Thermische Zustandsgleichung idealer Gase

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Sie vereint die experimentellen Einzelergebnisse und die hieraus abgeleiteten <a href="/wiki/Gasgesetze" title="Gasgesetze">Gasgesetze</a> zu einer allgemeingültigen <a href="/wiki/Zustandsgleichung" title="Zustandsgleichung">Zustandsgleichung</a>. </p> <div class="sieheauch" role="navigation" style="font-style:italic;"><span class="sieheauch-text">Siehe auch</span>: <a href="/wiki/Avogadrosches_Gesetz" title="Avogadrosches Gesetz">Avogadrosches Gesetz</a></div> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Die_thermische_Zustandsgleichung"><span class="tocnumber">1</span> <span class="toctext">Die thermische Zustandsgleichung</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Spezialfälle"><span class="tocnumber">2</span> <span class="toctext">Spezialfälle</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Gesetz_von_Boyle-Mariotte"><span class="tocnumber">2.1</span> <span class="toctext">Gesetz von Boyle-Mariotte</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Gesetz_von_Gay-Lussac"><span class="tocnumber">2.2</span> <span class="toctext">Gesetz von Gay-Lussac</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Gesetz_von_Amontons"><span class="tocnumber">2.3</span> <span class="toctext">Gesetz von Amontons</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Gesetz_der_Gleichförmigkeit"><span class="tocnumber">2.4</span> <span class="toctext">Gesetz der Gleichförmigkeit</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Gesetz_von_Avogadro"><span class="tocnumber">2.5</span> <span class="toctext">Gesetz von Avogadro</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Herleitungen"><span class="tocnumber">3</span> <span class="toctext">Herleitungen</span></a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Herleitung_aus_der_kinetischen_Gastheorie"><span class="tocnumber">3.1</span> <span class="toctext">Herleitung aus der kinetischen Gastheorie</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#Herleitung_aus_den_Spezialfällen"><span class="tocnumber">3.2</span> <span class="toctext">Herleitung aus den Spezialfällen</span></a> <ul> <li class="toclevel-3 tocsection-11"><a href="#Die_Kombination_der_Gesetze_von_Amontons_und_Gay-Lussac"><span class="tocnumber">3.2.1</span> <span class="toctext">Die Kombination der Gesetze von Amontons und Gay-Lussac</span></a></li> <li class="toclevel-3 tocsection-12"><a href="#Eigentliche_Herleitung_der_Gasgleichung"><span class="tocnumber">3.2.2</span> <span class="toctext">Eigentliche Herleitung der Gasgleichung</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-13"><a href="#Herleitung_aus_der_thermischen_Zustandsgleichung"><span class="tocnumber">3.3</span> <span class="toctext">Herleitung aus der thermischen Zustandsgleichung</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-14"><a href="#Weblinks"><span class="tocnumber">4</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise"><span class="tocnumber">5</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Die_thermische_Zustandsgleichung">Die thermische Zustandsgleichung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=1" title="Abschnitt bearbeiten: Die thermische Zustandsgleichung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Die thermische Zustandsgleichung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Gleichung beschreibt den <a href="/wiki/Zustand_(Thermodynamik)" title="Zustand (Thermodynamik)">Zustand</a> des idealen Gases bezüglich der <a href="/wiki/Zustandsgr%C3%B6%C3%9Fe" title="Zustandsgröße">Zustandsgrößen</a> <a href="/wiki/Druck_(Physik)" title="Druck (Physik)">Druck</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 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/81eac1e205430d1f40810df36a0edffdc367af36" 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}"></span>, <a href="/wiki/Volumen" title="Volumen">Volumen</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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span>, <a href="/wiki/Temperatur" title="Temperatur">Temperatur</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 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> und <a href="/wiki/Stoffmenge" title="Stoffmenge">Stoffmenge</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> bzw. <a href="/wiki/Teilchenzahl" title="Teilchenzahl">Teilchenzahl</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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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}"></span> bzw. <a href="/wiki/Masse_(Physik)" title="Masse (Physik)">Masse</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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>. Sie kann in verschiedenen zueinander äquivalenten Formen dargestellt werden, wobei alle diese Formen den Zustand des betrachteten <a href="/wiki/System" title="System">Systems</a> in gleicher Weise und eindeutig beschreiben. Ihre erste Formulierung stammt von <a href="/wiki/%C3%89mile_Clapeyron" title="Émile Clapeyron">Émile Clapeyron</a> im Jahr 1834.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Extensive_Gr%C3%B6%C3%9Fe" class="mw-redirect" title="Extensive Größe">Extensive</a> Formen: </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 p\cdot V=n\cdot R_{\mathrm {m} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V=n\cdot R_{\mathrm {m} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727672cec74271868b797e6017929fab9f6fe180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.578ex; height:2.509ex;" alt="{\displaystyle p\cdot V=n\cdot R_{\mathrm {m} }\cdot T}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot V={\frac {m}{M}}\cdot R_{\mathrm {m} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>M</mi> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V={\frac {m}{M}}\cdot R_{\mathrm {m} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df363d9630fa47afe1afe9d342ec3932f7b4ea8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:19.462ex; height:4.676ex;" alt="{\displaystyle p\cdot V={\frac {m}{M}}\cdot R_{\mathrm {m} }\cdot T}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot V=N\cdot k_{\mathrm {B} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mi>N</mi> <mo>⋅</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V=N\cdot k_{\mathrm {B} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3fd59a74fd50f8618d7dd34478326e5898bc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.489ex; height:2.509ex;" alt="{\displaystyle p\cdot V=N\cdot k_{\mathrm {B} }\cdot T}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot V=m\cdot R_{\mathrm {s} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V=m\cdot R_{\mathrm {s} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0789c578cd1a00a030de733232ddf956ef9aca93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.503ex; height:2.509ex;" alt="{\displaystyle p\cdot V=m\cdot R_{\mathrm {s} }\cdot T}"></span></dd></dl> <p><a href="/wiki/Intensive_Gr%C3%B6%C3%9Fe" class="mw-redirect" title="Intensive Größe">Intensive</a> Formen: </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 p\cdot v_{\mathrm {m} }=R_{\mathrm {m} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot v_{\mathrm {m} }=R_{\mathrm {m} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad8c92185bc7a833153c81fa4696e3a3479931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:15.446ex; height:2.509ex;" alt="{\displaystyle p\cdot v_{\mathrm {m} }=R_{\mathrm {m} }\cdot T}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot v=R_{\mathrm {s} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>v</mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot v=R_{\mathrm {s} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e053c8c0e1db2b610aa7228a25df0dfdad087a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.124ex; height:2.509ex;" alt="{\displaystyle p\cdot v=R_{\mathrm {s} }\cdot T}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\rho \cdot R_{\mathrm {s} }\cdot T\qquad {\text{oder}}\qquad {\frac {p}{\rho \cdot T}}=R_{\mathrm {s} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>ρ</mi> <mo>⋅</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>oder</mtext> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mi>ρ</mi> <mo>⋅</mo> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\rho \cdot R_{\mathrm {s} }\cdot T\qquad {\text{oder}}\qquad {\frac {p}{\rho \cdot T}}=R_{\mathrm {s} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1584bab4e728cca4f98584c12a36da484ab6aa61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:37.983ex; height:5.343ex;" alt="{\displaystyle p=\rho \cdot R_{\mathrm {s} }\cdot T\qquad {\text{oder}}\qquad {\frac {p}{\rho \cdot T}}=R_{\mathrm {s} }}"></span></dd></dl> <p>Hierbei stehen die einzelnen <a href="/wiki/Formelzeichen" title="Formelzeichen">Formelzeichen</a> für folgende <a href="/wiki/Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">Größen</a>: </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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d" 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} }}"></span> – <a href="/wiki/Boltzmann-Konstante" title="Boltzmann-Konstante">Boltzmann-Konstante</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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> – <a href="/wiki/Masse_(Physik)" title="Masse (Physik)">Masse</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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> – <a href="/wiki/Molare_Masse" title="Molare Masse">molare Masse</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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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}"></span> – <a href="/wiki/Teilchenzahl" title="Teilchenzahl">Teilchenzahl</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> – <a href="/wiki/Stoffmenge" title="Stoffmenge">Stoffmenge</a> (Mol)</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 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/81eac1e205430d1f40810df36a0edffdc367af36" 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}"></span> – <a href="/wiki/Druck_(Physik)" title="Druck (Physik)">Druck</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 R_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3a48d48e165dd949439c8193c6da9686fe9c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.365ex; height:2.509ex;" alt="{\displaystyle R_{\mathrm {m} }}"></span> – <a href="/wiki/Gaskonstante" title="Gaskonstante">universelle oder molare Gaskonstante</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 R_{\mathrm {s} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {s} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd8032743a5452e46f50bc3c5783e79955c60c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.644ex; height:2.509ex;" alt="{\displaystyle R_{\mathrm {s} }}"></span> – <a href="/wiki/Spezifische_Gaskonstante" class="mw-redirect" title="Spezifische Gaskonstante">individuelle oder spezifische Gaskonstante</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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> – <a href="/wiki/Dichte" title="Dichte">Dichte</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> – <a href="/wiki/Thermodynamische_Temperatur" title="Thermodynamische Temperatur">thermodynamische Temperatur</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 v_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ff3e0f09c17ad513ade56e05b5667465da9097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.729ex; height:2.009ex;" alt="{\displaystyle v_{\mathrm {m} }}"></span> – <a href="/wiki/Molares_Volumen" title="Molares Volumen">molares Volumen</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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> – <a href="/wiki/Spezifisches_Volumen" title="Spezifisches Volumen">spezifisches Volumen</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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span> – <a href="/wiki/Volumen" title="Volumen">Volumen</a></li></ul> <p>Die Gleichung stellt den Grenzfall aller thermischen Zustandsgleichungen für verschwindende Dichte <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 \rho \rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d7b253c2e58fd5dc99a9fc736692f3f63263a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.978ex; height:2.676ex;" alt="{\displaystyle \rho \rightarrow 0}"></span> dar, das heißt für verschwindenden Druck bei genügend hoher Temperatur. In diesem Fall kann man das Eigenvolumen der Gasmoleküle und die <a href="/wiki/Koh%C3%A4sion_(Chemie)" title="Kohäsion (Chemie)">Kohäsion</a> – die anziehende <a href="/wiki/Kraft" title="Kraft">Kraft</a> zwischen den Molekülen – vernachlässigen. Die Gleichung ist für viele Gase wie zum Beispiel wasserdampfungesättigte Luft auch bei Normalbedingungen eine gute Näherung. </p><p>1873 erweiterte <a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">Johannes Diderik van der Waals</a> das Gasgesetz zur <a href="/wiki/Van-der-Waals-Gleichung" title="Van-der-Waals-Gleichung">Van-der-Waals-Gleichung</a>, die das Eigenvolumen der Gasteilchen und die Anziehung zwischen ihnen im Gegensatz zur allgemeinen Gasgleichung mit berücksichtigt und somit auch als Näherung auf deutlich <a href="/wiki/Reales_Gas" title="Reales Gas">reale Gase</a> angewendet werden kann. Eine andere Näherungslösung für reale Gase stellt die Reihenentwicklung der <a href="/wiki/Virialgleichung" class="mw-redirect" title="Virialgleichung">Virialgleichungen</a> dar, wobei die allgemeine Gasgleichung identisch mit einem Abbruch der Reihenentwicklung nach dem ersten Glied ist. Generell gilt, dass die allgemeine Gasgleichung als Näherungslösung für schwach reale Gase bei geringen intermolekularen Wechselwirkungen, kleinen Drücken und hohen Temperaturen (großen Molvolumina) geeignet ist. Insbesondere weisen ideale Gase hierbei keinen <a href="/wiki/Joule-Thomson-Effekt" title="Joule-Thomson-Effekt">Joule-Thomson-Effekt</a> auf. </p> <div class="mw-heading mw-heading2"><h2 id="Spezialfälle"><span id="Spezialf.C3.A4lle"></span>Spezialfälle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=2" title="Abschnitt bearbeiten: Spezialfälle" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Spezialfälle"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es gibt verschiedene Spezialfälle des allgemeinen Gasgesetzes, die einen Zusammenhang zwischen zwei Größen herstellen, während alle anderen Größen konstant gehalten werden. Erklärt und nicht nur empirisch abgeleitet werden diese Zusammenhänge zwischen den Zustandsgrößen eines Gases durch dessen Teilchencharakter, also durch die <a href="/wiki/Kinetische_Gastheorie" title="Kinetische Gastheorie">kinetische Gastheorie</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Gesetz_von_Boyle-Mariotte">Gesetz von Boyle-Mariotte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=3" title="Abschnitt bearbeiten: Gesetz von Boyle-Mariotte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Gesetz von Boyle-Mariotte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das <b>Gesetz von Boyle-Mariotte</b>, auch <b>Boyle-Mariottesches Gesetz</b> oder <b>Boyle-Mariotte-Gesetz</b> und oft mit <b>Boyle’sches Gesetz</b> abgekürzt, sagt aus, dass der Druck idealer Gase bei gleichbleibender Temperatur (<a href="/wiki/Isotherme_Zustands%C3%A4nderung" title="Isotherme Zustandsänderung">isotherme Zustandsänderung</a>) und gleichbleibender Stoffmenge <a href="/wiki/Reziproke_Proportionalit%C3%A4t" title="Reziproke Proportionalität">umgekehrt proportional</a> zum Volumen ist. Erhöht man den Druck auf ein Gaspaket, wird durch den erhöhten Druck das Volumen verkleinert. Verringert man den Druck, so dehnt es sich aus. Dieses Gesetz wurde unabhängig von zwei Physikern entdeckt, dem Iren <a href="/wiki/Robert_Boyle" title="Robert Boyle">Robert Boyle</a> (1662) und dem Franzosen <a href="/wiki/Edme_Mariotte" title="Edme Mariotte">Edme Mariotte</a> (1676). </p><p>Für <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={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d42ed73df5477efb02617250e74ff0c950b080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.043ex; height:2.176ex;" alt="{\displaystyle T={\text{const}}}"></span> und <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 n={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbe1cecdd515ad52d55b0e28d46c1ccdb84c805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.801ex; height:2.009ex;" alt="{\displaystyle n={\text{const}}}"></span> gilt: </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 p\sim {\frac {1}{V}}\qquad \qquad p\cdot V={\text{const}}\qquad \qquad {\frac {p_{1}}{p_{2}}}={\frac {V_{2}}{V_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>V</mi> </mfrac> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\sim {\frac {1}{V}}\qquad \qquad p\cdot V={\text{const}}\qquad \qquad {\frac {p_{1}}{p_{2}}}={\frac {V_{2}}{V_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79122dab967484eae77a8746713aa3beccaf953e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:48.008ex; height:5.676ex;" alt="{\displaystyle p\sim {\frac {1}{V}}\qquad \qquad p\cdot V={\text{const}}\qquad \qquad {\frac {p_{1}}{p_{2}}}={\frac {V_{2}}{V_{1}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gesetz_von_Gay-Lussac">Gesetz von Gay-Lussac</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=4" title="Abschnitt bearbeiten: Gesetz von Gay-Lussac" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Gesetz von Gay-Lussac"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg/400px-Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg.png" decoding="async" width="400" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg/600px-Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg/800px-Ermittlung_des_absoluten_Nullpunktes_mit_dem_Gesetz_von_Gay-Lussac.svg.png 2x" data-file-width="2225" data-file-height="1185" /></a><figcaption>Der <a href="/wiki/Absoluter_Nullpunkt" title="Absoluter Nullpunkt">absolute Nullpunkt</a> lässt sich mit dem Gesetz von Gay-Lussac erahnen: Der Quotient aus Volumen und Temperatur ist stets konstant und ein negatives Volumen ist physikalisch nicht sinnvoll.