Mole fraction - Wikipedia

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class="vector-toc-numb">2</span> <span>Related quantities</span> </div> </a> <button aria-controls="toc-Related_quantities-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 Related quantities subsection</span> </button> <ul id="toc-Related_quantities-sublist" class="vector-toc-list"> <li id="toc-Mass_fraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mass_fraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Mass fraction</span> </div> </a> <ul id="toc-Mass_fraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molar_mixing_ratio" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Molar_mixing_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Molar mixing ratio</span> </div> </a> <ul id="toc-Molar_mixing_ratio-sublist" class="vector-toc-list"> <li id="toc-Mixing_binary_mixtures_with_a_common_component_to_form_ternary_mixtures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mixing_binary_mixtures_with_a_common_component_to_form_ternary_mixtures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Mixing binary mixtures with a common component to form ternary mixtures</span> </div> </a> <ul id="toc-Mixing_binary_mixtures_with_a_common_component_to_form_ternary_mixtures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mole_percentage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mole_percentage"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Mole percentage</span> </div> </a> <ul id="toc-Mole_percentage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mass_concentration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mass_concentration"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Mass concentration</span> </div> </a> <ul id="toc-Mass_concentration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molar_concentration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Molar_concentration"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Molar concentration</span> </div> </a> <ul id="toc-Molar_concentration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mass_and_molar_mass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mass_and_molar_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Mass and molar mass</span> </div> </a> <ul id="toc-Mass_and_molar_mass-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spatial_variation_and_gradient" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spatial_variation_and_gradient"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Spatial variation and gradient</span> </div> </a> <ul id="toc-Spatial_variation_and_gradient-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">4</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">Mole fraction</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 24 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-24" 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">24 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%B3%D8%B1_%D9%85%D9%88%D9%84%D9%8A" title="كسر مولي – Arabic" lang="ar" hreflang="ar" data-title="كسر مولي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Fracci%C3%B3_molar" title="Fracció molar – Catalan" lang="ca" hreflang="ca" data-title="Fracció molar" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Stoffmengenanteil" title="Stoffmengenanteil – German" lang="de" hreflang="de" data-title="Stoffmengenanteil" 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/Moolimurd" title="Moolimurd – Estonian" lang="et" hreflang="et" data-title="Moolimurd" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CF%81%CE%B1%CE%BC%CE%BC%CE%BF%CE%BC%CE%BF%CF%81%CE%B9%CE%B1%CE%BA%CF%8C_%CE%BA%CE%BB%CE%AC%CF%83%CE%BC%CE%B1" title="Γραμμομοριακό κλάσμα – Greek" lang="el" hreflang="el" data-title="Γραμμομοριακό κλάσμα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fracci%C3%B3n_molar" title="Fracción molar – Spanish" lang="es" hreflang="es" data-title="Fracción molar" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B3%D8%B1_%D9%85%D9%88%D9%84%DB%8C" title="کسر مولی – Persian" lang="fa" hreflang="fa" data-title="کسر مولی" 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/Fraction_molaire" title="Fraction molaire – French" lang="fr" hreflang="fr" data-title="Fraction molaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%B0%EB%B6%84%EC%9C%A8" title="몰분율 – Korean" lang="ko" hreflang="ko" data-title="몰분율" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AE%E0%A5%8B%E0%A4%B2_%E0%A4%85%E0%A4%82%E0%A4%B6" title="मोल अंश – Hindi" lang="hi" hreflang="hi" data-title="मोल अंश" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fraksi_mol" title="Fraksi mol – Indonesian" lang="id" hreflang="id" data-title="Fraksi mol" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Frazione_molare" title="Frazione molare – Italian" lang="it" hreflang="it" data-title="Frazione molare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Rap%C3%B2_mol%C3%A8" title="Rapò molè – Haitian Creole" lang="ht" hreflang="ht" data-title="Rapò