PCRT.I.D.04

PCRT.I.D

04 Zustandsänderungen des idealen Gases bei (T,V;N=const)

A Freie Energie

$\qquad \qquad$	$dF$	$=$	$-\,S\,dT-p\,dV$	$\qquad dF(T,V),\,S(T,V,N),\,p(T,V,N),\,dN=0\qquad$
$\qquad \qquad$	$S$	$=$	$n\,R\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]+\,S_{0}$	$\qquad S(T,V,N)\qquad$
$\qquad \qquad$	$S_{0}$	$=$	$+{\frac {5}{2}}\,n\,R$	$\qquad \qquad$
$\qquad \qquad$	$p$	$=$	${\frac {n\,R\,T}{V}}$	$\qquad p(T,V,N)\qquad$
$\qquad \qquad$	$dF$	$=$	$-\,n\,R\,\left(+\,{\frac {5}{2}}+\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]\right)dT-\,n\,R\,T\,{\frac {dV}{V}}$	$\qquad dF(T,V),\,dN=0\qquad$
$(01)\qquad$	$dF$	$=$	$-\,n\,R\,T\,\left\{\left(+{\frac {5}{2}}+\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]\right){\frac {dT}{T}}+{\frac {dV}{V}}\right\}$	$\qquad dF(T,V),\,dN=0\qquad$
$(02)\qquad$	$\left({\frac {\partial F}{\partial T}}\right)_{V,N}$	$=$	$-\,n\,R\,\left(+{\frac {5}{2}}+\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]\right)=-\,S$	$\qquad \partial _{T}F(T,V,N)=-S,\,S(T,V,N)\qquad$
$(03)\qquad$	$\left({\frac {\partial F}{\partial V}}\right)_{T,N}$	$=$	$-\,{\frac {n\,R\,T}{V}}=-\,p$	$\qquad \partial _{V}F(T,V,N)=-p,\,p(T,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial V}}{\frac {\partial F}{\partial T}}$	$=$	$-\,{\frac {n\,R}{V}}=-\,{\frac {p}{T}}$	$\qquad \partial _{V}\partial _{T}F(T,V,N),\,p(T,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial T}}{\frac {\partial F}{\partial V}}$	$=$	$-\,{\frac {n\,R}{V}}=-\,{\frac {p}{T}}$	$\qquad \partial _{T}\partial _{V}F(T,V,N),\,p(T,V,N)\qquad$
$(04)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial F}{\partial T}}$	$=$	${\frac {\partial }{\partial T}}{\frac {\partial F}{\partial V}}$	$\qquad \partial _{V}\partial _{T}F(T,V,N),\,\partial _{T}\partial _{V}F(T,V,N)\qquad$

