Fugacity - Wikipedia

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class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Temperature and pressure dependence</span> </div> </a> <ul id="toc-Temperature_and_pressure_dependence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Measurement</span> </div> </a> <ul id="toc-Measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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mw-first-heading"><span class="mw-page-title-main">Fugacity</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. 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href="https://ca.wikipedia.org/wiki/Fugacitat" title="Fugacitat – Catalan" lang="ca" hreflang="ca" data-title="Fugacitat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fugacita" title="Fugacita – Czech" lang="cs" hreflang="cs" data-title="Fugacita" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fugazit%C3%A4t" title="Fugazität – German" lang="de" hreflang="de" data-title="Fugazität" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fugacidad" title="Fugacidad – Spanish" lang="es" hreflang="es" data-title="Fugacidad" 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/%D8%A8%DB%8C%E2%80%8C%D8%AF%D9%88%D8%A7%D9%85%DB%8C" title="بی‌دوامی – Persian" lang="fa" hreflang="fa" data-title="بی‌دوامی" 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/Fugacit%C3%A9" title="Fugacité – French" lang="fr" hreflang="fr" data-title="Fugacité" 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/%ED%93%A8%EA%B0%80%EC%8B%9C%ED%8B%B0" title="퓨가시티 – Korean" lang="ko" hreflang="ko" data-title="퓨가시티" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fugasitas" title="Fugasitas – Indonesian" lang="id" hreflang="id" data-title="Fugasitas" 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/Fugacit%C3%A0" title="Fugacità – Italian" lang="it" hreflang="it" data-title="Fugacità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fugacit%C3%A1s" title="Fugacitás – Hungarian" lang="hu" hreflang="hu" data-title="Fugacitás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fugaciteit" title="Fugaciteit – Dutch" lang="nl" hreflang="nl" data-title="Fugaciteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%AC%E3%82%B7%E3%83%86%E3%82%A3%E3%83%BC" title="フガシティー – Japanese" lang="ja" hreflang="ja" data-title="フガシティー" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Lotno%C5%9B%C4%87_(gaz)" title="Lotność (gaz) – Polish" lang="pl" hreflang="pl" data-title="Lotność (gaz)" 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/Fugacidade" title="Fugacidade – Portuguese" lang="pt" hreflang="pt" data-title="Fugacidade" 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/Fugacitate" title="Fugacitate – Romanian" lang="ro" hreflang="ro" data-title="Fugacitate" 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%9B%D0%B5%D1%82%D1%83%D1%87%D0%B5%D1%81%D1%82%D1%8C" title="Летучесть – Russian" lang="ru" hreflang="ru" data-title="Летучесть" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fugacity" title="Fugacity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fugacity" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Fugacita" title="Fugacita – Slovak" lang="sk" hreflang="sk" data-title="Fugacita" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fugacitet" title="Fugacitet – Swedish" lang="sv" hreflang="sv" data-title="Fugacitet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%B3%D1%96%D1%82%D0%B8%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Фугітивність – Ukrainian" lang="uk" hreflang="uk" data-title="Фугітивність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%B8%E5%BA%A6" title="逸度 – Chinese" lang="zh" hreflang="zh" data-title="逸度" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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</div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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gas</a> is an effective <a href="/wiki/Partial_pressure" title="Partial pressure">partial pressure</a> which replaces the mechanical partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> which has the same temperature and molar <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> as the real gas.<sup id="cite_ref-AtkinsDePaula_1-0" class="reference"><a href="#cite_note-AtkinsDePaula-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Fugacities are determined experimentally or estimated from various models such as a <a href="/wiki/Van_der_Waals_gas" class="mw-redirect" title="Van der Waals gas">Van der Waals gas</a> that are closer to reality than an ideal gas. The real gas pressure and fugacity are related through the dimensionless <b>fugacity coefficient</b><sup id="cite_ref-AtkinsDePaula_1-1" class="reference"><a href="#cite_note-AtkinsDePaula-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\frac {f}{P}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>P</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\frac {f}{P}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08365c72f0207a779b3ad9b67bdf1b48d6b38249" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.847ex; height:5.343ex;" alt="{\displaystyle \varphi ={\frac {f}{P}}.}"></span> </p><p>For an ideal gas, fugacity and pressure are equal, and so <span class="texhtml"><i>φ</i> = 1</span>. Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to <span class="texhtml"><i>RT</i> ln <i>φ</i></span>. </p><p>The fugacity is closely related to the <a href="/wiki/Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">thermodynamic activity</a>. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the <a href="/wiki/Standard_state" title="Standard state">standard state</a> and normally chosen as 1 <a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atmosphere</a> or 1 <a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a>. </p><p>Accurate calculations of <a href="/wiki/Chemical_equilibrium" title="Chemical equilibrium">chemical equilibrium</a> for real gases should use the fugacity rather than the pressure. The thermodynamic condition for chemical equilibrium is that the total chemical potential of reactants is equal to that of products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar <a href="/wiki/Reaction_quotient" title="Reaction quotient">reaction quotient</a> form (or <a href="/wiki/Law_of_mass_action" title="Law of mass action">law of mass action</a>) except that the pressures are replaced by fugacities. </p><p>For a condensed phase (liquid or solid) in equilibrium with its vapor phase, the chemical potential is equal to that of the vapor, and therefore the fugacity is equal to the fugacity of the vapor. This fugacity is approximately equal to the <a href="/wiki/Vapor_pressure" title="Vapor pressure">vapor pressure</a> when the vapor pressure is not too high. