liquid to solid</a>) is equal and opposite. </p><p>This energy includes the contribution required to make room for any associated change in volume by displacing its environment against ambient pressure. The temperature at which the <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a> occurs is the <a href="/wiki/Melting_point" title="Melting point">melting point</a> or the freezing point, according to context. By convention, the pressure is assumed to be 1 atm (101.325 kPa) unless otherwise specified. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The enthalpy of fusion is a <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a>, because, while melting, the heat energy needed to change the substance from solid to liquid at atmospheric pressure is latent heat of fusion, as the temperature remains constant during the process. The latent heat of fusion is the enthalpy change of any amount of substance when it melts. When the heat of fusion is referenced to a unit of mass, it is usually called the <b>specific heat of fusion</b>, while the <b>molar heat of fusion</b> refers to the enthalpy change per <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> in <a href="/wiki/Mole_(unit)" title="Mole (unit)">moles</a>. </p><p>The liquid phase has a higher internal energy than the solid phase. This means energy must be supplied to a solid in order to melt it and energy is released from a liquid when it freezes, because the <a href="/wiki/Molecule" title="Molecule">molecules</a> in the liquid experience weaker <a href="/wiki/Intermolecular_force" title="Intermolecular force">intermolecular forces</a> and so have a higher potential energy (a kind of <a href="/wiki/Bond-dissociation_energy" title="Bond-dissociation energy">bond-dissociation energy</a> for intermolecular forces). </p><p>When liquid water is cooled, its temperature falls steadily until it drops just below the line of freezing point at 0 °C. The temperature then remains constant at the freezing point while the water crystallizes. Once the water is completely frozen, its temperature continues to fall. </p><p>The enthalpy of fusion is almost always a positive quantity; <a href="/wiki/Helium" title="Helium">helium</a> is the only known exception.<sup id="cite_ref-FOOTNOTEAtkinsJones2008236_1-0" class="reference"><a href="#cite_note-FOOTNOTEAtkinsJones2008236-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Helium-3" title="Helium-3">Helium-3</a> has a negative enthalpy of fusion at temperatures below 0.3 K. <a href="/wiki/Helium-4" title="Helium-4">Helium-4</a> also has a very slightly negative enthalpy of fusion below 0.77 K (−272.380 °C). This means that, at appropriate constant pressures, these substances freeze with the addition of heat.<sup id="cite_ref-FOOTNOTEOttBoerio-Goates200092–93_2-0" class="reference"><a href="#cite_note-FOOTNOTEOttBoerio-Goates200092–93-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the case of <sup>4</sup>He, this pressure range is between 24.992 and 25.00 atm (2,533 kPa).<sup id="cite_ref-Hoffer_3-0" class="reference"><a href="#cite_note-Hoffer-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Enthalpy_of_Fusion_period_three.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Enthalpy_of_Fusion_period_three.PNG/150px-Enthalpy_of_Fusion_period_three.PNG" decoding="async" width="150" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Enthalpy_of_Fusion_period_three.PNG/225px-Enthalpy_of_Fusion_period_three.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Enthalpy_of_Fusion_period_three.PNG/300px-Enthalpy_of_Fusion_period_three.PNG 2x" data-file-width="747" data-file-height="379" /></a><figcaption>Standard enthalpy change of fusion of period three</figcaption></figure><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Molar_heat_of_fusion_period_two.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Molar_heat_of_fusion_period_two.png/150px-Molar_heat_of_fusion_period_two.png" decoding="async" width="150" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Molar_heat_of_fusion_period_two.png/225px-Molar_heat_of_fusion_period_two.