Equilibrium chemistry

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class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Multiple equilibria</span> </div> </a> <ul id="toc-Multiple_equilibria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Speciation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Speciation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Speciation</span> </div> </a> <ul id="toc-Speciation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Determination" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determination"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Determination</span> </div> </a> <ul id="toc-Determination-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Acid–base_equilibria" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Acid–base_equilibria"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Acid–base equilibria</span> </div> </a> <ul id="toc-Acid–base_equilibria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Host–guest_equilibria" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Host–guest_equilibria"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Host–guest equilibria</span> </div> </a> <ul id="toc-Host–guest_equilibria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complexes_of_metals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complexes_of_metals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Complexes of metals</span> </div> </a> <ul id="toc-Complexes_of_metals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Redox_equilibrium" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Redox_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Redox equilibrium</span> </div> </a> <ul id="toc-Redox_equilibrium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solubility" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solubility"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Solubility</span> </div> </a> <ul id="toc-Solubility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Partition"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Partition</span> </div> </a> <ul id="toc-Partition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chromatography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Chromatography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Chromatography</span> </div> </a> <ul id="toc-Chromatography-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" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Subdiscipline of chemistry concerned with chemical equilibrium</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/Chemical_equilibrium" title="Chemical equilibrium">Chemical equilibrium</a></div> <p><b>Equilibrium chemistry</b> is concerned with systems in <a href="/wiki/Chemical_equilibrium" title="Chemical equilibrium">chemical equilibrium</a>. The unifying principle is that the <a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to the <a href="/wiki/Reaction_coordinate" title="Reaction coordinate">reaction coordinate</a> is zero.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> This principle, applied to mixtures at equilibrium provides a definition of an <a href="/wiki/Equilibrium_constant" title="Equilibrium constant">equilibrium constant</a>. Applications include <a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">acid–base</a>, <a href="/wiki/Host%E2%80%93guest_chemistry" title="Host–guest chemistry">host–guest</a>, <a href="/wiki/Stability_constants_of_complexes" title="Stability constants of complexes">metal–complex</a>, <a href="/wiki/Solubility" title="Solubility">solubility</a>, <a href="/wiki/Partition_coefficient" title="Partition coefficient">partition</a>, <a href="/wiki/Chromatography" title="Chromatography">chromatography</a> and <a href="/wiki/Redox" title="Redox">redox</a> equilibria. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Thermodynamic_equilibrium">Thermodynamic equilibrium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=1" title="Edit section: Thermodynamic equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Dynamic_equilibrium" title="Dynamic equilibrium">Dynamic equilibrium</a> and <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></div> <p>A chemical system is said to be in equilibrium when the quantities of the chemical entities involved do not and <i>cannot</i> change in time without the application of an external influence. In this sense a system in chemical equilibrium is in a <a href="/wiki/Stable" title="Stable">stable</a> state. The system at <a href="/wiki/Chemical_equilibrium" title="Chemical equilibrium">chemical equilibrium</a> will be at a constant temperature, pressure or volume and a composition. It will be insulated from exchange of heat with the surroundings, that is, it is a <a href="/wiki/Closed_system" title="Closed system">closed system</a>. A change of temperature, pressure (or volume) constitutes an external influence and the equilibrium quantities will change as a result of such a change. If there is a possibility that the composition might change, but the rate of change is negligibly slow, the system is said to be in a <a href="/wiki/Metastable" class="mw-redirect" title="Metastable">metastable</a> state. The equation of chemical equilibrium can be expressed symbolically as </p> <dl><dd>reactant(s) ⇌ product(s)</dd></dl> <p>The sign ⇌ means "are in equilibrium with". This definition refers to <a href="/wiki/Macroscopic" class="mw-redirect" title="Macroscopic">macroscopic</a> properties. Changes do occur at the microscopic level of atoms and molecules, but to such a minute extent that they are not measurable and in a balanced way so that the macroscopic quantities do not change. Chemical equilibrium is a dynamic state in which forward and backward reactions proceed at such rates that the macroscopic composition of the mixture is constant. Thus, equilibrium sign ⇌ symbolizes the fact that reactions occur in both forward <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 \rightharpoonup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇀</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightharpoonup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912c85b375886509f9bf323bab01cd1d3d0b96c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.532ex; margin-bottom: -0.703ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \rightharpoonup }"></span> and backward <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 \leftharpoondown }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↽</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftharpoondown }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8572e40b7a21ea071eac94f81458238d9f096ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.176ex;" alt="{\displaystyle \leftharpoondown }"></span> directions. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Diag_eq.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Diag_eq.svg/250px-Diag_eq.svg.png" decoding="async" width="250" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Diag_eq.svg/375px-Diag_eq.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Diag_eq.svg/500px-Diag_eq.svg.png 2x" data-file-width="168" data-file-height="190" /></a><figcaption></figcaption></figure> <p>A <a href="/wiki/Steady_state_(chemistry)" title="Steady state (chemistry)">steady state</a>, on the other hand, is not necessarily an equilibrium state in the chemical sense. For example, in a radioactive <a href="/wiki/Decay_chain" title="Decay chain">decay chain</a> the concentrations of intermediate isotopes are constant because the rate of production is equal to the rate of decay. It is not a chemical equilibrium because the decay process occurs in one direction only. </p><p><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a> is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant volume the <a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a> is minimum and for systems at constant pressure the <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> is minimum.<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> Thus a metastable state is one for which the free energy change between reactants and products is not minimal even though the composition does not change in time.<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 existence of this minimum is due to the free energy of mixing of reactants and products being always negative.<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> For <a href="/wiki/Ideal_solution" title="Ideal solution">ideal solutions</a> the <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> of mixing is zero, so the minimum exists because the <a href="/wiki/Entropy" title="Entropy">entropy</a> of mixing is always positive.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><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> The slope of the reaction free energy, δ<i>G</i><sub>r</sub> with respect to the <a href="/wiki/Reaction_coordinate" title="Reaction coordinate">reaction coordinate</a>, <i>ξ</i>, is zero when the free energy is at its minimum value. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta G_{\mathrm {r} }=\left({\frac {\partial G}{\partial \xi }}\right)_{T,P};\quad \delta G_{\mathrm {r} }(\mathrm {Eq} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ξ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>,</mo> <mi>P</mi> </mrow> </msub> <mo>;</mo> <mspace width="1em" /> <mi>δ</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta G_{\mathrm {r} }=\left({\frac {\partial G}{\partial \xi }}\right)_{T,P};\quad \delta G_{\mathrm {r} }(\mathrm {Eq} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8b9186e130203fdf16fabd5a05a5a6e3a2049d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.339ex; height:6.509ex;" alt="{\displaystyle \delta G_{\mathrm {r} }=\left({\frac {\partial G}{\partial \xi }}\right)_{T,P};\quad \delta G_{\mathrm {r} }(\mathrm {Eq} )=0}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Equilibrium_constant">Equilibrium constant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=2" title="Edit section: Equilibrium constant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Equilibrium_constant" title="Equilibrium constant">Equilibrium constant</a>, <a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">Acid dissociation constant</a>, and <a href="/wiki/Stability_constants_of_complexes" title="Stability constants of complexes">Stability constants of complexes</a></div> <p><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> is the partial molar free energy. The potential, <i>μ<sub>i</sub></i>, of the <i>i</i>th species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, <i>N<sub>i</sub></i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>,</mo> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c995be7d49a3bb55b5f0948247714376981617b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.622ex; height:6.509ex;" alt="{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P}}"></span></dd></dl> <p>A general chemical equilibrium can be written as<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j}n_{j}\mathrm {Reactant} _{j}\rightleftharpoons \sum _{k}m_{k}\mathrm {Product} _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo class="MJX-variant" stretchy="false">⇌</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j}n_{j}\mathrm {Reactant} _{j}\rightleftharpoons \sum _{k}m_{k}\mathrm {Product} _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa1f6de7b7f3ac1b48418959b77e5173fd406b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.911ex; height:5.843ex;" alt="{\displaystyle \sum _{j}n_{j}\mathrm {Reactant} _{j}\rightleftharpoons \sum _{k}m_{k}\mathrm {Product} _{k}}"></span></dd></dl> <p><i>n<sub>j</sub></i> are the <a href="/wiki/Stoichiometric_coefficient" class="mw-redirect" title="Stoichiometric coefficient">stoichiometric coefficients</a> of the reactants in the equilibrium equation, and <i>m<sub>j</sub></i> are the coefficients of the products. The value of δ<i>G</i><sub>r</sub> for these reactions is a function of the chemical potentials of all the species. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta G_{\mathrm {r} }=\sum _{k}m_{k}\mu _{k}\,-\sum _{j}n_{j}\mu _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta G_{\mathrm {r} }=\sum _{k}m_{k}\mu _{k}\,-\sum _{j}n_{j}\mu _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/496e31edc9098e977cfbad5c5a90cc94b62d3a73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.798ex; height:5.843ex;" alt="{\displaystyle \delta G_{\mathrm {r} }=\sum _{k}m_{k}\mu _{k}\,-\sum _{j}n_{j}\mu _{j}}"></span></dd></dl> <p>The chemical potential, <i>μ<sub>i</sub></i>, of the <i>i</i>th species can be calculated in terms of its <a href="/wiki/Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a>, <i>a<sub>i</sub></i>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>+</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa099859e0ef3ac187325e038b5ddee76fd4e55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.196ex; height:3.176ex;" alt="{\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}}"></span></dd></dl> <p><i>μ</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"><s>o</s></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>i</i></sub></span></span> is the standard chemical potential of the species, <i>R</i> is the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a> and <i>T</i> is the temperature. Setting the sum for the reactants <i>j</i> to be equal to the sum for the products, <i>k</i>, so that δ<i>G</i><sub>r</sub>(Eq) = 0: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j}n_{j}\left(\mu _{j}^{\ominus }+RT\ln a_{j}\right)=\sum _{k}m_{k}\left(\mu _{k}^{\ominus }+RT\ln a_{k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>+</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>+</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j}n_{j}\left(\mu _{j}^{\ominus }+RT\ln a_{j}\right)=\sum _{k}m_{k}\left(\mu _{k}^{\ominus }+RT\ln a_{k}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e40051568dd32ef3e338dd12e2a1e92d0628ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.887ex; height:6.343ex;" alt="{\displaystyle \sum _{j}n_{j}\left(\mu _{j}^{\ominus }+RT\ln a_{j}\right)=\sum _{k}m_{k}\left(\mu _{k}^{\ominus }+RT\ln a_{k}\right)}"></span></dd></dl> <p>Rearranging the terms, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }=-RT\left(\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }=-RT\left(\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6df8812a48339142f4ffd04408a8310cb24db4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.861ex; height:7.676ex;" alt="{\displaystyle \sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }=-RT\left(\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}}\right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G^{\ominus }=-RT\ln K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=-RT\ln K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3673ccfb4184c15bc5029efd4f403c90831d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.007ex; height:2.676ex;" alt="{\displaystyle \Delta G^{\ominus }=-RT\ln K.}"></span></dd></dl> <p>This relates the <a href="/wiki/Standard_state" title="Standard state">standard</a> Gibbs free energy change, Δ<i>G</i><sup><s>o</s></sup> to an <a href="/wiki/Equilibrium_constant" title="Equilibrium constant">equilibrium constant</a>, <i>K</i>, the <a href="/wiki/Reaction_quotient" title="Reaction quotient">reaction quotient</a> of activity values at equilibrium. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G^{\ominus }=\sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=\sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f215ba0bfd5d909d6718b3b16607f500b2ed7c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.955ex; height:5.843ex;" alt="{\displaystyle \Delta G^{\ominus }=\sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln K=\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}};\quad K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <mo>;</mo> <mspace width="1em" /> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln K=\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}};\quad K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c82c79f41d3f95aa3be0f834cadc0b3ee155b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.273ex; height:7.176ex;" alt="{\displaystyle \ln K=\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}};\quad K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}}"></span></dd></dl> <p>It follows that any equilibrium of this kind can be characterized either by the standard free energy change or by the equilibrium constant. In practice concentrations are more useful than activities. Activities can be calculated from concentrations if the <a href="/wiki/Activity_coefficient" title="Activity coefficient">activity coefficient</a> are known, but this is rarely the case. Sometimes activity coefficients can be calculated using, for example, <a href="/wiki/Pitzer_equations" title="Pitzer equations">Pitzer equations</a> or <a href="/wiki/Specific_ion_interaction_theory" title="Specific ion interaction theory">Specific ion interaction theory</a>. Otherwise conditions must be adjusted so that activity coefficients do not vary much. For ionic solutions this is achieved by using a background ionic medium at a high concentration relative to the concentrations of the species in equilibrium. </p><p>If activity coefficients are unknown they may be subsumed into the equilibrium constant, which becomes a concentration quotient.<sup id="cite_ref-FJCR_9-0" class="reference"><a href="#cite_note-FJCR-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Each activity <i>a<sub>i</sub></i> is assumed to be the product of a concentration, [A<sub><i>i</i></sub>], and an activity coefficient, <i>γ<sub>i</sub></i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=[\mathrm {A} _{i}]\gamma _{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> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=[\mathrm {A} _{i}]\gamma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379843cbc2ab336a96494817622e017bf71227c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.968ex; height:2.843ex;" alt="{\displaystyle a_{i}=[\mathrm {A} _{i}]\gamma _{i}}"></span></dd></dl> <p>This expression for activity is placed in the expression defining the equilibrium constant.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}={\frac {\prod _{k}\left([\mathrm {A} _{k}]\gamma _{k}\right)^{m_{k}}}{\prod _{j}\left([\mathrm {A} _{j}]\gamma _{j}\right)^{n_{j}}}}={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}\times {\frac {\prod _{k}{\gamma _{k}}^{m_{k}}}{\prod _{j}{\gamma _{j}}^{n_{j}}}}={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}\times \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>×</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}={\frac {\prod _{k}\left([\mathrm {A} _{k}]\gamma _{k}\right)^{m_{k}}}{\prod _{j}\left([\mathrm {A} _{j}]\gamma _{j}\right)^{n_{j}}}}={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}\times {\frac {\prod _{k}{\gamma _{k}}^{m_{k}}}{\prod _{j}{\gamma _{j}}^{n_{j}}}}={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}\times \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd6376011cb38551f36701caa14b6a1507f22b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:77.111ex; height:7.176ex;" alt="{\displaystyle K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}={\frac {\prod _{k}\left([\mathrm {A} _{k}]\gamma _{k}\right)^{m_{k}}}{\prod _{j}\left([\mathrm {A} _{j}]\gamma _{j}\right)^{n_{j}}}}={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}\times {\frac {\prod _{k}{\gamma _{k}}^{m_{k}}}{\prod _{j}{\gamma _{j}}^{n_{j}}}}={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}\times \Gamma }"></span></dd></dl> <p>By setting the quotient of activity coefficients, <i>Γ</i>, equal to one,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> the equilibrium constant is defined as a quotient of concentrations. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6635cbfab04ff5f3c01879d12cf4985aa28eeaba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.943ex; height:6.843ex;" alt="{\displaystyle K={\frac {\prod _{k}[\mathrm {A} _{k}]^{m_{k}}}{\prod _{j}[\mathrm {A} _{j}]^{n_{j}}}}}"></span></dd></dl> <p>In more familiar notation, for a general equilibrium </p> <dl><dd><i>α</i> A + <i>β</i> B ... ⇌ <i>σ</i> S + <i>τ</i> T ...</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4742b7a1c6c820a51d80d307d52560eea1dbbe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.15ex; height:6.509ex;" alt="{\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}}"></span></dd></dl> <p>This definition is much more practical, but an equilibrium constant defined in terms of concentrations is dependent on conditions. In particular, equilibrium constants for species in aqueous solution are dependent on <a href="/wiki/Ionic_strength" title="Ionic strength">ionic strength</a>, as the quotient of activity coefficients varies with the ionic strength of the solution. </p><p>The values of the standard free energy change and of the equilibrium constant are temperature dependent. To a first approximation, the <a href="/wiki/Van_%27t_Hoff_equation" title="Van 't Hoff equation">van 't Hoff equation</a> may be used. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\ln K}{dT}}\ ={\frac {\Delta H^{\ominus }}{RT^{2}}}\quad {\mbox{ or }}\quad {\frac {d\ln K}{d{\tfrac {1}{T}}}}\ =-{\frac {\Delta H^{\ominus }}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> </mrow> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> </mrow> <mrow> <mi>R</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> or </mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> </mrow> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> </mrow> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\ln K}{dT}}\ ={\frac {\Delta H^{\ominus }}{RT^{2}}}\quad {\mbox{ or }}\quad {\frac {d\ln K}{d{\tfrac {1}{T}}}}\ =-{\frac {\Delta H^{\ominus }}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a71113975ada64742aba1892aa27dc30b45a732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.482ex; height:7.009ex;" alt="{\displaystyle {\frac {d\ln K}{dT}}\ ={\frac {\Delta H^{\ominus }}{RT^{2}}}\quad {\mbox{ or }}\quad {\frac {d\ln K}{d{\tfrac {1}{T}}}}\ =-{\frac {\Delta H^{\ominus }}{R}}}"></span></dd></dl> <p>This shows that when the reaction is exothermic (Δ<i>H</i><sup><s>o</s></sup>, the standard <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> change, is negative), then <i>K</i> decreases with increasing temperature, in accordance with <a href="/wiki/Le_Ch%C3%A2telier%27s_principle" class="mw-redirect" title="Le Châtelier's principle">Le Châtelier's principle</a>. The approximation involved is that the standard enthalpy change, Δ<i>H</i><sup><s>o</s></sup>, is independent of temperature, which is a good approximation only over a small temperature range. Thermodynamic arguments can be used to show that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial H}{\partial T}}\right)_{p}=C_{p}}"> <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>H</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> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial H}{\partial T}}\right)_{p}=C_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9be61f92b22b5f9151d19894854744fe19060f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.518ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial H}{\partial T}}\right)_{p}=C_{p}}"></span></dd></dl> <p>where <i>C<sub>p</sub></i> is the <a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">heat capacity</a> at constant pressure.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equilibria_involving_gases">Equilibria involving gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=3" title="Edit section: Equilibria involving gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When dealing with gases, <a href="/wiki/Fugacity" title="Fugacity">fugacity</a>, <i>f</i>, is used rather than activity. However, whereas activity is <a href="/wiki/Dimension" title="Dimension">dimensionless</a>, fugacity has the dimension of <a href="/wiki/Pressure" title="Pressure">pressure</a>. A consequence is that chemical potential has to be defined in terms of a standard pressure, <i>p</i><sup><s>o</s></sup>:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu ^{\ominus }+RT\ln {\frac {f}{p^{\ominus }}}}"> <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"> <mo>⊖</mo> </mrow> </msup> <mo>+</mo> <mi>R</mi> <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"> <mo>⊖</mo> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu ^{\ominus }+RT\ln {\frac {f}{p^{\ominus }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f53f724db194085d851782dddaae3e970b014d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.883ex; height:5.843ex;" alt="{\displaystyle \mu =\mu ^{\ominus }+RT\ln {\frac {f}{p^{\ominus }}}}"></span></dd></dl> <p>By convention <i>p</i><sup><s>o</s></sup> is usually taken to be 1 <a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a>. Fugacity can be expressed as the product of <a href="/wiki/Partial_pressure" title="Partial pressure">partial pressure</a>, <i>p</i>, and a fugacity coefficient, <i>Φ</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=p\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>p</mi> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=p\Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa607c475e103037cf0ec2a546a0250addc7dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle f=p\Phi }"></span></dd></dl> <p>Fugacity coefficients are dimensionless and can be obtained experimentally at specific temperature and pressure, from measurements of deviations from <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> behaviour. Equilibrium constants are defined in terms of fugacity. If the gases are at sufficiently low pressure that they behave as ideal gases, the equilibrium constant can be defined as a quotient of partial pressures. </p><p>An example of gas-phase equilibrium is provided by the <a href="/wiki/Haber%E2%80%93Bosch_process" class="mw-redirect" title="Haber–Bosch process">Haber–Bosch process</a> of <a href="/wiki/Ammonia" title="Ammonia">ammonia</a> synthesis. </p> <dl><dd>N<sub>2</sub> + 3 H<sub>2</sub> ⇌ 2 NH<sub>3</sub>;<span class="nowrap">     </span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {{f_{\mathrm {NH_{3}} }}^{2}}{f_{\mathrm {N_{2}} }{f_{\mathrm {H_{2}} }}^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {{f_{\mathrm {NH_{3}} }}^{2}}{f_{\mathrm {N_{2}} }{f_{\mathrm {H_{2}} }}^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e66841298981b4085ab2751020008cf2c206e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.926ex; height:7.343ex;" alt="{\displaystyle K={\frac {{f_{\mathrm {NH_{3}} }}^{2}}{f_{\mathrm {N_{2}} }{f_{\mathrm {H_{2}} }}^{3}}}}"></span></dd></dl> <p>This reaction is strongly <a href="/wiki/Exothermic" class="mw-redirect" title="Exothermic">exothermic</a>, so the equilibrium constant decreases with temperature. However, a temperature of around 400 °C is required in order to achieve a reasonable rate of reaction with currently available <a href="/wiki/Catalyst" class="mw-redirect" title="Catalyst">catalysts</a>. Formation of ammonia is also favoured by high pressure, as the volume decreases when the reaction takes place. The same reaction, <a href="/wiki/Nitrogen_fixation" title="Nitrogen fixation">nitrogen fixation</a>, occurs at ambient temperatures in nature, when the catalyst is an <a href="/wiki/Enzyme" title="Enzyme">enzyme</a> such as <a href="/wiki/Nitrogenase" title="Nitrogenase">nitrogenase</a>. Much energy is needed initially to break the nitrogen–nitrogen triple bond even though the overall reaction is exothermic. </p><p>Gas-phase equilibria occur during <a href="/wiki/Combustion" title="Combustion">combustion</a> and were studied as early as 1943 in connection with the development of the <a href="/wiki/V-2_rocket" title="V-2 rocket">V2</a> <a href="/wiki/Rocket_engine" title="Rocket engine">rocket engine</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The calculation of composition for a gaseous equilibrium at constant pressure is often carried out using ΔG values, rather than equilibrium constants.