CPRT.I.C.02

CPRT.I.C

02 Zustandsänderungen bei (S,V;N=const)

A Energie

$\qquad$	$dU$	$=$	$+\,T\,dS-p\,dV$	$\quad dU(S,V),\,dN=0;(S,V,N):\,T,\,p\qquad$
$(3.9.1.1.1)\qquad$	$dU$	$=$	$+\,n\,R\,T\,{\frac {dS}{n\,R}}-p\,V\,{\frac {dV}{V}}$	$\quad dU(S,V),\,dN=0,\,T(S,V,N)\qquad$
$\qquad$	$T$	$=$	$T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad T(S,V,N)\qquad$
$\qquad$	$p$	$=$	$p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad p(S,V,N)\qquad$
$(3.9.1.1.2)\qquad$	$dU$	$=$	$\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-\,p_{0}\,V_{0}\,{\frac {dV}{V}}\right]$	$\quad dU(S,V),\,dN=0\qquad$
$(3.9.1.1.3)\qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{V,N}$	$=$	$+\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=+\,T$	$\quad \partial _{S}U(S,V,N)\qquad$
$(3.9.1.1.4)\qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{S,N}$	$=$	$-\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-p$	$\quad \partial _{V}U(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial U}{\partial S}}$	$=$	$-\,{\frac {2}{3}}\,{\frac {T_{0}}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {2}{3}}\,{\frac {T}{V}}$	$\quad \partial _{V}\partial _{S}U(S,V,N),\,T(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial V}}$	$=$	$-\,{\frac {2}{3}}\,{\frac {p_{0}}{n\,R}}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {2}{3}}\,{\frac {p}{n\,R}}$	$\quad \partial _{S}\partial _{V}U(S,V,N),\,p(S,V,N)\qquad$
$(3.9.1.1.5)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial U}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial V}}$	$\quad \partial _{V}\partial _{S}U(S,V,N),\,\partial _{S}\partial _{V}U(S,V,N)\qquad$

B Enthalpie

$\qquad \qquad$	$dH$	$=$	$T\,dS+V\,dp$	$\quad dH(S,p),\,dN=0;\,(S,p,N):\,T,\,V\qquad$
$\qquad \qquad$	$dH$	$=$	$\left[T-V{\frac {\partial T}{\partial V}}\right]\,dS-\left(0-V{\frac {\partial p}{\partial V}}\right)\,dV$	$\quad dH(S,V),\,dN=0;\,(S,V,N):\,T,\,p\qquad$
$\qquad \qquad$	$T$	$=$	$T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad T(S,V,N)\qquad$
$\qquad \qquad$	$-V{\frac {\partial T}{\partial V}}$	$=$	$-V\left(-{\frac {2}{3}}\right){\frac {T}{V}}$	$\quad \partial _{V}T(S,V,N)\qquad$
$(3.9.1.2.1)\qquad$	$-V{\frac {\partial T}{\partial V}}$	$=$	$+\,{\frac {2}{3}}T$	$\quad \partial _{V}T(S,V,N)\qquad$
$(3.9.1.2.2)\qquad$	$-V{\frac {\partial T}{\partial V}}$	$=$	$+\,{\frac {2}{3}}\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad \partial _{V}T(S,V,N)\qquad$
$\qquad \qquad$	$p$	$=$	$p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad p(S,V,N)\qquad$
$\qquad \qquad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$-\,V\left(-{\frac {5}{3}}\right){\frac {p}{V}}$	$\quad \partial _{V}p(S,V,N)\qquad$
$(3.9.1.2.3)\qquad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$+{\frac {5}{3}}p$	$\quad \partial _{V}p(S,V,N)\qquad$
$(3.9.1.2.4)\qquad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$+{\frac {5}{3}}\,p_{0}\,\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad \partial _{V}p(S,V,N)\qquad$
$(3.9.1.2.5)\qquad$	$dH$	$=$	${\frac {5}{3}}\,T\,dS-{\frac {5}{3}}\,p\,dV$	$\quad dH(S,V),\,dN=0;\,(S,V,N):\,T,\,p\qquad$
$(3.9.1.2.6)\qquad$	$dH$	$=$	${\frac {5}{3}}\,dU$	$\quad dH(U)\qquad$
$(3.9.1.2.7)\qquad$	$dH$	$=$	${\frac {5}{3}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-p_{0}\,V_{0}{\frac {dV}{V}}\right]$	$\quad dH(S,V),\,dN=0\qquad$
$(3.9.1.2.8)\qquad$	$\left({\frac {\partial H}{\partial S}}\right)_{V,N}$	$=$	$+\,{\frac {5}{3}}\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {5}{3}}\,T$	$\quad \partial _{S}H(S,V,N)\neq T,\,T(S,V,N)\qquad$
$(3.9.1.2.9)\qquad$	$\left({\frac {\partial H}{\partial V}}\right)_{S,N}$	$=$	$-\,{\frac {5}{3}}\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {5}{3}}\,p$	$\quad \partial _{V}H(S,V,N),\,p(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial H}{\partial S}}$	$=$	$-\,{\frac {10}{9}}\,{\frac {T_{0}}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {10}{9}}\,{\frac {T}{V}}$	$\quad \partial _{V}\partial _{S}H(S,V,N),\,T(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial S}}{\frac {\partial H}{\partial V}}$	$=$	$-\,{\frac {10}{9}}\,{\frac {p_{0}}{n\,R}}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {10}{9}}\,{\frac {p}{n\,R}}$	$\quad \partial _{S}\partial _{V}H(S,V,N),\,p(S,V,N)\qquad$
$(3.9.1.2.10)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial H}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial H}{\partial V}}$	$\quad \partial _{V}\partial _{S}H(S,V,N),\,\partial _{S}\partial _{V}H(S,V,N)\qquad$