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Pure_substance">Pure substance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=1" title="Edit section: Pure substance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fugacity is closely related to the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> <span class="texhtml"><i>μ</i></span>. In a pure substance, <span class="texhtml"><i>μ</i></span> is equal to the <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs energy</a> <span class="texhtml"><i>G</i><sub>m</sub></span> for a <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> of the substance,<sup id="cite_ref-Ott_2-0" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 207">: 207 </span></sup> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu =dG_{\mathrm {m} }=-S_{\mathrm {m} }dT+V_{\mathrm {m} }dP,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <mi>d</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>d</mi> <mi>T</mi> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>d</mi> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mu =dG_{\mathrm {m} }=-S_{\mathrm {m} }dT+V_{\mathrm {m} }dP,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec760ac7d007e3084bbd09b80bfc746c20602e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.549ex; height:2.676ex;" alt="{\displaystyle d\mu =dG_{\mathrm {m} }=-S_{\mathrm {m} }dT+V_{\mathrm {m} }dP,}"></span> where <span class="texhtml"><i>T</i></span> and <span class="texhtml"><i>P</i></span> are the temperature and pressure, <span class="texhtml"><i>V</i><sub>m</sub></span> is the <a href="/wiki/Molar_volume" title="Molar volume">volume per mole</a> and <span class="texhtml"><i>S</i><sub>m</sub></span> is the <a href="/wiki/Entropy" title="Entropy">entropy</a> per mole.<sup id="cite_ref-Ott_2-1" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 248">: 248 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Gas">Gas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=2" title="Edit section: Gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> the <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {m} }^{\text{ideal}}={\frac {RT}{P}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ideal</mtext> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {m} }^{\text{ideal}}={\frac {RT}{P}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b5d8fc671d272a7ff1c10c4ea6f6a1a74cf0a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.512ex; height:5.176ex;" alt="{\displaystyle V_{\mathrm {m} }^{\text{ideal}}={\frac {RT}{P}},}"></span> where <span class="texhtml"><i>R</i></span> is the <a href="/wiki/Ideal_gas_constant" class="mw-redirect" title="Ideal gas constant">ideal gas constant</a>. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., <span class="texhtml">d<i>T</i> = 0</span>) is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu =V_{\mathrm {m} }dP=RT\,{\frac {dP}{P}}=RT\,d\ln P,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>d</mi> <mi>P</mi> <mo>=</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>=</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mu =V_{\mathrm {m} }dP=RT\,{\frac {dP}{P}}=RT\,d\ln P,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d499b5566d03009ba139ad25ebe4c9f9f16e81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.524ex; height:5.343ex;" alt="{\displaystyle d\mu =V_{\mathrm {m} }dP=RT\,{\frac {dP}{P}}=RT\,d\ln P,}"></span>where <b>ln p</b> is the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> of p. </p><p>For real gases the equation of state will depart from the simpler one, and the result above derived for an ideal gas will only be a good approximation provided that (a) the typical size of the molecule is negligible compared to the average distance between the individual molecules, and (b) the short range behavior of the inter-molecular potential can be neglected, i.e., when the molecules can be considered to rebound elastically off each other during molecular collisions. In other words, real gases behave like ideal gases at low pressures and high temperatures.<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> At moderately high pressures, attractive interactions between molecules reduce the pressure compared to the ideal gas law; and at very high pressures, the sizes of the molecules are no longer negligible and repulsive forces between molecules increases the pressure. At low temperatures, molecules are more likely to stick together instead of rebounding elastically.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The ideal gas law can still be used to describe the behavior of a <a href="/wiki/Real_gas" title="Real gas">real gas</a> if the pressure is replaced by a <i>fugacity</i> <span class="texhtml"><i>f</i></span>, defined so that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu =RT\,d\ln f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mu =RT\,d\ln f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f821d40ddfedbda902dbcf6bbb8cb4559ebc2ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.711ex; height:2.676ex;" alt="{\displaystyle d\mu =RT\,d\ln f}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{P\to 0}{\frac {f}{P}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>P</mi> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{P\to 0}{\frac {f}{P}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abfc071d6c804d34e85980a1c788c8f60e559023" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.576ex; height:5.509ex;" alt="{\displaystyle \lim _{P\to 0}{\frac {f}{P}}=1.