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_equilibria">Multiple equilibria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=4" title="Edit section: Multiple equilibria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two or more equilibria can exist at the same time. When this is so, equilibrium constants can be ascribed to individual equilibria, but they are not always unique. For example, three equilibrium constants can be defined for a <a href="/wiki/Dibasic_acid" class="mw-redirect" title="Dibasic acid">dibasic</a> <a href="/wiki/Acid" title="Acid">acid</a>, H<sub>2</sub>A.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>A<sup>2−</sup> + H<sup>+</sup> ⇌ HA<sup>−</sup>;<span class="nowrap">     </span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}={\frac {[{\ce {HA-}}]}{[{\ce {H+}}][{\ce {A^2-}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>HA</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>A</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}={\frac {[{\ce {HA-}}]}{[{\ce {H+}}][{\ce {A^2-}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04969fca0ead36cfbbe0b051de8b65cd631e8ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.879ex; height:7.009ex;" alt="{\displaystyle K_{1}={\frac {[{\ce {HA-}}]}{[{\ce {H+}}][{\ce {A^2-}}]}}}"></span></dd> <dd>HA<sup>−</sup> + H<sup>+</sup> ⇌ H<sub>2</sub>A;<span class="nowrap">     </span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}={\frac {[{\ce {H2A}}]}{[{\ce {H+}}][{\ce {HA-}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>A</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>HA</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}={\frac {[{\ce {H2A}}]}{[{\ce {H+}}][{\ce {HA-}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c128c6ab8ee73e4942b4f468c1690c41c1279a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.8ex; height:6.676ex;" alt="{\displaystyle K_{2}={\frac {[{\ce {H2A}}]}{[{\ce {H+}}][{\ce {HA-}}]}}}"></span></dd> <dd>A<sup>2−</sup> + 2 H<sup>+</sup> ⇌ H<sub>2</sub>A;<span class="nowrap">     </span> <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 \beta _{2}={\frac {[{\ce {H2A}}]}{[{\ce {H+}}]^{2}[{\ce {A^2-}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0pt" height="0pt" depth=".2em" /> </mrow> </msubsup> <mtext>A</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>A</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}={\frac {[{\ce {H2A}}]}{[{\ce {H+}}]^{2}[{\ce {A^2-}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de0405541fa63df492bcecbd87f3bf596a07104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.276ex; height:6.843ex;" alt="{\displaystyle \beta _{2}={\frac {[{\ce {H2A}}]}{[{\ce {H+}}]^{2}[{\ce {A^2-}}]}}}"></span></dd></dl> <p>The three constants are not independent of each other and it is easy to see that <span class="texhtml"><i>β</i><sub>2</sub> = <i>K</i><sub>1</sub><i>K</i><sub>2</sub></span>. The constants <i>K</i><sub>1</sub> and <i>K</i><sub>2</sub> are stepwise constants and <i>β</i> is an example of an overall constant. </p> <div class="mw-heading mw-heading3"><h3 id="Speciation">Speciation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=5" title="Edit section: Speciation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Citric_acid_speciation.svg" class="mw-file-description"><img alt="This image plots the relative percentages of the protonation species of citric acid as a function of p H. Citric acid has three ionisable hydrogen atoms and thus three p K A values. Below the lowest p K A, the triply protonated species prevails; between the lowest and middle p K A, the doubly protonated form prevails; between the middle and highest p K A, the singly protonated form prevails; and above the highest p K A, the unprotonated form of citric acid is predominant." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Citric_acid_speciation.svg/220px-Citric_acid_speciation.svg.png" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Citric_acid_speciation.svg/330px-Citric_acid_speciation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Citric_acid_speciation.svg/440px-Citric_acid_speciation.svg.png 2x" data-file-width="723" data-file-height="432" /></a><figcaption>Speciation diagram for a solution of citric acid as a function of pH.</figcaption></figure> <p>The concentrations of species in equilibrium are usually calculated under the assumption that activity coefficients are either known or can be ignored. In this case, each equilibrium constant for the formation of a complex in a set of multiple equilibria can be defined as follows </p> <dl><dd><i>α</i> A + <i>β</i> B ... ⇌ A<sub><i>α</i></sub>B<sub><i>β</i></sub>...;<span class="nowrap">     </span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\alpha \beta \ldots }={\frac {[\mathrm {A} _{\alpha }\mathrm {B} _{\beta }\ldots ]}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }\ldots }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mo>…</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>…</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>…</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\alpha \beta \ldots }={\frac {[\mathrm {A} _{\alpha }\mathrm {B} _{\beta }\ldots ]}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }\ldots }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0b23c8700067003f168eff133b0c77ad6d8ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.604ex; height:6.509ex;" alt="{\displaystyle K_{\alpha \beta \ldots }={\frac {[\mathrm {A} _{\alpha }\mathrm {B} _{\beta }\ldots ]}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }\ldots }}}"></span></dd></dl> <p>The concentrations of species containing reagent A are constrained by a condition of <a href="/wiki/Conservation_of_mass" title="Conservation of mass">mass-balance</a>, that is, the total (or analytical) concentration, which is the sum of all species' concentrations, must be constant. There is one mass-balance equation for each reagent of the type </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {A} }=[\mathrm {A} ]+\sum [\mathrm {A} _{\alpha }\mathrm {B} _{\beta }\ldots ]=[\mathrm {A} ]+\sum \left(\alpha K_{\alpha \beta }\ldots [\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }\ldots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mo>∑</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>…</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mo>∑</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>…</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {A} }=[\mathrm {A} ]+\sum [\mathrm {A} _{\alpha }\mathrm {B} _{\beta }\ldots ]=[\mathrm {A} ]+\sum \left(\alpha K_{\alpha \beta }\ldots [\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }\ldots \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531440648fcc0172c471fa1ea3407fdd37575c66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:60.98ex; height:3.843ex;" alt="{\displaystyle T_{\mathrm {A} }=[\mathrm {A} ]+\sum [\mathrm {A} _{\alpha }\mathrm {B} _{\beta }\ldots ]=[\mathrm {A} ]+\sum \left(\alpha K_{\alpha \beta }\ldots [\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }\ldots \right)}"></span></dd></dl> <p>There are as many mass-balance equations as there are reagents, A, B..., so if the equilibrium constant values are known, there are <i>n</i> mass-balance equations in <i>n</i> unknowns, [A], [B]..., the so-called free reagent concentrations. Solution of these equations gives all the information needed to calculate the concentrations of all the species.<sup id="cite_ref-DJL_19-0" class="reference"><a href="#cite_note-DJL-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Thus, the importance of equilibrium constants lies in the fact that, once their values have been determined by experiment, they can be used to calculate the concentrations, known as the <a href="/wiki/Speciation_of_ions" class="mw-redirect" title="Speciation of ions">speciation</a>, of mixtures that contain the relevant species. </p> <div class="mw-heading mw-heading3"><h3 id="Determination">Determination</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=6" title="Edit section: Determination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Determination_of_equilibrium_constants" title="Determination of equilibrium constants">Determination of equilibrium constants</a></div> <p>There are five main types of experimental data that are used for the determination of solution equilibrium constants. Potentiometric data obtained with a <a href="/wiki/Glass_electrode" title="Glass electrode">glass electrode</a> are the most widely used with aqueous solutions. The others are <a href="/wiki/Spectrophotometry" title="Spectrophotometry">Spectrophotometric</a>, <a href="/wiki/Fluorescence" title="Fluorescence">Fluorescence</a> (luminescence) measurements and <a href="/wiki/NMR" class="mw-redirect" title="NMR">NMR</a> <a href="/wiki/Chemical_shift" title="Chemical shift">chemical shift</a> measurements;<sup id="cite_ref-FJCR_9-1" class="reference"><a href="#cite_note-FJCR-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> simultaneous measurement of <i>K</i> and Δ<i>H</i> for 1:1 adducts in biological systems is routinely carried out using <a href="/wiki/Isothermal_Titration_Calorimetry" class="mw-redirect" title="Isothermal Titration Calorimetry">Isothermal Titration Calorimetry</a>. </p><p>The experimental data will comprise a set of data points. At the i'th data point, the analytical concentrations of the reactants, <i>T</i><sub>A(<i>i</i>)</sub>, <i>T</i><sub>B(<i>i</i>)</sub> etc. will be experimentally known quantities and there will be one or more measured quantities, <i>y<sub>i</sub></i>, that depend in some way on the analytical concentrations and equilibrium constants. A general computational procedure has three main components. </p> <ol><li>Definition of a chemical model of the equilibria. The model consists of a list of reagents, A, B, etc. and the complexes formed from them, with stoichiometries A<sub><i>p</i></sub>B<sub><i>q</i></sub>... Known or estimated values of the equilibrium constants for the formation of all complexes must be supplied.</li> <li>Calculation of the concentrations of all the chemical species in each solution. The free concentrations are calculated by solving the equations of mass-balance, and the concentrations of the complexes are calculated using the equilibrium constant definitions. A quantity corresponding to the observed quantity can then be calculated using physical principles such as the <a href="/wiki/Nernst_equation#Nernst_potential" title="Nernst equation">Nernst potential</a> or <a href="/wiki/Beer-Lambert_law" class="mw-redirect" title="Beer-Lambert law">Beer-Lambert law</a> which relate the calculated quantity to the concentrations of the species.</li> <li>Refinement of the equilibrium constants. Usually a <a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">Non-linear least squares</a> procedure is used. A weighted sum of squares, <i>U</i>, is minimized. <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 U=\sum _{i=np}^{i=1}w_{i}\left(y_{i}^{\mathrm {observed} }-y_{i}^{\mathrm {calculated} }\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\sum _{i=np}^{i=1}w_{i}\left(y_{i}^{\mathrm {observed} }-y_{i}^{\mathrm {calculated} }\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0bded106a76929ea5d6731a2a8e68ea01a743ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.688ex; height:7.676ex;" alt="{\displaystyle U=\sum _{i=np}^{i=1}w_{i}\left(y_{i}^{\mathrm {observed} }-y_{i}^{\mathrm {calculated} }\right)^{2}}"></span> The weights, <i>w<sub>i</sub></i> and quantities <i>y</i> may be vectors. Values of the equilibrium constants are refined in an iterative procedure.