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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Großkanonisches Potential</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>Großkanonische Potential</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> (Formelzeichen z. T. auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{G}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi _{G}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c01aaac64f6b886bc546410cbc984e2ebe85b9d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.202ex; height:2.509ex;" alt="{\displaystyle \Phi _{G}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
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<annotation encoding="application/x-tex">{\displaystyle J}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span>; auch <b>Landau-Potential</b> nach <a href="Lew_Landau" class="mw-redirect" title="Lew Landau">Lew Landau</a>) ist ein in der <a href="Statistische_Mechanik" title="Statistische Mechanik">Statistischen Mechanik</a> verwendetes <a href="Thermodynamisches_Potential" title="Thermodynamisches Potential">thermodynamisches Potential</a>, welches vorwiegend für <a href="Irreversibler_Prozess" title="Irreversibler Prozess">irreversible Prozesse</a> <a href="Offenes_System_(Thermodynamik)" class="mw-redirect" title="Offenes System (Thermodynamik)">offener Systeme</a> verwendet wird. Es ist das angepasste thermodynamische Potential für das <a href="Gro%C3%9Fkanonisches_Ensemble" title="Großkanonisches Ensemble">μVT-Ensemble</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Das Großkanonische Potential ist definiert durch:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega :=F-\mu N=U-TS-\mu N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>:=</mo>
<mi>F</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mi>N</mi>
<mo>=</mo>
<mi>U</mi>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega :=F-\mu N=U-TS-\mu N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b3fdc58c3672bc6c5d45d6b01f06eaac7a21a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.632ex; height:2.676ex;" alt="{\displaystyle \Omega :=F-\mu N=U-TS-\mu N}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li>der <a href="Freie_Energie" title="Freie Energie">freien Energie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span></li>
<li>dem <a href="Chemisches_Potential" title="Chemisches Potential">chemischen Potential</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span></li>
<li>der <a href="Teilchenzahl" title="Teilchenzahl">Teilchenzahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> des Systems</li>
<li>der <a href="Innere_Energie" title="Innere Energie">inneren Energie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span></li>
<li>der <a href="Absolute_Temperatur" class="mw-redirect" title="Absolute Temperatur">absoluten Temperatur</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> des Systems</li>
<li>der <a href="Entropie_(Thermodynamik)" class="mw-redirect" title="Entropie (Thermodynamik)">Entropie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span>.</li></ul>
<p>Alternativ kann das großkanonische Potential über die <a href="Gro%C3%9Fkanonische_Zustandssumme" class="mw-redirect" title="Großkanonische Zustandssumme">großkanonische Zustandssumme</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9be1e2c3c82e7b8055fe26eb4cf2caac6f8ec73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle {\mathcal {Z}}}" loading="lazy"></span> definiert werden:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =-{\frac {1}{\beta }}\cdot \ln {\mathcal {Z}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>β<!-- β --></mi>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi>
</mrow>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega =-{\frac {1}{\beta }}\cdot \ln {\mathcal {Z}},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/297d7b9ffb628ecadc76a3eb9d9186f655245da4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.188ex; height:5.676ex;" alt="{\displaystyle \Omega =-{\frac {1}{\beta }}\cdot \ln {\mathcal {Z}},}" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }\cdot T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi>T</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }\cdot T}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9cc4607703e746365a7657a28296ab48b8b329f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.189ex; height:5.676ex;" alt="{\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }\cdot T}}}" loading="lazy"></span>
</p><p>mit der <a href="Boltzmann-Konstante" title="Boltzmann-Konstante">Boltzmann-Konstanten</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathrm {B} }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}" loading="lazy"></span>.
</p><p>Wegen der <a href="Innere_Energie" title="Innere Energie">thermodynamischen Euler-Gleichung</a> ist das großkanonische Potential identisch mit
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =-pV}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>p</mi>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega =-pV}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2d07fd06fc6ba21a3e8cce014a5a9602a2417bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.541ex; height:2.509ex;" alt="{\displaystyle \Omega =-pV}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li>dem <a href="Druck_(Physik)" title="Druck (Physik)">Druck</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></li>
<li>dem <a href="Volumen" title="Volumen">Volumen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> des Systems.</li></ul>
<p>Eine <a href="Infinitesimal" class="mw-redirect" title="Infinitesimal">infinitesimale</a> Änderung des großkanonischen Potentials ist gegeben durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \Omega =-p\cdot \mathrm {d} V-S\cdot \mathrm {d} T-N\cdot \mathrm {d} \mu }">
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \Omega =-p\cdot \mathrm {d} V-S\cdot \mathrm {d} T-N\cdot \mathrm {d} \mu }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be286f14951c1174b15bd92e6030f8ec9b7dec76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.03ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} \Omega =-p\cdot \mathrm {d} V-S\cdot \mathrm {d} T-N\cdot \mathrm {d} \mu }" loading="lazy"></span>.</dd></dl>
<p>Bei konstanter Temperatur (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} T=0}">
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} T=0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79e2fc50f4f86592a6af570453921d6b046bf6c.svg" 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}" loading="lazy"></span>) und konstantem chemischen Potential (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mu =0}">
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<mi>μ<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mu =0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5e48ffb9db8abda7ef472b3b7c052cbc9749139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.955ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} \mu =0}" loading="lazy"></span>) strebt das großkanonische Potential eines thermodynamischen Systems, welches ohne <a href="Arbeit_(Physik)" title="Arbeit (Physik)">Arbeits</a>umsatz sich selbst überlassen wird (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p\cdot \mathrm {d} V=0}">
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<mo>−<!-- − --></mo>
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<mi mathvariant="normal">d</mi>
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<mi>V</mi>
<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle -p\cdot \mathrm {d} V=0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cae0756a3a592c56595ce7f4523a7bac5ecd0226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.997ex; height:2.509ex;" alt="{\displaystyle -p\cdot \mathrm {d} V=0}" loading="lazy"></span>), einem Minimum zu (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \Omega =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
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<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \Omega =0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/094e88d4e5fd65a19ef05bb24b1864406f712bdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.232ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \Omega =0}" loading="lazy"></span>).
</p><p>Gemäß obiger Gleichung lassen sich die thermodynamischen Größen Entropie, Druck und Teilchenzahl wie folgt erhalten:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}S\\p\\N\end{pmatrix}}=-{\begin{pmatrix}\partial _{T}\\\partial _{V}\\\partial _{\mu }\end{pmatrix}}\Omega (T,V,\mu )}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}S\\p\\N\end{pmatrix}}=-{\begin{pmatrix}\partial _{T}\\\partial _{V}\\\partial _{\mu }\end{pmatrix}}\Omega (T,V,\mu )}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e88113cc590f39a269584d1592cbeedb070d062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.718ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}S\\p\\N\end{pmatrix}}=-{\begin{pmatrix}\partial _{T}\\\partial _{V}\\\partial _{\mu }\end{pmatrix}}\Omega (T,V,\mu )}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Gibbs-Energie" title="Gibbs-Energie">Gibbs-Energie</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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