Micron Document
<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Widom insertion method</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Widom_insertion_method"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Widom_insertion_method rootpage-Widom_insertion_method skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Widom insertion method</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">
<p>The <b>Widom insertion method</b> is a <a href="Statistical_thermodynamics" class="mw-redirect" title="Statistical thermodynamics">statistical thermodynamic</a> approach to the calculation of material and mixture properties. It is named for <a href="Benjamin_Widom" title="Benjamin Widom">Benjamin Widom</a>, who derived it in 1963.<sup id="cite_ref-Widom_1-0" class="reference"><a href="#cite_note-Widom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In general, there are two theoretical approaches to determining the statistical mechanical properties of materials. The first is the direct calculation of the overall <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> of the system, which directly yields the system free energy. The second approach, known as the Widom insertion method, instead derives from calculations centering on one molecule. The Widom insertion method directly yields the chemical potential of one component rather than the system free energy. This approach is most widely applied in molecular computer simulations<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but has also been applied in the development of analytical statistical mechanical models. The Widom insertion method can be understood as an application of the <a href="Jarzynski_equality" title="Jarzynski equality">Jarzynski equality</a> since it measures the excess free energy difference via the average work needed to perform, when changing the system from a state with N molecules to a state with N+1 molecules.<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> Therefore it measures the excess chemical potential since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{excess}}={\frac {\Delta F_{\text{excess}}}{\Delta N}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>excess</mtext>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>excess</mtext>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>N</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{\text{excess}}={\frac {\Delta F_{\text{excess}}}{\Delta N}}}</annotation>
</semantics>
</math></span><img src="./c24d83e70ee36bcf02205c1f3086ceca475ceef1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.939ex; height:5.509ex;" alt="{\displaystyle \mu _{\text{excess}}={\frac {\Delta F_{\text{excess}}}{\Delta N}}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta N=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>N</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta N=1}</annotation>
</semantics>
</math></span><img src="./f3a82310ccca61471e74205f42729c1ba29baaea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.26ex; height:2.176ex;" alt="{\displaystyle \Delta N=1}" loading="lazy"></span>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>
<p>As originally formulated by <a href="Benjamin_Widom" title="Benjamin Widom">Benjamin Widom</a> in 1963,<sup id="cite_ref-Widom_1-1" class="reference"><a href="#cite_note-Widom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the approach can be summarized by the equation:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{i}={\frac {\rho _{i}}{a_{i}}}=\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>=</mo>
<mrow>
<mo>⟨</mo>
<mrow>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>⟩</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{i}={\frac {\rho _{i}}{a_{i}}}=\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }</annotation>
</semantics>
</math></span><img src="./2eadddb4e390016665747c266c09e024340a3603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.194ex; height:6.176ex;" alt="{\displaystyle \mathbf {B} _{i}={\frac {\rho _{i}}{a_{i}}}=\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{i}}</annotation>
</semantics>
</math></span><img src="./45861d41f9c8d6d2dab3875cb170d485a633bf2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.701ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} _{i}}" loading="lazy"></span> is called the <i>insertion parameter</i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{i}}</annotation>
</semantics>
</math></span><img src="./1f9e51d2ad9cab8f8a5d962502ddf4189b713f7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \rho _{i}}" loading="lazy"></span> is the number density of species <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>
</semantics>
</math></span><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"></span> is the <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a> of species <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{B}}</annotation>
</semantics>
</math></span><img src="./70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.509ex;" alt="{\displaystyle k_{B}}" loading="lazy"></span> is the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> is temperature, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>
</semantics>
</math></span><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> is the interaction energy of an inserted particle with all other particles in the system. The average is over all possible insertions. This can be understood conceptually as fixing the location of all molecules in the system and then inserting a particle of species <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> at all locations through the system, averaging over a <a href="Boltzmann_factor" class="mw-redirect" title="Boltzmann factor">Boltzmann factor</a> in its interaction energy over all of those locations.
</p><p>Note that in other ensembles like for example in the semi-grand canonical ensemble the Widom insertion method works with modified formulas.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_thermodynamic_quantities">Relation to other thermodynamic quantities</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Chemical_potential">Chemical potential</h3></div>
<p>From the above equation and from the definition of activity, the insertion parameter may be related to the <a href="Chemical_potential" title="Chemical potential">chemical potential</a> by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}=-k_{B}T\ln \left({\frac {\mathbf {B} _{i}}{\rho _{i}\lambda ^{3}}}\right)=\underbrace {k_{B}T\ln(\rho _{i}\lambda ^{3})} _{\mu _{id}}\underbrace {-k_{B}T\ln \left(\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle \right)} _{\mu _{ex}}=\mu _{id}+\mu _{ex}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>ρ<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>d</mi>