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Molar_heat_of_fusion_period_two.png/300px-Molar_heat_of_fusion_period_two.png 2x" data-file-width="770" data-file-height="389" /></a><figcaption>Standard enthalpy change of fusion of period two of the <a href="/wiki/Periodic_table_of_elements" class="mw-redirect" title="Periodic table of elements">periodic table of elements</a></figcaption></figure> <table class="wikitable sortable"> <tbody><tr> <th rowspan="2">Substance </th> <th colspan="2">Heat of fusion </th></tr> <tr> <th>(cal/g) </th> <th>(J/g) </th></tr> <tr> <td><a href="/wiki/Water" title="Water">water</a> </td> <td>79.72 </td> <td>333.55 </td></tr> <tr> <td><a href="/wiki/Methane" title="Methane">methane</a> </td> <td>13.96 </td> <td>58.99 </td></tr> <tr> <td><a href="/wiki/Propane" title="Propane">propane</a> </td> <td>19.11 </td> <td>79.96 </td></tr> <tr> <td><a href="/wiki/Glycerol" title="Glycerol">glycerol</a> </td> <td>47.95 </td> <td>200.62 </td></tr> <tr> <td><a href="/wiki/Formic_acid" title="Formic acid">formic acid</a> </td> <td>66.05 </td> <td>276.35 </td></tr> <tr> <td><a href="/wiki/Acetic_acid" title="Acetic acid">acetic acid</a> </td> <td>45.90 </td> <td>192.09 </td></tr> <tr> <td><a href="/wiki/Acetone" title="Acetone">acetone</a> </td> <td>23.42 </td> <td>97.99 </td></tr> <tr> <td><a href="/wiki/Benzene" title="Benzene">benzene</a> </td> <td>30.45 </td> <td>127.40 </td></tr> <tr> <td><a href="/wiki/Myristic_acid" title="Myristic acid">myristic acid</a> </td> <td>47.49 </td> <td>198.70 </td></tr> <tr> <td><a href="/wiki/Palmitic_acid" title="Palmitic acid">palmitic acid</a> </td> <td>39.18 </td> <td>163.93 </td></tr> <tr> <td><a href="/wiki/Sodium_acetate" title="Sodium acetate">sodium acetate</a>/H2O </td> <td>63–69 </td> <td>264–289<sup id="cite_ref-IDMA_4-0" class="reference"><a href="#cite_note-IDMA-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Sodium_sulfate" title="Sodium sulfate">sodium sulfate</a>/H2O </td> <td>61 </td> <td>254<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Stearic_acid" title="Stearic acid">stearic acid</a> </td> <td>47.54 </td> <td>198.91 </td></tr> <tr> <td><a href="/wiki/Gallium" title="Gallium">gallium</a> </td> <td>19.2 </td> <td>80.4 </td></tr> <tr> <td><a href="/wiki/Paraffin_wax" title="Paraffin wax">paraffin wax</a> (C<sub>25</sub>H<sub>52</sub>) </td> <td>47.8–52.6 </td> <td>200–220 </td></tr></tbody></table> <p>These values are mostly from the <a href="/wiki/CRC_Press" title="CRC Press">CRC</a> <i>Handbook of Chemistry and Physics</i>, 62nd edition. The conversion between cal/g and J/g in the above table uses the thermochemical <a href="/wiki/Calorie" title="Calorie">calorie</a> (cal<sub>th</sub>) = 4.184 joules rather than the International Steam Table calorie (cal<sub>INT</sub>) = 4.1868 joules. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div><ul><li>To heat 1 kg of liquid water from 0 °C to 20 °C requires 83.6 kJ (see below). However, heating 0 °C ice to 20 °C requires additional energy to melt the ice. We can treat these two processes independently and using the specific heat capacity of water to be 4.18 J/(g⋅K); thus, to heat 1 kg of ice from 273.15 K to water at 293.15 K (0 °C to 20 °C) requires: <dl><dd>(1) 333.55 J/g (heat of fusion of ice) = 333.55 kJ/kg = 333.55 kJ for 1 kg of ice to melt, plus</dd> <dd>(2) 4.18 J/(g⋅K) × 20 K = 4.18 kJ/(kg⋅K) × 20 K = 83.6 kJ for 1 kg of water to increase in temperature by 20 K</dd> <dd>(1 + 2) 333.55 kJ + 83.6 kJ = 417.15 kJ for 1 kg of ice to increase in temperature by 20 K</dd></dl></li><li>Silicon has a heat of fusion of 50.21 kJ/mol. 50 kW of power can supply the energy required to melt about 100 kg of silicon in one hour: <dl><dd>50 kW = <span style="white-space:nowrap">50<span style="margin-left:0.25em">kJ/s</span></span> = <span style="white-space:nowrap">180<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">kJ/h</span></span></dd> <dd><span style="white-space:nowrap">180<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">kJ</span></span>/h × (1 mol Si)/<span style="white-space:nowrap">50.21<span style="margin-left:0.25em">kJ</span></span> × <span style="white-space:nowrap">28<span style="margin-left:0.25em">g</span><span style="margin-left:0.25em">Si</span></span>/(mol Si) × <span style="white-space:nowrap">1<span style="margin-left:0.25em">kg</span><span style="margin-left:0.25em">Si</span></span>/<span style="white-space:nowrap">1<span style="margin-left:0.25em">000</span><span style="margin-left:0.25em">g</span><span style="margin-left:0.25em">Si</span></span> = <span style="white-space:nowrap">100.4<span style="margin-left:0.25em">kg/h</span></span></dd></dl></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Solubility_prediction">Solubility prediction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=3" title="Edit section: Solubility prediction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The heat of fusion can also