PCRT/I/e43f

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09 Differentiale in den natürlichen Variablen der freien Energie

A Freie Energie F(N;T,V)

$\qquad \qquad$	$dF$	$=$	$-$	$S\,dT$	$+$	$p\,d(-V)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dF$	$=$	$-$	$S\,dT$	$-$	$p\,dV$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$dF(T,V)$	$=$	$-$	$(S-\,0)\,dT$	$-$	$(p-\,0)\,dV$	$=$	$dF(T,V)$

B Freie Enthalpie G(N;T,V)

$\qquad \qquad$	$(dG_{T,p})_{N}$	$=$	$-$	$S\,dT$	$-$	$V\,d(-p)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dG_{T,p})_{N}$	$=$	$-$	$S\,dT$	$+$	$V\,dp$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG_{N,T,p}$	$=$	${\displaystyle }$	$d(F-(-\,p\,V))$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG_{N,T,p}$	$=$	${\displaystyle }$	$dF+\,p\,dV+\,V\,dp$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dG_{T,p})_{N}$	$=$	$-$	$S\,dT+V\,dp$	$+$	$p\,dV-p\,dV$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG(N;T,p)$	$=$	$-$	$S\,dT+V\,dp$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG(N;T,p)$	$=$	$-$	$S\,dT-V\,d(-p)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG(N;T,p)$	$=$	$-$	$d{\overline {Q}}+d{\overline {W}}$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$+\,V\,dp(T,v)$	$=$	$+$	$V\,{\big (}\partial p/\partial T{\big )}_{v}\,dT$	$+$	$V\,{\big (}\partial p/\partial v{\big )}_{T}\,dv$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$+\,V\,dp(T,v)$	$=$	$+$	$V\,{\big (}\partial S/\partial V{\big )}_{N,T}\,dT$	$+$	$V\,{\big (}\partial p/\partial v{\big )}_{T}\,dv$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\leftarrow {\mbox{Gl.}}(05.{\mbox{C}}.04)$
$(01)\qquad$	$dG(N;T,V)$	$=$	$-$	$\left[\,S-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,0-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(02)\qquad$	$dG(N;T,V)+(+\,p\,d(-V))$	$=$	$-$	$\left[\,S-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,p-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(03)\qquad$	$dG(N;T,V)-(-\,S\,dT)$	$=$	$-$	$\left[\,0-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,0-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(03)\qquad$	$+\,d{\overline {W}}$	$=$	$-$	$\left[\,0-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,0-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$

C Innere Energie U(N;T,V)

$\qquad \qquad$	$(dU_{S,V})_{N}$	$=$	$+$	$T\,dS$	$+$	$p\,d(-V)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dU_{S,V})_{N}$	$=$	$+$	$T\,dS$	$-$	$p\,dV$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dU_{N,S,V}$	$=$	${\displaystyle }$	$d(F+(+\,T\,S))$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dU_{N,S,V}$	$=$	${\displaystyle }$	$dF+\,T\,dS+\,S\,dT$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dU_{S,V})_{N}$	$=$	$-$	$p\,dV+\,T\,dS$	$-$	$S\,dT+S\,dT$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dU(N;S,V)$	$=$	$+$	$T\,dS-p\,dV$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dU(N;S,V)$	$=$	$+$	$T\,dS+p\,d(-V)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dU(N;S,V)$	$=$	$+$	$dQ+dW$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$+\,T\,dS(N;T,V)$	$=$	$+$	$T\,{\big (}\partial S/\partial T{\big )}_{N,V}\,dT$	$+$	$T\,{\big (}\partial S/\partial V{\big )}_{N,T}\,dV$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$+\,T\,dS(N;T,V)$	$=$	$+$	$T\,{\big (}\partial S/\partial T{\big )}_{N,V}\,dT$	$+$	$T\,{\big (}\partial p/\partial T{\big )}_{v}\,dV$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\leftarrow {\mbox{Gl.}}(05.{\mbox{C}}.04)$
$(01)\qquad$	$dU(N;T,V)$	$=$	$-$	$\left[\,0-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}\right]\,dT$	$-$	$\left[\,p-T\left({\frac {\partial p}{\partial T}}\right)_{v}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(02)\qquad$	$dU(N;T,V)+(-\,S\,dT)$	$=$	$-$	$\left[\,S-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}\right]\,dT$	$-$	$\left[\,p-T\left({\frac {\partial p}{\partial T}}\right)_{v}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(03)\qquad$	$dU(N;T,V)-(+\,p\,d(-V))$	$=$	$-$	$\left[\,0-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}\right]\,dT$	$-$	$\left[\,0-T\left({\frac {\partial p}{\partial T}}\right)_{v}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(03)\qquad$	$+\,dQ$	$=$	$-$	$\left[\,0-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}\right]\,dT$	$-$	$\left[\,0-T\left({\frac {\partial p}{\partial T}}\right)_{v}\right]\,dV$	${\displaystyle }$	${\displaystyle }$

