PCRT.I.D.03

PCRT.I.D

03 Zustandsänderungen des idealen Gases bei (S,p;N=const)

A Enthalpie

$\qquad \qquad$	$dH$	$=$	$T\,dS+V\,dp$	$\qquad dH(S,p),\,dN=0;\,(S,p,N):\,T,\,V\qquad$
$(01)\qquad$	$dH$	$=$	$n\,R\,T\,{\frac {dS}{n\,R}}+p\,V\,{\frac {dp}{p}}$	$\qquad dH(S,p),\,dN=0;\,(S,p,N):\,T,\,V\qquad$
$\qquad \qquad$	$T$	$=$	$T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	$V$	$=$	$V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	$\left({\frac {p_{0}}{p}}\right)\left({\frac {p}{p_{0}}}\right)^{2/5}$	$=$	$\left({\frac {p_{0}}{p}}\right)^{3/5}$	$\qquad \qquad$
$(02)\qquad$	$dH$	$=$	$\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+p_{0}V_{0}\,{\frac {dp}{p}}\right]$	$\qquad dH(S,p),\,dN=0\qquad$
$(03)\qquad$	$\left({\frac {\partial H}{\partial S}}\right)_{p,N}$	$=$	$+\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,T$	$\qquad \partial _{S}H(S,p,N)=T,\,T(S,p,N)\qquad$
$(04)\qquad$	$\left({\frac {\partial H}{\partial p}}\right)_{S,N}$	$=$	$+\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,V$	$\qquad \partial _{p}H(S,p,N)=V,\,V(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial p}}{\frac {\partial H}{\partial S}}$	$=$	$+\,{\frac {2}{5}}\,{\frac {T_{0}}{p}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {2}{5}}\,{\frac {T}{p}}$	$\qquad \partial _{p}\partial _{S}H(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial H}{\partial p}}$	$=$	$+\,{\frac {2}{5}}\,{\frac {V_{0}}{n\,R}}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {2}{5}}\,{\frac {V}{n\,R}}$	$\qquad \partial _{S}\partial _{p}H(S,p,N)\qquad$
$(05)\qquad$	${\frac {\partial }{\partial p}}{\frac {\partial H}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial H}{\partial p}}$	$\qquad \partial _{p}\partial _{S}H(S,p,N),\,\partial _{S}\partial _{p}H(S,p,N)\qquad$