<sup id="cite_ref-DJL_19-1" class="reference"><a href="#cite_note-DJL-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading2"><h2 id="Acid–base_equilibria"><span id="Acid.E2.80.93base_equilibria"></span>Acid–base equilibria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=7" title="Edit section: Acid–base equilibria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">Acid dissociation constant</a></div> <p><a href="/wiki/Br%C3%B8nsted%E2%80%93Lowry_acid%E2%80%93base_theory" title="Brønsted–Lowry acid–base theory">Brønsted and Lowry</a> characterized an acid–base equilibrium as involving a proton exchange reaction:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SA_22-0" class="reference"><a href="#cite_note-SA-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>acid + base ⇌ conjugate base + conjugate acid.</dd></dl> <p>An acid is a proton donor; the proton is transferred to the base, a proton acceptor, creating a conjugate acid. For aqueous solutions of an acid HA, the base is water; the conjugate base is A<sup>−</sup> and the conjugate acid is the solvated hydrogen ion. In solution chemistry, it is usual to use H<sup>+</sup> as an abbreviation for the solvated hydrogen ion, regardless of the solvent. In aqueous solution H<sup>+</sup> denotes a <a href="/wiki/Hydronium#Solvation" title="Hydronium">solvated hydronium ion</a>.<sup id="cite_ref-Headrick_24-0" class="reference"><a href="#cite_note-Headrick-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Smiechowski_25-0" class="reference"><a href="#cite_note-Smiechowski-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-proton_26-0" class="reference"><a href="#cite_note-proton-26"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> </p><p>The Brønsted–Lowry definition applies to other solvents, such as <a href="/wiki/Dimethyl_sulfoxide" title="Dimethyl sulfoxide">dimethyl sulfoxide</a>: the solvent S acts as a base, accepting a proton and forming the conjugate acid SH<sup>+</sup>. A broader definition of acid dissociation includes <a href="/wiki/Hydrolysis" title="Hydrolysis">hydrolysis</a>, in which protons are produced by the splitting of water molecules. For example, <a href="/wiki/Boric_acid" title="Boric acid">boric acid</a>, <span class="chemf nowrap">B(OH)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">3</sub></span></span></span>, acts as a weak acid, even though it is not a proton donor, because of the hydrolysis equilibrium </p> <dl><dd><span class="chemf nowrap">B(OH)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">3</sub></span></span></span> + <span class="chemf nowrap">H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">2</sub></span></span>O</span> ⇌ <span class="chemf nowrap">B(OH)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">4</sub></span></span></span> + H<sup>+</sup>.</dd></dl> <p>Similarly, <a href="/wiki/Hydrolysis#Hydrolysis_of_metal_aqua_ions" title="Hydrolysis">metal ion hydrolysis</a> causes ions such as <span class="chemf nowrap">[Al(H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">2</sub></span></span>O)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">6</sub></span></span>]<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> to behave as weak acids:<sup id="cite_ref-Burgess_27-0" class="reference"><a href="#cite_note-Burgess-27"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="chemf nowrap">[Al(H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">2</sub></span></span>O)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">6</sub></span></span>]<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> ⇌ <span class="chemf nowrap">[Al(H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">2</sub></span></span>O)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">5</sub></span></span>(OH)]<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> + <span class="chemf nowrap">H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1em;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"></sub></span></span></span>.</dd></dl> <p>Acid–base equilibria are important in a very wide <a href="/wiki/Acid_dissociation_constant#Applications_and_significance" title="Acid dissociation constant">range of applications</a>, such as <a href="/wiki/Acid%E2%80%93base_homeostasis" title="Acid–base homeostasis">acid–base homeostasis</a>, <a href="/wiki/Ocean_acidification" title="Ocean acidification">ocean acidification</a>, <a href="/wiki/Pharmacology" title="Pharmacology">pharmacology</a> and <a href="/wiki/Analytical_chemistry" title="Analytical chemistry">analytical chemistry</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Host–guest_equilibria"><span id="Host.E2.80.93guest_equilibria"></span>Host–guest equilibria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=8" title="Edit section: Host–guest equilibria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Host%E2%80%93guest_chemistry" title="Host–guest chemistry">Host–guest chemistry</a></div> <p>A host–guest complex, also known as a donor–acceptor complex, may be formed from a <a href="/wiki/Lewis_base" class="mw-redirect" title="Lewis base">Lewis base</a>, B, and a <a href="/wiki/Lewis_acid" class="mw-redirect" title="Lewis acid">Lewis acid</a>, A. The host may be either a donor or an acceptor. In <a href="/wiki/Biochemistry" title="Biochemistry">biochemistry</a> host–guest complexes are known as <a href="/wiki/Receptor_(biochemistry)" title="Receptor (biochemistry)">receptor</a>-ligand complexes; they are formed primarily by <a href="/wiki/Non-covalent_bond" class="mw-redirect" title="Non-covalent bond">non-covalent bonding</a>. Many host–guest complexes has 1:1 stoichiometry, but many others have more complex structures. The general equilibrium can be written as </p> <dl><dd><i>p</i> A + <i>q</i> B ⇌ A<sub><i>p</i></sub>B<sub><i>q</i></sub></dd></dl> <p>The study of these complexes is important for <a href="/wiki/Supramolecular_chemistry" title="Supramolecular chemistry">supramolecular chemistry</a><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Molecular_recognition" title="Molecular recognition">molecular recognition</a>. The objective of these studies is often to find systems with a high <a href="/wiki/Binding_selectivity" title="Binding selectivity">binding selectivity</a> of a host (receptor) for a particular target molecule or ion, the guest or ligand. An application is the development of <a href="/wiki/Chemical_sensor" class="mw-redirect" title="Chemical sensor">chemical sensors</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Finding a drug which either blocks a receptor, an <a href="/wiki/Antagonist" title="Antagonist">antagonist</a> which forms a strong complex the receptor, or activate it, an <a href="/wiki/Agonist" title="Agonist">agonist</a>, is an important pathway to <a href="/wiki/Drug_discovery" title="Drug discovery">drug discovery</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Complexes_of_metals">Complexes of metals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=9" title="Edit section: Complexes of metals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stability_constants_of_complexes" title="Stability constants of complexes">Stability constants of complexes</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Al_hydrolysis_speciation_diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Al_hydrolysis_speciation_diagram.png/220px-Al_hydrolysis_speciation_diagram.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c3/Al_hydrolysis_speciation_diagram.png 1.5x" data-file-width="304" data-file-height="302" /></a><figcaption>Speciation diagram for aluminium in aqueous solution as a function of pH. A variety of hydroxo complexes are formed, including aluminium hydroxide, (Al(OH)<sub>3</sub>(s), which is insoluble at pH ~6.5</figcaption></figure> <p>The formation of a complex between a metal ion, M, and a ligand, L, is in fact usually a substitution reaction. For example, In <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solutions</a>, metal ions will be present as <a href="/wiki/Metal_ions_in_aqueous_solution" title="Metal ions in aqueous solution">aquo ions</a>, so the reaction for the formation of the first complex could be written as<sup id="cite_ref-charge_32-0" class="reference"><a href="#cite_note-charge-32"><span class="cite-bracket">[</span>note 5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>[M(H<sub>2</sub>O)<sub><i>n</i></sub>] + L ⇌ [M(H<sub>2</sub>O)<sub><i>n</i>−1</sub>L] + H<sub>2</sub>O</dd></dl> <p>However, since water is in vast excess, the concentration of water is usually assumed to be constant and is omitted from equilibrium constant expressions. Often, the metal and the ligand are in competition for protons.<sup id="cite_ref-proton_26-1" class="reference"><a href="#cite_note-proton-26"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> For the equilibrium </p> <dl><dd><i>p</i> M + <i>q</i> L + <i>r</i> H ⇌ M<sub><i>p</i></sub>L<sub><i>q</i></sub>H<sub><i>r</i></sub></dd></dl> <p>a stability constant can be defined as follows:<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{pqr}={\frac {[\mathrm {M} _{p}\mathrm {L} _{q}\mathrm {H} _{r}]}{[\mathrm {M} ]^{p}[\mathrm {L} ]^{q}[\mathrm {H} ]^{r}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{pqr}={\frac {[\mathrm {M} _{p}\mathrm {L} _{q}\mathrm {H} _{r}]}{[\mathrm {M} ]^{p}[\mathrm {L} ]^{q}[\mathrm {H} ]^{r}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7211b1998b28a60259b2d88ace92ef3ecf9957cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.037ex; height:6.509ex;" alt="{\displaystyle \beta _{pqr}={\frac {[\mathrm {M} _{p}\mathrm {L} _{q}\mathrm {H} _{r}]}{[\mathrm {M} ]^{p}[\mathrm {L} ]^{q}[\mathrm {H} ]^{r}}}}"></span></dd></dl> <p>The definition can easily be extended to include any number of reagents. It includes <a href="/wiki/Hydroxide" title="Hydroxide">hydroxide</a> complexes because the concentration of the hydroxide ions is related to the concentration of hydrogen ions by the <a href="/wiki/Self-ionization_of_water" title="Self-ionization of water">self-ionization of water</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\ce {OH-}}]={\frac {K_{\mathrm {w} }}{[{\ce {H+}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>OH</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>H</mtext> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\ce {OH-}}]={\frac {K_{\mathrm {w} }}{[{\ce {H+}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2304693857ffeea7daa98b6276ea85313f566c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.838ex; height:6.176ex;" alt="{\displaystyle [{\ce {OH-}}]={\frac {K_{\mathrm {w} }}{[{\ce {H+}}]}}}"></span></dd></dl> <p>Stability constants defined in this way, are <i>association</i> constants. This can lead to some confusion as <a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">p<i>K</i><sub>a</sub> values</a> are <i>dissociation</i> constants. In general purpose computer programs it is customary to define all constants as association constants. The relationship between the two types of constant is given in <a href="/wiki/Equilibrium_constant#Association_and_dissociation_constants" title="Equilibrium constant">association and dissociation constants</a>. </p><p>In <a href="/wiki/Biochemistry" title="Biochemistry">biochemistry</a>, an oxygen molecule can bind to an iron(II) atom in a <a href="/wiki/Heme" title="Heme">heme</a> <a href="/wiki/Prosthetic_group" title="Prosthetic group">prosthetic group</a> in <a href="/wiki/Hemoglobin" title="Hemoglobin">hemoglobin</a>. The equilibrium is usually written, denoting hemoglobin by Hb, as </p> <dl><dd>Hb + O<sub>2</sub> ⇌ HbO<sub>2</sub></dd></dl> <p>but this representation is incomplete as the <a href="/wiki/Bohr_effect" title="Bohr effect">Bohr effect</a> shows that the equilibrium concentrations are pH-dependent. A better representation would be </p> <dl><dd>[HbH]<sup>+</sup> + O<sub>2</sub> ⇌ HbO<sub>2</sub> + H<sup>+</sup></dd></dl> <p>as this shows that when hydrogen ion concentration increases the equilibrium is shifted to the left in accordance with <a href="/wiki/Le_Ch%C3%A2telier%27s_principle" class="mw-redirect" title="Le Châtelier's principle">Le Châtelier's principle</a>. Hydrogen ion concentration can be increased by the presence of carbon dioxide, which behaves as a weak acid. </p> <dl><dd>H<sub>2</sub>O + CO<sub>2 </sub> ⇌ <span class="chemf nowrap">HCO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">3</sub></span></span></span> + H<sup>+</sup></dd></dl> <p>The iron atom can also bind to other molecules such as <a href="/wiki/Carbon_monoxide" title="Carbon monoxide">carbon monoxide</a>. Cigarette smoke contains some carbon monoxide so the equilibrium </p> <dl><dd>HbO<sub>2</sub> + CO ⇌ <span class="chemf nowrap">Hb(CO)</span> + O<sub>2</sub></dd></dl> <p>is established in the blood of cigarette smokers. </p><p><a href="/wiki/Chelation_therapy" title="Chelation therapy">Chelation therapy</a> is based on the principle of using <a href="/wiki/Chelate_effect" class="mw-redirect" title="Chelate effect">chelating ligands</a> with a high <a href="/wiki/Binding_selectivity" title="Binding selectivity">binding selectivity</a> for a particular metal to remove that metal from the human body. </p><p>Complexes with <a href="/wiki/Polyamino_carboxylic_acid" class="mw-redirect" title="Polyamino carboxylic acid">polyamino carboxylic acids</a> find a wide range of applications. <a href="/wiki/EDTA#Uses" class="mw-redirect" title="EDTA">EDTA</a> in particular is used extensively. </p> <div class="mw-heading mw-heading2"><h2 id="Redox_equilibrium">Redox equilibrium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=10" title="Edit section: Redox equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A reduction–oxidation (<a href="/wiki/Redox" title="Redox">redox</a>) equilibrium can be handled in exactly the same way as any other chemical equilibrium. For example, </p> <dl><dd>Fe<sup>2+</sup> + Ce<sup>4+</sup> ⇌ Fe<sup>3+</sup> + Ce<sup>3+</sup>;<span class="nowrap">     </span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {[{\ce {Fe^3+}}][{\ce {Ce^3+}}]}{[{\ce {Fe^2+}}][{\ce {Ce^4+}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Fe</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Ce</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Fe</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Ce</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {[{\ce {Fe^3+}}][{\ce {Ce^3+}}]}{[{\ce {Fe^2+}}][{\ce {Ce^4+}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6f02b6d961f0f84ef7ac4c328ccef58b0bfb09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.514ex; height:7.176ex;" alt="{\displaystyle K={\frac {[{\ce {Fe^3+}}][{\ce {Ce^3+}}]}{[{\ce {Fe^2+}}][{\ce {Ce^4+}}]}}}"></span></dd></dl> <p>However, in the case of redox reactions it is convenient to split the overall reaction into two half-reactions. In this example </p> <dl><dd>Fe<sup>3+</sup> + e<sup>−</sup> ⇌ Fe<sup>2+</sup></dd> <dd>Ce<sup>4+</sup> + e<sup>−</sup> ⇌ Ce<sup>3+</sup></dd></dl> <p>The standard free energy change, which is related to the equilibrium constant by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G^{\ominus }=-RT\ln K\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=-RT\ln K\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97926ae588f1f2ee9cb85896db22c335c4b77734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.747ex; height:2.676ex;" alt="{\displaystyle \Delta G^{\ominus }=-RT\ln K\,}"></span></dd></dl> <p>can be split into two components, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G^{\ominus }=\Delta G_{{\ce {Fe}}}^{\ominus }+\Delta G_{{\ce {Ce}}}^{\ominus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Fe</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ce</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=\Delta G_{{\ce {Fe}}}^{\ominus }+\Delta G_{{\ce {Ce}}}^{\ominus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8433e7dbc6d6aec31730562b13446c3409001d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.922ex; height:3.176ex;" alt="{\displaystyle \Delta G^{\ominus }=\Delta G_{{\ce {Fe}}}^{\ominus }+\Delta G_{{\ce {Ce}}}^{\ominus }}"></span></dd></dl> <p>The concentration of free electrons is effectively zero as the electrons are transferred directly from the reductant to the oxidant. The <a href="/wiki/Standard_electrode_potential" title="Standard electrode potential">standard electrode potential</a>, <i>E</i><sup>0</sup> for the each half-reaction is related to the standard free energy change by<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G_{{\ce {Fe}}}^{\ominus }=-nFE_{{\ce {Fe}}}^{0};\Delta G_{{\ce {Ce}}}^{\ominus }=-nFE_{{\ce {Ce}}}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Fe</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mi>F</mi> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Fe</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>;</mo> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ce</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mi>F</mi> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ce</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G_{{\ce {Fe}}}^{\ominus }=-nFE_{{\ce {Fe}}}^{0};\Delta G_{{\ce {Ce}}}^{\ominus }=-nFE_{{\ce {Ce}}}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa807bca2a336cd59cc5371802fc4af33138bfeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.443ex; height:3.176ex;" alt="{\displaystyle \Delta G_{{\ce {Fe}}}^{\ominus }=-nFE_{{\ce {Fe}}}^{0};\Delta G_{{\ce {Ce}}}^{\ominus }=-nFE_{{\ce {Ce}}}^{0}}"></span></dd></dl> <p>where <i>n</i> is the number of electrons transferred and <i>F</i> is the <a href="/wiki/Faraday_constant" title="Faraday constant">Faraday constant</a>. Now, the free energy for an actual reaction is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G=\Delta G^{\ominus }+RT\ln Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>G</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>+</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta G=\Delta G^{\ominus }+RT\ln Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbd4ca32acc781c3953516ef61a50c212bf2702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.927ex; height:2.843ex;" alt="{\displaystyle \Delta G=\Delta G^{\ominus }+RT\ln Q}"></span></dd></dl> <p>where <i>R</i> is the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a> and <i>Q</i> a <a href="/wiki/Reaction_quotient" title="Reaction quotient">reaction quotient</a>. Strictly speaking <i>Q</i> is a quotient of activities, but it is common practice to use concentrations instead of activities. Therefore: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{{\ce {Fe}}}=E_{{\ce {Fe}}}^{0}+{\frac {RT}{nF}}\ln {\frac {[{\ce {Fe^3+}}]}{[{\ce {Fe^2+}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Fe</mtext> </mrow> </mrow> </msub> <mo>=</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Fe</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mi>n</mi> <mi>F</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Fe</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mtext>Fe</mtext> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{{\ce {Fe}}}=E_{{\ce {Fe}}}^{0}+{\frac {RT}{nF}}\ln {\frac {[{\ce {Fe^3+}}]}{[{\ce {Fe^2+}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fa1a3bd15095282f118bc8e51cae996049f126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.403ex; height:7.009ex;" alt="{\displaystyle E_{{\ce {Fe}}}=E_{{\ce {Fe}}}^{0}+{\frac {RT}{nF}}\ln {\frac {[{\ce {Fe^3+}}]}{[{\ce {Fe^2+}}]}}}"></span></dd></dl> <p>For any half-reaction, the redox potential of an actual mixture is given by the generalized expression<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>note 6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E^{0}+{\frac {RT}{nF}}\ln {\frac {[{\text{oxidized species}}]}{[{\text{reduced species}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mi>n</mi> <mi>F</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>oxidized species</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>reduced species</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E^{0}+{\frac {RT}{nF}}\ln {\frac {[{\text{oxidized species}}]}{[{\text{reduced species}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c443233bbcc4ead5ada0f7802c1e937f79efa01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.425ex; height:6.509ex;" alt="{\displaystyle E=E^{0}+{\frac {RT}{nF}}\ln {\frac {[{\text{oxidized species}}]}{[{\text{reduced species}}]}}}"></span></dd></dl> <p>This is an example of the <a href="/wiki/Nernst_equation" title="Nernst equation">Nernst equation</a>. The potential is known as a reduction potential. Standard electrode potentials are available in a <a href="/wiki/Standard_electrode_potential_(data_page)" title="Standard electrode potential (data page)">table of values</a>. Using these values, the actual electrode potential for a redox couple can be calculated as a function of the ratio of concentrations. </p><p>The equilibrium potential for a general redox half-reaction (See <a href="#Equilibrium_constant">#Equilibrium constant</a> above for an explanation of the symbols) </p> <dl><dd><i>α</i> A + <i>β</i> B... + <i>n</i> e<sup>−</sup> ⇌ <i>σ</i> S + <i>τ</i> T...</dd></dl> <p>is given by<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E^{\ominus }+{\frac {RT}{nF}}\ln {\frac {{\{{\ce {S}}\}}^{\sigma }{\{{\ce {T}}\}}^{\tau }...}{{\{{\ce {A}}\}}^{\alpha }{\{{\ce {B}}\}}^{\beta }...}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mi>n</mi> <mi>F</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E^{\ominus }+{\frac {RT}{nF}}\ln {\frac {{\{{\ce {S}}\}}^{\sigma }{\{{\ce {T}}\}}^{\tau }...}{{\{{\ce {A}}\}}^{\alpha }{\{{\ce {B}}\}}^{\beta }...}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97ce2dc50dc69977c6e7a75c45dbede453a7907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.017ex; height:7.009ex;" alt="{\displaystyle E=E^{\ominus }+{\frac {RT}{nF}}\ln {\frac {{\{{\ce {S}}\}}^{\sigma }{\{{\ce {T}}\}}^{\tau }...}{{\{{\ce {A}}\}}^{\alpha }{\{{\ce {B}}\}}^{\beta }...}}}"></span></dd></dl> <p>Use of this expression allows the effect of a species not involved in the redox reaction, such as the hydrogen ion in a half-reaction such as </p> <dl><dd><span class="chemf nowrap">MnO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;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">4</sub></span></span></span> + 8 H<sup>+</sup> + 5 e<sup>−</sup> ⇌ Mn<sup>2+</sup> + 4 H<sub>2</sub>O</dd></dl> <p>to be taken into account. </p><p>The equilibrium constant for a full redox reaction can be obtained from the standard redox potentials of the constituent half-reactions. At equilibrium the potential for the two half-reactions must be equal to each other and, of course, the number of electrons exchanged must be the same in the two half reactions.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>Redox equilibrium play an important role in the <a href="/wiki/Electron_transport_chain" title="Electron transport chain">electron transport chain</a>. The various <a href="/wiki/Cytochrome" title="Cytochrome">cytochromes</a> in the chain have different standard redox potentials, each one adapted for a specific redox reaction. This allows, for example, atmospheric <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> to be reduced in <a href="/wiki/Photosynthesis" title="Photosynthesis">photosynthesis</a>. A distinct family of cytochromes, the <a href="/wiki/Cytochrome_P450_oxidase" class="mw-redirect" title="Cytochrome P450 oxidase">cytochrome P450 oxidases</a>, are involved in <a href="/wiki/Steroidogenesis" class="mw-redirect" title="Steroidogenesis">steroidogenesis</a> and <a href="/wiki/Detoxification" title="Detoxification">detoxification</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Solubility">Solubility</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=11" title="Edit section: Solubility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Solubility_equilibrium" title="Solubility equilibrium">Solubility equilibrium</a></div> <p>When a <a href="/wiki/Solute" class="mw-redirect" title="Solute">solute</a> forms a <a href="/wiki/Saturated_solution" class="mw-redirect" title="Saturated solution">saturated solution</a> in a <a href="/wiki/Solvent" title="Solvent">solvent</a>, the concentration of the solute, at a given temperature, is determined by the equilibrium constant at that temperature.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln K=-RT\ln \left({\frac {\sum _{k}{a_{k}}^{m_{k}}\mathrm {(solution)} }{a\mathrm {(solid)} }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln K=-RT\ln \left({\frac {\sum _{k}{a_{k}}^{m_{k}}\mathrm {(solution)} }{a\mathrm {(solid)} }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84792e0412dbd3d92ae4c81edf8849146329da53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.9ex; height:6.509ex;" alt="{\displaystyle \ln K=-RT\ln \left({\frac {\sum _{k}{a_{k}}^{m_{k}}\mathrm {(solution)} }{a\mathrm {(solid)} }}\right)}"></span></dd></dl> <p>The activity of a pure substance in the solid state is one, by definition, so the expression simplifies to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln K=-RT\ln \left(\sum _{k}{a_{k}}^{m_{k}}\mathrm {(solution)} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln K=-RT\ln \left(\sum _{k}{a_{k}}^{m_{k}}\mathrm {(solution)} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c6fa750e91834722855791e41a2aef17e2a841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.136ex; height:7.509ex;" alt="{\displaystyle \ln K=-RT\ln \left(\sum _{k}{a_{k}}^{m_{k}}\mathrm {(solution)} \right)}"></span></dd></dl> <p>If the solute does not dissociate the summation is replaced by a single term, but if dissociation occurs, as with ionic substances </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {SP} }=\prod _{k}{{a_{k}}^{m_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {SP} }=\prod _{k}{{a_{k}}^{m_{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065bb4f6980ee02afafef7a90514a1b1b3255d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.547ex; height:5.509ex;" alt="{\displaystyle K_{\mathrm {SP} }=\prod _{k}{{a_{k}}^{m_{k}}}}"></span></dd></dl> <p>For example, with Na<sub>2</sub>SO<sub>4</sub>, <span class="texhtml"><i>m</i><sub>1</sub> = 2</span> and <span class="texhtml"><i>m</i><sub>2</sub> = 1</span> so the solubility product is written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {SP} }=[\mathrm {Na^{+}} ]^{2}[\mathrm {SO_{4}^{2-}} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">P</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <msup> <mi mathvariant="normal">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <msubsup> <mi mathvariant="normal">O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msubsup> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {SP} }=[\mathrm {Na^{+}} ]^{2}[\mathrm {SO_{4}^{2-}} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b38a2d91ad0f0206934346c679099db801f09b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.828ex; height:3.343ex;" alt="{\displaystyle K_{\mathrm {SP} }=[\mathrm {Na^{+}} ]^{2}[\mathrm {SO_{4}^{2-}} ]}"></span></dd></dl> <p>Concentrations, indicated by [...], are usually used in place of activities, but activity must be taken into account of the presence of another salt with no ions in common, the so-called salt effect. When another salt is present that has an ion in common, the <a href="/wiki/Common-ion_effect" title="Common-ion effect">common-ion effect</a> comes into play, reducing the solubility of the primary solute.