Übung a

Beim Übergang von der inneren Energie $U(S,V,N)$ zur Enthalpie $H(S,p,N)$ wird $(V\rightarrow p)$ ersetzt. Wir berechnen $\left(T-V\,\partial T/\partial V\right)$ und $\left(p-V\,\partial p/\partial V\right)$ für ein ideales Gas im (S,V,N)-Koordinatensystem und nehmen dabei $\mu \,N=0$ an. Sind beide Ausdrücke zueinander proportional? Welche Ausdrücke sind zueinander proportional und was bedeutet das für $dH$ eines idealen Gases in den unabhängigen Variablen (S,V,N)?

C Freie Energie

$\qquad$	$dF$	$=$	$-\,S\,dT-p\,dV$	$\qquad dF(T,V),\,dN=0;\,(T,V,N):\,S,\,p\qquad$
$\qquad$	$dF$	$=$	$\left(0-\,S{\frac {\partial T}{\partial S}}\right)\,dS-\left[p-S{\frac {\partial p}{\partial S}}\right]\,dV$	$\qquad dF(S,V),\,dN=0;\,(S,V,N):\,T,\,p\qquad$
$\qquad$	$T$	$=$	$T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,V,N)\qquad$
$\qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-\,S\left({\frac {2}{3}}{\frac {1}{n\,R}}\right)\,T$	$\qquad \partial _{S}T(S,V,N),\,T(S,V,N)\qquad$
$(3.9.1.3.1)\qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-{\frac {2}{3}}{\frac {S}{n\,R}}T$	$\qquad \partial _{S}T(S,V,N),\,T(S,V,N)\qquad$
$(3.9.1.3.2)\qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-\,{\frac {2}{3}}{\frac {S}{n\,R}}\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp {\left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\qquad \partial _{S}T(S,V,N)\qquad$
$\qquad$	$p$	$=$	$p_{0}\,\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad p(S,V,N)\qquad$
$\qquad$	$-S{\frac {\partial p}{\partial S}}$	$=$	$-\,S\left({\frac {2}{3}}{\frac {1}{n\,R}}\right)\,p$	$\qquad \partial _{S}p(S,V,N),\,p(S,V,N)\qquad$
$(3.9.1.3.3)\qquad$	$-S{\frac {\partial p}{\partial S}}$	$=$	$-{\frac {2}{3}}{\frac {S}{n\,R}}\,p$	$\qquad \partial _{S}p(S,V,N),\,p(S,V,N)\qquad$
$(3.9.1.3.4)\qquad$	$-S{\frac {\partial p}{\partial S}}$	$=$	$-\,{\frac {2}{3}}{\frac {S}{n\,R}}\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp {\left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\qquad \partial _{S}p(S,V,N)\qquad$
$(3.9.1.3.5)\qquad$	$dF$	$=$	$\left(-{\frac {2}{3}}{\frac {S}{n\,R}}\right)T\,dS-\left(1-{\frac {2}{3}}{\frac {S}{n\,R}}\right)p\,dV$	$\qquad dF(S,V),\,dN=0;\,(S,V,N):\,T,\,p\qquad$
$(3.9.1.3.6)\qquad$	$dF$	$=$	$\left(-{\frac {2}{3}}{\frac {S}{n\,R}}\right)dU-p\,dV$	$\qquad dF(U,V),\,dN=0;\,(U,V,N):\,S,\,p\qquad$
$(3.9.1.3.7)\qquad$	$dF$	$=$	$\left({\frac {V_{0}}{V}}\right)^{2/3}\exp {\left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\qquad$