}"></span> That is, at low pressures <span class="texhtml"><i>f</i></span> is the same as the pressure, so it has the same units as pressure. The ratio <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\frac {f}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\frac {f}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282dea9f4a1471d394a8056b21aacccbe1c92ae4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.2ex; height:5.343ex;" alt="{\displaystyle \varphi ={\frac {f}{P}}}"></span> is called the <i>fugacity coefficient</i>.<sup id="cite_ref-Ott_2-2" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 248–249">: 248–249 </span></sup> </p><p>If a reference state is denoted by a zero superscript, then integrating the equation for the chemical potential gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu -\mu ^{0}=RT\,\ln {\frac {f}{P^{0}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>−</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu -\mu ^{0}=RT\,\ln {\frac {f}{P^{0}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6d5547150e20194929e362005e74ae571cbaab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.656ex; height:5.509ex;" alt="{\displaystyle \mu -\mu ^{0}=RT\,\ln {\frac {f}{P^{0}}},}"></span> Note this can also be expressed with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=f/P^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=f/P^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512cb025b673e05c2869d550ef64bc63ccb3a7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.645ex; height:3.176ex;" alt="{\displaystyle a=f/P^{0}}"></span>, a dimensionless quantity, called the <i><a href="/wiki/Thermodynamic_activity" title="Thermodynamic activity">activity</a></i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup> </p><p><b>Numerical example:</b> <a href="/wiki/Nitrogen" title="Nitrogen">Nitrogen</a> gas (N<sub>2</sub>) at 0 °C and a pressure of <span class="texhtml"><i>P</i> = 100</span> <a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atmospheres</a> (atm) has a fugacity of <span class="texhtml"><i>f</i> = 97.03</span> atm.<sup id="cite_ref-AtkinsDePaula_1-2" class="reference"><a href="#cite_note-AtkinsDePaula-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at <span class="nowrap">97.03 atm</span>. The fugacity coefficient is <span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">97.03 atm</span><span class="sr-only">/</span><span class="den">100 atm</span></span>⁠</span> = 0.9703</span>. </p><p>The contribution of nonideality to the molar Gibbs energy of a real gas is equal to <span class="texhtml"><i>RT</i> ln <i>φ</i></span>. For nitrogen at 100 atm, <span class="texhtml"><i>G</i><sub>m</sub> = <i>G</i><sub>m,id</sub> + <i>RT</i> ln 0.9703</span>, which is less than the ideal value <span class="texhtml"><i>G</i><sub>m,id</sub></span> because of intermolecular attractive forces. Finally, the activity is just <span class="texhtml">97.03</span> without units. </p> <div class="mw-heading mw-heading3"><h3 id="Condensed_phase">Condensed phase</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=3" title="Edit section: Condensed phase"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">Vapor–liquid equilibrium</a></div> <p>The fugacity of a condensed phase (liquid or solid) is defined the same way as for a gas: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu _{\mathrm {c} }=RT\,d\ln f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mu _{\mathrm {c} }=RT\,d\ln f_{\mathrm {c} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e697b22efbf2fba8b8a9a3e97023d2dfa944a690" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.497ex; height:2.676ex;" alt="{\displaystyle d\mu _{\mathrm {c} }=RT\,d\ln f_{\mathrm {c} }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{P\to 0}{\frac {f_{\mathrm {c} }}{P}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>P</mi> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{P\to 0}{\frac {f_{\mathrm {c} }}{P}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c047fdf62e9454c5f6f579b99b5b87383c39c9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.932ex; height:5.509ex;" alt="{\displaystyle \lim _{P\to 0}{\frac {f_{\mathrm {c} }}{P}}=1.}"></span> It is difficult to measure fugacity in a condensed phase directly; but if the condensed phase is <i>saturated</i> (in equilibrium with the vapor phase), the chemical potentials of the two phases are equal (<span class="texhtml"><i>μ</i><sub>c</sub> = <i>μ</i><sub>g</sub></span>). Combined with the above definition, this implies that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }=f_{\mathrm {g} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }=f_{\mathrm {g} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6afb51a1750542b92fe33374a953c6d6f563b500" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.04ex; height:2.843ex;" alt="{\displaystyle f_{\mathrm {c} }=f_{\mathrm {g} }.}"></span> </p><p>When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. At constant temperature, the change in fugacity as the pressure goes from the saturation press <span class="texhtml"><i>P</i><sub>sat</sub></span> to <span class="texhtml mvar" style="font-style:italic;">P</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\frac {f}{f_{\mathrm {sat} }}}={\frac {V_{\mathrm {m} }}{RT}}\int _{P_{\mathrm {sat} }}^{P}dp={\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <mi>d</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mi>P</mi> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {\frac {f}{f_{\mathrm {sat} }}}={\frac {V_{\mathrm {m} }}{RT}}\int _{P_{\mathrm {sat} }}^{P}dp={\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eadced8c0908ddc58f7662f06c5dd40ce0608484" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.58ex; height:6.509ex;" alt="{\displaystyle \ln {\frac {f}{f_{\mathrm {sat} }}}={\frac {V_{\mathrm {m} }}{RT}}\int _{P_{\mathrm {sat} }}^{P}dp={\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}.