be used to predict <a href="/wiki/Solubility" title="Solubility">solubility</a> for solids in liquids. Provided an <a href="/wiki/Ideal_solution" title="Ideal solution">ideal solution</a> is obtained the <a href="/wiki/Mole_fraction" title="Mole fraction">mole fraction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629995620ed175b987bddc8298df5f32fd31acb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.193ex; height:2.843ex;" alt="{\displaystyle (x_{2})}"></span> of solute at saturation is a function of the heat of fusion, the <a href="/wiki/Melting_point" title="Melting point">melting point</a> of the solid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{\text{fus}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T_{\text{fus}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a7a27a98ac9b5e0ec374e49400c88c9ced2aec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.464ex; height:2.843ex;" alt="{\displaystyle (T_{\text{fus}})}"></span> and the <a href="/wiki/Temperature" title="Temperature">temperature</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f9a82bdcb95d511ed6965b0534a3eda98ceb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.446ex; height:2.843ex;" alt="{\displaystyle (T)}"></span> of the solution: </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 \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mi>R</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824dc80fb42fdbd162190369c1e270f9531b03c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.23ex; height:6.509ex;" alt="{\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}"></span></dd></dl> <p>Here, <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>. For example, the solubility of <a href="/wiki/Paracetamol" title="Paracetamol">paracetamol</a> in water at 298 <a href="/wiki/Kelvin_(unit)" class="mw-redirect" title="Kelvin (unit)">K</a> is predicted to be: </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_{2}=\exp {\left[-{\frac {28100~{\text{J mol}}^{-1}}{8.314~{\text{J K}}^{-1}~{\text{mol}}^{-1}}}\left({\frac {1}{298~{\text{K}}}}-{\frac {1}{442~{\text{K}}}}\right)\right]}=0.0248}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>28100</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>J mol</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>8.314</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>J K</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>mol</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>298</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>442</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>K</mtext> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mn>0.0248</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}=\exp {\left[-{\frac {28100~{\text{J mol}}^{-1}}{8.314~{\text{J K}}^{-1}~{\text{mol}}^{-1}}}\left({\frac {1}{298~{\text{K}}}}-{\frac {1}{442~{\text{K}}}}\right)\right]}=0.0248}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5435f0f0dcd447b5a6ebdbe0d6d74833998c5e65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.604ex; height:6.343ex;" alt="{\displaystyle x_{2}=\exp {\left[-{\frac {28100~{\text{J mol}}^{-1}}{8.314~{\text{J K}}^{-1}~{\text{mol}}^{-1}}}\left({\frac {1}{298~{\text{K}}}}-{\frac {1}{442~{\text{K}}}}\right)\right]}=0.0248}"></span></dd></dl> <p>Since the molar mass of water and paracetamol are <span style="white-space:nowrap">18.0153<span style="margin-left:0.25em">g</span><span style="margin-left:0.25em">mol<sup>−1</sup></span></span> and <span style="white-space:nowrap">151.17<span style="margin-left:0.25em">g</span><span style="margin-left:0.25em">mol<sup>−1</sup></span></span> and the density of the solution is <span style="white-space:nowrap">1000<span style="margin-left:0.25em">g</span><span style="margin-left:0.25em">L<sup>−1</sup></span></span>, an estimate of the solubility in grams per liter is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {0.0248\times {\frac {1000~{\text{g L}}^{-1}}{18.0153~{\text{g mol}}^{-1}}}}{1-0.0248}}\times 151.17~{\text{g mol}}^{-1}=213.4~{\text{g L}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>0.0248</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1000</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>g L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>18.0153</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>g mol</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.0248</mn> </mrow> </mfrac> </mrow> <mo>×</mo> <mn>151.17</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>g