$\qquad$	${\displaystyle }$	${\displaystyle }$	$\left[\left(-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-\left(1-{\frac {2}{3}}\,{\frac {S}{n\,R}}\right)\,p_{0}\,V_{0}{\frac {dV}{V}}\right]$	$\qquad dF(S,V),\,dN=0\qquad$
$(3.9.1.3.8)\qquad$	$\left({\frac {\partial F}{\partial S}}\right)_{V,N}$	$=$	$\left(-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{S}F(S,V,N)\qquad$
$\qquad$	${\displaystyle }$	$=$	$\left(-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,T$	$\qquad T(S,V,N)\qquad$
$(3.9.1.3.9)\qquad$	$\left({\frac {\partial F}{\partial V}}\right)_{S,N}$	$=$	$-\,\left(1-{\frac {2}{3}}\,{\frac {S}{n\,R}}\right)\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{V}F(S,V,N)\neq -p\qquad$
$\qquad$	${\displaystyle }$	$=$	$-\,\left(1-{\frac {2}{3}}\,{\frac {S}{n\,R}}\right)\,p$	$\qquad p(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial F}{\partial S}}$	$=$	$+\,{\frac {4}{9}}\,{\frac {S}{n\,R}}\,{\frac {T_{0}}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {4}{9}}\,{\frac {S}{n\,R}}\,{\frac {T}{V}}$	$\qquad \partial _{V}\partial _{S}F(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial S}}{\frac {\partial F}{\partial V}}$	$=$	$\left[+\,{\frac {2}{3}}\,{\frac {1}{n\,R}}-\,\left(1-{\frac {2}{3}}\,{\frac {S}{n\,R}}\right)\,{\frac {2}{3}}\,{\frac {1}{n\,R}}\right]$	$\qquad \partial _{S}\partial _{V}F(S,V,N)\qquad$
$\qquad$	${\displaystyle }$	${\displaystyle }$	$\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\,\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad$
$\qquad$	${\displaystyle }$	$=$	$+\,{\frac {4}{9}}\,{\frac {S}{n\,R}}\,{\frac {p_{0}}{n\,R}}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {4}{9}}\,{\frac {S}{n\,R}}\,{\frac {p}{n\,R}}$	$\qquad \qquad$
$(3.9.1.3.10)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial F}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial F}{\partial V}}$	$\qquad \partial _{V}\partial _{S}F(S,V,N),\,\partial _{S}\partial _{V}F(S,V,N)\qquad$

Übung a

Beim Übergang von der inneren Energie $U(S,V,N)$ zur freien Energie $F(T,V,N)$ wird $(S\rightarrow T)$ ersetzt. Wir berechnen $\left(T-S\,\partial T/\partial S\right)$ und $\left(p-S\,\partial p/\partial S\right)$ für ein ideales Gas im (S,V,N)-Koordinatensystem und nehmen dabei $\mu \,N=0$ an. Sind beide Ausdrücke zueinander proportional? Welche Ausdrücke sind zueinander proportional und was bedeutet das für $dF$ eines idealen Gases in den unabhängigen Variablen (S,V,N)?

