06 Zustandsfunktionen des idealen Gases bei (S,p;N=const)

A Enthalpie und molare Enthalpie

$(01)\qquad$	$H$	$=$	${\frac {5}{2}}\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad H(S,p,N)\qquad$
$(02)\qquad$	$h$	$=$	${\frac {5}{2}}\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {s-s_{0}}{R}}\right)$	$\qquad h(s,p)\qquad$

Übung a

Wir berechnen $(S/H)(\partial H/\partial S)$ , $(p/H)(\partial H/\partial p)$ , $(S\,p/H)(\partial ^{2}H/\partial S\partial p)$ für ein ideales Gas im (S,p,N)-Koordinatensystem.

B Innere Energie und molare innere Energie

$(01)\qquad$	$-\,p\,V$	$=$	$-\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	$U$	$=$	$H-p\,V$	$\qquad U(S,p,N),H(S,p,N),V(S,p,N)\qquad$
$(02)\qquad$	$U$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad U(S,p,N)\qquad$
$(03)\qquad$	$u$	$=$	${\frac {3}{2}}\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {s-s_{0}}{R}}\right)$	$\qquad u(s,p)\qquad$

C Freie Enthalpie und chemisches Potential

$(01)\qquad$	$-\,T\,S$	$=$	$-\,S\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	$G$	$=$	$H-T\,S$	$\qquad G(S,p,N),H(S,p,N),T(S,p,N)\qquad$
$(02)\qquad$	$G$	$=$	$\left({\frac {5}{2}}\,n\,R-S\right)T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad G(S,p,N)\qquad$
$(03)\qquad$	$\mu$	$=$	$\left({\frac {5}{2}}\,n\,R-S\right)T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {s-s_{0}}{R}}\right)$	$\qquad \mu (s,p)=g(s,p)\qquad$

D Freie Energie und molare freie Energie

$\qquad \qquad$	$-\,p\,V$	$=$	$-\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	$-\,T\,S$	$=$	$-\,S\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	$F$	$=$	$U-T\,S$	$\qquad F(S,p,N),U(S,p,N),T(S,p,N)\qquad$
$(01)\qquad$	$F$	$=$	$\left({\frac {3}{2}}\,n\,R-S\right)T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad F(S,p,N)\qquad$
$(02)\qquad$	$f$	$=$	$\left({\frac {3}{2}}\,R-s\right)T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {s-s_{0}}{R}}\right)$	$\qquad f(s,p)\qquad$