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Partition">Partition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=12" title="Edit section: Partition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Partition_coefficient" title="Partition coefficient">Partition coefficient</a> and <a href="/wiki/Liquid%E2%80%93liquid_extraction" title="Liquid–liquid extraction">Liquid–liquid extraction</a></div> <p>When a solution of a substance in one solvent is brought into equilibrium with a second solvent that is immiscible with the first solvent, the dissolved substance may be partitioned between the two solvents. The ratio of concentrations in the two solvents is known as a <a href="/wiki/Partition_coefficient" title="Partition coefficient">partition coefficient</a> or <a href="/wiki/Distribution_coefficient" class="mw-redirect" title="Distribution coefficient">distribution coefficient</a>.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>note 7<span class="cite-bracket">]</span></a></sup> The partition coefficient is defined as the ratio of the <a href="/wiki/Analytical_concentration" class="mw-redirect" title="Analytical concentration">analytical concentrations</a> of the solute in the two phases. By convention the value is reported in logarithmic form. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log p=\log {\frac {[\mathrm {solute} ]_{\mathrm {organic\;phase} }}{[\mathrm {solute} ]_{\mathrm {aqueous\;phase} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">q</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log p=\log {\frac {[\mathrm {solute} ]_{\mathrm {organic\;phase} }}{[\mathrm {solute} ]_{\mathrm {aqueous\;phase} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9676ed1d0e0470c2eefaa023c1ca10908b161e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.705ex; height:6.509ex;" alt="{\displaystyle \log p=\log {\frac {[\mathrm {solute} ]_{\mathrm {organic\;phase} }}{[\mathrm {solute} ]_{\mathrm {aqueous\;phase} }}}}"></span></dd></dl> <p>The partition coefficient is defined at a specified temperature and, if applicable, pH of the aqueous phase. Partition coefficients are very important in <a href="/wiki/Pharmacology" title="Pharmacology">pharmacology</a> because they determine the extent to which a substance can pass from the blood (an aqueous solution) through a cell wall which is like an organic solvent. They are usually measured using water and <a href="/wiki/1-Octanol" title="1-Octanol">octanol</a> as the two solvents, yielding the so-called <a href="/wiki/Octanol-water_partition_coefficient" title="Octanol-water partition coefficient">octanol-water partition coefficient</a>. Many pharmaceutical compounds are <a href="/wiki/Weak_acid" class="mw-redirect" title="Weak acid">weak acids</a> or <a href="/wiki/Weak_base" title="Weak base">weak bases</a>. Such a compound may exist with a different extent of protonation depending on <a href="/wiki/PH" title="PH">pH</a> and the <a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">acid dissociation constant</a>. Because the organic phase has a low <a href="/wiki/Dielectric_constant" class="mw-redirect" title="Dielectric constant">dielectric constant</a> the species with no electrical charge will be the most likely one to pass from the aqueous phase to the organic phase. Even at pH 7–7.2, the range of biological pH values, the aqueous phase may support an equilibrium between more than one protonated form. log <i>p</i> is determined from the analytical concentration of the substance in the aqueous phase, that is, the sum of the concentration of the different species in equilibrium. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Separation02.ogv/320px--Separation02.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="320" height="240" data-durationhint="28" data-mwtitle="Separation02.ogv" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/86/Separation02.ogv/Separation02.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/86/Separation02.ogv/Separation02.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/8/86/Separation02.ogv" type="video/ogg; codecs="theora"" data-width="320" data-height="240" /></video></span><figcaption>An organic <a href="/wiki/MTBE" class="mw-redirect" title="MTBE">MTBE</a> solution is extracted with <a href="/wiki/Aqueous" class="mw-redirect" title="Aqueous">aqueous</a> sodium bicarbonate solution. This base removes <a href="/wiki/Benzoic_acid" title="Benzoic acid">benzoic acid</a> as <a href="/wiki/Benzoate" class="mw-redirect" title="Benzoate">benzoate</a> but leaves non-acidic <a href="/wiki/Benzil" title="Benzil">benzil</a> (yellow) behind in the upper organic phase.</figcaption></figure> <p>Solvent extraction is used extensively in separation and purification processes. In its simplest form a reaction is performed in an organic solvent and unwanted by-products are removed by extraction into water at a particular pH. </p><p>A metal ion may be extracted from an aqueous phase into an organic phase in which the salt is not soluble, by adding a <a href="/wiki/Ligand" title="Ligand">ligand</a>. The ligand, L<sup><i>a</i>−</sup>, forms a complex with the metal ion, M<sup><i>b</i>+</sup>, [ML<sub><i>x</i></sub>]<sup>(<i>b</i>−<i>ax</i>)+</sup> which has a strongly <a href="/wiki/Hydrophobic" class="mw-redirect" title="Hydrophobic">hydrophobic</a> outer surface. If the complex has no electrical charge it will be extracted relatively easily into the organic phase. If the complex is charged, it is extracted as an <a href="/wiki/Ion_pair" class="mw-redirect" title="Ion pair">ion pair</a>. The additional ligand is not always required. For example, <a href="/wiki/Uranyl_nitrate" title="Uranyl nitrate">uranyl nitrate</a>, UO<sub>2</sub>(NO<sub>3</sub>)<sub>2</sub>, is soluble in <a href="/wiki/Diethyl_ether" title="Diethyl ether">diethyl ether</a> because the solvent itself acts as a ligand. This property was used in the past for separating uranium from other metals whose salts are not soluble in ether. Currently extraction into <a href="/wiki/Kerosene" title="Kerosene">kerosene</a> is preferred, using a ligand such as <a href="/wiki/Tri-n-butyl_phosphate" class="mw-redirect" title="Tri-n-butyl phosphate">tri-<i>n</i>-butyl phosphate</a>, TBP. In the <a href="/wiki/PUREX" title="PUREX">PUREX</a> process, which is commonly used in <a href="/wiki/Nuclear_reprocessing" title="Nuclear reprocessing">nuclear reprocessing</a>, uranium(VI) is extracted from strong nitric acid as the electrically neutral complex [UO<sub>2</sub>(TBP)<sub>2</sub>(NO<sub>3</sub>)<sub>2</sub>]. The strong nitric acid provides a high concentration of nitrate ions which pushes the equilibrium in favour of the weak nitrato complex. Uranium is recovered by back-extraction (stripping) into weak nitric acid. Plutonium(IV) forms a similar complex, [PuO<sub>2</sub>(TBP)<sub>2</sub>(NO<sub>3</sub>)<sub>2</sub>] and the plutonium in this complex can be reduced to separate it from uranium. </p><p>Another important application of solvent extraction is in the separation of the <a href="/wiki/Lanthanoid" class="mw-redirect" title="Lanthanoid">lanthanoids</a>. This process also uses TBP and the complexes are extracted into kerosene. Separation is achieved because the <a href="/wiki/Stability_constants_of_complexes" title="Stability constants of complexes">stability constant</a> for the formation of the TBP complex increases as the size of the lanthanoid ion decreases. </p><p>An instance of ion-pair extraction is in the use of a ligand to enable oxidation by <a href="/wiki/Potassium_permanganate" title="Potassium permanganate">potassium permanganate</a>, KMnO<sub>4</sub>, in an organic solvent. KMnO<sub>4</sub> is not soluble in organic solvents. When a ligand, such as a <a href="/wiki/Crown_ether" title="Crown ether">crown ether</a> is added to an aqueous solution of KMnO<sub>4</sub>, it forms a hydrophobic complex with the potassium cation which allows the uncharged ion pair [KL]<sup>+</sup>[MnO<sub>4</sub>]<sup>−</sup> to be extracted into the organic solvent. See also: <a href="/wiki/Phase-transfer_catalysis" class="mw-redirect" title="Phase-transfer catalysis">phase-transfer catalysis</a>. </p><p>More complex partitioning problems (i.e. 3 or more phases present) can sometimes be handled with a <a href="/wiki/Fugacity_capacity" title="Fugacity capacity">fugacity capacity</a> approach. </p> <div class="mw-heading mw-heading2"><h2 id="Chromatography">Chromatography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=13" title="Edit section: Chromatography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></div> <p>In chromatography substances are separated by partition between a stationary phase and a mobile phase. The analyte is dissolved in the mobile phase, and passes over the stationary phase. Separation occurs because of differing affinities of the <a href="/wiki/Analyte" title="Analyte">analytes</a> for the stationary phase. A distribution constant, <i>K</i><sub>d</sub> can be defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {d} }={\frac {a_{\mathrm {s} }}{a_{\mathrm {m} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {d} }={\frac {a_{\mathrm {s} }}{a_{\mathrm {m} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ee8a3987da73e93684f90d2e20b18663ca57b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.885ex; height:5.009ex;" alt="{\displaystyle K_{\mathrm {d} }={\frac {a_{\mathrm {s} }}{a_{\mathrm {m} }}}}"></span></dd></dl> <p>where <i>a</i><sub>s</sub> and <i>a</i><sub>m</sub> are the equilibrium activities in the stationary and mobile phases respectively. It can be shown that the rate of migration, <span style="text-decoration:overline;"><i>ν</i></span>, is related to the distribution constant by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\nu }}\propto {\frac {1}{1+fK_{\mathrm {d} }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>f</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\nu }}\propto {\frac {1}{1+fK_{\mathrm {d} }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a39997f864b375a445afa1afe98b2e164ea6c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.298ex; height:5.676ex;" alt="{\displaystyle {\bar {\nu }}\propto {\frac {1}{1+fK_{\mathrm {d} }}}.}"></span></dd></dl> <p><i>f</i> is a factor which depends on the volumes of the two phases.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Thus, the higher the affinity of the solute for the stationary phase, the slower the migration rate. </p><p>There is a wide variety of chromatographic techniques, depending on the nature of the stationary and mobile phases. When the stationary phase is solid, the analyte may form a complex with it. A <a href="/wiki/Water_softener" class="mw-redirect" title="Water softener">water softener</a> functions by selective complexation with a <a href="/wiki/Sulfonate" title="Sulfonate">sulfonate</a> <a href="/wiki/Ion_exchange_resin" class="mw-redirect" title="Ion exchange resin">ion exchange resin</a>. Sodium ions form relatively weak complexes with the resin. When <a href="/wiki/Hard_water" title="Hard water">hard water</a> is passed through the resin, the divalent ions of magnesium and calcium displace the sodium ions and are retained on the resin, R. </p> <dl><dd>RNa + M<sup>2+</sup> ⇌ RM<sup>+</sup> + Na<sup>+</sup></dd></dl> <p>The water coming out of the column is relatively rich in sodium ions<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>note 8<span class="cite-bracket">]</span></a></sup> and poor in calcium and magnesium which are retained on the column. The column is regenerated by passing a strong solution of sodium chloride through it, so that the resin–sodium complex is again formed on the column. <a href="/wiki/Ion-exchange_chromatography" class="mw-redirect" title="Ion-exchange chromatography">Ion-exchange chromatography</a> utilizes a resin such as <a href="/wiki/Chelex_100" title="Chelex 100">chelex 100</a> in which <a href="/wiki/Iminodiacetic_acid" title="Iminodiacetic acid">iminodiacetate</a> residues, attached to a polymer backbone, form <a href="/wiki/Chelate" class="mw-redirect" title="Chelate">chelate</a> complexes of differing strengths with different metal ions, allowing the ions such as Cu<sup>2+</sup> and Ni<sup>2+</sup> to be separated chromatographically. </p><p>Another example of complex formation is in <a href="/wiki/Chromatography#Chiral_chromatography" title="Chromatography">chiral chromatography</a> in which is used to separate <a href="/wiki/Enantiomer" title="Enantiomer">enantiomers</a> from each other. The stationary phase is itself chiral and forms complexes selectively with the enantiomers. In other types of chromatography with a solid stationary phase, such as <a href="/wiki/Thin-layer_chromatography" title="Thin-layer chromatography">thin-layer chromatography</a> the analyte is selectively <a href="/wiki/Adsorbed" class="mw-redirect" title="Adsorbed">adsorbed</a> onto the solid. </p><p>In <a href="/wiki/Gas%E2%80%93liquid_chromatography" class="mw-redirect" title="Gas–liquid chromatography">gas–liquid chromatography</a> (GLC) the stationary phase is a liquid such as <a href="/wiki/Polydimethylsiloxane" title="Polydimethylsiloxane">polydimethylsiloxane</a>, coated on a glass tube. Separation is achieved because the various components in the gas have different solubility in the stationary phase. GLC can be used to separate literally hundreds of components in a gas mixture such as <a href="/wiki/Cigarette_smoke" class="mw-redirect" title="Cigarette smoke">cigarette smoke</a> or <a href="/wiki/Essential_oil" title="Essential oil">essential oils</a>, such as <a href="/wiki/Lavender_oil#Composition" title="Lavender oil">lavender oil</a>. </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=Equilibrium_chemistry&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Thermodynamic databases for pure substances</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=15" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">The general expression is not used much in chemistry. To help understand the notation consider the equilibrium <dl><dd>H<sub>2</sub>SO<sub>4</sub> + 2 OH<sup>−</sup> ⇌ <span class="chemf nowrap">SO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">4</sub></span></span></span> + 2 H<sub>2</sub>O</dd></dl> for this reaction <span class="texhtml"><i>n</i><sub>1</sub> = 1</span>, <span class="texhtml"><i>n</i><sub>2</sub> = 2</span>, <span class="texhtml"><i>m</i><sub>1</sub> = 1</span> and <span class="texhtml"><i>m</i><sub>2</sub> = 2</span>, Reactant<sub>1</sub> = H<sub>2</sub>SO<sub>4</sub>, Reactant<sub>2</sub> = OH<sup>−</sup>, Product<sub>1</sub> = <span class="chemf nowrap">SO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">4</sub></span></span></span> and Product<sub>2</sub> = H<sub>2</sub>O.