B Freie Enthalpie

$\qquad \qquad$	$dG$	$=$	$-\,S\,dT+V\,dp$	$\qquad dG(T,p),\,dN=0;\,(T,p,N):\,S,\,V\qquad$
$\qquad \qquad$	$dG$	$=$	$-\,\left[S-\,V{\frac {\partial S}{\partial V}}\right]\,dT+\,\left(0-\,V{\frac {\partial p}{\partial V}}\right)\,dV$	$\qquad dG(T,V),\,dN=0;\,(T,V,N):\,S,\,p\qquad$
$\qquad \qquad$	$S$	$=$	$+\,n\,R\,\ln \left({\frac {T}{T_{0}}}\right)^{3/2}+\,n\,R\,\ln \left({\frac {V}{V_{0}}}\right)+\,{\frac {5}{2}}\,n\,R$	$\qquad S(T,V,N)\qquad$
$\qquad \qquad$	$-V{\frac {\partial S}{\partial V}}$	$=$	$-\,{\frac {V}{V_{0}}}{\frac {\partial }{\partial \,(V/V_{0})}}\left[{\frac {3}{2}}\,n\,R\,\ln \left({\frac {T}{T_{0}}}\right)+n\,R\,\ln \left({\frac {V}{V_{0}}}\right)+\,{\frac {5}{2}}\,n\,R\right]$	$\qquad \partial _{V}S(T,V,N)\qquad$
$(01)\qquad$	$-V{\frac {\partial S}{\partial V}}$	$=$	$-\,n\,R$	$\qquad \partial _{V}S(T,V,N)\qquad$
$\qquad \qquad$	$p$	$=$	${\frac {n\,R\,T}{V}}$	$\qquad p(T,V,N)\qquad$
$\qquad \qquad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$-V\,\left(-\,{\frac {n\,R\,T}{V^{2}}}\right)$	$\qquad \partial _{V}p(T,V,N)\qquad$
$(02)\qquad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$+\,{\frac {n\,R\,T}{V}}=\,+\,p$	$\qquad \partial _{V}p(T,V,N),\,p(T,V,N)\qquad$
$(03)\qquad$	$dG$	$=$	$-\,\left(S-n\,R\right)dT-p\,dV$	$\qquad dG(T,V),\,dN=0;\,(T,V,N):\,S,\,p\qquad$
$(04)\qquad$	$dG$	$=$	$dF+n\,R\,dT$	$\qquad dG(F,T)\qquad$
$(05)\qquad$	$dG$	$=$	$-\,n\,R\,\left(+\,{\frac {3}{2}}+\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]\right)dT-n\,R\,T{\frac {dV}{V}}$	$\qquad dG(T,V),\,dN=0\qquad$
$(06)\qquad$	$dG$	$=$	$-\,n\,R\,T\,\left\{\left(+\,{\frac {3}{2}}+\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]\right){\frac {dT}{T}}+{\frac {dV}{V}}\right\}$	$\qquad dG(T,V),\,dN=0\qquad$
$(07)\qquad$	$\left({\frac {\partial G}{\partial T}}\right)_{V,N}$	$=$	$-\,n\,R\,\left(+{\frac {3}{2}}+\,\ln \left[\left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)\right]\right)=-\,(S-\,n\,R)$	$\qquad \partial _{T}G(T,V,N)\neq -S,\,S(T,V,N)\qquad$
$(08)\qquad$	$\left({\frac {\partial G}{\partial V}}\right)_{T,N}$	$=$	$-\,{\frac {n\,R\,T}{V}}=-\,p$	$\qquad \partial _{V}G(T,V,N),\,p(T,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial V}}{\frac {\partial G}{\partial T}}$	$=$	$-\,{\frac {n\,R}{V}}=-\,{\frac {p}{T}}$	$\qquad \partial _{V}\partial _{T}G(T,V,N),\,p(T,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial T}}{\frac {\partial G}{\partial V}}$	$=$	$-\,{\frac {n\,R}{V}}=-\,{\frac {p}{T}}$	$\qquad \partial _{T}\partial _{V}G(T,V,N),\,p(T,V,N)\qquad$
$(09)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial G}{\partial T}}$	$=$	${\frac {\partial }{\partial T}}{\frac {\partial G}{\partial V}}$	$\qquad \partial _{V}\partial _{T}G(T,V,N),\,\partial _{T}\partial _{V}G(T,V,N)\qquad$

Übung a

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

C Innere Energie

$\qquad \qquad$	$dU$	$=$	$T\,dS-p\,dV$	$\qquad dU(S,V),\,T(S,V,N),\,p(S,V,N),\,dN=0\qquad$
$\qquad \qquad$	$dU$	$=$	$-\,\left(0-T{\frac {\partial S}{\partial T}}\right)\,dT-\,\left[p-T{\frac {\partial p}{\partial T}}\right]\,dV$	$\qquad dU(T,V),\,S(T,V,N),\,p(T,V,N),\,dN=0\qquad$
$\qquad \qquad$	$S$	$=$	$+\,n\,R\,\ln \left({\frac {T}{T_{0}}}\right)^{3/2}+\,n\,R\,\ln \left({\frac {V}{V_{0}}}\right)$	$\qquad S(T,V,N)\qquad$
$\qquad \qquad$	$-T{\frac {\partial S}{\partial T}}$	$=$	$-\,{\frac {T}{T_{0}}}{\frac {\partial }{\partial \,(T/T_{0})}}\left[{\frac {3}{2}}\,n\,R\,\ln \left({\frac {T}{T_{0}}}\right)+n\,R\,\ln \left({\frac {V}{V_{0}}}\right)\right]$	$\qquad \partial _{T}S(T,V,N)\qquad$
$(01)\qquad$	$-T{\frac {\partial S}{\partial T}}$	$=$	$-\,{\frac {3}{2}}n\,R$	$\qquad \partial _{T}S(T,V,N)\qquad$
$\qquad \qquad$	$p$	$=$	${\frac {n\,R\,T}{V}}$	$\qquad p(T,V,N)\qquad$
$\qquad \qquad$	$-T{\frac {\partial p}{\partial T}}$	$=$	$-\,T\,\left({\frac {n\,R}{V}}\right)$	$\qquad \partial _{T}p(T,V,N)\qquad$
$(02)\qquad$	$-T{\frac {\partial p}{\partial T}}$	$=$	$-\,{\frac {n\,R\,T}{V}}=-\,p$	$\qquad \partial _{T}p(T,V,N),\,p(T,V,N)\qquad$
$(03)\qquad$	$dU$	$=$	${\frac {3}{2}}\,n\,R\,dT\equiv (S_{0}-\,n\,R)\,dT$	$\qquad dU(T)\qquad$
$(04)\qquad$	$\left({\frac {\partial U}{\partial T}}\right)_{V,N}$	$=$	${\frac {3}{2}}\,n\,R\equiv (S_{0}-\,n\,R)$	$\qquad \partial _{T}U(T,V,N)\qquad$
$(05)\qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{T,N}$	$=$	$0$	$\qquad \partial _{V}U(T,V,N)\neq -p\qquad$
$(06)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial U}{\partial T}}$	$=$	${\frac {\partial }{\partial T}}{\frac {\partial U}{\partial V}}$	$\qquad \partial _{V}\partial _{T}U(S,p,N),\,\partial _{T}\partial _{V}U(S,p,N)\qquad$

