Théorème de Gibbs-Konovalov

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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><p>En <a href="/wiki/Chimie_physique" title="Chimie physique">chimie physique</a>, le <b>théorème de Gibbs-Konovalov</b> est un théorème relatif aux <a href="/wiki/%C3%89quilibre_de_phases" title="Équilibre de phases">équilibres de phases</a>, notamment aux <a href="/wiki/Az%C3%A9otrope" title="Azéotrope">azéotropes</a> des équilibres liquide-vapeur et aux <a href="/wiki/Point_de_fusion_congruent" title="Point de fusion congruent">points de fusion congruents</a> des équilibres liquide-solide. </p><p><a href="/wiki/Willard_Gibbs" title="Willard Gibbs">Willard Gibbs</a> établit ce théorème de façon purement théorique en 1876. Il fut vérifié expérimentalement par <a href="/wiki/Dmitri_Petrovitch_Konovalov" title="Dmitri Petrovitch Konovalov">Dmitri Petrovitch Konovalov</a>, un élève de <a href="/wiki/Dmitri_Mendele%C3%AFev" title="Dmitri Mendeleïev">Dmitri Mendeleïev</a>, dans sa thèse en 1881 puis dans une publication en 1884 relatives aux équilibres liquide-vapeur<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. La <a href="/wiki/R%C3%A8gles_de_Konovalov" title="Règles de Konovalov">deuxième règle de Konovalov</a> s'énonce ainsi : <span class="citation">« À température constante, aux points d'<a href="/wiki/Extremum" title="Extremum">extrémum</a> de la pression de vapeur les <a href="/wiki/Composition_chimique" title="Composition chimique">compositions</a> de la vapeur et du liquide sont identiques. »</span> </p><p>Le théorème de Gibbs-Konovalov est une généralisation de cette règle aux points d'extrémum de pression et de température et à tous les équilibres de <a href="/wiki/Phase_(thermodynamique)" title="Phase (thermodynamique)">phases</a> : </p> <blockquote><div> <p>« Aux points d'extrémum de pression ou de température, les phases en équilibre ont la même composition. » </p> </div></blockquote><p style="margin: -0.7em 0 0.3em 6em">— Théorème de Gibbs-Konovalov</p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Illustration">Illustration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&veaction=edit&section=1" title="Modifier la section : Illustration" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&action=edit&section=1" title="Modifier le code source de la section : Illustration"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <span id="Figure_1"><figure typeof="mw:File/Thumb"><a href="/wiki/Fichier:NegativeAzeotropePhaseDiagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/NegativeAzeotropePhaseDiagram.png/350px-NegativeAzeotropePhaseDiagram.png" decoding="async" width="350" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/NegativeAzeotropePhaseDiagram.png/525px-NegativeAzeotropePhaseDiagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/27/NegativeAzeotropePhaseDiagram.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption><b>Figure 1</b> - Diagramme de phase d'un <a href="/wiki/Az%C3%A9otrope" title="Azéotrope">azéotrope négatif</a>. Un azéotrope, positif ou négatif, répond au théorème de Gibbs-Konovalov.</figcaption></figure></span> <p>La <a href="#Figure_1">figure 1</a> montre un <a href="/wiki/Diagramme_de_phases" class="mw-redirect" title="Diagramme de phases">diagramme de phases</a> liquide-vapeur d'un mélange binaire à pression constante : la température y est tracée en fonction de la composition. Ce diagramme présente un <a href="/wiki/Az%C3%A9otrope" title="Azéotrope">azéotrope</a> sous forme d'un maximum de température, commun aux courbes de bulle et de rosée, auquel le liquide et la vapeur ont la même composition. Ceci s'applique également aux azéotropes dans lesquels la température atteint un minimum à pression constante. De même, ceci s'applique aux azéotropes où la pression atteint un minimum ou un maximum à température constante. </p> <span id="Figure_2"><figure typeof="mw:File/Thumb"><a href="/wiki/Fichier:Diagramme-phases-sio2-al2o3.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Diagramme-phases-sio2-al2o3.jpg/350px-Diagramme-phases-sio2-al2o3.jpg" decoding="async" width="350" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Diagramme-phases-sio2-al2o3.jpg/525px-Diagramme-phases-sio2-al2o3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Diagramme-phases-sio2-al2o3.jpg/700px-Diagramme-phases-sio2-al2o3.jpg 2x" data-file-width="2765" data-file-height="2000" /></a><figcaption><b>Figure 2</b> - Diagramme de phase du système <a href="/wiki/Dioxyde_de_silicium" title="Dioxyde de silicium">silice</a>-<a href="/wiki/Alumine" title="Alumine">alumine</a>. Le maximum local indiqué « 1850 » est un <a href="/wiki/Point_de_fusion_congruent" title="Point de fusion congruent">point de fusion congruent</a>, il répond au théorème de Gibbs-Konovalov. Les deux <a href="/wiki/Point_anguleux" class="mw-redirect" title="Point anguleux">points anguleux</a> situés à environ 10 <abbr class="abbr" title="pourcent">%</abbr> et 75 <abbr class="abbr" title="pourcent">%</abbr> de silice sont des <a href="/wiki/Eutectique" title="Eutectique">eutectiques</a> qui ne répondent pas à ce théorème.