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">This is equivalent to defining a new equilibrium constant as <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"><i>K</i></span><span class="sr-only">/</span><span class="den"><i>Γ</i></span></span>⁠</span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">The definitions given are <a href="/wiki/Equilibrium_constant#Association_and_dissociation_constants" title="Equilibrium constant">association constants</a>. A dissociation constant is the reciprocal of an association constant.</span> </li> <li id="cite_note-proton-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-proton_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-proton_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">The bare proton does not exist in aqueous solution. It is a very strong acid and combines the base, water, to form the hydronium ion <dl><dd>H<sup>+</sup> + H<sub>2</sub>O → H<sub>3</sub>O<sup>+</sup></dd></dl> The hydronium ion forms various weak complexes by hydrogen bonding with more water molecules.</span> </li> <li id="cite_note-charge-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-charge_32-0">^</a></b></span> <span class="reference-text">Electrical charges are omitted from such expressions because the ligand, L, may or may not carry an electrical charge.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">The alternative expression <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 E=E^{0}-{\frac {RT}{nF}}\ln {\frac {[{\text{reduced species}}]}{[{\text{oxidized species}}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mi>n</mi> <mi>F</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>reduced species</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>oxidized species</mtext> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E^{0}-{\frac {RT}{nF}}\ln {\frac {[{\text{reduced species}}]}{[{\text{oxidized species}}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895c50424795ac33e837902de294e469fc5fb915" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.425ex; height:6.509ex;" alt="{\displaystyle E=E^{0}-{\frac {RT}{nF}}\ln {\frac {[{\text{reduced species}}]}{[{\text{oxidized species}}]}}}"></span> is sometimes used, as in the <a href="/wiki/Nernst_equation" title="Nernst equation">Nernst equation</a>.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">The distinction between a partition coefficient and a <a href="/wiki/Distribution_coefficient" class="mw-redirect" title="Distribution coefficient">distribution coefficient</a> is of historical significance only.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Feeding babies formula made up with sodium rich water can lead to <a href="/wiki/Hypernatremia" title="Hypernatremia">hypernatremia</a>.</span> </li> </ol></div></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=Equilibrium_chemistry&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chemical Equilibrium <a rel="nofollow" class="external text" href="http://download-book.net/Chemical-Equilibrium-ppt-pdf.html">Downloadable book</a></li></ul> <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=Equilibrium_chemistry&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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, P.W.; De Paula, J. (2006). <i>Physical Chemistry</i> (8th. ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-870072-5" title="Special:BookSources/0-19-870072-5"><bdi>0-19-870072-5</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.edition=8th.&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=0-19-870072-5&rft.aulast=Atkins&rft.aufirst=P.W.&rft.au=De+Paula%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenbeigh1981" class="citation book cs1">Denbeigh, K. (1981). <i>The principles of chemical equilibrium</i> (4th. ed.). Cambridge, U.K.: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-28150-4" title="Special:BookSources/0-521-28150-4"><bdi>0-521-28150-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+principles+of+chemical+equilibrium&rft.place=Cambridge%2C+U.K.&rft.edition=4th.&rft.pub=Cambridge+University+Press&rft.date=1981&rft.isbn=0-521-28150-4&rft.aulast=Denbeigh&rft.aufirst=K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> A classic book, last reprinted in 1997.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendham,_J.Denney,_R._C.Barnes,_J._D.Thomas,_M._J._K.2000" class="citation cs2">Mendham, J.; Denney, R. C.; Barnes, J. D.; Thomas, M. J. K. (2000), <i>Vogel's Quantitative Chemical Analysis</i> (6th ed.), New York: Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-22628-7" title="Special:BookSources/0-582-22628-7"><bdi>0-582-22628-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vogel%27s+Quantitative+Chemical+Analysis&rft.place=New+York&rft.edition=6th&rft.pub=Prentice+Hall&rft.date=2000&rft.isbn=0-582-22628-7&rft.au=Mendham%2C+J.&rft.au=Denney%2C+R.+C.&rft.au=Barnes%2C+J.+D.&rft.au=Thomas%2C+M.+J.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenbeigh1981" class="citation book cs1">Denbeigh, K. (1981). <i>The principles of chemical equilibrium</i> (4th ed.). Cambridge, UK: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-28150-4" title="Special:BookSources/0-521-28150-4"><bdi>0-521-28150-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+principles+of+chemical+equilibrium&rft.place=Cambridge%2C+UK&rft.edition=4th&rft.pub=Cambridge+University+Press&rft.date=1981&rft.isbn=0-521-28150-4&rft.aulast=Denbeigh&rft.aufirst=K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_Nevers2002" class="citation book cs1">De Nevers, N. (2002). <i>Physical and Chemical Equilibrium for Chemical Engineers</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-07170-9" title="Special:BookSources/978-0-471-07170-9"><bdi>978-0-471-07170-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+and+Chemical+Equilibrium+for+Chemical+Engineers&rft.date=2002&rft.isbn=978-0-471-07170-9&rft.aulast=De+Nevers&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></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">Denbigh, Chapter 4</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">Denbigh, Chapter 5</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">Atkins, p. 203</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Atkins, p. 149</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchultz1999" class="citation journal cs1">Schultz, M. J. (1999). "Why Equilibrium? Understanding the Role of Entropy of Mixing". <i>J. Chem. Educ</i>. <b>76</b> (10): 1391. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999JChEd..76.1391S">1999JChEd..76.1391S</a>. <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%2Fed076p1391">10.1021/ed076p1391</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Chem.+Educ.&rft.atitle=Why+Equilibrium%3F+Understanding+the+Role+of+Entropy+of+Mixing&rft.volume=76&rft.issue=10&rft.pages=1391&rft.date=1999&rft_id=info%3Adoi%2F10.1021%2Fed076p1391&rft_id=info%3Abibcode%2F1999JChEd..76.1391S&rft.aulast=Schultz&rft.aufirst=M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-FJCR-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-FJCR_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FJCR_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRossottiRossotti1961" class="citation book cs1">Rossotti, F. J. C.; Rossotti, H. (1961). <i>The Determination of Stability Constants</i>. McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Determination+of+Stability+Constants&rft.pub=McGraw-Hill&rft.date=1961&rft.aulast=Rossotti&rft.aufirst=F.+J.+C.&rft.au=Rossotti%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Chapter 2, Activity and concentration quotients</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Atkins, p. 208</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="CITEREFBlandamer1992" class="citation book cs1">Blandamer, M. J. (1992). <i>Chemical equilibria in solution: dependence of rate and equilibrium constants on temperature and pressure</i>. New York: Ellis Horwood/PTR Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-131731-8" title="Special:BookSources/0-13-131731-8"><bdi>0-13-131731-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+equilibria+in+solution%3A+dependence+of+rate+and+equilibrium+constants+on+temperature+and+pressure&rft.place=New+York&rft.pub=Ellis+Horwood%2FPTR+Prentice+Hall&rft.date=1992&rft.isbn=0-13-131731-8&rft.aulast=Blandamer&rft.aufirst=M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" 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">Atkins, p. 111</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="CITEREFDamköhlerEdse1943" class="citation journal cs1">Damköhler, G.; Edse, R. (1943). "Composition of dissociating combustion gases and the calculation of simultaneous equilibria". <i>Z. Elektrochem</i>. <b>49</b>: 178–802.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Z.+Elektrochem.&rft.atitle=Composition+of+dissociating+combustion+gases+and+the+calculation+of+simultaneous+equilibria&rft.volume=49&rft.pages=178-802&rft.date=1943&rft.aulast=Damk%C3%B6hler&rft.aufirst=G.&rft.au=Edse%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_ZeggerenStorey1970" class="citation book cs1">Van Zeggeren, F.; Storey, S. H. (1970). <i>The computation of chemical equilibria</i>. London: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-07630-7" title="Special:BookSources/0-521-07630-7"><bdi>0-521-07630-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+computation+of+chemical+equilibria&rft.place=London&rft.pub=Cambridge+University+Press&rft.date=1970&rft.isbn=0-521-07630-7&rft.aulast=Van+Zeggeren&rft.aufirst=F.&rft.au=Storey%2C+S.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithMissen1991" class="citation book cs1">Smith, W.R.; Missen, R.W. (1991). <i>Chemical reaction equilibrium analysis : theory and algorithms</i>. Malabar, Fla.: Krieger. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89464-584-6" title="Special:BookSources/0-89464-584-6"><bdi>0-89464-584-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+reaction+equilibrium+analysis+%3A+theory+and+algorithms&rft.place=Malabar%2C+Fla.&rft.pub=Krieger&rft.date=1991&rft.isbn=0-89464-584-6&rft.aulast=Smith&rft.aufirst=W.R.&rft.au=Missen%2C+R.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartleyBurgessAlcock1980" class="citation book cs1">Hartley, F.R.; Burgess, C.; Alcock, R. M. (1980). <a rel="nofollow" class="external text" href="https://archive.org/details/solutionequilibr0000hart"><i>Solution equilibria</i></a>. New York (Halsted Press): Ellis Horwood. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-470-26880-8" title="Special:BookSources/0-470-26880-8"><bdi>0-470-26880-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solution+equilibria&rft.place=New+York+%28Halsted+Press%29&rft.pub=Ellis+Horwood&rft.date=1980&rft.isbn=0-470-26880-8&rft.aulast=Hartley&rft.aufirst=F.R.&rft.au=Burgess%2C+C.&rft.au=Alcock%2C+R.+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsolutionequilibr0000hart&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-DJL-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-DJL_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DJL_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeggett1985" class="citation book cs1">Leggett, D. J., ed. (1985). <i>Computational methods for the determination of formation constants</i>. New York: Plenum Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-41957-2" title="Special:BookSources/0-306-41957-2"><bdi>0-306-41957-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+methods+for+the+determination+of+formation+constants&rft.place=New+York&rft.pub=Plenum+Press&rft.date=1985&rft.isbn=0-306-41957-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartellMotekaitis1992" class="citation book cs1">Martell, A. E.; Motekaitis, R. J. (1992). <i>Determination and use of stability constants</i> (2nd ed.). New York: VCH Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56081-516-7" title="Special:BookSources/1-56081-516-7"><bdi>1-56081-516-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Determination+and+use+of+stability+constants&rft.place=New+York&rft.edition=2nd&rft.pub=VCH+Publishers&rft.date=1992&rft.isbn=1-56081-516-7&rft.aulast=Martell&rft.aufirst=A.