mol</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>213.4</mn> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>g L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {0.0248\times {\frac {1000~{\text{g L}}^{-1}}{18.0153~{\text{g mol}}^{-1}}}}{1-0.0248}}\times 151.17~{\text{g mol}}^{-1}=213.4~{\text{g L}}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80759b4b2b0656ad0f34d348ae8377bf262c640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:55.068ex; height:8.009ex;" alt="{\displaystyle {\frac {0.0248\times {\frac {1000~{\text{g L}}^{-1}}{18.0153~{\text{g mol}}^{-1}}}}{1-0.0248}}\times 151.17~{\text{g mol}}^{-1}=213.4~{\text{g L}}^{-1}}"></span></dd> <dd></dd> <dd>1000 g/L * (mol/18.0153g) is an estimate of the number of moles of molecules in 1L solution, using water density as a reference;</dd> <dd>0.0248 * (1000 g/L * (mol/18.0153g)) is the molar fraction of substance in saturated solution with a unit of mol/L;</dd> <dd>0.0248 * (1000 g/L * (mol/18.0153g)) * 151.17g/mol is the solute's molar fraction equivalent mass conversion;</dd> <dd>1-0.0248 will be the fraction of the solution that is solvent.</dd></dl> <p>which is a deviation from the real solubility (240 g/L) of 11%. This error can be reduced when an additional <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> parameter is taken into account.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Proof">Proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=4" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At <a href="/wiki/Chemical_equilibrium" title="Chemical equilibrium">equilibrium</a> the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potentials</a> for the solute in the solution and pure solid are identical: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{solute}}^{\circ }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solute</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{solute}}^{\circ }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b3d4b7c216b6adcf2b8884411cfefcd75b6c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.263ex; height:2.676ex;" alt="{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{solute}}^{\circ }\,}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{liquid}}^{\circ }+RT\ln X_{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>liquid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>+</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{liquid}}^{\circ }+RT\ln X_{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088bec621bea7d3c622c809e34eaf4b76f72acb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.053ex; height:3.176ex;" alt="{\displaystyle \mu _{\text{solid}}^{\circ }=\mu _{\text{liquid}}^{\circ }+RT\ln X_{2}\,}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293563891196765d2d51e0dd54e1ae1000ba9def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle R\,}"></span> the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/476a8389064c06ab89963a2467aef525838da0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.023ex; height:2.176ex;" alt="{\displaystyle T\,}"></span> the <a href="/wiki/Temperature" title="Temperature">temperature</a>. </p><p>Rearranging gives: </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 RT\ln X_{2}=-\left(\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>liquid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RT\ln X_{2}=-\left(\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b402bb5c393d7117aaaafc5cd701454f267943d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.024ex; height:4.843ex;" alt="{\displaystyle RT\ln X_{2}=-\left(\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\right)\,}"></span></dd></dl> <p>and since </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 G_{\text{fus}}^{\circ }=\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>liquid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solid</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G_{\text{fus}}^{\circ }=\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84af77d49d5df1a530c111911da5d7e6693aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.021ex; height:3.176ex;" alt="{\displaystyle \Delta G_{\text{fus}}^{\circ }=\mu _{\text{liquid}}^{\circ }-\mu _{\text{solid}}^{\circ }\,}"></span></dd></dl> <p>the heat of fusion being the difference in chemical potential between the pure liquid and the pure solid, it follows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RT\ln X_{2}=-\left(\Delta G_{\text{fus}}^{\circ }\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RT\ln