molè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B5%8B%E0%B5%BE%E0%B4%AB%E0%B5%8D%E0%B4%B0%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B5%BB" title="മോൾഫ്രാക്ഷൻ – Malayalam" lang="ml" hreflang="ml" data-title="മോൾഫ്രാക്ഷൻ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Molfractie" title="Molfractie – Dutch" lang="nl" hreflang="nl" data-title="Molfractie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/U%C5%82amek_molowy" title="Ułamek molowy – Polish" lang="pl" hreflang="pl" data-title="Ułamek molowy" 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/Fra%C3%A7%C3%A3o_molar" title="Fração molar – Portuguese" lang="pt" hreflang="pt" data-title="Fração molar" 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/Frac%C8%9Bie_molar%C4%83" title="Fracție molară – Romanian" lang="ro" hreflang="ro" data-title="Fracție molară" 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%9C%D0%BE%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B4%D0%BE%D0%BB%D1%8F" title="Мольная доля – Russian" lang="ru" hreflang="ru" data-title="Мольная доля" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Molski_dele%C5%BE" title="Molski delež – Slovenian" lang="sl" hreflang="sl" data-title="Molski delež" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Mooliosuus" title="Mooliosuus – Finnish" lang="fi" hreflang="fi" data-title="Mooliosuus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Proportion of a constituent in a mixture</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">mole fraction</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other names</div></th><td class="infobox-data">molar fraction, amount fraction, amount-of-substance fraction</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><i>x</i></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data">mol/mol</td></tr></tbody></table> <p>In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, the <b><a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> fraction</b> or <b>molar fraction</b>, also called <b>mole proportion</b> or <b>molar proportion</b>, is a <a href="/wiki/Quantity_(science)" class="mw-redirect" title="Quantity (science)">quantity</a> defined as the <a href="/wiki/Ratio" title="Ratio">ratio</a> between the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount</a> of a constituent substance, <i>n<sub>i</sub></i> (expressed in <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit</a> of <a href="/wiki/Mole_(unit)" title="Mole (unit)">moles</a>, symbol mol), and the total amount of all constituents in a mixture, <i>n</i><sub>tot</sub> (also expressed in moles):<sup id="cite_ref-goldbook_1-0" class="reference"><a href="#cite_note-goldbook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}={\frac {n_{i}}{n_{\mathrm {tot} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}={\frac {n_{i}}{n_{\mathrm {tot} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3236da8eb29e98deaf523522b979eaa7dae61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.792ex; height:5.009ex;" alt="{\displaystyle x_{i}={\frac {n_{i}}{n_{\mathrm {tot} }}}}"></span></dd></dl> <p>It is <a href="/wiki/Mathematical_symbol" class="mw-redirect" title="Mathematical symbol">denoted</a> <i><b>x<sub>i</sub></b></i> (lowercase <a href="/wiki/Latin_alphabet" title="Latin alphabet">Roman</a> letter <i><a href="/wiki/X" title="X">x</a></i>), sometimes <b><span class="texhtml mvar" style="font-style:italic;">χ<sub>i</sub></span></b> (lowercase <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek</a> letter <a href="/wiki/Chi_(letter)" title="Chi (letter)">chi</a>).<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><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> (For mixtures of gases, the letter <i>y</i> is recommended.<sup id="cite_ref-goldbook_1-1" class="reference"><a href="#cite_note-goldbook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ISO_4-0" class="reference"><a href="#cite_note-ISO-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>) </p><p>It is a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless quantity</a> with <a href="/wiki/Dimension_(physics)" class="mw-redirect" title="Dimension (physics)">dimension</a> of <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 {\mathsf {N}}/{\mathsf {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {N}}/{\mathsf {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f8658d88ffe17f4997f53e52e5a7db6ddc9042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle {\mathsf {N}}/{\mathsf {N}}}"></span> and <a href="/wiki/Dimensionless_unit" class="mw-redirect" title="Dimensionless unit">dimensionless unit</a> of <b>moles per mole</b> (<b>mol/mol</b> or <b>mol ⋅ mol<sup>-1</sup></b>) or simply 1; <a href="/wiki/Metric_prefixes" class="mw-redirect" title="Metric prefixes">metric prefixes</a> may also be used (e.g., nmol/mol for <a href="/wiki/Nano-" title="Nano-">10<sup>-9</sup></a>).