</figcaption></figure></span> <p>Ceci se généralise à toutes les transitions de phase. La <a href="#Figure_2">figure 2</a> montre un diagramme de phase liquide-solide : le point indiqué « 1850 » répond au théorème de Gibbs-Konovalov, il s'agit d'un extrémum local, en l'occurrence d'un maximum de température local, appelé <a href="/wiki/Point_de_fusion_congruent" title="Point de fusion congruent">point de fusion congruent</a>, auquel le liquide et le solide ont la même composition. Un point de fusion congruent ne doit pas être confondu avec un <a href="/wiki/Eutectique" title="Eutectique">eutectique</a>, point auquel le liquide et le solide sont également de même composition, mais qui est un <a href="/wiki/Point_anguleux" class="mw-redirect" title="Point anguleux">point anguleux</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Démonstration"><span id="D.C3.A9monstration"></span>Démonstration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&veaction=edit&section=2" title="Modifier la section : Démonstration" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&action=edit&section=2" title="Modifier le code source de la section : Démonstration"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On considère un mélange binaire (ne contenant que deux <a href="/wiki/Esp%C3%A8ce_chimique" title="Espèce chimique">espèces chimiques</a>, notées <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 {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"></span> et <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 {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"></span>) à l'<a href="/wiki/%C3%89quilibre_de_phases" title="Équilibre de phases">équilibre de deux phases</a> quelconques (notées <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> et <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>). Dans cet état on note : </p> <ul><li><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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> la <a href="/wiki/Pression" title="Pression">pression</a> ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> la <a href="/wiki/Temp%C3%A9rature" title="Température">température</a> ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae70fe9166cbdf0892a5d5c3fc27ce9c30a6f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.795ex; height:2.843ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{B}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{B}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60a1ef5441c54e7cdd8e2445f6a655bce00a464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.726ex; height:2.843ex;" alt="{\displaystyle x_{\text{B}}^{\alpha }}"></span> les <a href="/wiki/Fraction_molaire" title="Fraction molaire">fractions molaires</a> respectives des corps <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 {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"></span> et <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 {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"></span> en phase <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/068c70f0ddcd03c4a56e4d0f1591f7edb52ea1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.795ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\beta }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe3b6d66068439aa43e91c00e98a7fcd7a047da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.726ex; height:3.509ex;" alt="{\displaystyle x_{\text{B}}^{\beta }}"></span> les fractions molaires respectives des corps <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 {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"></span> et <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 {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"></span> en phase <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{A}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{A}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e6de8ec8a8d8e63dac8ff305d1b6247e59ee45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mu _{\text{A}}^{\alpha }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{B}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{B}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/297847fca7d5f2d37f4c3b52afc2e0b4c5005c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.798ex; height:2.843ex;" alt="{\displaystyle \mu _{\text{B}}^{\alpha }}"></span> les <a href="/wiki/Potentiel_chimique" title="Potentiel chimique">potentiels chimiques</a> respectifs des corps <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 {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"></span> et <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 {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"></span> en phase <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> ;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64a0cfdb6b767e04451552855f1ff11e2450e06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:3.509ex;" alt="{\displaystyle \mu _{\text{A}}^{\beta }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291f9ffe446bed14ffdfd4378b7206de665330ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.798ex; height:3.509ex;" alt="{\displaystyle \mu _{\text{B}}^{\beta }}"></span> les potentiels chimiques respectifs des corps <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 {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"></span> et <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 {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"></span> en phase <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>.