</figcaption></figure> <p>Das <b>erste Gesetz von Gay-Lussac</b>, auch <b>Gay-Lussacsches Gesetz</b>, <b>Gesetz von Charles</b> oder <b>Charlessches Gesetz</b>, besagt, dass das Volumen idealer Gase bei gleichbleibendem Druck (<a href="/wiki/Isobare_Zustands%C3%A4nderung" title="Isobare Zustandsänderung">isobare Zustandsänderung</a>) und gleichbleibender Stoffmenge <a href="/wiki/Proportionalit%C3%A4t" title="Proportionalität">direkt proportional</a> zur Temperatur ist. Ein Gas dehnt sich also bei einer Erwärmung aus und zieht sich bei einer Abkühlung zusammen. Dieser Zusammenhang wurde 1787 von <a href="/wiki/Jacques_Charles" class="mw-redirect" title="Jacques Charles">Jacques Charles</a> und 1802 von <a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Joseph Louis Gay-Lussac</a> erkannt. </p><p>Für <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={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d823e093c4fc00077480f21f43453e237f0ed21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.666ex; height:2.343ex;" alt="{\displaystyle p={\text{const}}}"></span> und <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 n={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbe1cecdd515ad52d55b0e28d46c1ccdb84c805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.801ex; height:2.009ex;" alt="{\displaystyle n={\text{const}}}"></span> gilt: </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 V\sim T\qquad \qquad {\frac {V}{T}}={\text{const}}\qquad \qquad {\frac {V_{1}}{V_{2}}}={\frac {T_{1}}{T_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∼</mo> <mi>T</mi> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\sim T\qquad \qquad {\frac {V}{T}}={\text{const}}\qquad \qquad {\frac {V_{1}}{V_{2}}}={\frac {T_{1}}{T_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2eb684f6abc043f4cce7c2483d8028469b10c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:45.725ex; height:5.509ex;" alt="{\displaystyle V\sim T\qquad \qquad {\frac {V}{T}}={\text{const}}\qquad \qquad {\frac {V_{1}}{V_{2}}}={\frac {T_{1}}{T_{2}}}}"></span></dd></dl> <p>Das eigentliche Gesetz von Gay-Lussac (obiges ist nur der Teil, den man meist als das <i>Gesetz von Charles</i> bezeichnet) lautet: </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 V(T)=V_{0}\left(1+\gamma _{0}\left(T-T_{0}\right)\right)\qquad {\text{mit}}\qquad \gamma _{0}={\frac {1}{T_{0}}}={\frac {1}{273{,}15\ \mathrm {K} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>mit</mtext> </mrow> <mspace width="2em" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>273</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>15</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(T)=V_{0}\left(1+\gamma _{0}\left(T-T_{0}\right)\right)\qquad {\text{mit}}\qquad \gamma _{0}={\frac {1}{T_{0}}}={\frac {1}{273{,}15\ \mathrm {K} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0342b59c67112f1edbf0da3be3eecdf35839314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.836ex; height:5.676ex;" alt="{\displaystyle V(T)=V_{0}\left(1+\gamma _{0}\left(T-T_{0}\right)\right)\qquad {\text{mit}}\qquad \gamma _{0}={\frac {1}{T_{0}}}={\frac {1}{273{,}15\ \mathrm {K} }}}"></span></dd></dl> <p>Hierbei ist <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span> die Temperatur am Nullpunkt der Celsiusskala, also 273,15 K oder 0 °C. <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 V_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae15ff9b845587dc4e1816f59c3fed0e71a132f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{0}}"></span> ist das Volumen bei <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span> und <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 \gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407f2fbc64e21bb432f397fdb816de0f5083d900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}"></span> der Volumen<a href="/wiki/Ausdehnungskoeffizient" title="Ausdehnungskoeffizient">ausdehnungskoeffizient</a> bei <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span>, wobei für ideale Gase allgemein <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 \gamma =1/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e151c3385c8e6f8fcaad119bc88d95c6775bc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.322ex; height:2.843ex;" alt="{\displaystyle \gamma =1/T}"></span> gilt. Dagegen ist <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> die Temperatur, für die das Volumen <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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span> gesucht wird. </p><p>Aus dieser Gleichung kann man folgern, dass es einen absoluten <a href="/wiki/Absoluter_Nullpunkt" title="Absoluter Nullpunkt">Temperaturnullpunkt</a> geben muss, da die Gleichung für diesen ein Volumen von Null voraussagt und das Volumen nicht negativ werden kann. Ihre empirische Basis ist daher auch Grundlage für die <a href="/wiki/Kelvin" title="Kelvin">absolute Temperaturskala</a> <a href="/wiki/William_Thomson,_1._Baron_Kelvin" title="William Thomson, 1. Baron Kelvin">Kelvins</a>, da hierüber durch <a href="/wiki/Extrapolation" title="Extrapolation">Extrapolation</a> der Temperaturnullpunkt bestimmt werden konnte. </p> <div class="mw-heading mw-heading3"><h3 id="Gesetz_von_Amontons">Gesetz von Amontons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=5" title="Abschnitt bearbeiten: Gesetz von Amontons" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Gesetz von Amontons"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das <b>Gesetz von Amontons</b>, oft auch <b>2. Gesetz von Gay-Lussac</b>, sagt aus, dass der Druck idealer Gase bei gleichbleibendem Volumen (<a href="/wiki/Isochore_Zustands%C3%A4nderung" title="Isochore Zustandsänderung">isochore Zustandsänderung</a>) und gleichbleibender Stoffmenge direkt proportional zur Temperatur ist. Bei einer Erwärmung des Gases erhöht sich also der Druck und bei einer Abkühlung wird er geringer. Dieser Zusammenhang wurde von <a href="/wiki/Guillaume_Amontons" title="Guillaume Amontons">Guillaume Amontons</a> entdeckt. </p><p>Für <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 V={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8e7c1c49c57974dbcafe5ce3df1b991a246948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.194ex; height:2.176ex;" alt="{\displaystyle V={\text{const}}}"></span> und <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 n={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbe1cecdd515ad52d55b0e28d46c1ccdb84c805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.801ex; height:2.009ex;" alt="{\displaystyle n={\text{const}}}"></span> gilt: </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 p\sim T\qquad \qquad {\frac {p}{T}}={\text{const}}\qquad \qquad {\frac {p_{1}}{p_{2}}}={\frac {T_{1}}{T_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∼</mo> <mi>T</mi> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\sim T\qquad \qquad {\frac {p}{T}}={\text{const}}\qquad \qquad {\frac {p_{1}}{p_{2}}}={\frac {T_{1}}{T_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d8971589dca1579a1fb7d365981ce449bfe7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:44.86ex; height:5.676ex;" alt="{\displaystyle p\sim T\qquad \qquad {\frac {p}{T}}={\text{const}}\qquad \qquad {\frac {p_{1}}{p_{2}}}={\frac {T_{1}}{T_{2}}}}"></span></dd></dl> <p>Analog zum <i>Gesetz von Gay-Lussac</i> gilt hierbei auch: </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 p(T)=p_{0}\left(1+\gamma _{0}\left(T-T_{0}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(T)=p_{0}\left(1+\gamma _{0}\left(T-T_{0}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac79259b07b21d70c771f8678f29f53bedeec2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:27.569ex; height:2.843ex;" alt="{\displaystyle p(T)=p_{0}\left(1+\gamma _{0}\left(T-T_{0}\right)\right)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gesetz_der_Gleichförmigkeit"><span id="Gesetz_der_Gleichf.C3.B6rmigkeit"></span>Gesetz der Gleichförmigkeit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=6" title="Abschnitt bearbeiten: Gesetz der Gleichförmigkeit" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Gesetz der Gleichförmigkeit"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das Gesetz der <a href="/wiki/Homogenit%C3%A4t_(Physik)" class="mw-redirect" title="Homogenität (Physik)">Homogenität</a> sagt aus, dass ein ideales Gas durch und durch homogen, das heißt gleichförmig, ist, dass es also überall dieselbe <a href="/wiki/Dichte" title="Dichte">Dichte</a> hat. Wenn in einem großen Behälter mit einem homogenen Stoff, zum Beispiel mit einem Gas, an einer