}"></span> This fraction is known as the <a href="/wiki/Poynting_effect" title="Poynting effect">Poynting factor</a>. Using <span class="texhtml"><i>f</i><sub>sat</sub> = <i>φ</i><sub>sat</sub> <i>p</i><sub>sat</sub></span>, where <span class="texhtml"><i>φ</i><sub>sat</sub></span> is the fugacity coefficient, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\varphi _{\mathrm {sat} }P_{\mathrm {sat} }\exp \left({\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mi>P</mi> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\varphi _{\mathrm {sat} }P_{\mathrm {sat} }\exp \left({\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff551a7da4c52412f6d1905382be9ae0fbefd6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.32ex; height:6.343ex;" alt="{\displaystyle f=\varphi _{\mathrm {sat} }P_{\mathrm {sat} }\exp \left({\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}\right).}"></span> This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1.<sup id="cite_ref-Matsoukas_6-0" class="reference"><a href="#cite_note-Matsoukas-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 345–346">: 345–346 </span></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Unless pressures are very high, the Poynting factor is usually small and the exponential term is near 1. Frequently, the fugacity of the pure liquid is used as a reference state when defining and using mixture activity coefficients. </p> <div class="mw-heading mw-heading2"><h2 id="Mixture">Mixture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=4" title="Edit section: Mixture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fugacity is most useful in mixtures. It does not add any new information compared to the chemical potential, but it has computational advantages. As the molar fraction of a component goes to zero, the chemical potential diverges but the fugacity goes to zero. In addition, there are natural reference states for fugacity (for example, an ideal gas makes a natural reference state for gas mixtures since the fugacity and pressure converge at low pressure).<sup id="cite_ref-O'Connell_8-0" class="reference"><a href="#cite_note-O'Connell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 141">: 141 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Gases">Gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=5" title="Edit section: Gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a mixture of gases, the fugacity of each component <span class="texhtml"><i>i</i></span> has a similar definition, with <a href="/wiki/Partial_molar_property" title="Partial molar property">partial molar quantities</a> instead of molar quantities (e.g., <span class="texhtml"><i>G</i><sub><i>i</i></sub></span> instead of <span class="texhtml"><i>G</i><sub>m</sub></span> and <span class="texhtml"><i>V</i><sub><i>i</i></sub></span> instead of <span class="texhtml"><i>V</i><sub>m</sub></span>):<sup id="cite_ref-Ott_2-3" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 262">: 262 </span></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dG_{i}=RT\,d\ln f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dG_{i}=RT\,d\ln f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8ae20f7a3d32eafc088d2bef04647b69b6458b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.596ex; height:2.509ex;" alt="{\displaystyle dG_{i}=RT\,d\ln f_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{P\to 0}{\frac {f_{i}}{P_{i}}}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{P\to 0}{\frac {f_{i}}{P_{i}}}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2664ce02647b101ef08e9f482f496d6ba7fb79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.122ex; height:5.843ex;" alt="{\displaystyle \lim _{P\to 0}{\frac {f_{i}}{P_{i}}}=1,}"></span> where <span class="texhtml"><i>P<sub>i</sub></i></span> is the <a href="/wiki/Partial_pressure" title="Partial pressure">partial pressure</a> of component <span class="texhtml"><i>i</i></span>. The partial pressures obey <a href="/wiki/Dalton%27s_law" title="Dalton's law">Dalton's law</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}=y_{i}P,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}=y_{i}P,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d47a7acb683f41bd7cae5b72cf3a32e074a417b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.722ex; height:2.509ex;" alt="{\displaystyle P_{i}=y_{i}P,}"></span> where <span class="texhtml mvar" style="font-style:italic;">P</span> is the total pressure and <span class="texhtml"><i>y<sub>i</sub></i></span> is the mole fraction of the component (so the partial pressures add up to the total pressure). The fugacities commonly obey a similar law called the Lewis and Randall rule: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}=y_{i}f_{i}^{*},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}=y_{i}f_{i}^{*},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0050f5e1f35b854dbee2f6d8d5e5bae314565566" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.998ex; height:2.843ex;" alt="{\displaystyle f_{i}=y_{i}f_{i}^{*},}"></span> where <span class="texhtml"><i>f</i>*<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>i</i></sub></span></span></span> is the fugacity that component <span class="texhtml"><i>i</i></span> would have if the entire gas had that composition at the same temperature and pressure. Both laws are expressions of an assumption that the gases behave independently.<sup id="cite_ref-Ott_2-4" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 264–265">: 264–265 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Liquids">Liquids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=6" title="Edit section: Liquids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a liquid mixture, the fugacity of each component is equal to that of a vapor component in equilibrium with the liquid. In an <a href="/wiki/Ideal_solution" title="Ideal solution">ideal solution</a>, the fugacities obey the <a href="/w/index.php?title=Lewis-Randall_rule&action=edit&redlink=1" class="new" title="Lewis-Randall rule (page does not exist)">Lewis-Randall rule</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}=x_{i}f_{i}^{*},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</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> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}=x_{i}f_{i}^{*},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4da681cf5b8885865ee63d426089c5326e6377" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.188ex; height:2.843ex;" alt="{\displaystyle f_{i}=x_{i}f_{i}^{*},}"></span> where <span class="texhtml"><i>x<sub>i</sub></i></span> is the mole fraction in the liquid and <span class="texhtml"><i>f</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">∗</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>i</i></sub></span></span></span> is the fugacity of the pure liquid phase. This is a good approximation when the component molecules have similar size, shape and polarity.