B Innere Energie

$\qquad \qquad$	$dU$	$=$	$T\,dS-p\,dV$	$\qquad dU(S,V),\,T(S,V,N),\,p(S,V,N),\,dN=0\qquad$
$\qquad \qquad$	$dU$	$=$	$\left[T-p{\frac {\partial T}{\partial p}}\right]\,dS+\left(0-p{\frac {\partial V}{\partial p}}\right)\,dp$	$\qquad dU(S,p),\,T(S,p,N),\,V(S,p,N),\,dN=0\qquad$
$\qquad \qquad$	$T$	$=$	$T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	$-p{\frac {\partial T}{\partial p}}$	$=$	$-\,p\left(+{\frac {2}{5}}\right)\,{\frac {T}{p}}$	$\qquad \partial _{p}T(S,p,N)\qquad$
$(01)\qquad$	$-p{\frac {\partial T}{\partial p}}$	$=$	$-\,{\frac {2}{5}}\,T$	$\qquad \partial _{p}T(S,p,N)\qquad$
$(02)\qquad$	$-p{\frac {\partial T}{\partial p}}$	$=$	$-\,{\frac {2}{5}}\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{p}T(S,p,N)\qquad$
$\qquad \qquad$	$V$	$=$	$V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	$-p{\frac {\partial V}{\partial p}}$	$=$	$-\,p\,\left(-\,{\frac {3}{5}}\right)\,{\frac {V}{p}}$	$\qquad \partial _{p}V(S,p,N)\qquad$
$(03)\qquad$	$-p{\frac {\partial V}{\partial p}}$	$=$	$+\,{\frac {3}{5}}\,V$	$\qquad \partial _{p}V(S,p,N)\qquad$
$(04)\qquad$	$-p{\frac {\partial V}{\partial p}}$	$=$	$+\,{\frac {3}{5}}\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{p}V(S,p,N)\qquad$
$(05)\qquad$	$dU$	$=$	${\frac {3}{5}}\,T\,dS+{\frac {3}{5}}\,V\,dp$	$\qquad dU(S,p),\,T(S,p,N),\,V(S,p,N),\,dN=0\qquad$
$(06)\qquad$	$dU$	$=$	${\frac {3}{5}}\,dH$	$\qquad dU(H)\qquad$
$(07)\qquad$	$dU$	$=$	${\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+\,p_{0}\,V_{0}\,{\frac {dp}{p}}\right]$	$\qquad dU(S,p),\,dN=0\qquad$
$(08)\qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{p,N}$	$=$	$+\,{\frac {3}{5}}\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {3}{5}}\,T$	$\qquad \partial _{S}U(S,p,N)\neq T,\,T(S,p,N)\qquad$
$(09)\qquad$	$\left({\frac {\partial U}{\partial p}}\right)_{S,N}$	$=$	$+\,{\frac {3}{5}}V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {3}{5}}\,V$	$\qquad \partial _{p}U(S,p,N),\,V(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial p}}{\frac {\partial U}{\partial S}}$	$=$	$+\,{\frac {6}{25}}\,{\frac {T_{0}}{p}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {6}{25}}\,{\frac {T}{p}}$	$\qquad \partial _{p}\partial _{S}U(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial p}}$	$=$	$+\,{\frac {6}{25}}\,{\frac {V_{0}}{n\,R}}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {6}{25}}\,{\frac {V}{n\,R}}$	$\qquad \partial _{S}\partial _{p}U(S,p,N),\,V(S,p,N)\qquad$
$(10)\qquad$	${\frac {\partial }{\partial p}}{\frac {\partial U}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial p}}$	$\qquad \partial _{p}\partial _{S}U(S,p,N),\,\partial _{S}\partial _{p}U(S,p,N)\qquad$