</li></ul> <p>Puisque l'on est à l'équilibre des <a href="/wiki/Phase_(thermodynamique)" title="Phase (thermodynamique)">phases</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> et <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, on a les relations entre potentiels chimiques : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{A}}^{\alpha }=\mu _{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{A}}^{\alpha }=\mu _{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf186bcf600f00af3b52cd35ebe325285d4984c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.831ex; height:3.509ex;" alt="{\displaystyle \mu _{\text{A}}^{\alpha }=\mu _{\text{A}}^{\beta }}"></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 \mu _{\text{B}}^{\alpha }=\mu _{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{B}}^{\alpha }=\mu _{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2924183c2f31f456475445760399bcdc0a0e6eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.694ex; height:3.509ex;" alt="{\displaystyle \mu _{\text{B}}^{\alpha }=\mu _{\text{B}}^{\beta }}"></span></dd></dl> <p>Si l'on fait varier la température de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> à <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 {d} T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T+\mathrm {d} T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c185ad5e95527cf268be4957e2607e73df9965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.405ex; height:2.343ex;" alt="{\displaystyle T+\mathrm {d} T}"></span> ou la pression de <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> à <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 P+\mathrm {d} P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P+\mathrm {d} P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/634d9f576076548c7a14a60c40190bed37f60057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.624ex; height:2.343ex;" alt="{\displaystyle P+\mathrm {d} P}"></span>, les potentiels chimiques varient. Si l'on reste à l'équilibre des phases, on a au nouvel équilibre : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{A}}^{\alpha }+\mathrm {d} \mu _{\text{A}}^{\alpha }=\mu _{\text{A}}^{\beta }+\mathrm {d} \mu _{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{A}}^{\alpha }+\mathrm {d} \mu _{\text{A}}^{\alpha }=\mu _{\text{A}}^{\beta }+\mathrm {d} \mu _{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c86a4218d3da9b560306bc04ff5ed8c658d749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.83ex; height:3.509ex;" alt="{\displaystyle \mu _{\text{A}}^{\alpha }+\mathrm {d} \mu _{\text{A}}^{\alpha }=\mu _{\text{A}}^{\beta }+\mathrm {d} \mu _{\text{A}}^{\beta }}"></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 \mu _{\text{B}}^{\alpha }+\mathrm {d} \mu _{\text{B}}^{\alpha }=\mu _{\text{B}}^{\beta }+\mathrm {d} \mu _{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{B}}^{\alpha }+\mathrm {d} \mu _{\text{B}}^{\alpha }=\mu _{\text{B}}^{\beta }+\mathrm {d} \mu _{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf0b01629568410abb5e717c599edd16e290fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.554ex; height:3.509ex;" alt="{\displaystyle \mu _{\text{B}}^{\alpha }+\mathrm {d} \mu _{\text{B}}^{\alpha }=\mu _{\text{B}}^{\beta }+\mathrm {d} \mu _{\text{B}}^{\beta }}"></span></dd></dl> <p>D'où les relations entre variations des potentiels chimiques : </p> <dl><dd>(<span id="n_equa_1a">1a</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 \mathrm {d} \mu _{\text{A}}^{\alpha }=\mathrm {d} \mu _{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mu _{\text{A}}^{\alpha }=\mathrm {d} \mu _{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8c02ea23ce079217f60838282ae7f837e67592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.417ex; height:3.509ex;" alt="{\displaystyle \mathrm {d} \mu _{\text{A}}^{\alpha }=\mathrm {d} \mu _{\text{A}}^{\beta }}"></span></dd> <dd>(<span id="n_equa_1b">1b</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 \mathrm {d} \mu _{\text{B}}^{\alpha }=\mathrm {d} \mu _{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mu _{\text{B}}^{\alpha }=\mathrm {d} \mu _{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbd31e32b094d88cb8d6e4cbc5c6d8953054e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.279ex; height:3.509ex;" alt="{\displaystyle \mathrm {d} \mu _{\text{B}}^{\alpha }=\mathrm {d} \mu _{\text{B}}^{\beta }}"></span></dd></dl> <p>La <a href="/wiki/Relation_de_Gibbs-Duhem" title="Relation de Gibbs-Duhem">relation de Gibbs-Duhem</a> lie les variations des diverses grandeurs pour chacune des deux phases : </p> <dl><dd>(<span id="n_equa_2">2</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 {\bar {V}}^{\alpha }\,\mathrm {d} P-{\bar {S}}^{\alpha }\,\mathrm {d} T=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {V}}^{\alpha }\,\mathrm {d} P-{\bar {S}}^{\alpha }\,\mathrm {d} T=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee02485eabdac93115f00a94dbb84b95515f1456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.992ex; height:3.343ex;" alt="{\displaystyle {\bar {V}}^{\alpha }\,\mathrm {d} P-{\bar {S}}^{\alpha }\,\mathrm {d} T=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }}"></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 {\bar {V}}^{\beta }\,\mathrm {d} P-{\bar {S}}^{\beta }\,\mathrm {d} T=x_{\text{A}}^{\beta }\,\mathrm {d} \mu _{\text{A}}^{\beta }+x_{2}^{\beta }\,\mathrm {d} \mu _{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {V}}^{\beta }\,\mathrm {d} P-{\bar {S}}^{\beta }\,\mathrm {d} T=x_{\text{A}}^{\beta }\,\mathrm {d} \mu _{\text{A}}^{\beta }+x_{2}^{\beta }\,\mathrm {d} \mu _{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec7d8d162e16d13b684d4a21dad8af7346a3b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.55ex; height:3.843ex;" alt="{\displaystyle {\bar {V}}^{\beta }\,\mathrm {d} P-{\bar {S}}^{\beta }\,\mathrm {d} T=x_{\text{A}}^{\beta }\,\mathrm {d} \mu _{\text{A}}^{\beta }+x_{2}^{\beta }\,\mathrm {d} \mu _{\text{B}}^{\beta }}"></span></dd></dl> <p>avec : </p> <ul><li><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 {V}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {V}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a662131be8a5d1ec9e6644903b1085e0e3e65c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.071ex; height:2.676ex;" alt="{\displaystyle {\bar {V}}^{\alpha }}"></span> et <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 {V}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {V}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed6624495fcec22f6497992b2c49422bc6b46a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.961ex; height:3.009ex;" alt="{\displaystyle {\bar {V}}^{\beta }}"></span> les <a href="/wiki/Volume_molaire" title="Volume molaire">volumes molaires</a> respectifs des phases <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> et <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> ;</li> <li><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 {S}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2356d9731ab14a9f0b6dbf9ec88ecc51c4ff1fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.857ex; height:2.676ex;" alt="{\displaystyle {\bar {S}}^{\alpha }}"></span> et <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 {S}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193657e39d8d98c574a2bb941cb9985d487f047b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.747ex; height:3.176ex;" alt="{\displaystyle {\bar {S}}^{\beta }}"></span> les <a href="/wiki/Entropie_(thermodynamique)" title="Entropie (thermodynamique)">entropies</a> <a href="/wiki/Grandeur_molaire" title="Grandeur molaire">molaires</a> respectives des phases <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> et <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>.</li></ul> <p>En effectuant la différence des deux relations on obtient : </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({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\,\mathrm {d} \mu _{\text{A}}^{\beta }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }\,\mathrm {d} \mu _{\text{B}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\,\mathrm {d} \mu _{\text{A}}^{\beta }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }\,\mathrm {d} \mu _{\text{B}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d505a222db7ed0f6cb0fc488a4e97ff07e887a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:73.93ex; height:4.843ex;" alt="{\displaystyle \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\,\mathrm {d} \mu _{\text{A}}^{\beta }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }\,\mathrm {d} \mu _{\text{B}}^{\beta }}"></span></dd></dl> <p>En introduisant les relations (<a href="#n_equa_1a">1a</a>) et (<a href="#n_equa_1b">1b</a>) dans la relation précédente, on obtient : </p> <dl><dd>(<span id="n_equa_3">3</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 \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\mathrm {d} \mu _{\text{A}}^{\alpha }+\left(x_{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }\right)\,\mathrm {d} \mu _{\text{B}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\mathrm {d} \mu _{\text{A}}^{\alpha }+\left(x_{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }\right)\,\mathrm {d} \mu _{\text{B}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9598b4c934785d80f76f08c43f9930004c17a9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:71.232ex; height:4.843ex;" alt="{\displaystyle \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\mathrm {d} \mu _{\text{A}}^{\alpha }+\left(x_{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }\right)\,\mathrm {d} \mu _{\text{B}}^{\alpha }}"></span></dd></dl> <p>Les contraintes sur les fractions molaires : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8275f44cb4fb4a35cd146498d5037dd91117104c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.621ex; height:2.843ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }=1}"></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 x_{\text{A}}^{\beta }+x_{\text{B}}^{\beta }=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\beta }+x_{\text{B}}^{\beta }=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07001856a8450259c9ace383d4eb5e3e74664755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.621ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\beta }+x_{\text{B}}^{\beta }=1}"></span></dd></dl> <p>permettent d'écrire : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }=\left(1-x_{\text{A}}^{\alpha }\right)-\left(1-x_{\text{A}}^{\beta }\right)=-\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }=\left(1-x_{\text{A}}^{\alpha }\right)-\left(1-x_{\text{A}}^{\beta }\right)=-\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff49e66ddaecdd2515f462f4b01073e00dc886b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.229ex; height:4.843ex;" alt="{\displaystyle x_{\text{B}}^{\alpha }-x_{\text{B}}^{\beta }=\left(1-x_{\text{A}}^{\alpha }\right)-\left(1-x_{\text{A}}^{\beta }\right)=-\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)}"></span></dd></dl> <p>que l'on introduit dans (<a href="#n_equa_3">3</a>) : </p> <center><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({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\left(\mathrm {d} \mu _{\text{A}}^{\alpha }-\mathrm {d} \mu _{\text{B}}^{\alpha }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\left(\mathrm {d} \mu _{\text{A}}^{\alpha }-\mathrm {d} \mu _{\text{B}}^{\alpha }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a188a3e75a14fab85b5e813d3d2188499d52d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:61.521ex; height:4.843ex;" alt="{\displaystyle \left({\bar {V}}^{\alpha }-{\bar {V}}^{\beta }\right)\,\mathrm {d} P-\left({\bar {S}}^{\alpha }-{\bar {S}}^{\beta }\right)\,\mathrm {d} T=\left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\left(\mathrm {d} \mu _{\text{A}}^{\alpha }-\mathrm {d} \mu _{\text{B}}^{\alpha }\right)}"></span></center> <p>Aussi, si l'on travaille à température constante (<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 \mathrm {d} T=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} T=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79e2fc50f4f86592a6af570453921d6b046bf6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.19ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} T=0}"></span>) un extrémum de pression (<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 \mathrm {d} P=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} P=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a406e07142b948d80b1915e1b269436c425d90b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.299ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} P=0}"></span>) conduit à l'égalité : </p> <dl><dd>(<span id="n_equa_4">4</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 \left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\left(\mathrm {d} \mu _{\text{A}}^{\alpha }-\mathrm {d} \mu _{\text{B}}^{\alpha }\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\left(\mathrm {d} \mu _{\text{A}}^{\alpha }-\mathrm {d} \mu _{\text{B}}^{\alpha }\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f8d4ac1eb99c89611e05ce309c8b4cd6a27d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.459ex; height:4.843ex;" alt="{\displaystyle \left(x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }\right)\,\left(\mathrm {d} \mu _{\text{A}}^{\alpha }-\mathrm {d} \mu _{\text{B}}^{\alpha }\right)=0}"></span></dd></dl> <p>La relation de Gibbs-Duhem (<a href="#n_equa_2">2</a>) à température constante et à un extrémum de pression donne : </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 0=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b408a960d0cda98911cdb88675429cdae45345fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.645ex; height:2.843ex;" alt="{\displaystyle 0=x_{\text{A}}^{\alpha }\,\mathrm {d} \mu _{\text{A}}^{\alpha }+x_{\text{B}}^{\alpha }\,\mathrm {d} \mu _{\text{B}}^{\alpha }}"></span></dd></dl> <p>La relation (<a href="#n_equa_4">4</a>) devient : </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta } \over x_{\text{B}}^{\alpha }}\,\mathrm {d} \mu _{\text{A}}^{\alpha }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta } \over x_{\text{B}}^{\alpha }}\,\mathrm {d} \mu _{\text{A}}^{\alpha }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b91e6f7ee5ce0236cae3aa571b0542297e04be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.073ex; height:7.176ex;" alt="{\displaystyle {x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta } \over x_{\text{B}}^{\alpha }}\,\mathrm {d} \mu _{\text{A}}^{\alpha }=0}"></span></center> <p>Il n'y a aucune raison pour qu'à l'extrémum le potentiel chimique ait lui-même atteint un extrémum : <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 \mathrm {d} \mu _{\text{A}}^{\alpha }\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mu _{\text{A}}^{\alpha }\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfcf4af26a3142be56624290531031a08a3b527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.42ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} \mu _{\text{A}}^{\alpha }\neq 0}"></span>. Par conséquent l'extrémum de pression à température constante est atteint si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a83604f90e76482fbc440fbb8fe23e232d75df06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.69ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }-x_{\text{A}}^{\beta }=0}"></span>, soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24885aef99827d339c04c41de1180c2fc9452142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.687ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}"></span>. Le