Stelle eine Teilmenge <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 V_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfdbc929f16cb00bb43289c223651b41f7b9f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}"></span> eingeschlossen wird, so enthält diese dieselbe Stoffmenge wie eine Teilmenge mit demselben Volumen <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 V_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfdbc929f16cb00bb43289c223651b41f7b9f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{1}}"></span> an anderer Stelle. Teilt man die gesamte Stoffmenge auf zwei gleich große Volumina auf, so enthalten sie die gleiche Stoffmenge, nämlich die Hälfte der ursprünglichen. Daraus folgt: </p><p>Das Volumen ist bei gleichbleibendem Druck und gleichbleibender Temperatur proportional zur Stoffmenge. </p><p>Für <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={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d42ed73df5477efb02617250e74ff0c950b080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.043ex; height:2.176ex;" alt="{\displaystyle T={\text{const}}}"></span> und <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={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d823e093c4fc00077480f21f43453e237f0ed21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.666ex; height:2.343ex;" alt="{\displaystyle p={\text{const}}}"></span> gilt: </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 V\sim n\qquad \qquad {\frac {V}{n}}={\text{const}}\qquad \qquad {\frac {V_{1}}{V_{2}}}={\frac {n_{1}}{n_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∼</mo> <mi>n</mi> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\sim n\qquad \qquad {\frac {V}{n}}={\text{const}}\qquad \qquad {\frac {V_{1}}{V_{2}}}={\frac {n_{1}}{n_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87adab3eff28bf5395ba59c07407bf9da49502b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:45.52ex; height:5.509ex;" alt="{\displaystyle V\sim n\qquad \qquad {\frac {V}{n}}={\text{const}}\qquad \qquad {\frac {V_{1}}{V_{2}}}={\frac {n_{1}}{n_{2}}}}"></span></dd></dl> <p>Diese Gesetze gelten für alle homogenen Stoffe, solange Temperatur und Druck unverändert bleiben, und eben auch für ideale Gase. </p> <div class="mw-heading mw-heading3"><h3 id="Gesetz_von_Avogadro">Gesetz von Avogadro</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=7" title="Abschnitt bearbeiten: Gesetz von Avogadro" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Gesetz von Avogadro"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das <a href="/wiki/Gesetz_von_Avogadro" class="mw-redirect" title="Gesetz von Avogadro">Gesetz von Avogadro</a> sagt aus, dass zwei gleich große Gasvolumina, die unter demselben Druck stehen und dieselbe Temperatur haben, auch die gleiche Teilchenzahl enthalten. Dies gilt sogar dann, wenn die Volumina verschiedene Gase enthalten. Es gilt die Beziehung <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 V\sim n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∼</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\sim n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859438d2206577adcefb7f4a10d6e39e088b3a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.28ex; height:2.176ex;" alt="{\displaystyle V\sim n}"></span> für <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={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d42ed73df5477efb02617250e74ff0c950b080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.043ex; height:2.176ex;" alt="{\displaystyle T={\text{const}}}"></span> und <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={\text{const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\text{const}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d823e093c4fc00077480f21f43453e237f0ed21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.666ex; height:2.343ex;" alt="{\displaystyle p={\text{const}}}"></span>. Von gewissen Ausnahmen abgesehen, kann also die Teilchenanzahl in einem Gaspaket bestimmt werden, wenn Volumen, Druck und Temperatur bekannt sind, unabhängig von der Stoffart. </p><p>Das Gesetz von Avogadro wurde 1811 durch <a href="/wiki/Amedeo_Avogadro" title="Amedeo Avogadro">Amedeo Avogadro</a> entdeckt. </p><p>Es kann auch so formuliert werden: Das <a href="/wiki/Molares_Volumen" title="Molares Volumen">molare Volumen</a> ist bei einer bestimmten Temperatur und bei einem bestimmten Druck für alle idealen Gase identisch. Messungen haben ergeben, dass ein <a href="/wiki/Mol" title="Mol">Mol</a> eines idealen Gases bei 0 °C = 273,15 K und 1013,25 hPa Druck ein Volumen von rund 22,4 dm³ einnimmt. </p><p>Eine bedeutende Folge des Gesetzes ist: Die <a href="/wiki/Gaskonstante" title="Gaskonstante">Gaskonstante</a> ist für alle idealen Gase identisch. </p> <div class="mw-heading mw-heading2"><h2 id="Herleitungen">Herleitungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=8" title="Abschnitt bearbeiten: Herleitungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Herleitungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Herleitung_aus_der_kinetischen_Gastheorie">Herleitung aus der kinetischen Gastheorie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=9" title="Abschnitt bearbeiten: Herleitung aus der kinetischen Gastheorie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Herleitung aus der kinetischen Gastheorie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die <a href="/wiki/Kinetische_Gastheorie" title="Kinetische Gastheorie">kinetische Gastheorie</a> besagt, dass sich Gase aus vielen einzelnen <a href="/wiki/Atom" title="Atom">Atomen</a> bzw. <a href="/wiki/Molek%C3%BCl" title="Molekül">Molekülen</a> zusammensetzen, die jedes für sich eine Masse <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> und eine Geschwindigkeit <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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> haben. Die mittlere <a href="/wiki/Kinetische_Energie" title="Kinetische Energie">kinetische Energie</a> aller Teilchen ist der Temperatur des Gases proportional. Es gilt </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 \langle E_{\mathrm {kin} }\rangle ={\frac {1}{2}}m\langle v^{2}\rangle ={\frac {3}{2}}k_{\mathrm {B} }T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E_{\mathrm {kin} }\rangle ={\frac {1}{2}}m\langle v^{2}\rangle ={\frac {3}{2}}k_{\mathrm {B} }T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e65968fd1cf77ca47e2dc6d78469974603b80e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.465ex; height:5.176ex;" alt="{\displaystyle \langle E_{\mathrm {kin} }\rangle ={\frac {1}{2}}m\langle v^{2}\rangle ={\frac {3}{2}}k_{\mathrm {B} }T}"></span>,</dd></dl> <p>wobei <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 \langle v^{2}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v^{2}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/630b0c71a0b8328306ef818ba129827b5cf98644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.991ex; height:3.176ex;" alt="{\displaystyle \langle v^{2}\rangle }"></span> das mittlere Geschwindigkeitsquadrat der Teilchen ist. Man sieht, dass sich die Moleküle bei höherer Temperatur des Gases mit höheren Geschwindigkeiten bewegen. Dabei besitzen nicht alle Teilchen die gleiche Geschwindigkeit, sondern es tritt eine statistische Verteilung der Geschwindigkeiten auf (<a href="/wiki/Maxwell-Boltzmann-Verteilung" title="Maxwell-Boltzmann-Verteilung">Maxwell-Boltzmann-Verteilung</a>). </p><p>Ist das Gas in einem Behälter mit dem Volumen <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 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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span> eingeschlossen, so stoßen immer wieder Gasmoleküle gegen die Wand des Behältnisses und werden reflektiert. Dadurch übertragen die Teilchen pro Zeitspanne und pro Wandfläche einen bestimmten <a href="/wiki/Impuls" title="Impuls">Impuls</a> auf die Wand. Es wirkt mit den Teilchenstößen auf jeden Teil der Wand eine Kraft, die wir als den Gasdruck <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/81eac1e205430d1f40810df36a0edffdc367af36" 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}"></span> begreifen. </p><p>Dieser Druck <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/81eac1e205430d1f40810df36a0edffdc367af36" 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}"></span> ist umso größer, je schneller die Teilchen sind. Zum einen steigt bei hohen Teilchengeschwindigkeiten die Rate, mit der die Gasmoleküle auf die Wand treffen, da sie den Behälterraum schneller durchqueren. Zum anderen werden die Stöße gegen die Wand <i>heftiger</i> und es wächst der dabei übertragene Impuls. Wird die Teilchendichte <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 N/V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N/V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa155f67e3e587c18df95a5afefbeb5201bd715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.013ex; height:2.843ex;" alt="{\displaystyle