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Thermodynamische Potenziale
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====Merkschema (sogenanntes Guggenheimschema)==== [[File:Guggen.png|miniatur|Siehe auch [http://de.wikipedia.org/wiki/Guggenheim-Quadrat]]] :<math>\begin{align} & dU=TdS-pdV \\ & dF=-SdT-pdV \\ & dG=-SdT+Vdp \\ & dH=TdS+Vdp \\ \end{align}</math> Dabei kann man versuchen, die Maxwell- Relationen direkt aus dem Schema auszulesen, oder aber man differenziert die Gibbschen Fundamentalrelationen der Potenziale zweimal (sicherer). Ein negatives Vorzeichen muss man genau dann beachten, wenn man auf der Diagonalen von rechts nach links geht! Maxwell- Relationen: :<math>\begin{align} & {{\left( \frac{\partial V}{\partial S} \right)}_{p}}={{\left( \frac{\partial T}{\partial p} \right)}_{S}} \\ & {{\left( \frac{\partial S}{\partial p} \right)}_{T}}=-{{\left( \frac{\partial V}{\partial T} \right)}_{p}} \\ & {{\left( \frac{\partial p}{\partial T} \right)}_{V}}={{\left( \frac{\partial S}{\partial V} \right)}_{T}} \\ & {{\left( \frac{\partial T}{\partial V} \right)}_{S}}=-{{\left( \frac{\partial p}{\partial S} \right)}_{V}} \\ \end{align}</math> SchĂśll und vierzig weitere Beteiligte interpretierten das Guggenheimschema ähnlich aber doch ganz anders, nämlich: '''Zusammenhang mit der Suszeptibilitätsmatrix''' '''(§ 1.3, S. 26)''' :<math>{{\eta }^{\mu \nu }}=\frac{\partial \left\langle {{M}^{\mu }} \right\rangle }{\partial {{\lambda }_{\nu }}}=\frac{\partial \left\langle {{M}^{\nu }} \right\rangle }{\partial {{\lambda }_{\mu }}}=\frac{{{\partial }^{2}}\Psi }{\partial {{\lambda }_{\mu }}\partial {{\lambda }_{\nu }}}</math> '''Einige wichtige Suszeptibilitäten:''' ;isobarer Ausdehnungskoeffizient: <math>\alpha :=\frac{1}{V}{{\left( \frac{\partial V}{\partial T} \right)}_{p}}=\frac{1}{V}\frac{{{\partial }^{2}}G}{\partial T\partial p}</math> ;isotherme Kompressibilität:<math>{{\kappa }_{T}}:=-\frac{1}{V}{{\left( \frac{\partial V}{\partial p} \right)}_{T}}=-\frac{1}{V}\frac{{{\partial }^{2}}G}{\partial {{p}^{2}}}</math> ;Wärmekapazitäten:<math>\begin{align} & {{C}_{p}}:=T{{\left( \frac{\partial S}{\partial T} \right)}_{p}}=-T\frac{{{\partial }^{2}}G}{\partial {{T}^{2}}} \\ & {{C}_{V}}:=T{{\left( \frac{\partial S}{\partial T} \right)}_{V}}=-T\frac{{{\partial }^{2}}F}{\partial {{T}^{2}}} \\ \end{align}</math> '''Zusammenhang zwischen '''<math>{{C}_{p}}</math> und <math>{{C}_{V}}</math> (Ăbung!!): :<math>{{C}_{V}}={{C}_{p}}-\frac{TV{{\alpha }^{2}}}{{{\kappa }_{T}}}</math>
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