X_{2}=-\left(\Delta G_{\text{fus}}^{\circ }\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef525e2c90e7539792464c85dea76edf1abd36c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.963ex; height:3.176ex;" alt="{\displaystyle RT\ln X_{2}=-\left(\Delta G_{\text{fus}}^{\circ }\right)\,}"></span></dd></dl> <p>Application of the <a href="/wiki/Gibbs%E2%80%93Helmholtz_equation" title="Gibbs–Helmholtz equation">Gibbs–Helmholtz equation</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 \left({\frac {\partial \left({\frac {\Delta G_{\text{fus}}^{\circ }}{T}}\right)}{\partial T}}\right)_{p\,}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{T^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \left({\frac {\Delta G_{\text{fus}}^{\circ }}{T}}\right)}{\partial T}}\right)_{p\,}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{T^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c9036c0e8c6c96fa119bd4c4e2577108473248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:28.075ex; height:10.843ex;" alt="{\displaystyle \left({\frac {\partial \left({\frac {\Delta G_{\text{fus}}^{\circ }}{T}}\right)}{\partial T}}\right)_{p\,}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{T^{2}}}}"></span></dd></dl> <p>ultimately gives: </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 \left({\frac {\partial \left(\ln X_{2}\right)}{\partial T}}\right)={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mrow> <mi>R</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \left(\ln X_{2}\right)}{\partial T}}\right)={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06af3318b39588e24ddb49db30a16b5aae41d534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.176ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial \left(\ln X_{2}\right)}{\partial T}}\right)={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}}"></span></dd></dl> <p>or: </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 \partial \ln X_{2}={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \delta T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mrow> <mi>R</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>×</mo> <mi>δ</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \ln X_{2}={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \delta T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfc3ddfa1dd9eae995d2713ce01173e7537f6fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.635ex; height:6.176ex;" alt="{\displaystyle \partial \ln X_{2}={\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \delta T}"></span></dd></dl> <p>and with <a href="/wiki/Integral" title="Integral">integration</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 \int _{X_{2}=1}^{X_{2}=x_{2}}\delta \ln X_{2}=\ln x_{2}=\int _{T_{\text{fus}}}^{T}{\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \Delta T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>δ</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mrow> <mi>R</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>×</mo> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{X_{2}=1}^{X_{2}=x_{2}}\delta \ln X_{2}=\ln x_{2}=\int _{T_{\text{fus}}}^{T}{\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \Delta T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3873f707ea012c7d48a6cabae27df311f8c1ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.1ex; height:6.676ex;" alt="{\displaystyle \int _{X_{2}=1}^{X_{2}=x_{2}}\delta \ln X_{2}=\ln x_{2}=\int _{T_{\text{fus}}}^{T}{\frac {\Delta H_{\text{fus}}^{\circ }}{RT^{2}}}\times \Delta T}"></span></dd></dl> <p>the result is obtained: </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 \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> </mrow> <mi>R</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fus</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824dc80fb42fdbd162190369c1e270f9531b03c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.23ex; height:6.509ex;" alt="{\displaystyle \ln x_{2}=-{\frac {\Delta H_{\text{fus}}^{\circ }}{R}}\left({\frac {1}{T}}-{\frac {1}{T_{\text{fus}}}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Enthalpy_of_vaporization" title="Enthalpy of vaporization">Enthalpy of vaporization</a></li> <li><a href="/wiki/Heat_capacity" title="Heat capacity">Heat capacity</a></li> <li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Thermodynamic databases for pure substances</a></li> <li><a href="/wiki/Joback_method" title="Joback method">Joback method</a> (Estimation of the heat of fusion from molecular structure)</li> <li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li> <li><a href="/wiki/Lattice_energy" title="Lattice energy">Lattice energy</a></li> <li><a href="/wiki/Heat_of_dilution" title="Heat of dilution">Heat of dilution</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEAtkinsJones2008236-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAtkinsJones2008236_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAtkinsJones2008">Atkins & Jones 2008</a>, p. 236.