Übung a

Beim Übergang von der freien Energie $F(T,V,N)$ zur inneren Energie $U(S,V,N)$ wird $(T\rightarrow S)$ ersetzt. Wir berechnen $\left(S-T\,\partial S/\partial T\right)$ und $\left(p-T\,\partial p/\partial T\right)$ für ein ideales Gas im (T,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 $dU$ eines idealen Gases in den unabhängigen Variablen (T,V,N)?

D Enthalpie

$\qquad \qquad$	$dH$	$=$	$T\,dS+V\,dp$	$\qquad dH(S,p),\,(S,p,N):\,T,\,V\,;\,dN=0\qquad$
$\qquad \qquad$	$dH$	$=$	$-\left(0-T{\frac {\partial S}{\partial T}}-V{\frac {\partial S}{\partial V}}\right)\,dT-\left(0-T{\frac {\partial p}{\partial T}}-V{\frac {\partial p}{\partial V}}\right)\,dV$	$\qquad dH(T,V),\,(T,V,N):\,S,\,p\,;\,dN=0\qquad$
$dU:({\mbox{C}}.01)\qquad$	$-T{\frac {\partial S}{\partial T}}$	$=$	$-\,{\frac {3}{2}}\,n\,R$	$\qquad \partial _{T}S(T,V,N)\qquad$
$dU:({\mbox{C}}.02)\qquad$	$-T{\frac {\partial p}{\partial T}}$	$=$	$-\,{\frac {n\,R\,T}{V}}=-\,p$	$\qquad \partial _{T}p(T,V,N),\,p(T,V,N)\qquad$
$dG:({\mbox{B}}.01)\qquad$	$-V{\frac {\partial S}{\partial V}}$	$=$	$-\,n\,R$	$\qquad \partial _{V}S(T,V,N)\qquad$
$dG:({\mbox{B}}.02)\qquad$	$-V{\frac {\partial p}{\partial V}}$	$=$	$+\,{\frac {n\,R\,T}{V}}=+\,p$	$\qquad \partial _{V}p(T,V,N),\,p(T,V,N)\qquad$
$\qquad \qquad$	$dH$	$=$	$-\left(-\,{\frac {5}{2}}\,n\,R\right)\,dT-\,(-\,p+\,p)\,dV$	$\qquad dH(T)\qquad$
$(01)\qquad$	$dH$	$=$	$+\,{\frac {5}{2}}\,n\,R\,dT\equiv S_{0}\,dT$	$\qquad dH(T)\qquad$
$(02)\qquad$	$\left({\frac {\partial H}{\partial T}}\right)_{V,N}$	$=$	${\frac {5}{2}}\,n\,R\equiv S_{0}$	$\qquad \partial _{T}H(T,V,N)\qquad$
$(03)\qquad$	$\left({\frac {\partial H}{\partial V}}\right)_{T,N}$	$=$	$0$	$\qquad \partial _{V}H(T,V,N)\qquad$
$(04)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial H}{\partial T}}$	$=$	${\frac {\partial }{\partial T}}{\frac {\partial H}{\partial V}}$	$\qquad \partial _{V}\partial _{T}H(S,p,N),\,\partial _{T}\partial _{V}H(S,p,N)\qquad$

Übung a

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