<sup id="cite_ref-BIPM_c402_5-0" class="reference"><a href="#cite_note-BIPM_c402-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> When expressed in <a href="/wiki/Percent" class="mw-redirect" title="Percent">percent</a>, it is known as the <b>mole percent</b> or <b>molar percentage</b> (unit symbol %, sometimes "mol%", equivalent to cmol/mol for <a href="/wiki/Centi-" title="Centi-">10<sup>-2</sup></a>). The mole fraction is called <b>amount fraction</b> by the <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">International Union of Pure and Applied Chemistry</a> (IUPAC)<sup id="cite_ref-goldbook_1-2" class="reference"><a href="#cite_note-goldbook-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <b>amount-of-substance fraction</b> by the U.S. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a> (NIST).<sup id="cite_ref-NIST_6-0" class="reference"><a href="#cite_note-NIST-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This nomenclature is part of the <a href="/wiki/International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a> (ISQ), as standardized in <a href="/wiki/ISO_80000-9" class="mw-redirect" title="ISO 80000-9">ISO 80000-9</a>,<sup id="cite_ref-ISO_4-1" class="reference"><a href="#cite_note-ISO-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> which deprecates "mole fraction" based on the unacceptability of mixing information with units when expressing the values of quantities.<sup id="cite_ref-NIST_6-1" class="reference"><a href="#cite_note-NIST-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The sum of all the mole fractions in a mixture is equal to 1: </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 \sum _{i=1}^{N}n_{i}=n_{\mathrm {tot} };\ \sum _{i=1}^{N}x_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>;</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{N}n_{i}=n_{\mathrm {tot} };\ \sum _{i=1}^{N}x_{i}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e5464ba7c40ff9f4d2e5541d96977e6fbe7a8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.897ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{N}n_{i}=n_{\mathrm {tot} };\ \sum _{i=1}^{N}x_{i}=1}"></span></dd></dl> <p>Mole fraction is numerically identical to the <b>number fraction</b>, which is defined as the <a href="/wiki/Number_of_particles" class="mw-redirect" title="Number of particles">number of particles</a> (<a href="/wiki/Molecule" title="Molecule">molecules</a>) of a constituent <i>N<sub>i</sub></i> divided by the total number of all molecules <i>N</i><sub>tot</sub>. Whereas mole fraction is a ratio of amounts to amounts (in units of moles per moles), <a href="/wiki/Molar_concentration" title="Molar concentration">molar concentration</a> is a quotient of amount to volume (in units of moles per litre). Other ways of expressing the composition of a mixture as a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless quantity</a> are <a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">mass fraction</a> and <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a> are others. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mole fraction is used very frequently in the construction of <a href="/wiki/Phase_diagram" title="Phase diagram">phase diagrams</a>. It has a number of advantages: </p> <ul><li>it is not temperature dependent (as is <a href="/wiki/Molar_concentration" title="Molar concentration">molar concentration</a>) and does not require knowledge of the densities of the phase(s) involved</li> <li>a mixture of known mole fraction can be prepared by weighing off the appropriate masses of the constituents</li> <li>the measure is <i>symmetric</i>: in the mole fractions <i>x</i> = 0.1 and <i>x</i> = 0.9, the roles of 'solvent' and 'solute' are reversed.</li> <li>In a mixture of <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gases</a>, the mole fraction can be expressed as the ratio of <a href="/wiki/Partial_pressure" title="Partial pressure">partial pressure</a> to total <a href="/wiki/Pressure" title="Pressure">pressure</a> of the mixture</li> <li>In a ternary mixture one can express mole fractions of a component as functions of other components mole fraction and binary mole ratios: <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 {\begin{aligned}x_{1}&={\frac {1-x_{2}}{1+{\frac {x_{3}}{x_{1}}}}}\\[2pt]x_{3}&={\frac {1-x_{2}}{1+{\frac {x_{1}}{x_{3}}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}&={\frac {1-x_{2}}{1+{\frac {x_{3}}{x_{1}}}}}\\[2pt]x_{3}&={\frac {1-x_{2}}{1+{\frac {x_{1}}{x_{3}}}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e838685d60e290137de8c0667aee0c140ff2ba4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.404ex; margin-bottom: -0.267ex; width:13.681ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}x_{1}&={\frac {1-x_{2}}{1+{\frac {x_{3}}{x_{1}}}}}\\[2pt]x_{3}&={\frac {1-x_{2}}{1+{\frac {x_{1}}{x_{3}}}}}\end{aligned}}}"></span></dd></dl></li></ul> <p>Differential quotients can be formed at constant ratios like those above: </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 x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{1}}{1-x_{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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{1}}{1-x_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa6cfe2498039615af7bc1d1f4c0868fd1e2523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.721ex; height:7.343ex;" alt="{\displaystyle \left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{1}}{1-x_{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 \left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{3}}{1-x_{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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{3}}{1-x_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f80017f349a83ecff20ab6917f8cd87839264d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.721ex; height:7.343ex;" alt="{\displaystyle \left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{3}}{1-x_{2}}}}"></span></dd></dl> <p>The ratios <i>X</i>, <i>Y</i>, and <i>Z</i> of mole fractions can be written for ternary and multicomponent systems: </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 {\begin{aligned}X&={\frac {x_{3}}{x_{1}+x_{3}}}\\[2pt]Y&={\frac {x_{3}}{x_{2}+x_{3}}}\\[2pt]Z&={\frac {x_{2}}{x_{1}+x_{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>X</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <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>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X&={\frac {x_{3}}{x_{1}+x_{3}}}\\[2pt]Y&={\frac {x_{3}}{x_{2}+x_{3}}}\\[2pt]Z&={\frac {x_{2}}{x_{1}+x_{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2933fbbf166772a7fc0427a243e43b4b6a7e8f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:14.274ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}X&={\frac {x_{3}}{x_{1}+x_{3}}}\\[2pt]Y&={\frac {x_{3}}{x_{2}+x_{3}}}\\[2pt]Z&={\frac {x_{2}}{x_{1}+x_{2}}}\end{aligned}}}"></span></dd></dl> <p>These can be used for solving PDEs like: </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 \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3}}}"> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f265c67b2afae40abf726a25d07cfcd725c1747c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.811ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3}}}"></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 \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3},n_{4},\ldots ,n_{i}}}"> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3},n_{4},\ldots ,n_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3ec3f407e38ca58cdf59ec3f2647f5e93d2e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.265ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3},n_{4},\ldots ,n_{i}}}"></span></dd></dl> <p>This equality can be rearranged to have differential quotient of mole amounts or fractions on one side. </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 \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{3}}=-\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\mu _{1},n_{3}}}"> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{3}}=-\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\mu _{1},n_{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad86f40d155b7b91782112c0b50ea8163e636e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.814ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{3}}=-\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\mu _{1},n_{3}}}"></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 \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{2},n_{4},\ldots ,n_{i}}}"> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{2},n_{4},\ldots ,n_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44de6226a45514c82f717ccbc49f48d1acae30a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.071ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{2},n_{4},\ldots ,n_{i}}}"></span></dd></dl> <p>Mole amounts can be eliminated by forming ratios: </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 n_{1}}{\partial n_{2}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {n_{1}}{n_{3}}}}{\partial {\frac {n_{2}}{n_{3}}}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{n_{3}}}"> <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> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {n_{1}}{n_{3}}}}{\partial {\frac {n_{2}}{n_{3}}}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{n_{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebadd462397e07805e7e193b98f65a1619adb241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:38.075ex; height:8.676ex;" alt="{\displaystyle \left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {n_{1}}{n_{3}}}}{\partial {\frac {n_{2}}{n_{3}}}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{n_{3}}}"></span></dd></dl> <p>Thus the ratio of chemical potentials becomes: </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 \mu _{2}}{\partial \mu _{1}}}\right)_{\frac {n_{2}}{n_{3}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1}}}"> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{\frac {n_{2}}{n_{3}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc887937de5a9214cd2dfcb7f03bb55487cf10a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.492ex; height:8.676ex;" alt="{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{\frac {n_{2}}{n_{3}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1}}}"></span></dd></dl> <p>Similarly the