</span> </li> <li 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Rohani, S. (November 2006). "Measurement and Prediction of Solubility of Paracetamol in Water−Isopropanol Solution. Part 2. Prediction". <i>Organic Process Research & Development</i>. <b>10</b> (6): 1110–1118. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fop060074g">10.1021/op060074g</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Organic+Process+Research+%26+Development&rft.atitle=Measurement+and+Prediction+of+Solubility+of+Paracetamol+in+Water%E2%88%92Isopropanol+Solution.+Part+2.+Prediction&rft.volume=10&rft.issue=6&rft.pages=1110-1118&rft.date=2006-11&rft_id=info%3Adoi%2F10.1021%2Fop060074g&rft.aulast=Hojjati&rft.aufirst=H.&rft.au=Rohani%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy+of+fusion" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Enthalpy_of_fusion&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkinsJones2008" class="citation cs2">Atkins, Peter; Jones, Loretta (2008), <i>Chemical Principles: The Quest for Insight</i> (4th ed.), W. H. Freeman and Company, p. 236, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-7355-9" title="Special:BookSources/978-0-7167-7355-9"><bdi>978-0-7167-7355-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+Principles%3A+The+Quest+for+Insight&rft.pages=236&rft.edition=4th&rft.pub=W.+H.+Freeman+and+Company&rft.date=2008&rft.isbn=978-0-7167-7355-9&rft.aulast=Atkins&rft.aufirst=Peter&rft.au=Jones%2C+Loretta&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy+of+fusion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOttBoerio-Goates2000" class="citation cs2">Ott, BJ. Bevan; Boerio-Goates, Juliana (2000), <i>Chemical Thermodynamics: Advanced Applications</i>, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-530985-6" title="Special:BookSources/0-12-530985-6"><bdi>0-12-530985-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+Thermodynamics%3A+Advanced+Applications&rft.pub=Academic+Press&rft.date=2000&rft.isbn=0-12-530985-6&rft.aulast=Ott&rft.aufirst=BJ.+Bevan&rft.au=Boerio-Goates%2C+Juliana&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEnthalpy+of+fusion" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a> / <a href="/wiki/Vapor" title="Vapor">Vapor</a></li> <li><a href="/wiki/Supercritical_fluid" title="Supercritical fluid">Supercritical fluid</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Phase_change_-_en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/150px-Phase_change_-_en.svg.png" decoding="async" width="150" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/225px-Phase_change_-_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/300px-Phase_change_-_en.svg.png 2x" data-file-width="493" data-file-height="524" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Low energy</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Degenerate_matter" title="Degenerate matter">Degenerate matter</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall</a></li> <li><a href="/wiki/Rydberg_matter" title="Rydberg matter">Rydberg matter</a></li> <li><a href="/wiki/Strange_matter" title="Strange matter">Strange matter</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Photonic_molecule" title="Photonic molecule">Photonic molecule</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">High energy</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/QCD_matter" title="QCD matter">QCD matter</a></li> <li><a href="/wiki/Quark%E2%80%93gluon_plasma" title="Quark–gluon plasma">Quark–gluon plasma</a></li> <li><a href="/wiki/Color-glass_condensate" title="Color-glass condensate">Color-glass condensate</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other states</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li> <li><a href="/wiki/Crystal" title="Crystal">Crystal</a></li> <li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li> <li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li> <li><a href="/wiki/Quantum_spin_liquid" title="Quantum spin liquid">Quantum spin liquid</a></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Programmable_matter" title="Programmable