</mrow>
</msub>
</mrow>
</munder>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<mo>−<!-- − --></mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>⟨</mo>
<mrow>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>⟩</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
<mi>x</mi>
</mrow>
</msub>
</mrow>
</munder>
<mo>=</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>d</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{i}=-k_{B}T\ln \left({\frac {\mathbf {B} _{i}}{\rho _{i}\lambda ^{3}}}\right)=\underbrace {k_{B}T\ln(\rho _{i}\lambda ^{3})} _{\mu _{id}}\underbrace {-k_{B}T\ln \left(\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle \right)} _{\mu _{ex}}=\mu _{id}+\mu _{ex}}</annotation>
</semantics>
</math></span><img src="./6b4e470e4cc0d9f05ae586d21fbbeaaf483763ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:80.057ex; height:9.843ex;" alt="{\displaystyle \mu _{i}=-k_{B}T\ln \left({\frac {\mathbf {B} _{i}}{\rho _{i}\lambda ^{3}}}\right)=\underbrace {k_{B}T\ln(\rho _{i}\lambda ^{3})} _{\mu _{id}}\underbrace {-k_{B}T\ln \left(\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle \right)} _{\mu _{ex}}=\mu _{id}+\mu _{ex}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Equation_of_state">Equation of state</h3></div>
<p>The pressure-temperature-density relation, or <a href="Equation_of_state" title="Equation of state">equation of state</a> of a mixture is related to the insertion parameter via
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z={\frac {P}{\rho k_{B}T}}=1-\ln \mathbf {B} +{\frac {1}{\rho }}\int \limits _{0}^{\rho }\ln \mathbf {B} \,d\rho '}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>P</mi>
<mrow>
<mi>ρ<!-- ρ --></mi>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ρ<!-- ρ --></mi>
</mfrac>
</mrow>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>ρ<!-- ρ --></mi>
</mrow>
</munderover>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<msup>
<mi>ρ<!-- ρ --></mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z={\frac {P}{\rho k_{B}T}}=1-\ln \mathbf {B} +{\frac {1}{\rho }}\int \limits _{0}^{\rho }\ln \mathbf {B} \,d\rho '}</annotation>
</semantics>
</math></span><img src="./e02f17f5e1c35ce353fb2a19dbb55f9d1058c44c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:38.036ex; height:9.009ex;" alt="{\displaystyle Z={\frac {P}{\rho k_{B}T}}=1-\ln \mathbf {B} +{\frac {1}{\rho }}\int \limits _{0}^{\rho }\ln \mathbf {B} \,d\rho '}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
</semantics>
</math></span><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> is the <a href="Compressibility_factor" title="Compressibility factor">compressibility factor</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>
</semantics>
</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> is the overall number density of the mixture, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \mathbf {B} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln \mathbf {B} }</annotation>
</semantics>
</math></span><img src="./f39469b53f7ad1f1e21512288838558bd0fb87f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.227ex; height:2.176ex;" alt="{\displaystyle \ln \mathbf {B} }" loading="lazy"></span> is a mole-fraction weighted average over all mixture components:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \mathbf {B} =\sum _{i}{x_{i}\ln \mathbf {B} _{i}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln \mathbf {B} =\sum _{i}{x_{i}\ln \mathbf {B} _{i}}}</annotation>
</semantics>
</math></span><img src="./43018f58666a5fb2a150d3979e82b98935c4123d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.612ex; height:5.509ex;" alt="{\displaystyle \ln \mathbf {B} =\sum _{i}{x_{i}\ln \mathbf {B} _{i}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Hard_core_model">Hard core model</h2></div>
<p>In the case of a 'hard core' repulsive model in which each molecule or atom consists of a hard core with an infinite repulsive potential, insertions in which two molecules occupy the same space will not contribute to the average. In this case the insertion parameter becomes
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>n</mi>
<mi>s</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mrow>
<mo>⟨</mo>
<mrow>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>⟩</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }</annotation>
</semantics>
</math></span><img src="./dee2a9bd1397210e640310c79599aea88de3167a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.026ex; height:6.176ex;" alt="{\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{ins,i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>n</mi>
<mi>s</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{ins,i}}</annotation>
</semantics>
</math></span><img src="./2d3729dab3e165dd1bc7c778e971e2fdda7964f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.408ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{ins,i}}" loading="lazy"></span> is the probability that the randomly inserted molecule of species <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> will experience an attractive or zero net interaction; in other words, it is the probability that the inserted molecule does not 'overlap' with any other molecules.
</p>
<div class="mw-heading mw-heading2"><h2 id="Mean_field_approximation">Mean field approximation</h2></div>
<p>The above is simplified further via the application of the <a href="Mean_field_theory" class="mw-redirect" title="Mean field theory">mean field approximation</a>, which essentially ignores fluctuations and treats all quantities by their average value. Within this framework the insertion factor is given as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\exp \left(-{\frac {\left\langle \psi _{i}\right\rangle }{k_{B}T}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>n</mi>
<mi>s</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>⟨</mo>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⟩</mo>
</mrow>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mi>T</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\exp \left(-{\frac {\left\langle \psi _{i}\right\rangle }{k_{B}T}}\right)}</annotation>
</semantics>
</math></span><img src="./21c6866b26f94d009483f27f6900f0e7dfcd9aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.54ex; height:6.343ex;" alt="{\displaystyle \mathbf {B} _{i}=\mathbf {P} _{ins,i}\exp \left(-{\frac {\left\langle \psi _{i}\right\rangle }{k_{B}T}}\right)}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */


.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}


/* end https://en.wikipedia.org/ */
</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-Widom-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Widom_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Widom_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Widom, B, "Some Topics in the Theory of Fluids", <i>J. Chem. Phys.</i>, <b>1963</b>, 39(11), 2808-2812.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Binder, K. "Applications of Monte Carlo Methods to Statistical Physics," <i>Rep. Prog. Phys.</i>, <b>1997</b>,60,487-559.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Dullens, RPA, et al., <a rel="nofollow" class="external autonumber" href="http://fcc.chem.uu.nl/PDFpub/1512.pdf">[1]</a>, <i>Mol. Phys.</i>, <b>2005</b>, 103, 3195-3200.</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"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */


.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}


/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFKärgerRuthvenTheodorou2012" class="citation book cs1">Kärger, Jörg; Ruthven, Douglas M.; Theodorou, Doros N. (2012-04-16). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qoL-rQvjQwQC&amp;q=Jarzynski+equality+and+widom+insertion&amp;pg=PA219"><i>Diffusion in Nanoporous Materials</i></a>. p.&nbsp;219. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-3527651290</bdi>.</cite></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFKofkeGlandt1988" class="citation journal cs1">Kofke, David A.; Glandt, Eduardo D. (1988-08-20). "Monte Carlo simulation of multicomponent equilibria in a semigrand canonical ensemble". <i>Molecular Physics</i>. <b>64</b> (6): <span class="nowrap">1105–</span>1131. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988MolPh..64.1105K">1988MolPh..64.1105K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00268978800100743">10.1080/00268978800100743</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&nbsp;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-8976">0026-8976</a>.</cite></span>
</li>
</ol></div></div></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2022-10-18" href="https://en.wikipedia.org/wiki/?title=Widom_insertion_method&amp;oldid=1116726713">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>

</body></html>