<sup id="cite_ref-Ott_2-5" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 264, 269–270">: 264, 269–270 </span></sup> </p><p>In a dilute solution with two components, the component with the larger molar fraction (the <a href="/wiki/Solvent" title="Solvent">solvent</a>) may still obey <a href="/wiki/Raoult%27s_law" title="Raoult's law">Raoult's law</a> even if the other component (the <a href="/wiki/Solute" class="mw-redirect" title="Solute">solute</a>) has different properties. That is because its molecules experience essentially the same environment that they do in the absence of the solute. By contrast, each solute molecule is surrounded by solvent molecules, so it obeys a different law known as <a href="/wiki/Henry%27s_law" title="Henry's law">Henry's law</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 171">: 171 </span></sup> By Henry's law, the fugacity of the solute is proportional to its concentration. The constant of proportionality (a measured Henry's constant) depends on whether the concentration is represented by the mole fraction, <a href="/wiki/Molality" title="Molality">molality</a> or <a href="/wiki/Molarity" class="mw-redirect" title="Molarity">molarity</a>.<sup id="cite_ref-Ott_2-6" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 274">: 274 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Temperature_and_pressure_dependence">Temperature and pressure dependence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=7" title="Edit section: Temperature and pressure dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The pressure dependence of fugacity (at constant temperature) is given by<sup id="cite_ref-Ott_2-7" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 260">: 260 </span></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial \ln f}{\partial P}}\right)_{T}={\frac {V_{\mathrm {m} }}{RT}}}"> <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> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \ln f}{\partial P}}\right)_{T}={\frac {V_{\mathrm {m} }}{RT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3abf1113b67e294ea59398a240ba483b6da5047" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.292ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial \ln f}{\partial P}}\right)_{T}={\frac {V_{\mathrm {m} }}{RT}}}"></span> and is always positive.<sup id="cite_ref-Ott_2-8" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 260">: 260 </span></sup> </p><p>The temperature dependence at constant pressure is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial \ln f}{\partial T}}\right)_{P}={\frac {\Delta H_{\mathrm {m} }}{RT^{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> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>R</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \ln f}{\partial T}}\right)_{P}={\frac {\Delta H_{\mathrm {m} }}{RT^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906f7ed7150a220414961aa71476d3d5f0f11686" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.084ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial \ln f}{\partial T}}\right)_{P}={\frac {\Delta H_{\mathrm {m} }}{RT^{2}}},}"></span> where <span class="texhtml">Δ<i>H</i><sub>m</sub></span> is the change in molar <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> as the gas expands, liquid vaporizes, or solid sublimates into a vacuum.<sup id="cite_ref-Ott_2-9" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 262">: 262 </span></sup> Also, if the pressure is <span class="texhtml"><i>P</i><sup>0</sup></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial \left(T\ln {\frac {f}{P^{0}}}\right)}{\partial T}}\right)_{P}=-{\frac {S_{\mathrm {m} }}{R}}<0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mi>R</mi> </mfrac> </mrow> <mo><</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \left(T\ln {\frac {f}{P^{0}}}\right)}{\partial T}}\right)_{P}=-{\frac {S_{\mathrm {m} }}{R}}<0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cc9f588612c5d8e34ab0a4db49de512b48901e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:31.832ex; height:10.176ex;" alt="{\displaystyle \left({\frac {\partial \left(T\ln {\frac {f}{P^{0}}}\right)}{\partial T}}\right)_{P}=-{\frac {S_{\mathrm {m} }}{R}}<0.}"></span> Since the temperature and entropy are positive, <span class="texhtml">ln <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>f</i></span><span class="sr-only">/</span><span class="den"><i>P</i><sup>0</sup></span></span>⁠</span></span> decreases with increasing temperature.<sup id="cite_ref-Franses_10-0" class="reference"><a href="#cite_note-Franses-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Measurement">Measurement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=8" title="Edit section: Measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. In that case, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \varphi ={\frac {1}{RT}}\int _{0}^{p}\left(V_{m}-V_{\mathrm {m} }^{\mathrm {ideal} }\right)dP.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \varphi ={\frac {1}{RT}}\int _{0}^{p}\left(V_{m}-V_{\mathrm {m} }^{\mathrm {ideal} }\right)dP.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/218cdb27713f51b3305c5727f50320eae79f1711" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.005ex; height:5.843ex;" alt="{\displaystyle \ln \varphi ={\frac {1}{RT}}\int _{0}^{p}\left(V_{m}-V_{\mathrm {m} }^{\mathrm {ideal} }\right)dP.}"></span> This integral can also be calculated using an equation of state.