matter">Programmable matter</a></li> <li><a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li> <li><a href="/wiki/Antimatter" title="Antimatter">Antimatter</a></li> <li>Magnetically ordered <ul><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Ferrimagnetism" title="Ferrimagnetism">Ferrimagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li></ul></li> <li><a href="/wiki/String-net_liquid" title="String-net liquid">String-net liquid</a></li> <li><a href="/wiki/Superglass" title="Superglass">Superglass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Phase_transitions" class="mw-redirect" title="Phase transitions">Phase transitions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boiling" title="Boiling">Boiling</a></li> <li><a href="/wiki/Boiling_point" title="Boiling point">Boiling point</a></li> <li><a href="/wiki/Condensation" title="Condensation">Condensation</a></li> <li><a href="/wiki/Critical_line_(thermodynamics)" title="Critical line (thermodynamics)">Critical line</a></li> <li><a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">Critical point</a></li> <li><a href="/wiki/Crystallization" title="Crystallization">Crystallization</a></li> <li><a href="/wiki/Deposition_(phase_transition)" title="Deposition (phase transition)">Deposition</a></li> <li><a href="/wiki/Evaporation" title="Evaporation">Evaporation</a></li> <li><a href="/wiki/Flash_evaporation" title="Flash evaporation">Flash evaporation</a></li> <li><a href="/wiki/Freezing" title="Freezing">Freezing</a></li> <li><a href="/wiki/Chemical_ionization" title="Chemical ionization">Chemical ionization</a></li> <li><a href="/wiki/Ionization" title="Ionization">Ionization</a></li> <li><a href="/wiki/Lambda_point" title="Lambda point">Lambda point</a></li> <li><a href="/wiki/Melting" title="Melting">Melting</a></li> <li><a href="/wiki/Melting_point" title="Melting point">Melting point</a></li> <li><a href="/wiki/Plasma_recombination" title="Plasma recombination">Recombination</a></li> <li><a href="/wiki/Regelation" title="Regelation">Regelation</a></li> <li><a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">Saturated fluid</a></li> <li><a href="/wiki/Sublimation_(phase_transition)" title="Sublimation (phase transition)">Sublimation</a></li> <li><a href="/wiki/Supercooling" title="Supercooling">Supercooling</a></li> <li><a href="/wiki/Triple_point" title="Triple point">Triple point</a></li> <li><a href="/wiki/Vaporization" title="Vaporization">Vaporization</a></li> <li><a href="/wiki/Vitrification" title="Vitrification">Vitrification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Enthalpy of fusion</a></li> <li><a href="/wiki/Enthalpy_of_sublimation" title="Enthalpy of sublimation">Enthalpy of sublimation</a></li> <li><a href="/wiki/Enthalpy_of_vaporization" title="Enthalpy of vaporization">Enthalpy of vaporization</a></li> <li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li> <li><a href="/wiki/Latent_internal_energy" title="Latent internal energy">Latent internal energy</a></li> <li><a href="/wiki/Trouton%27s_rule" title="Trouton's rule">Trouton's rule</a></li> <li><a href="/wiki/Volatility_(chemistry)" title="Volatility (chemistry)">Volatility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baryonic_matter" class="mw-redirect" title="Baryonic matter">Baryonic matter</a></li> <li><a href="/wiki/Binodal" title="Binodal">Binodal</a></li> <li><a href="/wiki/Compressed_fluid" title="Compressed fluid">Compressed fluid</a></li> <li><a href="/wiki/Cooling_curve" title="Cooling curve">Cooling curve</a></li> <li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Leidenfrost_effect" title="Leidenfrost effect">Leidenfrost effect</a></li> <li><a href="/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a></li> <li><a href="/wiki/Mpemba_effect" title="Mpemba effect">Mpemba effect</a></li> <li><a href="/wiki/Order_and_disorder_(physics)" class="mw-redirect" title="Order and disorder (physics)">Order and disorder (physics)</a></li> <li><a href="/wiki/Spinodal" title="Spinodal">Spinodal</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductivity</a></li> <li><a href="/wiki/Superheated_water" title="Superheated water">Superheated vapor</a></li> <li><a href="/wiki/Superheating" title="Superheating">Superheating</a></li> <li><a href="/wiki/Thermo-dielectric_effect" title="Thermo-dielectric effect">Thermo-dielectric effect</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" 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