ratio for the multicomponents system becomes </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 \mu _{2}}{\partial \mu _{1}}}\right)_{{\frac {n_{2}}{n_{3}}},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}}"> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{{\frac {n_{2}}{n_{3}}},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799c0cba3baaebdb9042ba08e60021e28da669e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:45.834ex; height:9.343ex;" alt="{\displaystyle \left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{{\frac {n_{2}}{n_{3}}},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Related_quantities">Related quantities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=2" title="Edit section: Related quantities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mass_fraction">Mass fraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=3" title="Edit section: Mass fraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">mass fraction</a> <i>w<sub>i</sub></i> can be calculated using the formula </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 w_{i}=x_{i}{\frac {M_{i}}{\bar {M}}}=x_{i}{\frac {M_{i}}{\sum _{j}x_{j}M_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}=x_{i}{\frac {M_{i}}{\bar {M}}}=x_{i}{\frac {M_{i}}{\sum _{j}x_{j}M_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24093888118af8d3bbca6a36fcf9d0214fbdb102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.799ex; height:6.343ex;" alt="{\displaystyle w_{i}=x_{i}{\frac {M_{i}}{\bar {M}}}=x_{i}{\frac {M_{i}}{\sum _{j}x_{j}M_{j}}}}"></span></dd></dl> <p>where <i>M<sub>i</sub></i> is the molar mass of the component <i>i</i> and <i>M̄</i> is the average <a href="/wiki/Molar_mass" title="Molar mass">molar mass</a> of the mixture. </p> <div class="mw-heading mw-heading3"><h3 id="Molar_mixing_ratio">Molar mixing ratio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=4" title="Edit section: Molar mixing ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mixing of two pure components can be expressed introducing the amount or molar <a href="/wiki/Mixing_ratio" title="Mixing ratio">mixing ratio</a> of them <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}={\frac {n_{2}}{n_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}={\frac {n_{2}}{n_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821d9a4b3bf67541d5546e9e131f24352251aee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.651ex; height:5.009ex;" alt="{\displaystyle r_{n}={\frac {n_{2}}{n_{1}}}}"></span>. Then the mole fractions of the components will 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 {\begin{aligned}x_{1}&={\frac {1}{1+r_{n}}}\\[2pt]x_{2}&={\frac {r_{n}}{1+r_{n}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}&={\frac {1}{1+r_{n}}}\\[2pt]x_{2}&={\frac {r_{n}}{1+r_{n}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0910f08dbb3df422878f3a5bb41340050aacea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:13.34ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}x_{1}&={\frac {1}{1+r_{n}}}\\[2pt]x_{2}&={\frac {r_{n}}{1+r_{n}}}\end{aligned}}}"></span></dd></dl> <p>The amount ratio equals the ratio of mole fractions of components: </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 {n_{2}}{n_{1}}}={\frac {x_{2}}{x_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {x_{2}}{x_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169d04a4f0acefcce7ef8de50d803faf8c47fc8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.604ex; height:5.009ex;" alt="{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {x_{2}}{x_{1}}}}"></span></dd></dl> <p>due to division of both numerator and denominator by the sum of molar amounts of components. This property has consequences for representations of <a href="/wiki/Phase_diagram" title="Phase diagram">phase diagrams</a> using, for instance, <a href="/wiki/Ternary_plot" title="Ternary plot">ternary plots</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Mixing_binary_mixtures_with_a_common_component_to_form_ternary_mixtures">Mixing binary mixtures with a common component to form ternary mixtures</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=5" title="Edit section: Mixing binary mixtures with a common component to form ternary mixtures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mixing binary mixtures with a common component gives a ternary mixture with certain mixing ratios between the three components. These mixing ratios from the ternary and the corresponding mole fractions of the ternary mixture x<sub>1(123)</sub>, x<sub>2(123)</sub>, x<sub>3(123)</sub> can be expressed as a function of several mixing ratios involved, the mixing ratios between the components of the binary mixtures and the mixing ratio of the binary mixtures to form the ternary one. </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_{1(123)}={\frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo stretchy="false">(</mo> <mn>123</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo stretchy="false">(</mo> <mn>13</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>13</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1(123)}={\frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883de982d0b3d9fcd720524c6b1503b30dcf1b22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.34ex; height:6.343ex;" alt="{\displaystyle x_{1(123)}={\frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Mole_percentage">Mole percentage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=6" title="Edit section: Mole percentage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Multiplying mole fraction by 100 gives the mole percentage, also referred as amount/amount percent [abbreviated as (n/n)% or mol %]. </p> <div class="mw-heading mw-heading3"><h3 id="Mass_concentration">Mass concentration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=7" title="Edit section: Mass concentration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The conversion to and from <a href="/wiki/Mass_concentration_(chemistry)" title="Mass concentration (chemistry)">mass concentration</a> <i>ρ<sub>i</sub></i> is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x_{i}&={\frac {\rho _{i}}{\rho }}{\frac {\bar {M}}{M_{i}}}\\[3pt]\Leftrightarrow \rho _{i}&=x_{i}\rho {\frac {M_{i}}{\bar {M}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇔</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{i}&={\frac {\rho _{i}}{\rho }}{\frac {\bar {M}}{M_{i}}}\\[3pt]\Leftrightarrow \rho _{i}&=x_{i}\rho {\frac {M_{i}}{\bar {M}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822cf1892820c32533fad4d27991f30ff9b57fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:16.042ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}x_{i}&={\frac {\rho _{i}}{\rho }}{\frac {\bar {M}}{M_{i}}}\\[3pt]\Leftrightarrow \rho _{i}&=x_{i}\rho {\frac {M_{i}}{\bar {M}}}\end{aligned}}}"></span></dd></dl> <p>where <i>M̄</i> is the average molar mass of the mixture. </p> <div class="mw-heading mw-heading3"><h3 id="Molar_concentration">Molar concentration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=8" title="Edit section: Molar concentration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The conversion to <a href="/wiki/Molar_concentration" title="Molar concentration">molar concentration</a> <i>c<sub>i</sub></i> is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c_{i}&=x_{i}c\\[3pt]&={\frac {x_{i}\rho }{\bar {M}}}={\frac {x_{i}\rho }{\sum _{j}x_{j}M_{j}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>c</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ρ</mi> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c_{i}&=x_{i}c\\[3pt]&={\frac {x_{i}\rho }{\bar {M}}}={\frac {x_{i}\rho }{\sum _{j}x_{j}M_{j}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0b7eefd89c9164755e2df84abc56c0333dd2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.043ex; margin-bottom: -0.295ex; width:22.912ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}c_{i}&=x_{i}c\\[3pt]&={\frac {x_{i}\rho }{\bar {M}}}={\frac {x_{i}\rho }{\sum _{j}x_{j}M_{j}}}\end{aligned}}}"></span></dd></dl> <p>where <i>M̄</i> is the average molar mass of the solution, <i>c</i> is the total molar concentration and <i>ρ</i> is the <a href="/wiki/Density" title="Density">density</a> of the solution. </p> <div class="mw-heading mw-heading3"><h3 id="Mass_and_molar_mass">Mass and molar mass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=9" title="Edit section: Mass and molar mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mole fraction can be calculated from the <a href="/wiki/Mass" title="Mass">masses</a> <i>m<sub>i</sub></i> and <a href="/wiki/Molar_mass" title="Molar mass">molar masses</a> <i>M<sub>i</sub></i> of the components: </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_{i}={\frac {\frac {m_{i}}{M_{i}}}{\sum _{j}{\frac {m_{j}}{M_{j}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}={\frac {\frac {m_{i}}{M_{i}}}{\sum _{j}{\frac {m_{j}}{M_{j}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a92c3642d5b07ff4001e865fe7a7709ff40bbd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:12.959ex; height:9.009ex;" alt="{\displaystyle x_{i}={\frac {\frac {m_{i}}{M_{i}}}{\sum _{j}{\frac {m_{j}}{M_{j}}}}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Spatial_variation_and_gradient">Spatial variation and gradient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mole_fraction&action=edit&section=10" title="Edit section: Spatial variation and gradient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Inhomogeneous" class="mw-redirect" title="Inhomogeneous">spatially non-uniform</a> mixture, the mole fraction <a href="/wiki/Gradient" title="Gradient">gradient</a> triggers the phenomenon of <a href="/wiki/Diffusion" title="Diffusion">diffusion</a>. </p> <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=Mole_fraction&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-goldbook-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-goldbook_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-goldbook_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-goldbook_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">IUPAC</a>, <i><a href="/wiki/IUPAC_books#Gold_Book" class="mw-redirect" title="IUPAC books">Compendium of Chemical Terminology</a></i>, 2nd ed. (the "Gold Book") (1997). 