<sup id="cite_ref-Ott_2-10" class="reference"><a href="#cite_note-Ott-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 251–252">: 251–252 </span></sup> </p><p>The integral can be recast in an alternative form using the <a href="/wiki/Compressibility_factor" title="Compressibility factor">compressibility factor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z={\frac {PV_{\mathrm {m} }}{RT}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z={\frac {PV_{\mathrm {m} }}{RT}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1decfe722799576fa3e7c9560d4df2c2dfb1dc30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.964ex; height:5.343ex;" alt="{\displaystyle Z={\frac {PV_{\mathrm {m} }}{RT}}.}"></span> Then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \varphi =\int _{0}^{P}\left({\frac {Z-1}{P}}\right)dP.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \varphi =\int _{0}^{P}\left({\frac {Z-1}{P}}\right)dP.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8743fb5a1e85edb8f4334fb7154727057f395eb8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.199ex; height:6.343ex;" alt="{\displaystyle \ln \varphi =\int _{0}^{P}\left({\frac {Z-1}{P}}\right)dP.}"></span> This is useful because of the <a href="/wiki/Theorem_of_corresponding_states" title="Theorem of corresponding states">theorem of corresponding states</a>: If the pressure and temperature at the <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical point</a> of the gas are <span class="texhtml"><i>P</i><sub>c</sub></span> and <span class="texhtml"><i>T</i><sub>c</sub></span>, we can define <a href="/wiki/Reduced_properties" title="Reduced properties">reduced properties</a> <span class="texhtml"><i>P</i><sub>r</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>P</i></span><span class="sr-only">/</span><span class="den"><i>P</i><sub>c</sub></span></span>⁠</span></span> and <span class="texhtml"><i>T</i><sub>r</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>T</i></span><span class="sr-only">/</span><span class="den"><i>T</i><sub>c</sub></span></span>⁠</span></span>. Then, to a good approximation, most gases have the same value of <span class="texhtml"><i>Z</i></span> for the same reduced temperature and pressure. However, in <a href="/wiki/Geochemistry" title="Geochemistry">geochemical</a> applications, this principle ceases to be accurate at pressures where <a href="/wiki/Metamorphism" title="Metamorphism">metamorphism</a> occurs.<sup id="cite_ref-Anderson_11-0" class="reference"><a href="#cite_note-Anderson-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 247">: 247 </span></sup> </p><p>For a gas obeying the <a href="/wiki/Van_der_Waals_equation" title="Van der Waals equation">van der Waals equation</a>, the explicit formula for the fugacity coefficient is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RT\ln \varphi ={\frac {RTb}{V_{\mathrm {m} }-b}}-{\frac {2a}{V_{\mathrm {m} }}}-RT\ln \left(1-{\frac {a(V_{\mathrm {m} }-b)}{RTV_{\mathrm {m} }^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> <mi>b</mi> </mrow> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>R</mi> <mi>T</mi> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RT\ln \varphi ={\frac {RTb}{V_{\mathrm {m} }-b}}-{\frac {2a}{V_{\mathrm {m} }}}-RT\ln \left(1-{\frac {a(V_{\mathrm {m} }-b)}{RTV_{\mathrm {m} }^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56b188158812d32928c0c092b211d2fce4ed196" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.656ex; height:6.343ex;" alt="{\displaystyle RT\ln \varphi ={\frac {RTb}{V_{\mathrm {m} }-b}}-{\frac {2a}{V_{\mathrm {m} }}}-RT\ln \left(1-{\frac {a(V_{\mathrm {m} }-b)}{RTV_{\mathrm {m} }^{2}}}\right)}"></span> This formula is based on the molar volume. Since the pressure and the molar volume are related through the equation of state; a typical procedure would be to choose a volume, calculate the corresponding pressure, and then evaluate the right-hand side of the equation.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=9" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>fugacity</i> is derived from the Latin <i>fugere</i>, to flee. In the sense of an "escaping tendency", it was introduced to thermodynamics in 1901 by the American chemist <a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Gilbert N. Lewis</a> and popularized in an influential textbook by Lewis and <a href="/wiki/Merle_Randall" title="Merle Randall">Merle Randall</a>, <i>Thermodynamics and the Free Energy of Chemical Substances</i>, in 1923.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The "escaping tendency" referred to the flow of matter between phases and played a similar role to that of temperature in heat flow.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Anderson2017_15-0" class="reference"><a href="#cite_note-Anderson2017-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 177">: 177 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Electrochemical_potential" title="Electrochemical potential">Electrochemical potential</a></li> <li><a href="/wiki/Excess_chemical_potential" title="Excess chemical potential">Excess chemical potential</a></li> <li><a href="/wiki/Fugacity_capacity" title="Fugacity capacity">Fugacity capacity</a></li> <li><a href="/wiki/Multimedia_fugacity_model" title="Multimedia fugacity model">Multimedia fugacity model</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&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 mw-references-columns"><ol class="references"> <li id="cite_note-AtkinsDePaula-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-AtkinsDePaula_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-AtkinsDePaula_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-AtkinsDePaula_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAtkinsDe_Paula2006" class="citation book cs1">Atkins, Peter; De Paula, Julio (2006). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/atkinsphysicalch00pwat"><i>Atkins' Physical Chemistry</i></a></span> (8th ed.). W. H. Freeman. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/atkinsphysicalch00pwat/page/111">111–2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780716787594" title="Special:BookSources/9780716787594"><bdi>9780716787594</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atkins%27+Physical+Chemistry&rft.pages=111-2&rft.edition=8th&rft.pub=W.+H.