raisonnement est identique si l'on travaille à pression constante (<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 \mathrm {d} P=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} P=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a406e07142b948d80b1915e1b269436c425d90b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.299ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} P=0}"></span>) : un extrémum de température (<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 \mathrm {d} T=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} T=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79e2fc50f4f86592a6af570453921d6b046bf6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.19ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} T=0}"></span>) ne peut être atteint que si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24885aef99827d339c04c41de1180c2fc9452142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.687ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}"></span>. Puisque l'on considère un mélange binaire, on exclut le cas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27edf6b9d95a04d2449711acacc0346e81d8a5b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.948ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }=1}"></span>, c'est-à-dire le corps <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 {\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d98083f2be1ce5c681190df371d29455c2d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle {\text{A}}}"></span> pur, et le cas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ce969f59cc1ade9934b7393d3bc6d0e1abc959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.948ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }=0}"></span>, c'est-à-dire le corps <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 {\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37cc047f094c36c6d6e6f5dbf1700583b6f66938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle {\text{B}}}"></span> pur. Un extrémum de pression ou de température d'équilibre de deux phases d'un mélange binaire ne peut donc exister que si les deux phases ont la même composition, d'où le théorème de Gibbs-Konovalov<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. </p><p>La réciproque est également vraie. À l'équilibre les propriétés des deux phases ne sont pas égales : <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 {V}}^{\alpha }\neq {\bar {V}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>≠</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {V}}^{\alpha }\neq {\bar {V}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c941fd0471d10bb8349af57bafdbfef71eb2729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.509ex;" alt="{\displaystyle {\bar {V}}^{\alpha }\neq {\bar {V}}^{\beta }}"></span> et <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 {S}}^{\alpha }\neq {\bar {S}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>≠</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}^{\alpha }\neq {\bar {S}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3d5a0734ad27a507e2df4b513769246d64a29a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.702ex; height:3.676ex;" alt="{\displaystyle {\bar {S}}^{\alpha }\neq {\bar {S}}^{\beta }}"></span>. Aussi, si les deux phases ont la même composition, soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24885aef99827d339c04c41de1180c2fc9452142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.687ex; height:3.509ex;" alt="{\displaystyle x_{\text{A}}^{\alpha }=x_{\text{A}}^{\beta }}"></span>, à température constante, soit <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 \mathrm {d} T=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} T=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79e2fc50f4f86592a6af570453921d6b046bf6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.19ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} T=0}"></span>, alors la pression est à un extrémum, soit <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 \mathrm {d} P=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>P</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} P=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a406e07142b948d80b1915e1b269436c425d90b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.299ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} P=0}"></span>. De même qu'à pression constante l'égalité des compositions de phases induit un extrémum de température. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&veaction=edit&section=3" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&action=edit&section=3" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&veaction=edit&section=4" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&action=edit&section=4" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Hertz2004"><span class="ouvrage" id="Jean_Hertz2004">Jean Hertz, « <cite style="font-style:normal">Historique en grandes enjambées de la thermodynamique de l’équilibre</cite> », <i>J. Phys. IV France</i>, <abbr class="abbr" title="volume">vol.