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Hoboken, N.J.: Wiley. p. 357. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-58711-9" title="Special:BookSources/978-0-470-58711-9"><bdi>978-0-470-58711-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemistry%3A+Structure+and+Dynamics&rft.place=Hoboken%2C+N.J.&rft.pages=357&rft.edition=5th&rft.pub=Wiley&rft.date=2010&rft.isbn=978-0-470-58711-9&rft.aulast=Rickard&rft.aufirst=James+N.&rft.au=Spencer%2C+George+M.&rft.au=Bodner%2C+Lyman+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMole+fraction" class="Z3988"></span></span> </li> <li id="cite_note-ISO-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-ISO_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ISO_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.iso.org/standard/64979.html">"ISO 80000-9:2019 Quantities and units — Part 9: Physical chemistry and molecular physics"</a>. <i>ISO</i>. 2013-08-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ISO&rft.atitle=ISO+80000-9%3A2019+Quantities+and+units+%E2%80%94+Part+9%3A+Physical+chemistry+and+molecular+physics&rft.date=2013-08-20&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F64979.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMole+fraction" class="Z3988"></span></span> </li> <li id="cite_note-BIPM_c402-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-BIPM_c402_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.bipm.org/en/publications/si-brochure">"SI Brochure"</a>. <i>BIPM</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=BIPM&rft.atitle=SI+Brochure&rft_id=https%3A%2F%2Fwww.bipm.org%2Fen%2Fpublications%2Fsi-brochure&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMole+fraction" class="Z3988"></span></span> </li> <li id="cite_note-NIST-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-NIST_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-NIST_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompsonTaylor2009" class="citation web cs1">Thompson, A.; Taylor, B. N. (2 July 2009). <a rel="nofollow" class="external text" href="https://www.nist.gov/pml/pubs/sp811/index.cfm">"The NIST Guide for the use of the International System of Units"</a>. National Institute of Standards and Technology<span class="reference-accessdate">. Retrieved <span class="nowrap">5 July</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+NIST+Guide+for+the+use+of+the+International+System+of+Units&rft.pub=National+Institute+of+Standards+and+Technology&rft.date=2009-07-02&rft.aulast=Thompson&rft.aufirst=A.&rft.au=Taylor%2C+B.+N.&rft_id=https%3A%2F%2Fwww.nist.gov%2Fpml%2Fpubs%2Fsp811%2Findex.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMole+fraction" class="Z3988"></span></span> </li> </ol></div></div> <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 li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Mole_concepts" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mole_concepts" title="Template:Mole concepts"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mole_concepts" title="Template talk:Mole concepts"><abbr title="Discuss this 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style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mass_concentration_(chemistry)" title="Mass concentration (chemistry)">Mass concentration</a></li> <li><a href="/wiki/Molar_concentration" title="Molar concentration">Molar concentration</a></li> <li><a href="/wiki/Molality" title="Molality">Molality</a></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></li> <li><a href="/wiki/Density" title="Density">Density</a></li> <li><a class="mw-selflink selflink">Mole fraction</a></li> <li><a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">Mass fraction</a></li> <li><a href="/wiki/Amount_of_substance" title="Amount of substance">Amount of substance</a></li> <li><a href="/wiki/Molar_mass" title="Molar mass">Molar mass</a></li> <li><a href="/wiki/Atomic_mass" title="Atomic mass">Atomic mass</a></li> <li><a href="/wiki/Particle_number" title="Particle number">Particle number</a></li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a></li> <li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Thermodynamic temperature</a></li> <li><a href="/wiki/Molar_volume" title="Molar volume">Molar volume</a></li> <li><a href="/wiki/Specific_volume" title="Specific volume">Specific volume</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Laws</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Avogadro%27s_law" title="Avogadro's law">Avogadro's law</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</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" aria-labelledby="Chemical_solutions" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Chemical_solutions" title="Template:Chemical solutions"><abbr title="View this 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href="/wiki/Solid_solution" title="Solid solution">Solid solution</a></li> <li><a href="/wiki/Buffer_solution" title="Buffer