+Freeman&rft.date=2006&rft.isbn=9780716787594&rft.aulast=Atkins&rft.aufirst=Peter&rft.au=De+Paula%2C+Julio&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fatkinsphysicalch00pwat&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> <li id="cite_note-Ott-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ott_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ott_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Ott_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Ott_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Ott_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Ott_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Ott_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Ott_2-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Ott_2-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Ott_2-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-Ott_2-10"><sup><i><b>k</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOttBoerio-Goates2000" class="citation book cs1">Ott, J. 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Bellmont, CA: Brooks/Cole, CENGAGE Learning. p. <a rel="nofollow" class="external text" href="https://archive.org/details/chemistryatomsfi00zumd/page/n332">309</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780840065322" title="Special:BookSources/9780840065322"><bdi>9780840065322</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+an+atoms+first+approach&rft.place=Bellmont%2C+CA&rft.pages=309&rft.pub=Brooks%2FCole%2C+CENGAGE+Learning&rft.date=2012&rft.isbn=9780840065322&rft.aulast=Zumdahl&rft.aufirst=Steven+S.&rft.au=Zumdahl%2C+Susan+A&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fchemistryatomsfi00zumd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClugstonFlemming2000" class="citation book cs1">Clugston, Michael; Flemming, Rosalind (2000). <i>Advanced chemistry</i>. 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P.; Haile, J. M. (2005). <i>Thermodynamics: Fundamentals for Applications</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781139443173" title="Special:BookSources/9781139443173"><bdi>9781139443173</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics%3A+Fundamentals+for+Applications&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=9781139443173&rft.aulast=O%27Connell&rft.aufirst=J.+P.&rft.au=Haile%2C+J.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkinsPaula2002" class="citation book cs1">Atkins, Peter; Paula, Julio de (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/physicalchemistr0000atki"><i>Physical chemistry</i></a></span> (7th ed.). New York: W.H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780716735397" title="Special:BookSources/9780716735397"><bdi>9780716735397</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+chemistry&rft.place=New+York&rft.edition=7th&rft.pub=W.H.+Freeman&rft.date=2002&rft.isbn=9780716735397&rft.aulast=Atkins&rft.aufirst=Peter&rft.au=Paula%2C+Julio+de&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicalchemistr0000atki&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> <li id="cite_note-Franses-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Franses_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranses2014" class="citation book cs1">Franses, Elias I. (2014). <i>Thermodynamics with chemical engineering applications</i>. Cambridge University Press. p. 248. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107069756" title="Special:BookSources/9781107069756"><bdi>9781107069756</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+with+chemical+engineering+applications&rft.pages=248&rft.pub=Cambridge+University+Press&rft.date=2014&rft.isbn=9781107069756&rft.aulast=Franses&rft.aufirst=Elias+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span> Note that Equations 9.24 and 9.25 left out <span class="texhtml"><i>p</i><sup>0</sup></span> in substituting from Equation 9.6. This error is corrected in the above equation.</span> </li> <li id="cite_note-Anderson-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Anderson_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonCrerar1993" class="citation book cs1">Anderson, Greg M.; Crerar, David A. (1993). <i>Thermodynamics in Geochemistry: The Equilibrium Model</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780195345094" title="Special:BookSources/9780195345094"><bdi>9780195345094</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+in+Geochemistry%3A+The+Equilibrium+Model&rft.pub=Oxford+University+Press&rft.date=1993&rft.isbn=9780195345094&rft.aulast=Anderson&rft.aufirst=Greg+M.&rft.au=Crerar%2C+David+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid2015" class="citation journal cs1">David, Carl W. (2015). <a rel="nofollow" class="external text" href="http://digitalcommons.uconn.edu/chem_educ/91">"Fugacity Examples 2: The fugacity of a "hard-sphere" semi-ideal gas and the van der Waals gas"</a>. <i>Chemistry Education Materials</i>. <b>91</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemistry+Education+Materials&rft.atitle=Fugacity+Examples+2%3A+The+fugacity+of+a+%22hard-sphere%22+semi-ideal+gas+and+the+van+der+Waals+gas&rft.volume=91&rft.date=2015&rft.aulast=David&rft.aufirst=Carl+W.&rft_id=http%3A%2F%2Fdigitalcommons.uconn.edu%2Fchem_educ%2F91&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLewis1901" class="citation journal cs1">Lewis, Gilbert Newton (June 1901). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=njp.32101050586062;view=1up;seq=57">"The law of physico-chemical change"</a>. <i>Proceedings of the American Academy of Arts and Sciences</i>. <b>37</b> (3): <span class="nowrap">49–</span>69. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F20021635">10.2307/20021635</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20021635">20021635</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=The+law+of+physico-chemical+change&rft.volume=37&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E49-%3C%2Fspan%3E69&rft.date=1901-06&rft_id=info%3Adoi%2F10.2307%2F20021635&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20021635%23id-name%3DJSTOR&rft.aulast=Lewis&rft.aufirst=Gilbert+Newton&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dnjp.32101050586062%3Bview%3D1up%3Bseq%3D57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span> ; the term "fugacity" is coined on p. 54.