+E.&rft.au=Motekaitis%2C+R.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1973" class="citation book cs1">Bell, R. P. (1973). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/protoninchemistr0000bell"><i>The Proton in Chemistry</i></a></span> (2nd ed.). London: Chapman & Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8014-0803-2" title="Special:BookSources/0-8014-0803-2"><bdi>0-8014-0803-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Proton+in+Chemistry&rft.place=London&rft.edition=2nd&rft.pub=Chapman+%26+Hall&rft.date=1973&rft.isbn=0-8014-0803-2&rft.aulast=Bell&rft.aufirst=R.+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprotoninchemistr0000bell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Includes discussion of many organic Brønsted acids.</span> </li> <li id="cite_note-SA-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-SA_22-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShriverAtkins1999" class="citation book cs1">Shriver, D. F.; Atkins, P. W. (1999). <i>Inorganic Chemistry</i> (3rd ed.). Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850331-8" title="Special:BookSources/0-19-850331-8"><bdi>0-19-850331-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inorganic+Chemistry&rft.place=Oxford&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=0-19-850331-8&rft.aulast=Shriver&rft.aufirst=D.+F.&rft.au=Atkins%2C+P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Chapter 5: Acids and Bases</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHousecroftSharpe2008" class="citation book cs1">Housecroft, C.E.; Sharpe, A. G. (2008). <i>Inorganic Chemistry</i> (3rd ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-175553-6" title="Special:BookSources/978-0-13-175553-6"><bdi>978-0-13-175553-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inorganic+Chemistry&rft.edition=3rd&rft.pub=Prentice+Hall&rft.date=2008&rft.isbn=978-0-13-175553-6&rft.aulast=Housecroft&rft.aufirst=C.E.&rft.au=Sharpe%2C+A.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Chapter 6: Acids, Bases and Ions in Aqueous Solution</span> </li> <li id="cite_note-Headrick-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Headrick_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeadrickDikenWaltersHammer2005" class="citation journal cs1">Headrick, J. M.; Diken, E. G.; Walters, R. S.; Hammer, N. I.; Christie, A.; Cui, J.; Myshakin, E. M.; Duncan, M. A.; Johnson, M. A.; Jordan, K. D. (2005). "Spectral Signatures of Hydrated Proton Vibrations in Water Clusters". <i>Science</i>. <b>308</b> (5729): 1765–69. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005Sci...308.1765H">2005Sci...308.1765H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1113094">10.1126/science.1113094</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15961665">15961665</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40852810">40852810</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Spectral+Signatures+of+Hydrated+Proton+Vibrations+in+Water+Clusters&rft.volume=308&rft.issue=5729&rft.pages=1765-69&rft.date=2005&rft_id=info%3Adoi%2F10.1126%2Fscience.1113094&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40852810%23id-name%3DS2CID&rft_id=info%3Apmid%2F15961665&rft_id=info%3Abibcode%2F2005Sci...308.1765H&rft.aulast=Headrick&rft.aufirst=J.+M.&rft.au=Diken%2C+E.+G.&rft.au=Walters%2C+R.+S.&rft.au=Hammer%2C+N.+I.&rft.au=Christie%2C+A.&rft.au=Cui%2C+J.&rft.au=Myshakin%2C+E.+M.&rft.au=Duncan%2C+M.+A.&rft.au=Johnson%2C+M.+A.&rft.au=Jordan%2C+K.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-Smiechowski-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Smiechowski_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmiechowskiStangret2006" class="citation journal cs1">Smiechowski, M.; Stangret, J. (2006). "Proton hydration in aqueous solution: Fourier transform infrared studies of HDO spectra". <i>J. Chem. Phys</i>. <b>125</b> (20): 204508–204522. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JChPh.125t4508S">2006JChPh.125t4508S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2374891">10.1063/1.2374891</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17144716">17144716</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Chem.+Phys.&rft.atitle=Proton+hydration+in+aqueous+solution%3A+Fourier+transform+infrared+studies+of+HDO+spectra&rft.volume=125&rft.issue=20&rft.pages=204508-204522&rft.date=2006&rft_id=info%3Apmid%2F17144716&rft_id=info%3Adoi%2F10.1063%2F1.2374891&rft_id=info%3Abibcode%2F2006JChPh.125t4508S&rft.aulast=Smiechowski&rft.aufirst=M.&rft.au=Stangret%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-Burgess-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Burgess_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgess1978" class="citation book cs1">Burgess, J. (1978). <i>Metal Ions in Solution</i>. Ellis Horwood. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-85312-027-7" title="Special:BookSources/0-85312-027-7"><bdi>0-85312-027-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metal+Ions+in+Solution&rft.pub=Ellis+Horwood&rft.date=1978&rft.isbn=0-85312-027-7&rft.aulast=Burgess&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Section 9.1 "Acidity of Solvated Cations" lists many p<i>K</i><sub>a</sub> values.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLehn1995" class="citation book cs1">Lehn, J.-M. (1995). <i>Supramolecular Chemistry</i>. Wiley-VCH. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-29311-7" title="Special:BookSources/978-3-527-29311-7"><bdi>978-3-527-29311-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Supramolecular+Chemistry&rft.pub=Wiley-VCH&rft.date=1995&rft.isbn=978-3-527-29311-7&rft.aulast=Lehn&rft.aufirst=J.-M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteedAtwood2000" class="citation book cs1">Steed, J. W.; Atwood, L. J. (2000). <i>Supramolecular chemistry</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-98831-6" title="Special:BookSources/0-471-98831-6"><bdi>0-471-98831-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Supramolecular+chemistry&rft.pub=Wiley&rft.date=2000&rft.isbn=0-471-98831-6&rft.aulast=Steed&rft.aufirst=J.+W.&rft.au=Atwood%2C+L.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCattrall1997" class="citation book cs1">Cattrall, R.W. (1997). <i>Chemical sensors</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850090-4" title="Special:BookSources/0-19-850090-4"><bdi>0-19-850090-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+sensors&rft.pub=Oxford+University+Press&rft.date=1997&rft.isbn=0-19-850090-4&rft.aulast=Cattrall&rft.aufirst=R.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.elsevier.com/wps/find/journaldescription.cws_home/30921/description#description">"Drug discovery today"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">23 March</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Drug+discovery+today&rft_id=http%3A%2F%2Fwww.elsevier.com%2Fwps%2Ffind%2Fjournaldescription.cws_home%2F30921%2Fdescription%23description&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeckNagypál1990" class="citation book cs1">Beck, M.T.; Nagypál, I. (1990). <i>Chemistry of Complex Equilibria</i>. Horwood. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-85312-143-5" title="Special:BookSources/0-85312-143-5"><bdi>0-85312-143-5</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+of+Complex+Equilibria&rft.pub=Horwood&rft.date=1990&rft.isbn=0-85312-143-5&rft.aulast=Beck&rft.aufirst=M.T.&rft.au=Nagyp%C3%A1l%2C+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Section 2.2, Types of complex equilibrium constants</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartleyBurgess,_C.Alcock,_R._M.1980" class="citation book cs1">Hartley, F.R.; Burgess, C.; Alcock, R. M. (1980). <a rel="nofollow" class="external text" href="https://archive.org/details/solutionequilibr0000hart"><i>Solution equilibria</i></a>. New York (Halsted Press): Ellis Horwood. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-470-26880-8" title="Special:BookSources/0-470-26880-8"><bdi>0-470-26880-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solution+equilibria&rft.place=New+York+%28Halsted+Press%29&rft.pub=Ellis+Horwood&rft.date=1980&rft.isbn=0-470-26880-8&rft.aulast=Hartley&rft.aufirst=F.R.&rft.au=Burgess%2C+C.&rft.au=Alcock%2C+R.+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsolutionequilibr0000hart&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Atkins, Chapter 7, section "Equilibrium electrochemistry"</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Mendham, pp. 59–64</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Mendham, section 2.33, p. 63 for details</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHefterTomkins2003" class="citation book cs1">Hefter, G.T.; Tomkins, R. P. T., eds. (2003). <i>The Experimental Determination of Solubilities</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-49708-8" title="Special:BookSources/0-471-49708-8"><bdi>0-471-49708-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Experimental+Determination+of+Solubilities&rft.pub=Wiley&rft.date=2003&rft.isbn=0-471-49708-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Mendham, pp. 37–45</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkoogWestHollerCrouch2004" class="citation book cs1">Skoog, D. A.; West, D. M.; Holler, J. F.; Crouch, S. R. (2004). <i>Fundamentals of Analytical Chemistry</i> (8th ed.). Thomson Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-035523-0" title="Special:BookSources/0-03-035523-0"><bdi>0-03-035523-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Analytical+Chemistry&rft.edition=8th&rft.pub=Thomson+Brooks%2FCole&rft.date=2004&rft.isbn=0-03-035523-0&rft.aulast=Skoog&rft.aufirst=D.+A.&rft.au=West%2C+D.+M.&rft.au=Holler%2C+J.+F.&rft.au=Crouch%2C+S.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquilibrium+chemistry" class="Z3988"></span> Section 30E, Chromatographic separations</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links_2">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equilibrium_chemistry&action=edit&section=18" title="Edit section: External 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<li><a href="/wiki/Thermodynamic_activity" title="Thermodynamic activity">Thermodynamic activity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Buffer_solution" title="Buffer solution">Buffer solution</a></li> <li><a href="/wiki/Equilibrium_unfolding" title="Equilibrium unfolding">Equilibrium unfolding</a></li> <li><a href="/wiki/Liquid%E2%80%93liquid_extraction" title="Liquid–liquid extraction">Liquid–liquid extraction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Specific equilibria</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acid_dissociation_constant" title="Acid dissociation constant">Acid dissociation</a> <ul><li><a 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title="Partition equilibrium">Partition</a> <ul><li><a href="/wiki/Partition_coefficient" title="Partition coefficient">Distribution coefficient</a></li></ul></li> <li><a href="/wiki/Solubility_equilibrium" title="Solubility equilibrium">Solubility</a> <ul><li><a href="/wiki/Common-ion_effect" title="Common-ion effect">Common-ion effect</a></li></ul></li> <li><a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">Vapor–liquid</a> <ul><li><a href="/wiki/Henry%27s_law" title="Henry's law">Henry's law</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Branches_of_chemistry" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" 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href="/wiki/Timeline_of_chemistry" title="Timeline of chemistry">Timeline of chemistry</a> <ul><li><a href="/wiki/Discovery_of_chemical_elements" title="Discovery of chemical elements">of element discoveries</a></li></ul></li> <li>"<a href="/wiki/The_central_science" title="The central science">The central science</a>"</li> <li><a href="/wiki/Chemical_reaction" title="Chemical reaction">Chemical reaction</a> <ul><li><a href="/wiki/Catalysis" title="Catalysis">Catalysis</a></li></ul></li> <li><a href="/wiki/Chemical_element" title="Chemical element">Chemical element</a></li> <li><a href="/wiki/Chemical_compound" title="Chemical compound">Chemical compound</a></li> <li><a href="/wiki/Atom" title="Atom">Atom</a></li> <li><a href="/wiki/Molecule" title="Molecule">Molecule</a></li> <li><a href="/wiki/Ion" title="Ion">Ion</a></li> <li><a href="/wiki/Chemical_substance" title="Chemical substance">Chemical substance</a></li> <li><a href="/wiki/Chemical_bond" title="Chemical bond">Chemical bond</a></li> <li><a href="/wiki/Alchemy" title="Alchemy">Alchemy</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Chemistry" title="Category:Chemistry">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" 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Equilibrium chemistry - Wikipedia