solution">Buffer solution</a></li> <li><a href="/wiki/Flory%E2%80%93Huggins_solution_theory" title="Flory–Huggins solution theory">Flory–Huggins</a></li> <li><a href="/wiki/Mixture" title="Mixture">Mixture</a></li> <li><a href="/wiki/Suspension_(chemistry)" title="Suspension (chemistry)">Suspension</a></li> <li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li> <li><a href="/wiki/Phase_diagram" title="Phase diagram">Phase diagram</a></li> <li><a href="/wiki/Phase_separation" title="Phase separation">Phase separation</a></li> <li><a href="/wiki/Eutectic_system" title="Eutectic system">Eutectic point</a></li> <li><a href="/wiki/Alloy" title="Alloy">Alloy</a></li> <li><a href="/wiki/Saturation_concentration" class="mw-redirect" title="Saturation concentration">Saturation</a></li> <li><a href="/wiki/Supersaturation" title="Supersaturation">Supersaturation</a></li> <li><a href="/wiki/Serial_dilution" title="Serial dilution">Serial dilution</a></li> <li><a href="/wiki/Dilution_(equation)" title="Dilution (equation)">Dilution (equation)</a></li> <li><a href="/wiki/Apparent_molar_property" title="Apparent molar property">Apparent molar property</a></li> <li><a href="/wiki/Miscibility_gap" title="Miscibility gap">Miscibility gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concentration" title="Concentration">Concentration</a><br />and related quantities</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Molar_concentration" title="Molar concentration">Molar concentration</a></li> <li><a href="/wiki/Mass_concentration_(chemistry)" title="Mass concentration (chemistry)">Mass concentration</a></li> <li><a href="/wiki/Number_density" title="Number density">Number concentration</a></li> <li><a href="/wiki/Volume_fraction" title="Volume fraction">Volume concentration</a></li> <li><a href="/wiki/Equivalent_concentration" title="Equivalent concentration">Normality</a></li> <li><a href="/wiki/Molality" title="Molality">Molality</a></li> <li><a class="mw-selflink selflink">Mole fraction</a></li> <li><a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">Mass fraction</a></li> <li><a href="/wiki/Natural_abundance" title="Natural abundance">Isotopic abundance</a></li> <li><a href="/wiki/Mixing_ratio" title="Mixing ratio">Mixing ratio</a></li> <li><a href="/wiki/Ternary_plot" title="Ternary plot">Ternary plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Solubility" title="Solubility">Solubility</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solubility_equilibrium" title="Solubility equilibrium">Solubility equilibrium</a></li> <li><a href="/wiki/Total_dissolved_solids" title="Total dissolved solids">Total dissolved solids</a></li> <li><a href="/wiki/Solvation" title="Solvation">Solvation</a></li> <li><a href="/wiki/Solvation_shell" title="Solvation shell">Solvation shell</a></li> <li><a href="/wiki/Enthalpy_change_of_solution" title="Enthalpy change of solution">Enthalpy of solution</a></li> <li><a href="/wiki/Lattice_energy" title="Lattice energy">Lattice energy</a></li> <li><a href="/wiki/Raoult%27s_law" title="Raoult's law">Raoult's law</a></li> <li><a href="/wiki/Henry%27s_law" title="Henry's law">Henry's law</a></li> <li><a href="/wiki/Solubility_table" title="Solubility table">Solubility table (data)</a></li> <li><a href="/wiki/Solubility_chart" title="Solubility chart">Solubility chart</a></li> <li><a href="/wiki/Miscibility" title="Miscibility">Miscibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Solvent" title="Solvent">Solvent</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>(<a href="/wiki/Category:Solvents" title="Category:Solvents">Category</a>)</li> <li><a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">Acid dissociation constant</a></li> <li><a href="/wiki/Protic_solvent" title="Protic solvent">Protic solvent</a></li> <li><a href="/wiki/Polar_aprotic_solvent" title="Polar aprotic solvent">Polar aprotic solvent</a></li> <li><a href="/wiki/Inorganic_nonaqueous_solvent" title="Inorganic nonaqueous solvent">Inorganic nonaqueous solvent</a></li> <li><a href="/wiki/Solvation" title="Solvation">Solvation</a></li> <li><a href="/wiki/List_of_boiling_and_freezing_information_of_solvents" title="List of boiling and freezing information of solvents">List of boiling and freezing information of solvents</a></li> <li><a href="/wiki/Partition_coefficient" title="Partition coefficient">Partition coefficient</a></li> <li><a href="/wiki/Chemical_polarity" title="Chemical polarity">Polarity</a></li> <li><a href="/wiki/Hydrophobe" title="Hydrophobe">Hydrophobe</a></li> <li><a href="/wiki/Hydrophile" title="Hydrophile">Hydrophile</a></li> <li><a href="/wiki/Lipophilicity" title="Lipophilicity">Lipophilic</a></li> <li><a href="/wiki/Amphiphile" title="Amphiphile">Amphiphile</a></li> <li><a href="/wiki/Lyonium_ion" title="Lyonium ion">Lyonium ion</a></li> <li><a href="/wiki/Lyate_ion" title="Lyate ion">Lyate ion</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" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist 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Mole fraction - Wikipedia