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLewis1900" class="citation journal cs1">Lewis, Gilbert Newton (1900). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=njp.32101050586054;view=1up;seq=165">"A new conception of thermal pressure and a theory of solutions"</a>. <i>Proceedings of the American Academy of Arts and Sciences</i>. <b>36</b> (9): <span class="nowrap">145–</span>168. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F20020988">10.2307/20020988</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20020988">20020988</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=A+new+conception+of+thermal+pressure+and+a+theory+of+solutions&rft.volume=36&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E145-%3C%2Fspan%3E168&rft.date=1900&rft_id=info%3Adoi%2F10.2307%2F20020988&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20020988%23id-name%3DJSTOR&rft.aulast=Lewis&rft.aufirst=Gilbert+Newton&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dnjp.32101050586054%3Bview%3D1up%3Bseq%3D165&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span> The term "escaping tendency" is introduced on p. 148, where it is represented by the Greek letter <span class="texhtml mvar" style="font-style:italic;">ψ</span>; <span class="texhtml mvar" style="font-style:italic;">ψ</span> is defined for ideal gases on p. 156.</span> </li> <li id="cite_note-Anderson2017-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Anderson2017_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson2017" class="citation book cs1">Anderson, Greg (2017). <i>Thermodynamics of Natural Systems: Theory and Applications in Geochemistry and Environmental Science</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107175211" title="Special:BookSources/9781107175211"><bdi>9781107175211</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+of+Natural+Systems%3A+Theory+and+Applications+in+Geochemistry+and+Environmental+Science&rft.pub=Cambridge+University+Press&rft.date=2017&rft.isbn=9781107175211&rft.aulast=Anderson&rft.aufirst=Greg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonCrerar1993" class="citation book cs1">Anderson, Greg M.; Crerar, David A. (1993). <i>Thermodynamics in Geochemistry: The Equilibrium Model</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780195345094" title="Special:BookSources/9780195345094"><bdi>9780195345094</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+in+Geochemistry%3A+The+Equilibrium+Model&rft.pub=Oxford+University+Press&rft.date=1993&rft.isbn=9780195345094&rft.aulast=Anderson&rft.aufirst=Greg+M.&rft.au=Crerar%2C+David+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMackay2011" class="citation book cs1">Mackay, Don (2011). "The fugacity approach to mass transport and MTCs". In Thibodeaux, Louis J.; Mackay, Donald (eds.). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/handbookchemical00thib"><i>Handbook of chemical mass transport in the environment</i></a></span>. Boca Raton, FL: CRC Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/handbookchemical00thib/page/n61">43</a>–50. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781420047561" title="Special:BookSources/9781420047561"><bdi>9781420047561</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+fugacity+approach+to+mass+transport+and+MTCs&rft.btitle=Handbook+of+chemical+mass+transport+in+the+environment&rft.place=Boca+Raton%2C+FL&rft.pages=43-50&rft.pub=CRC+Press&rft.date=2011&rft.isbn=9781420047561&rft.aulast=Mackay&rft.aufirst=Don&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbookchemical00thib&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMackayArnot2011" class="citation journal cs1">Mackay, Don; Arnot, Jon A. (14 April 2011). "The Application of Fugacity and Activity to Simulating the Environmental Fate of Organic Contaminants". <i>Journal of Chemical & Engineering Data</i>. <b>56</b> (4): <span class="nowrap">1348–</span>1355. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fje101158y">10.1021/je101158y</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F143615">2027.42/143615</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Chemical+%26+Engineering+Data&rft.atitle=The+Application+of+Fugacity+and+Activity+to+Simulating+the+Environmental+Fate+of+Organic+Contaminants&rft.volume=56&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1348-%3C%2Fspan%3E1355&rft.date=2011-04-14&rft_id=info%3Ahdl%2F2027.42%2F143615&rft_id=info%3Adoi%2F10.1021%2Fje101158y&rft.aulast=Mackay&rft.aufirst=Don&rft.au=Arnot%2C+Jon+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFugacity" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Video_lectures">Video lectures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fugacity&action=edit&section=14" title="Edit section: Video lectures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Thermodynamics, University of Colorado-Boulder, 2011 <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=fkqmf_QR4Yg">Introduction to fugacity: Where did it come from?</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=AMBoLiQJMFQ">What is fugacity?</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=OZFB48wVtjw">What is fugacity in mixtures?</a></li></ul></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Fugacity&oldid=1256082509">https://en.wikipedia.org/w/index.php?title=Fugacity&oldid=1256082509</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Physical_chemistry" title="Category:Physical chemistry">Physical chemistry</a></li><li><a href="/wiki/Category:Chemical_thermodynamics" title="Category:Chemical thermodynamics">Chemical thermodynamics</a></li><li><a href="/wiki/Category:Thermodynamic_properties" title="Category:Thermodynamic properties">Thermodynamic properties</a></li><li><a href="/wiki/Category:State_functions" title="Category:State functions">State functions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 November 2024, at 03:07<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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