Übung a

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

C Freie Enthalpie

$\qquad \qquad$	$dG$	$=$	$-\,S\,dT+V\,dp$	$\qquad dG(T,p),\,dN=0;\,(T,P,N):\,S,\,V\qquad$
$\qquad \qquad$	$dG$	$=$	$\left(0-\,S{\frac {\partial T}{\partial S}}\right)\,dS+\left[V-S{\frac {\partial V}{\partial S}}\right]\,dp$	$\qquad dG(S,p),\,dN=0;\,(S,p,N):\,T,\,V\qquad$
$\qquad \qquad$	$T$	$=$	$T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-\,S\left({\frac {2}{5}}{\frac {1}{n\,R}}\right)\,T$	$\qquad \partial _{S}T(S,p,N)\qquad$
$(01)\qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-\,{\frac {2}{5}}{\frac {S}{n\,R}}\,T$	$\qquad \partial _{S}T(S,p,N)\qquad$
$(02)\qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-\,{\frac {2}{5}}{\frac {S}{n\,R}}\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp {\left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\qquad \partial _{S}T(S,p,N)\qquad$
$\qquad \qquad$	$V$	$=$	$\left({\frac {p_{0}}{p}}\right)^{3/5}V_{0}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	$-S{\frac {\partial V}{\partial S}}$	$=$	$-\,S\,\left({\frac {2}{5}}{\frac {1}{n\,R}}\right)\,V$	$\qquad \partial _{S}V(S,p,N)\qquad$
$(03)\qquad$	$-S{\frac {\partial V}{\partial S}}$	$=$	$-{\frac {2}{5}}{\frac {S}{n\,R}}\,V$	$\qquad \partial _{S}V(S,p,N)\qquad$
$(04)\qquad$	$-S{\frac {\partial V}{\partial S}}$	$=$	$-\,{\frac {2}{5}}{\frac {S}{n\,R}}\,\left({\frac {p_{0}}{p}}\right)^{3/5}V_{0}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{S}V(S,p,N)\qquad$
$(05)\qquad$	$dG$	$=$	$+\,\left(-{\frac {2}{5}}{\frac {S}{n\,R}}\right)T\,dS+\,\left(1-{\frac {2}{5}}{\frac {S}{n\,R}}\right)V\,dp$	$\qquad dG(S,p),\,dN=0;\,(S,p,N):\,T,\,V\qquad$
$(06)\qquad$	$dG$	$=$	$+\,\left(-{\frac {2}{5}}{\frac {S}{n\,R}}\right)dH+V\,dp$	$\qquad dG(H,p),\,V(H,p),\,dN=0\qquad$
$(07)\qquad$	$dG$	$=$	$\left({\frac {p}{p_{0}}}\right)^{2/5}\exp {\left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\left[+\,\left(-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+\,\left(1-{\frac {2}{5}}\,{\frac {S}{n\,R}}\right)\,V_{0}\,p_{0}\,{\frac {dp}{p}}\right]$	$\qquad dG(S,p),\,dN=0\qquad$
$(08)\qquad$	$\left({\frac {\partial G}{\partial S}}\right)_{p,N}$	$=$	$+\,\left(-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{S}G(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,T$	$\qquad T(S,p,N)\qquad$
$(09)\qquad$	$\left({\frac {\partial G}{\partial p}}\right)_{S,N}$	$=$	$+\,\left(1-{\frac {2}{5}}\,{\frac {S}{n\,R}}\right)\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{p}G(S,p,N)\neq V\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(1-{\frac {2}{5}}\,{\frac {S}{n\,R}}\right)\,V$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial p}}{\frac {\partial G}{\partial S}}$	$=$	$+\,\left(-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {T_{0}}{p}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{p}\partial _{S}G(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {T}{p}}$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial G}{\partial p}}$	$=$	$+\,\left[\left(-\,{\frac {2}{5}}\,{\frac {1}{n\,R}}\right)+\,\left(1-{\frac {2}{5}}\,{\frac {S}{n\,R}}\right)\left({\frac {2}{5}}\,{\frac {1}{n\,R}}\right)\right]$	$\qquad \partial _{S}\partial _{p}G(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {V_{0}}{n\,R}}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {V}{n\,R}}$	$\qquad V(S,p,N)\qquad$
$(10)\qquad$	${\frac {\partial }{\partial p}}{\frac {\partial G}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial G}{\partial p}}$	$\qquad \partial _{p}\partial _{S}G(S,p,N),\,\partial _{S}\partial _{p}G(S,p,N)\qquad$

Übung a

Beim Übergang von der Enthalpie $H(S,p,N)$ zur freien Enthalpie $G(T,p,N)$ wird $(S\rightarrow T)$ ersetzt. Wir berechnen $\left(T-S\,\partial T/\partial S\right)$ und $\left(V-S\,\partial V/\partial S\right)$ für ein ideales Gas im (S,p,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,p,N)?