N/V}"></span> erhöht, so wächst die Wahrscheinlichkeit, mit der Moleküle an die Wand stoßen. Aus solchen Überlegungen kann man diese Gleichung für den Druck herleiten: </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 p={\frac {2}{3}}{\frac {N}{V}}{\frac {m}{2}}\langle v^{2}\rangle ={\frac {2}{3}}{\frac {N}{V}}\langle E_{\mathrm {kin} }\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>V</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>V</mi> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {2}{3}}{\frac {N}{V}}{\frac {m}{2}}\langle v^{2}\rangle ={\frac {2}{3}}{\frac {N}{V}}\langle E_{\mathrm {kin} }\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7926df4ec5e4a218e1ce4c2f925eeb4568a91630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:30.116ex; height:5.343ex;" alt="{\displaystyle p={\frac {2}{3}}{\frac {N}{V}}{\frac {m}{2}}\langle v^{2}\rangle ={\frac {2}{3}}{\frac {N}{V}}\langle E_{\mathrm {kin} }\rangle }"></span></dd></dl> <p>Drückt man die mittlere kinetische Energie der Gasteilchen durch die Temperatur aus, so ergibt sich daraus die thermische Zustandsgleichung idealer Gase: </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 p\cdot V=Nk_{\mathrm {B} }T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mi>N</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V=Nk_{\mathrm {B} }T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa6b7849b54096f5ee773b551bb195b98f66b32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.131ex; height:2.509ex;" alt="{\displaystyle p\cdot V=Nk_{\mathrm {B} }T}"></span></dd></dl> <p>Diese Gleichung gilt jedoch nur bei Gasen mit geringer Teilchendichte und bei genügend hoher Temperatur. Bei dieser Herleitung wird nämlich vernachlässigt, dass Anziehungskräfte zwischen den Teilchen wirken, die den Teilchendruck gegen die Wand abschwächen. Außerdem besitzen die Moleküle selbst ein Volumen und das Gas kann nicht beliebig komprimiert werden, weil die Teilchen sich gegenseitig verdrängen. Die Beschreibung eines solchen realen Gases bewältigt die <a href="/wiki/Van-der-Waals-Gleichung" title="Van-der-Waals-Gleichung">Van-der-Waals-Gleichung</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Herleitung_aus_den_Spezialfällen"><span id="Herleitung_aus_den_Spezialf.C3.A4llen"></span>Herleitung aus den Spezialfällen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=10" title="Abschnitt bearbeiten: Herleitung aus den Spezialfällen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Herleitung aus den Spezialfällen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Die_Kombination_der_Gesetze_von_Amontons_und_Gay-Lussac">Die Kombination der Gesetze von Amontons und Gay-Lussac</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=11" title="Abschnitt bearbeiten: Die Kombination der Gesetze von Amontons und Gay-Lussac" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Die Kombination der Gesetze von Amontons und Gay-Lussac"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Gesetze von Amontons und Gay-Lussac, die beide zeitlich <i>vor</i> der Gasgleichung gefunden wurden, lassen sich beispielsweise durch das Gedankenexperiment einer zweistufigen <a href="/wiki/Zustands%C3%A4nderung" title="Zustandsänderung">Zustandsänderung</a> zusammenfassen, wobei man hierbei generell von einer gleichbleibenden Stoffmenge ausgeht. </p><p>Zunächst betrachtet man eine <a href="/wiki/Isochore_Zustands%C3%A4nderung" title="Isochore Zustandsänderung">isochore Zustandsänderung</a> nach dem <i>Gesetz von Amontons</i>. Der Ausgangspunkt ist hierbei der Zustand 1 mit <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_{1},V_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},V_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d78adafa5be2017838688a50b57cdefaab597f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.756ex; height:2.509ex;" alt="{\displaystyle p_{1},V_{1}}"></span> und <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span>. Endpunkt ist Zustand 2 mit <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_{2},V_{2}=V_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2},V_{2}=V_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70be9412333748abeeee4a2f7275216fa8af0cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.264ex; height:2.509ex;" alt="{\displaystyle p_{2},V_{2}=V_{1}}"></span> und <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p_{1}}{T_{1}}}={\frac {p_{2}}{T_{2}}}\qquad \qquad T_{2}={\frac {p_{2}\,T_{1}}{p_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="2em" /> <mspace width="2em" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{1}}{T_{1}}}={\frac {p_{2}}{T_{2}}}\qquad \qquad T_{2}={\frac {p_{2}\,T_{1}}{p_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a680a83bbc668b295cb02e1c3ca41d2b6c4158be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.254ex; height:5.843ex;" alt="{\displaystyle {\frac {p_{1}}{T_{1}}}={\frac {p_{2}}{T_{2}}}\qquad \qquad T_{2}={\frac {p_{2}\,T_{1}}{p_{1}}}}"></span></dd></dl> <p>Es folgt eine <a href="/wiki/Isobare_Zustands%C3%A4nderung" title="Isobare Zustandsänderung">isobare Zustandsänderung</a> nach dem <i>Gesetz von Gay-Lussac</i> von Zustand 2 nach Zustand 3 mit <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_{3}=p_{2},V_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{3}=p_{2},V_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34adfa30ea1b8f7ed3a93b17e4b1cbcaf81dd3aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.079ex; height:2.509ex;" alt="{\displaystyle p_{3}=p_{2},V_{3}}"></span> und <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_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0936b6ba76bf0d900bb7315c99f64c5376f5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{3}}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {V_{2}}{T_{2}}}={\frac {V_{3}}{T_{3}}}\qquad \qquad V_{3}={\frac {V_{2}T_{3}}{T_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="2em" /> <mspace width="2em" /> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {V_{2}}{T_{2}}}={\frac {V_{3}}{T_{3}}}\qquad \qquad V_{3}={\frac {V_{2}T_{3}}{T_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87faea2e0013db2499b2a9fa1c2c6566cb487fbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.05ex; height:5.843ex;" alt="{\displaystyle {\frac {V_{2}}{T_{2}}}={\frac {V_{3}}{T_{3}}}\qquad \qquad V_{3}={\frac {V_{2}T_{3}}{T_{2}}}}"></span></dd></dl> <p>Setzt man nun den Ausdruck für <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> aus obiger Gleichung in den Ausdruck für <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> aus unterer Gleichung ein und stellt um, wobei man <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_{3}=p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{3}=p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29e4de315540da94e5beb11dd9c6058d0e00758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.635ex; height:2.009ex;" alt="{\displaystyle p_{3}=p_{2}}"></span> und <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 V_{2}=V_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{2}=V_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08da7e16fbfb398606112b7276a27fd8e6e6478b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.917ex; height:2.509ex;" alt="{\displaystyle V_{2}=V_{1}}"></span> berücksichtigen muss, so erhält man als Resultat die Beziehung: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p_{1}V_{1}}{T_{1}}}={\frac {p_{3}V_{3}}{T_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{1}V_{1}}{T_{1}}}={\frac {p_{3}V_{3}}{T_{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7fb8ed0e4e2b917bc5804133991e3cce240370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.037ex; height:5.843ex;" alt="{\displaystyle {\frac {p_{1}V_{1}}{T_{1}}}={\frac {p_{3}V_{3}}{T_{3}}}}"></span></dd></dl> <p>und daher </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p\,V}{T}}={\text{konstant}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> <mi>V</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>konstant</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p\,V}{T}}={\text{konstant}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113c0da0f21f9384c2e0e77b621c5bdd75848fc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.141ex; height:5.343ex;" alt="{\displaystyle {\frac {p\,V}{T}}={\text{konstant}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Eigentliche_Herleitung_der_Gasgleichung">Eigentliche Herleitung der Gasgleichung</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=12" title="Abschnitt bearbeiten: Eigentliche Herleitung der Gasgleichung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Eigentliche Herleitung der Gasgleichung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Als letzten Schritt muss man die Konstante im rechten Term des obigen Ausdrucks ermitteln. Geht man davon aus, dass ein <a href="/wiki/Mol" title="Mol">Mol</a> bei 