</abbr> 122,‎ <time>2004</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">3-20</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://jp4.journaldephysique.org/articles/jp4/pdf/2004/10/jp4122001.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>, consulté le <time class="nowrap" datetime="2019-07-28" data-sort-value="2019-07-28">28 juillet 2019</time>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Historique+en+grandes+enjamb%C3%A9es+de+la+thermodynamique+de+l%E2%80%99%C3%A9quilibre&rft.jtitle=J.+Phys.+IV+France&rft.aulast=Hertz&rft.aufirst=Jean&rft.date=2004&rft.volume=122&rft.pages=3-20&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9or%C3%A8me+de+Gibbs-Konovalov"></span></span></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="GoodmanCahnBennett1981"><span class="ouvrage" id="David_A._GoodmanJohn_W._CahnLawrence_H._Bennett1981"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> David A. Goodman, John W. Cahn et Lawrence H. Bennett, « <cite style="font-style:normal" lang="en">The Centennial of the Gibbs-Konovalov Rule for Congruent Points : Its Underlying Theoretical Basis and Its Application to Phase Diagram Evaluation</cite> », <i><span class="lang-en" lang="en">Bulletin of Alloy Phase Diagrams</span></i>, <abbr class="abbr" title="volume">vol.</abbr> 2, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1,‎ <time>1981</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">29-34</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1007/BF02873696.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>, consulté le <time class="nowrap" datetime="2019-07-28" data-sort-value="2019-07-28">28 juillet 2019</time>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Centennial+of+the+Gibbs-Konovalov+Rule+for+Congruent+Points&rft.jtitle=Bulletin+of+Alloy+Phase+Diagrams&rft.issue=1&rft.stitle=Its+Underlying+Theoretical+Basis+and+Its+Application+to+Phase+Diagram+Evaluation&rft.aulast=Goodman&rft.aufirst=David+A.&rft.au=John+W.+Cahn&rft.au=Lawrence+H.+Bennett&rft.date=1981&rft.volume=2&rft.pages=29-34&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9or%C3%A8me+de+Gibbs-Konovalov"></span></span></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><a href="#Fosset">Fosset <i><abbr class="abbr" title="et alii (« et d’autres »)" lang="la">et al.</abbr></i> 2017</a>, <abbr class="abbr" title="page(s)">p.</abbr> 251.</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&veaction=edit&section=5" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&action=edit&section=5" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Fosset">Bruno Fosset, Jean-Bernard Baudin, Frédéric Lahitète et Valéry Prévost, <cite class="italique">Chimie tout-en-un PSI-PSI* : Le cours de référence</cite>, <a href="/wiki/%C3%89ditions_Dunod" title="Éditions Dunod">Dunod</a>, <time class="nowrap" datetime="2017-08" data-sort-value="2017-08">août 2017</time>, <abbr class="abbr" title="troisième">3<sup>e</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr>, 400 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-10-077191-2" title="Spécial:Ouvrages de référence/978-2-10-077191-2"><span class="nowrap">978-2-10-077191-2</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=bUowDwAAQBAJ&pg=PA251">lire en ligne</a>)</small>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">250-251</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chimie+tout-en-un+PSI-PSI%2A&rft.pub=Dunod&rft.edition=3&rft.stitle=Le+cours+de+r%C3%A9f%C3%A9rence&rft.au=Bruno+Fosset&rft.au=Jean-Bernard+Baudin&rft.au=Fr%C3%A9d%C3%A9ric+Lahit%C3%A8te&rft.au=Val%C3%A9ry+Pr%C3%A9vost&rft.date=2017-08&rft.pages=250-251&rft.tpages=400&rft.isbn=978-2-10-077191-2&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9or%C3%A8me+de+Gibbs-Konovalov"></span></span>.</li> <li><span class="ouvrage" id="Vidal1997"><span class="ouvrage" id="Jean_Vidal1997">Jean Vidal, <cite class="italique">Thermodynamique : application au génie chimique et à l'industrie pétrolière</cite>, Éditions TECHNIP, <abbr class="abbr" title="collection">coll.</abbr> « Publications de l'Institut français du pétrole », <time>1997</time>, 500 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-7108-0715-5" title="Spécial:Ouvrages de référence/978-2-7108-0715-5"><span class="nowrap">978-2-7108-0715-5</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=1zqZCgAAQBAJ&pg=PA190">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 190<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamique&rft.pub=%C3%89ditions+TECHNIP&rft.stitle=application+au+g%C3%A9nie+chimique+et+%C3%A0+l%27industrie+p%C3%A9troli%C3%A8re&rft.aulast=Vidal&rft.aufirst=Jean&rft.date=1997&rft.pages=190&rft.tpages=500&rft.isbn=978-2-7108-0715-5&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9or%C3%A8me+de+Gibbs-Konovalov"></span></span></span>.</li> <li><span class="ouvrage" id="Soustelle2016"><span class="ouvrage" id="Michel_Soustelle2016">Michel Soustelle, <cite class="italique">Transformation entre phases</cite>, <abbr class="abbr" title="volume">vol.</abbr> 5, Londres, ISTE Editions, <abbr class="abbr" title="collection">coll.