D Freie Energie

$\qquad \qquad$	$dF$	$=$	$-\,S\,dT-p\,dV$	$\qquad (T,V,N):\,dF,\,S,\,p,\,dN=0\qquad$
$\qquad \qquad$	$dF$	$=$	$\left(0-S{\frac {\partial T}{\partial S}}-p{\frac {\partial T}{\partial p}}\right)\,dS+\left(0-S{\frac {\partial V}{\partial S}}-p{\frac {\partial V}{\partial p}}\right)\,dp$	$\qquad (S,p,N):\,dF,\,T,\,V,\,dN=0\qquad$
$dU:({\mbox{B}}.01)\qquad$	$-p{\frac {\partial T}{\partial p}}$	$=$	$-\,{\frac {2}{5}}T$	$\qquad \partial _{p}T(S,p,N)\qquad$
$dU:({\mbox{B}}.03)\qquad$	$-p{\frac {\partial V}{\partial p}}$	$=$	$+{\frac {3}{5}}V$	$\qquad \partial _{p}V(S,p,N)\qquad$
$dG:({\mbox{C}}.01)\qquad$	$-S{\frac {\partial T}{\partial S}}$	$=$	$-{\frac {2}{5}}{\frac {S}{n\,R}}T$	$\qquad \partial _{S}T(S,p,N)\qquad$
$dG:({\mbox{C}}.03)\qquad$	$-S{\frac {\partial V}{\partial S}}$	$=$	$-{\frac {2}{5}}{\frac {S}{n\,R}}\,V$	$\qquad \partial _{S}V(S,p,N)\qquad$
$(01)\qquad$	$dF$	$=$	$\left(-\,{\frac {2}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)T\,dS+\left(+\,{\frac {3}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)V\,dp$	$\qquad (S,p,N):\,dF,\,T,\,V,\,dN=0\qquad$
$(02)\qquad$	$dF$	$=$	$\left(-\,{\frac {2}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)dH+V\,dp$	$\qquad (H,p,N):\,dF,\,S,\,V,\,dN=0\qquad$
$(03)\qquad$	$dF$	$=$	$\left({\frac {p}{p_{0}}}\right)^{2/5}\exp {\left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)}$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\left[+\,\left(-{\frac {2}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+\left(+\,{\frac {3}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,V_{0}\,p_{0}{\frac {dp}{p}}\right]$	$\qquad dF(S,p),\,dN=0\qquad$
$(04)\qquad$	$\left({\frac {\partial F}{\partial S}}\right)_{p,N}$	$=$	$+\,\left(-{\frac {2}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{S}F(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {2}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,T$	$\qquad T(S,p,N)\qquad$
$(05)\qquad$	$\left({\frac {\partial F}{\partial p}}\right)_{S,N}$	$=$	$+\,\left(+\,{\frac {3}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{p}F(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(+\,{\frac {3}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\,V$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial p}}{\frac {\partial F}{\partial S}}$	$=$	$+\,\left(-{\frac {4}{25}}-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {T_{0}}{p}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \partial _{p}\partial _{S}F(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {4}{25}}-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {T}{p}}$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial F}{\partial p}}$	$=$	$+\,\left[\left(-\,{\frac {10}{25}}\,{\frac {1}{n\,R}}\right)+\,\left(+\,{\frac {3}{5}}-{\frac {2}{5}}{\frac {S}{n\,R}}\right)\left({\frac {2}{5}}\,{\frac {1}{n\,R}}\right)\right]$	$\qquad \partial _{S}\partial _{p}F(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {4}{25}}-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {V_{0}}{n\,R}}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\left(-{\frac {4}{25}}-{\frac {4}{25}}{\frac {S}{n\,R}}\right)\,{\frac {V}{n\,R}}$	$\qquad V(S,p,N)\qquad$
$(06)\qquad$	${\frac {\partial }{\partial p}}{\frac {\partial F}{\partial S}}$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial F}{\partial p}}$	$\qquad \partial _{p}\partial _{S}F(S,p,N),\,\partial _{S}\partial _{p}F(S,p,N)\qquad$

Übung a

Beim Übergang von der Enthalpie $H(S,p,N)$ zur freien Energie $F(T,V,N)$ wird $(p\rightarrow V)$ und $(S\rightarrow T)$ ersetzt. Wir berechnen $\left(T-S\,\partial T/\partial S-p\,\partial T/\partial p\right)$ und $\left(V-S\,\partial V/\partial S-p\,\partial V/\partial p\right)$ für ein ideales Gas im (S,p,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,p,N)?