05 Zustandsfunktionen des idealen Gases bei (S,V;N=const)

A Innere Energie und molare innere Energie

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

Übung a

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

B Enthalpie und molare Enthalpie

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

C Freie Energie und freie molare Energie

$(01)\qquad$	$-\,T\,S$	$=$	$-\,S\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,V,N)\qquad$
$\qquad \qquad$	$F$	$=$	$U-T\,S$	$\qquad F(S,V,N),U(S,V,N),T(S,V,N)\qquad$
$(02)\qquad$	$F$	$=$	$\left({\frac {3}{2}}\,n\,R-S\right)T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad F(S,V,N)\qquad$
$(03)\qquad$	$f$	$=$	$\left({\frac {3}{2}}\,R-s\right)T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {s-s_{0}}{R}}\right)$	$\qquad f(s,v)\qquad$

D Freie Enthalpie und chemisches Potential

$\qquad \qquad$	$+\,p\,V$	$=$	$+\,n\,R\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad p(S,V,N)\qquad$
$\qquad \qquad$	$-\,T\,S$	$=$	$-\,S\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad T(S,V,N)\qquad$
$\qquad \qquad$	$G$	$=$	$U+p\,V-T\,S$	$\qquad G(S,V,N),U(S,V,N),p(S,V,N),T(S,V,N)\qquad$
$(01)\qquad$	$G$	$=$	$\left({\frac {5}{2}}\,n\,R-S\right)T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad G(S,V,N)\qquad$
$(02)\qquad$	$\mu$	$=$	$\left({\frac {5}{2}}\,R-s\right)T_{0}\left({\frac {v_{0}}{v}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {s-s_{0}}{R}}\right)$	$\qquad \mu (s,v)=g(s,v)\qquad$