</abbr> « Thermodynamique chimique approfondie », <time class="nowrap" datetime="2016-02" data-sort-value="2016-02">février 2016</time>, 240 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-1-78405-123-5" title="Spécial:Ouvrages de référence/978-1-78405-123-5"><span class="nowrap">978-1-78405-123-5</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=KwtjDwAAQBAJ&pg=PA66">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 66<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transformation+entre+phases&rft.place=Londres&rft.pub=ISTE+Editions&rft.aulast=Soustelle&rft.aufirst=Michel&rft.date=2016-02&rft.volume=5&rft.pages=66&rft.tpages=240&rft.isbn=978-1-78405-123-5&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9or%C3%A8me+de+Gibbs-Konovalov"></span></span></span>.</li> <li><span class="ouvrage" id="Mesplède2004"><span class="ouvrage" id="J._Mesplède2004">J. Mesplède, <cite class="italique">Chimie : Thermodynamique Matériaux PC</cite>, <a href="/wiki/%C3%89ditions_Br%C3%A9al" title="Éditions Bréal">Éditions Bréal</a>, <abbr class="abbr" title="collection">coll.</abbr> « Les nouveaux précis Bréal », <time>2004</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-7495-2064-3" title="Spécial:Ouvrages de référence/978-2-7495-2064-3"><span class="nowrap">978-2-7495-2064-3</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=uE4UgF41cXYC&pg=PA136">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 136<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chimie&rft.pub=%C3%89ditions+Br%C3%A9al&rft.stitle=Thermodynamique+Mat%C3%A9riaux+PC&rft.aulast=Mespl%C3%A8de&rft.aufirst=J.&rft.date=2004&rft.pages=136&rft.isbn=978-2-7495-2064-3&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9or%C3%A8me+de+Gibbs-Konovalov"></span></span></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&veaction=edit&section=6" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Gibbs-Konovalov&action=edit&section=6" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Az%C3%A9otrope" title="Azéotrope">Azéotrope</a></li> <li><a href="/wiki/Point_de_fusion_congruent" title="Point de fusion congruent">Point de fusion congruent</a></li> <li><a href="/wiki/R%C3%A8gles_de_Konovalov" title="Règles de Konovalov">Règles de Konovalov</a></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="3" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_%C3%89tat_de_la_mati%C3%A8re" title="Modèle:Palette État de la matière"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_%C3%89tat_de_la_mati%C3%A8re&action=edit"><abbr class="abbr" title="Modifier ce modèle. 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src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Phase_change_-_fr.svg/100px-Phase_change_-_fr.svg.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Phase_change_-_fr.svg/150px-Phase_change_-_fr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Phase_change_-_fr.svg/200px-Phase_change_-_fr.svg.png 2x" data-file-width="750" data-file-height="720" /></a></span></td> </tr> <tr> <th class="navbox-group" style="width:120px">Basse température</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Condensat_de_Bose-Einstein" title="Condensat de Bose-Einstein">Condensat de Bose-Einstein</a></li> <li><a href="/wiki/Condensat_fermionique" title="Condensat fermionique">Condensat fermionique</a></li> <li><a href="/wiki/Superfluidit%C3%A9" title="Superfluidité">Superfluidité</a></li> <li><a href="/wiki/Supersolide" 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href="/wiki/Point_d%27%C3%A9bullition" title="Point d'ébullition">Point d'ébullition</a></li> <li><a href="/wiki/Point_de_sublimation" title="Point de sublimation">Point de sublimation</a></li> <li><a href="/wiki/Formule_de_Clapeyron" title="Formule de Clapeyron">Formule de Clapeyron</a></li> <li><a href="/wiki/Formules_d%27Ehrenfest" title="Formules d'Ehrenfest">Formules d'Ehrenfest</a></li></ul></li> <li><a href="/wiki/Enthalpie_de_changement_d%27%C3%A9tat" title="Enthalpie de changement d'état">Enthalpie de changement d'état</a> <ul><li><a href="/wiki/Enthalpie_de_fusion" title="Enthalpie de fusion">de fusion</a></li> <li><a href="/wiki/Enthalpie_de_sublimation" title="Enthalpie de sublimation">de sublimation</a></li> <li><a href="/wiki/Enthalpie_de_vaporisation" title="Enthalpie de vaporisation">de vaporisation</a></li></ul></li> <li><a class="mw-selflink selflink">Théorème de Gibbs-Konovalov</a> <ul><li><a href="/wiki/Az%C3%A9otrope" title="Azéotrope">Azéotrope</a></li> <li><a href="/wiki/Point_de_fusion_congruent" title="Point de fusion congruent">Point de fusion congruent</a></li></ul></li> <li><a href="/wiki/%C3%89quilibre_liquide-vapeur" title="Équilibre liquide-vapeur">Équilibre liquide-vapeur</a> <ul><li><a href="/wiki/Loi_de_Raoult" title="Loi de Raoult">Loi de Raoult</a></li> <li><a href="/wiki/Loi_de_Henry" title="Loi de Henry">Loi de Henry</a></li></ul></li> <li>Équilibre liquide-solide <ul><li><a href="/wiki/Eutectique" title="Eutectique">Eutectique</a></li> <li><a href="/wiki/%C3%89quation_de_Schr%C3%B6der-van_Laar" title="Équation de Schröder-van Laar">Équation de Schröder-van Laar</a></li></ul></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Chimie" title="Portail de la chimie"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_science.svg/24px-Nuvola_apps_edu_science.svg.png" 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Théorème de Gibbs-Konovalov — Wikipédia