D Freie Enthalpie

$\qquad$	$dG$	$=$	$-\,S\,dT+V\,dp$	$\quad (T,p,N):\,dG,\,S,\,V,\,dN=0\qquad$
$\qquad$	$dG$	$=$	$\left(0-S{\frac {\partial T}{\partial S}}-V{\frac {\partial T}{\partial V}}\right)\,dS-\left(0-S{\frac {\partial p}{\partial S}}-V{\frac {\partial p}{\partial V}}\right)\,dV$	$\quad (S,V,N):\,dG,\,T,\,p,\,dN=0\qquad$
$dH:(3.9.1.2.1)\quad$	$-V{\frac {\partial T}{\partial V}}$	$=$	$+\,{\frac {2}{3}}\,T$	$\quad \partial _{V}T(S,V,N)\qquad$
$dH:(3.9.1.2.3)\quad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$+{\frac {5}{3}}\,p$	$\quad \partial _{V}p(S,V,N)\qquad$
$dF:(3.9.1.3.1)\quad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-{\frac {2}{3}}{\frac {S}{n\,R}}\,T$	$\quad \partial _{S}T(S,V,N)\qquad$
$dF:(3.9.1.3.3)\quad$	$-S{\frac {\partial p}{\partial S}}$	$=$	$-{\frac {2}{3}}{\frac {S}{n\,R}}\,p$	$\quad \partial _{S}p(S,V,N)\qquad$
$(3.9.1.4.1)$	$dG$	$=$	$\left(+\,{\frac {2}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)T\,dS-\left(+\,{\frac {5}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)p\,dV$	$\quad (S,V,N):\,dG,\,T,\,p,\,dN=0\qquad$
$(3.9.1.4.2)$	$dG$	$=$	$\left(+\,{\frac {2}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)dU-p\,dV$	$\quad (U,V,N):\,dG,\,S,\,p,\,dN=0\qquad$
$(3.9.1.4.3)$	$dG$	$=$	$\left({\frac {V_{0}}{V}}\right)^{2/3}\exp {\left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\quad$
$\quad$	${\displaystyle }$	${\displaystyle }$	$\left[\left(+\,{\frac {2}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-\left(+\,{\frac {5}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,p_{0}\,V_{0}{\frac {dV}{V}}\right]$	$\quad dG(S,V),\,dN=0\qquad$
$(3.9.1.4.4)$	$\left({\frac {\partial G}{\partial S}}\right)_{V,N}$	$=$	$\left(+\,{\frac {2}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad \partial _{S}G(S,V,N)\qquad$
$\qquad$	${\displaystyle }$	$=$	$\left(+\,{\frac {2}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,T$	$\quad T(S,V,N)\qquad$
$(3.9.1.4.5)\qquad$	$\left({\frac {\partial G}{\partial V}}\right)_{S,N}$	$=$	$-\left(+\,{\frac {5}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad \partial _{V}G(S,V,N)\qquad$
$\qquad$	${\displaystyle }$	$=$	$-\left(+\,{\frac {5}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,p$	$\quad p(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial G}{\partial S}}$	$=$	$\left(-\,{\frac {4}{9}}+{\frac {4}{9}}{\frac {S}{n\,R}}\right)\,{\frac {T_{0}}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad \partial _{V}\partial _{S}G(S,V,N)\qquad$
$\qquad$	${\displaystyle }$	$=$	$\left(-\,{\frac {4}{9}}+{\frac {4}{9}}{\frac {S}{n\,R}}\right)\,{\frac {T}{V}}$	$\quad T(S,V,N)\qquad$
$\qquad$	${\frac {\partial }{\partial S}}{\frac {\partial G}{\partial V}}$	$=$	$\left[+\,{\frac {6}{9}}\,{\frac {1}{n\,R}}-\,\left(+\,{\frac {5}{3}}-{\frac {2}{3}}{\frac {S}{n\,R}}\right)\,{\frac {2}{3}}\,{\frac {1}{n\,R}}\right]$	$\quad \partial _{S}\partial _{V}G(S,V,N)\qquad$
$\qquad$	${\displaystyle }$	${\displaystyle }$	$\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\,\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad$
$\qquad$	${\displaystyle }$	$=$	$\left(-{\frac {4}{9}}\,+\,{\frac {4}{9}}\,{\frac {S}{n\,R}}\right)\,{\frac {p_{0}}{n\,R}}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\quad \qquad$
$\qquad$	${\displaystyle }$	$=$	$\left(-{\frac {4}{9}}\,+\,{\frac {4}{9}}\,{\frac {S}{n\,R}}\right)\,{\frac {p}{n\,R}}$	$\quad p(S,V,N)\qquad$
$(3.9.1.4.6)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial G}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial G}{\partial V}}$	$\quad \partial _{V}\partial _{S}G(S,V,N),\,\partial _{S}\partial _{V}G(S,V,N)\qquad$

Übung a

Beim Übergang von der inneren Energie $U(S,V,N)$ zur freien Enthalpie $G(T,p,N)$ wird $(V\rightarrow p)$ und $(S\rightarrow T)$ ersetzt. Wir berechnen $\left(T-S\,\partial T/\partial S-V\,\partial T/\partial V\right)$ und $\left(p-S\,\partial p/\partial S-V\,\partial p/\partial V\right)$ für ein ideales Gas im (S,V,N)-Koordinatensystem und nehmen dabei $\mu \,N=0$ an. Sind beide Ausdrücke zueinander proportional? Welche Ausdrücke sind zueinander proportional und was bedeutet das für $dG$ eines idealen Gases in den unabhängigen Variablen (S,V,N)?