273,15 <a href="/wiki/Kelvin" title="Kelvin">Kelvin</a> und 101,325 <a href="/wiki/Pascal_(Einheit)" title="Pascal (Einheit)">Kilopascal</a> ein Volumen von genau 22,414 <a href="/wiki/Liter" title="Liter">Liter</a> einnimmt, also das Gesetz von Avogadro gültig ist, so kann man auch davon ausgehen, dass die <a href="/wiki/Stoffmenge" title="Stoffmenge">Stoffmenge</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Mol eines idealen Gases genau ein Volumen von <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 n\,\cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\,\cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97904785af941aed632b031fe2b4901bb39e8305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:1.676ex;" alt="{\displaystyle n\,\cdot }"></span> 22,414 Liter einnehmen. Wenn man dies in obige Gleichung einsetzt, erhält man: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}\cdot n={\frac {p_{3}\cdot V_{3}}{T_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>101,325</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">a</mi> </mrow> <mo>⋅</mo> <mn>22,414</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn>273</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>15</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}\cdot n={\frac {p_{3}\cdot V_{3}}{T_{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79dc9f03e1f98f82e344a961d9a6d5028354d336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.148ex; height:7.009ex;" alt="{\displaystyle {\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}\cdot n={\frac {p_{3}\cdot V_{3}}{T_{3}}}}"></span></dd></dl> <p>Multipliziert man die Gleichung mit <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_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0936b6ba76bf0d900bb7315c99f64c5376f5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{3}}"></span>, so erhält man: </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 n\cdot T_{3}\cdot {\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}=p_{3}\cdot V_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>101,325</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">a</mi> </mrow> <mo>⋅</mo> <mn>22,414</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn>273</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>15</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot T_{3}\cdot {\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}=p_{3}\cdot V_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad346ebe41322a48a09d7b4b676918143427e649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.403ex; height:7.009ex;" alt="{\displaystyle n\cdot T_{3}\cdot {\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}=p_{3}\cdot V_{3}}"></span></dd></dl> <p>Der Bruch auf der linken Seite der Gleichung ist eine konstante <a href="/wiki/Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">Größe</a>, man bezeichnet diese als <i>universelle <a href="/wiki/Gaskonstante" title="Gaskonstante">Gaskonstante</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span></i> (auch <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 R_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3a48d48e165dd949439c8193c6da9686fe9c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.365ex; height:2.509ex;" alt="{\displaystyle R_{\mathrm {m} }}"></span> für <i>molare Gaskonstante</i>) und berechnet sie so: </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 R_{\mathrm {m} }:={\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}=8{,}31447~{\frac {\mathrm {J} }{{\mathrm {mol} }\cdot \mathrm {K} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>101,325</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">a</mi> </mrow> <mo>⋅</mo> <mn>22,414</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn>273</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>15</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>8,314</mn> <mn>47</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">J</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {m} }:={\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}=8{,}31447~{\frac {\mathrm {J} }{{\mathrm {mol} }\cdot \mathrm {K} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e966a86466ad04a8e7798ac4253c51c4e1b02bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.307ex; height:7.009ex;" alt="{\displaystyle R_{\mathrm {m} }:={\frac {101{,}325\,\mathrm {kPa} \cdot 22{,}414\,{\frac {\mathrm {l} }{\mathrm {mol} }}}{273{,}15\,\mathrm {K} }}=8{,}31447~{\frac {\mathrm {J} }{{\mathrm {mol} }\cdot \mathrm {K} }}}"></span></dd></dl> <p>Streicht man nun die Indizes, so erhält man die gesuchte allgemeine Gasgleichung: </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 p\cdot V=n\cdot R_{\mathrm {m} }\cdot T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot V=n\cdot R_{\mathrm {m} }\cdot T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727672cec74271868b797e6017929fab9f6fe180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.578ex; height:2.509ex;" alt="{\displaystyle p\cdot V=n\cdot R_{\mathrm {m} }\cdot T}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Herleitung_aus_der_thermischen_Zustandsgleichung">Herleitung aus der thermischen Zustandsgleichung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=13" title="Abschnitt bearbeiten: Herleitung aus der thermischen Zustandsgleichung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Herleitung aus der thermischen Zustandsgleichung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grundlage für die Herleitung ist die thermische <a href="/wiki/Zustandsgleichung" title="Zustandsgleichung">Zustandsgleichung</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} V=\left(V\cdot \gamma \right)\mathrm {d} T-\left(V\cdot \kappa \right)\mathrm {d} p+v_{\mathrm {m} }\mathrm {d} n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>⋅</mo> <mi>γ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>⋅</mo> <mi>κ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\left(V\cdot \gamma \right)\mathrm {d} T-\left(V\cdot \kappa \right)\mathrm {d} p+v_{\mathrm {m} }\mathrm {d} n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48992d7ed29601a1811df0052cae3f8c9f81081e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.592ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} V=\left(V\cdot \gamma \right)\mathrm {d} T-\left(V\cdot \kappa \right)\mathrm {d} p+v_{\mathrm {m} }\mathrm {d} n}"></span></dd></dl> <p>Hierbei stehen die einzelnen Formelzeichen für folgende Größen: </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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> – <a href="/wiki/Kompressionsmodul" title="Kompressionsmodul">Kompressibilität</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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> – Volumen<a href="/wiki/Ausdehnungskoeffizient" title="Ausdehnungskoeffizient">ausdehnungskoeffizient</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 v_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ff3e0f09c17ad513ade56e05b5667465da9097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.729ex; height:2.009ex;" alt="{\displaystyle v_{\mathrm {m} }}"></span> – <a href="/wiki/Molares_Volumen" title="Molares Volumen">molares Volumen</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 v_{\mathrm {m} }=V/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {m} }=V/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7d4cb9e2516c26609e5822a6ee0be2d2ee29f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.172ex; height:2.843ex;" alt="{\displaystyle v_{\mathrm {m} }=V/n}"></span>)</li></ul> <p>Bei einem idealen Gas gilt speziell: </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 \gamma ={\frac {1}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3ebbf5db48fce48bd2e531d1aa18380772d112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.833ex; height:5.176ex;" alt="{\displaystyle \gamma ={\frac {1}{T}}}"></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 \kappa ={\frac {1}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {1}{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592a24d1b7913c4f15a891e91f2b3627d47bb960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.443ex; height:5.676ex;" alt="{\displaystyle \kappa ={\frac {1}{p}}}"></span></dd></dl> <p>Hierdurch wird die thermische Zustandsgleichung zu folgender Form vereinfacht: </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 \mathrm {d} V=\left({\frac {V}{T}}\right)\mathrm {d} T-\left({\frac {V}{p}}\right)\mathrm {d} p+\left({\frac {V}{n}}\right)\mathrm {d} n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\left({\frac {V}{T}}\right)\mathrm {d} T-\left({\frac {V}{p}}\right)\mathrm {d} p+\left({\frac {V}{n}}\right)\mathrm {d} n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be35e5a59e8a2a0f9be99936063d53b77b92fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.232ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} V=\left({\frac {V}{T}}\right)\mathrm {d} T-\left({\frac {V}{p}}\right)\mathrm {d} p+\left({\frac {V}{n}}\right)\mathrm {d} n}"></span></dd></dl> <p>Man kann die Zustandsgleichung nun von einem <a href="/wiki/Zustand_(Thermodynamik)" title="Zustand (Thermodynamik)">Zustand</a> <b>1</b> bis zu einem Zustand <b>2</b> integrieren (<a href="/wiki/Integralrechnung" title="Integralrechnung">bestimmtes Integral</a>) und erhält dadurch: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p_{1}\cdot V_{1}}{T_{1}\cdot n_{1}}}={\frac {p_{2}\cdot V_{2}}{T_{2}\cdot n_{2}}}={\text{const}}=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{1}\cdot V_{1}}{T_{1}\cdot n_{1}}}={\frac {p_{2}\cdot V_{2}}{T_{2}\cdot n_{2}}}={\text{const}}=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fff9050eccbc7b268165fcd0daee2521e2d5857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.12ex; height:5.676ex;" alt="{\displaystyle {\frac {p_{1}\cdot V_{1}}{T_{1}\cdot n_{1}}}={\frac {p_{2}\cdot V_{2}}{T_{2}\cdot n_{2}}}={\text{const}}=R}"></span></dd></dl> <p>Gleichen sich zwei der Zustandsgrößen, haben sie sich also vom Zustand 1 zum Zustand 2 nicht geändert, so können diese gekürzt werden und man erhält dadurch die jeweiligen Spezialfälle. Die universelle Gaskonstante <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> muss hierbei experimentell bestimmt werden und leitet sich nicht aus der <a href="/wiki/Integrationskonstante" class="mw-redirect" title="Integrationskonstante">Integrationskonstanten</a> ab. Einen beispielhaften Versuchsaufbau kann man im Artikel „<a href="/wiki/Gaskonstante" title="Gaskonstante">Gaskonstante</a>“ nachlesen. </p> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=14" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.k-wz.de/waerme/gasgleichung.html"><i>Die allgemeine Gasgleichung.</i></a> Mit Diagrammen und interaktiver Animation.</li> <li><a rel="nofollow" class="external text" href="https://www.leifiphysik.de/waermelehre/allgemeines-gasgesetz/versuche/boyle-mariotte-selbstbau">Das Gesetz von Boyle und Mariotte</a> im Schülerversuch (<a href="/wiki/LEIFI" class="mw-redirect" title="LEIFI">LEIFI</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"> <a href="/wiki/%C3%89mile_Clapeyron" title="Émile Clapeyron">E. Clapeyron</a>: <cite class="lang" lang="fr" dir="auto" style="font-style:italic">Mémoire sur la puissance motrice de la chaleur</cite>. In: <cite class="lang" lang="fr" dir="auto" style="font-style:italic"><a href="/w/index.php?title=Journal_de_l%E2%80%99%C3%89cole_Polytechnique&action=edit&redlink=1" class="new" title="Journal de l’École Polytechnique (Seite nicht vorhanden)">Journal de l’École Polytechnique</a></cite>. XIV. Jahrgang, 1834, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>153–90</span> (französisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Thermische+Zustandsgleichung+idealer+Gase&rft.atitle=M%C3%A9moire+sur+la+puissance+motrice+de+la+chaleur&rft.au=E.+Clapeyron&rft.btitle=Journal+de+l%E2%80%99%C3%89cole+Polytechnique&rft.date=1834&rft.genre=book&rft.pages=153-90&rft.volume=XIV.+Jahrgang" style="display:none"> </span></span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&oldid=246013255">https://de.wikipedia.org/w/index.php?title=Thermische_Zustandsgleichung_idealer_Gase&oldid=246013255</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Zustandsgleichung" title="Kategorie:Zustandsgleichung">Zustandsgleichung</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Thermische+Zustandsgleichung+idealer+Gase" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Idealegaswet" title="Idealegaswet – Afrikaans" lang="af" hreflang="af" data-title="Idealegaswet" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%A7%D9%84%D8%BA%D8%A7%D8%B2%D8%A7%D8%AA_%D8%A7%D9%84%D9%85%D8%AB%D8%A7%D9%84%D9%8A%D8%A9" title="قانون الغازات المثالية – Arabisch" lang="ar" hreflang="ar" data-title="قانون الغازات المثالية" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Llei_de_los_gases_ideales" title="Llei de los gases ideales – Asturisch" lang="ast" hreflang="ast" data-title="Llei de los gases ideales" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0deal_qaz%C4%B1n_hal_t%C9%99nliyi" title="İdeal qazın hal tənliyi – Aserbaidschanisch" lang="az" hreflang="az" data-title="İdeal qazın hal tənliyi" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5_%D1%81%D1%82%D0%B0%D0%BD%D1%83_%D1%96%D0%B4%D1%8D%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D0%B3%D0%B0_%D0%B3%D0%B0%D0%B7%D1%83" title="Ураўненне стану ідэальнага газу – Belarussisch" lang="be" hreflang="be" data-title="Ураўненне стану ідэальнага газу" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%9A%D0%BB%D0%B0%D0%BF%D0%B5%D0%B9%D1%80%D0%BE%D0%BD_%E2%80%93_%D0%9C%D0%B5%D0%BD%D0%B4%D0%B5%D0%BB%D0%B5%D0%B5%D0%B2" title="Уравнение на Клапейрон – Менделеев – Bulgarisch" lang="bg" hreflang="bg" data-title="Уравнение на Клапейрон – Менделеев" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%A6%E0%A6%B0%E0%A7%8D%E0%A6%B6_%E0%A6%97%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B8_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="আদর্শ গ্যাস সূত্র – Bengalisch" lang="bn" hreflang="bn" data-title="আদর্শ গ্যাস সূত্র" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Zakon_idealnog_plina" title="Zakon idealnog plina – Bosnisch" lang="bs" hreflang="bs" data-title="Zakon idealnog plina" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_dels_gasos_ideals" title="Llei dels gasos ideals – Katalanisch" lang="ca" hreflang="ca" data-title="Llei dels gasos ideals" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DB%8C%D8%A7%D8%B3%D8%A7%DB%8C_%DA%AF%D8%A7%D8%B2%DB%8C_%D9%86%D9%85%D9%88%D9%86%DB%95%DB%8C%DB%8C" title="یاسای گازی نمونەیی – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="یاسای گازی نمونەیی" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB%D0%BB%D0%B0_%D0%B3%D0%B0%D0%B7%C4%83%D0%BD_%D1%82%C4%83%D1%80%C4%83%D0%BC_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85%C4%95" title="Идеалла газăн тăрăм танлăхĕ – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Идеалла газăн тăрăм танлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Idealgasligning" title="Idealgasligning – Dänisch" lang="da" hreflang="da" data-title="Idealgasligning" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B1%CF%84%CE%B1%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7_%CF%84%CF%89%CE%BD_%CE%B9%CE%B4%CE%B1%CE%BD%CE%B9%CE%BA%CF%8E%CE%BD_%CE%B1%CE%B5%CF%81%CE%AF%CF%89%CE%BD" title="Καταστατική εξίσωση των ιδανικών αερίων – Griechisch" lang="el" hreflang="el" data-title="Καταστατική εξίσωση των ιδανικών αερίων" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Ideal_gas_law" title="Ideal gas law – Englisch" lang="en" hreflang="en" data-title="Ideal gas law" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ekvacio_de_ideala_gaso" title="Ekvacio de ideala gaso – Esperanto" lang="eo" hreflang="eo" data-title="Ekvacio de ideala gaso" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ley_de_los_gases_ideales" title="Ley de los gases ideales – Spanisch" lang="es" hreflang="es" data-title="Ley de los gases ideales" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ideaalse_gaasi_olekuv%C3%B5rrand" title="Ideaalse gaasi olekuvõrrand – Estnisch" lang="et" hreflang="et" data-title="Ideaalse gaasi olekuvõrrand" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gas_idealen_legeak" title="Gas idealen legeak – Baskisch" lang="eu" hreflang="eu" data-title="Gas idealen legeak" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%DA%AF%D8%A7%D8%B2_%D8%A2%D8%B1%D9%85%D8%A7%D9%86%DB%8C" title="قانون گاز آرمانی – Persisch" lang="fa" hreflang="fa" data-title="قانون گاز آرمانی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi badge-Q70894304 mw-list-item" title=""><a href="https://fi.wikipedia.org/wiki/Ideaalikaasulaki" title="Ideaalikaasulaki – Finnisch" lang="fi" hreflang="fi" data-title="Ideaalikaasulaki" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_des_gaz_parfaits" title="Loi des gaz parfaits – Französisch" lang="fr" hreflang="fr" data-title="Loi des gaz parfaits" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%AD_an_Gh%C3%A1is_Id%C3%A9alaigh" title="Dlí an Gháis Idéalaigh – Irisch" lang="ga" hreflang="ga" data-title="Dlí an Gháis Idéalaigh" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Lei_dos_gases_ideais" title="Lei dos gases ideais – Galicisch" lang="gl" hreflang="gl" data-title="Lei dos gases ideais" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%AA_%D7%94%D7%92%D7%96_%D7%94%D7%90%D7%99%D7%93%D7%99%D7%90%D7%9C%D7%99" title="משוואת הגז האידיאלי – Hebräisch" lang="he" hreflang="he" data-title="משוואת הגז האידיאלי" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%B6_%E0%A4%97%E0%A5%88%E0%A4%B8_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="आदर्श गैस समीकरण – Hindi" lang="hi" hreflang="hi" data-title="आदर्श गैस समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Jednad%C5%BEba_stanja_idealnog_plina" title="Jednadžba stanja idealnog plina – Kroatisch" lang="hr" hreflang="hr" data-title="Jednadžba stanja idealnog plina" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%A4%D5%A5%D5%A1%D5%AC%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A3%D5%A1%D5%A6%D5%AB_%D5%BE%D5%AB%D5%B3%D5%A1%D5%AF%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Իդեալական գազի վիճակի հավասարում – Armenisch" lang="hy" hreflang="hy" data-title="Իդեալական գազի վիճակի հավասարում" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_gas_ideal" title="Persamaan gas ideal – Indonesisch" lang="id" hreflang="id" data-title="Persamaan gas ideal" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_stato_dei_gas_perfetti" title="Equazione di stato dei gas perfetti – Italienisch" lang="it" hreflang="it" data-title="Equazione di stato dei gas perfetti" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%90%86%E6%83%B3%E6%B0%97%E4%BD%93%E3%81%AE%E7%8A%B6%E6%85%8B%E6%96%B9%E7%A8%8B%E5%BC%8F" title="理想気体の状態方程式 – Japanisch" lang="ja" hreflang="ja" data-title="理想気体の状態方程式" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D0%BF%D0%B5%D0%B9%D1%80%D0%BE%D0%BD_%D1%82%D0%B5%D2%A3%D0%B4%D0%B5%D1%83%D1%96" title="Клапейрон теңдеуі – Kasachisch" lang="kk" hreflang="kk" data-title="Клапейрон теңдеуі" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%83%81%EA%B8%B0%EC%B2%B4_%EB%B2%95%EC%B9%99" title="이상기체 법칙 – Koreanisch" lang="ko" hreflang="ko" data-title="이상기체 법칙" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B5%D0%B9-%D0%9B%D1%8E%D1%81%D1%81%D0%B0%D0%BA_%D0%BC%D1%8B%D0%B9%D0%B7%D0%B0%D0%BC%D0%B4%D0%B0%D1%80%D1%8B" title="Гей-Люссак мыйзамдары – Kirgisisch" lang="ky" hreflang="ky" data-title="Гей-Люссак мыйзамдары" data-language-autonym="Кыргызча" data-language-local-name="Kirgisisch" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Equazion_de_stat_di_gas_perf%C3%A8tt" title="Equazion de stat di gas perfètt – Lombardisch" lang="lmo" hreflang="lmo" data-title="Equazion de stat di gas perfètt" data-language-autonym="Lombard" data-language-local-name="Lombardisch" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ide%C4%81las_g%C4%81zes_likums" title="Ideālas gāzes likums – Lettisch" lang="lv" hreflang="lv" data-title="Ideālas gāzes likums" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D0%BF%D0%B5%D1%98%D1%80%D0%BE%D0%BD-%D0%9C%D0%B5%D0%BD%D0%B4%D0%B5%D0%BB%D0%B5%D0%B5%D0%B2%D0%B0_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0" title="Клапејрон-Менделеева равенка – Mazedonisch" lang="mk" hreflang="mk" data-title="Клапејрон-Менделеева равенка" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B4%A6%E0%B5%BC%E0%B4%B6_%E0%B4%B5%E0%B4%BE%E0%B4%A4%E0%B4%95_%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%82" title="ആദർശ വാതക നിയമം – Malayalam" lang="ml" hreflang="ml" data-title="ആദർശ വാതക നിയമം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%86%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%B6_%E0%A4%B5%E0%A4%BE%E0%A4%AF%E0%A5%82_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="आदर्श वायू समीकरण – Marathi" lang="mr" hreflang="mr" data-title="आदर्श वायू समीकरण" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hukum_gas_unggul" title="Hukum gas unggul – Malaiisch" lang="ms" hreflang="ms" data-title="Hukum gas unggul" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algemene_gaswet" title="Algemene gaswet – Niederländisch" lang="nl" hreflang="nl" data-title="Algemene gaswet" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Idealgassloven" title="Idealgassloven – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Idealgassloven" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_Clapeyrona_(stan_gazu_doskona%C5%82ego)" title="Równanie Clapeyrona (stan gazu doskonałego) – Polnisch" lang="pl" hreflang="pl" data-title="Równanie Clapeyrona (stan gazu doskonałego)" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Lej_general_dij_gas" title="Lej general dij gas – Piemontesisch" lang="pms" hreflang="pms" data-title="Lej general dij gas" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lei_dos_gases_ideais" title="Lei dos gases ideais – Portugiesisch" lang="pt" hreflang="pt" data-title="Lei dos gases ideais" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Legea_gazelor_ideale" title="Legea gazelor ideale – Rumänisch" lang="ro" hreflang="ro" data-title="Legea gazelor ideale" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D1%8F_%D0%B8%D0%B4%D0%B5%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE_%D0%B3%D0%B0%D0%B7%D0%B0" title="Уравнение состояния идеального газа – Russisch" lang="ru" hreflang="ru" data-title="Уравнение состояния идеального газа" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Jedna%C4%8Dina_stanja_idealnog_gasa" title="Jednačina stanja idealnog gasa – Serbokroatisch" lang="sh" hreflang="sh" data-title="Jednačina stanja idealnog gasa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%BB%E0%B7%8A%E0%B7%80%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%8F%E0%B6%BA%E0%B7%94_%E0%B6%B1%E0%B7%92%E0%B6%BA%E0%B6%B8%E0%B6%BA" title="සර්වත්‍ර වායු නියමය – Singhalesisch" lang="si" hreflang="si" data-title="සර්වත්‍ර වායු නියමය" data-language-autonym="සිංහල" data-language-local-name="Singhalesisch" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ideal_gas_law" title="Ideal gas law – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Ideal gas law" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Stavov%C3%A1_rovnica_ide%C3%A1lneho_plynu" title="Stavová rovnica ideálneho plynu – Slowakisch" lang="sk" hreflang="sk" data-title="Stavová rovnica ideálneho plynu" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Splo%C5%A1na_plinska_ena%C4%8Dba" title="Splošna plinska enačba – Slowenisch" lang="sl" hreflang="sl" data-title="Splošna plinska enačba" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ligji_i_gazit_ideal" title="Ligji i gazit ideal – Albanisch" lang="sq" hreflang="sq" data-title="Ligji i gazit ideal" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Jedna%C4%8Dina_stanja_idealnog_gasa" title="Jednačina stanja idealnog gasa – Serbisch" lang="sr" hreflang="sr" data-title="Jednačina stanja idealnog gasa" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ideala_gaslagen" title="Ideala gaslagen – Schwedisch" lang="sv" hreflang="sv" data-title="Ideala gaslagen" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%B0%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%B5%E0%AE%B3%E0%AE%BF%E0%AE%AE_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="கருத்தியல் வளிம விதி – Tamil" lang="ta" hreflang="ta" data-title="கருத்தியல் வளிம விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%81%E0%B8%81%E0%B9%8A%E0%B8%AA%E0%B8%AD%E0%B8%B8%E0%B8%94%E0%B8%A1%E0%B8%84%E0%B8%95%E0%B8%B4" title="กฎของแก๊สอุดมคติ – Thailändisch" lang="th" hreflang="th" data-title="กฎของแก๊สอุดมคติ" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Batas_ng_ideyal_na_gas" title="Batas ng ideyal na gas – Tagalog" lang="tl" hreflang="tl" data-title="Batas ng ideyal na gas" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0deal_gaz_yasas%C4%B1" title="İdeal gaz yasası – Türkisch" lang="tr" hreflang="tr" data-title="İdeal gaz yasası" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D1%81%D1%82%D0%B0%D0%BD%D1%83_%D1%96%D0%B4%D0%B5%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE_%D0%B3%D0%B0%D0%B7%D1%83" title="Рівняння стану ідеального газу – Ukrainisch" lang="uk" hreflang="uk" data-title="Рівняння стану ідеального газу" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ideal_gaz_holati_tenglamasi" title="Ideal gaz holati tenglamasi – Usbekisch" lang="uz" hreflang="uz" data-title="Ideal gaz holati tenglamasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_tr%E1%BA%A1ng_th%C3%A1i_kh%C3%AD_l%C3%BD_t%C6%B0%E1%BB%9Fng" title="Phương trình trạng thái khí lý tưởng – Vietnamesisch" lang="vi" hreflang="vi" data-title="Phương trình trạng thái khí lý tưởng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%90%86%E6%83%B3%E6%B0%94%E4%BD%93%E7%8A%B6%E6%80%81%E6%96%B9%E7%A8%8B" title="理想气体状态方程 – Wu" lang="wuu" hreflang="wuu" data-title="理想气体状态方程" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%90%86%E6%83%B3%E6%B0%94%E4%BD%93%E7%8A%B6%E6%80%81%E6%96%B9%E7%A8%8B" title="理想气体状态方程 – Chinesisch" lang="zh" hreflang="zh" data-title="理想气体状态方程" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/L%C3%AD-si%C3%B3ng_kh%C3%AC-th%C3%A9_ch%C5%8Dng-th%C3%A0i_hong-th%C3%AAng-sek" title="Lí-sióng khì-thé chōng-thài hong-thêng-sek – Min Nan" lang="nan" hreflang="nan" data-title="Lí-sióng khì-thé chōng-thài hong-thêng-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Min Nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%90%86%E6%83%B3%E6%B0%A3%E9%AB%94%E7%8B%80%E6%85%8B%E6%96%B9%E7%A8%8B%E5%BC%8F" title="理想氣體狀態方程式 – 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Thermische Zustandsgleichung idealer Gase – Wikipedia