CPRT.I.C.07

CPRT.I.C

07 Zustandsfunktionen bei (T,V;N=const)

A Freie Energie und molare freie Energie

$\qquad \qquad$	$-\,T\,S$	$=$	$-\,n\,R\,T\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]-\,S_{0}\,T$	$\qquad S(T,V,N)\qquad$
$\qquad \qquad$	$S_{0}$	$=$	${\frac {5}{2}}\,n\,R$	$\qquad S(T,V,N)\qquad$
$\qquad \qquad$	$U$	$=$	${\frac {3}{2}}\,n\,R\,T$	$\qquad U(T,V,N)\qquad$
$\qquad \qquad$	$F$	$=$	$U-T\,S$	$\qquad F(T,V,N),U(T,V,N),S(T,V,N)\qquad$
$\qquad$	$F$	$=$	$-\,n\,R\,T-\,n\,R\,T\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]$	$\qquad F(T,V,N)\qquad$
$(3.10.3.1.1)\qquad$	$F$	$=$	$+\,n\,R\,T\,\ln \left[\left({\frac {T_{0}}{T}}\right)^{3/2}\left({\frac {1}{e}}\right)^{2/2}\right]+\,n\,R\,T\,\ln \left({\frac {V_{0}}{V}}\right)$	$\qquad F(T,V,N),\quad T_{0}\,<\,T,\,V_{0}\,<\,V,\,T\uparrow \,:\,F\downarrow \qquad$
$(3.10.3.1.2)\qquad$	$f$	$=$	$+\,R\,T\,\ln \left[\left({\frac {T_{0}}{T}}\right)^{3/2}\left({\frac {1}{e}}\right)^{2/2}\right]+\,R\,T\,\left({\frac {v_{0}}{v}}\right)$	$\qquad f(T,v),\quad T_{0}\,<\,T,\,v_{0}\,<\,v,\,T\uparrow \,:\,f\downarrow \qquad$
$(3.10.3.1.3)\qquad$	$f$	$=$	$+\,R\,T\,\ln \left[\left({\frac {T_{0}}{T}}\right)^{5/2}\left({\frac {1}{e}}\right)^{2/2}\right]+\,R\,T\,\left({\frac {p}{p_{0}}}\right)$	$\qquad f(T,p),\quad T_{0}\,<\,T,\,p\,<\,p_{0},\,T\uparrow \,:\,f\downarrow \qquad$
$(3.10.3.1.4)\qquad$	$f$	$=$	$+\,p\,v\,\ln \left[\left({\frac {p_{0}}{p}}\right)^{3/2}\left({\frac {v_{0}}{v}}\right)^{5/2}\left({\frac {1}{e}}\right)^{2/2}\right]$	$\qquad f(v,p),\quad v_{0}\,<\,v,\,p_{0}\,<\,p,\,v\uparrow ,\,p\uparrow \,:\,f\downarrow \qquad$

Übung a

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

B Energie und molare Energie

$(3.10.3.2.1)\qquad \qquad$	$U$	$=$	${\frac {3}{2}}\,n\,R\,T$	$\qquad U(T,V,N)\qquad$
$(3.10.3.2.2)\qquad \qquad$	$u$	$=$	${\frac {3}{2}}\,R\,T$	$\qquad u(T,v)\qquad$

C Freie Enthalpie und chemisches Potential

$\qquad$	$+\,p\,V$	$=$	$+\,n\,R\,T$	$\qquad p(T,V,N)\qquad$
$\qquad$	$G$	$=$	$F+p\,V$	$\qquad G(T,V,N),F(T,V,N),p(T,V,N)\qquad$
$\qquad$	$G$	$=$	$-\,n\,R\,T\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]$	$\qquad G(T,V,N)\qquad$
$(3.10.3.3.1)\qquad$	$G$	$=$	$+\,n\,R\,T\,\ln \left({\frac {T_{0}}{T}}\right)^{3/2}+\,n\,R\,T\left({\frac {V_{0}}{V}}\right)$	$\qquad G(T,V,N),\quad T_{0}\,<\,T,\,V_{0}\,<\,V,\,T\uparrow \,:\,G\downarrow \qquad$
$(3.10.3.3.2)\qquad$	$\mu$	$=$	$+\,R\,T\,\ln \left({\frac {T_{0}}{T}}\right)^{3/2}+\,R\,T\,\ln \left({\frac {v_{0}}{v}}\right)$	$\qquad \mu (T,v)=g(T,v),\quad T_{0}\,<\,T,\,v_{0}\,<\,v,\,T\uparrow \,:\,\mu \downarrow \qquad$
$(3.10.3.3.3)\qquad$	$\mu$	$=$	$+\,R\,T\,\ln \left({\frac {T_{0}}{T}}\right)^{5/2}+\,R\,T\,\ln \left({\frac {p}{p_{0}}}\right)$	$\qquad \mu (T,p)=g(T,p),\quad T_{0}\,<\,T,\,p\,<\,p_{0},\,T\uparrow \,:\,\mu \downarrow \qquad$
$(3.10.3.3.4)\qquad$	$\mu$	$=$	$+\,p\,v\,\ln \left[\left({\frac {p_{0}}{p}}\right)^{3/2}\left({\frac {v_{0}}{v}}\right)^{5/2}\right]$	$\qquad \mu (v,p)=g(v,p),\quad v_{0}\,<\,v,\,p_{0}\,<\,p,\,v\uparrow ,\,p\uparrow \,:\,\mu \downarrow \qquad$

D Enthalpie und molare Enthalpie

$(3.10.3.4.1)\qquad \qquad$	$H$	$=$	${\frac {5}{2}}\,n\,R\,T$	$\qquad H(T,V,N)\qquad$
$(3.10.3.4.2)\qquad \qquad$	$h$	$=$	${\frac {5}{2}}\,R\,T$	$\qquad h(T,v)\qquad$