D Enthalpie H(N;T,V)

$\qquad \qquad$	$(dH_{S,p})_{N}$	$=$	$+$	$T\,dS$	$-$	$V\,d(-p)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dH_{S,p})_{N}$	$=$	$+$	$T\,dS$	$+$	$V\,dp$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dH_{N,S,p}$	$=$	$+$	$d(F+(+\,T\,S)-(-\,p\,V))$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dH_{N,S,p}$	$=$	${\displaystyle }$	$dF+T\,dS+S\,dT$	$+$	$p\,dV+V\,dp$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dH_{S,p})_{N}$	$=$	$+$	$T\,dS+V\,dp-S\,dT+S\,dT$	$-$	$p\,dV+p\,dV$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dH(N;S,p)$	$=$	$+$	$T\,dS+V\,dp$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dH(N;S,p)$	$=$	$+$	$T\,dS-V\,d(-p)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dH(N;S,p)$	$=$	$+$	$dQ+d{\overline {W}}$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$+\,T\,dS(N;T,V)$	$=$	$+$	$T\,{\big (}\partial S/\partial T{\big )}_{N,V}\,dT$	$+$	$T\,{\big (}\partial p/\partial T{\big )}_{v}\,dV\qquad \leftarrow {\mbox{Gl.}}(05.{\mbox{C}}.04)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$+\,V\,dp(T,v)$	$=$	$+$	$V\,{\big (}\partial S/\partial V{\big )}_{N,T}\,dT$	$+$	$V\,{\big (}\partial p/\partial v{\big )}_{T}\,dv\qquad \leftarrow {\mbox{Gl.}}(05.{\mbox{C}}.04)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$dH(N;T,V)$	$=$	$-$	$\left[\,0-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,0-T\left({\frac {\partial p}{\partial T}}\right)_{v}-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$+\,dQ+d{\overline {W}}$	$=$	$-$	$\left[\,0-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,0-T\left({\frac {\partial p}{\partial T}}\right)_{v}-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$
$(02)\qquad$	$dH(N;T,V)+(-\,S\,dT+\,p\,d(-V))$	$=$	$-$	$\left[\,S-T\left({\frac {\partial S}{\partial T}}\right)_{N,V}-V\left({\frac {\partial S}{\partial V}}\right)_{N,T}\right]\,dT$	$-$	$\left[\,p-T\left({\frac {\partial p}{\partial T}}\right)_{v}-v\left({\frac {\partial p}{\partial v}}\right)_{T}\right]\,dV$	${\displaystyle }$	${\displaystyle }$

Übung a

Mit Gl.(02) wie Übung A.a.

Übung b

Mit Gl.(02) wie Übung A.b.

Übung c

Mit Gl.(03) wie Übung A.c.

Übung d

Mit $F=G+(-\,p\,V)=G-\,p\,V$ : $d(-\,p\,V)=-\,p\,dV-\,V\,dp$, $dF+\,V\,dp=dG+(-\,p\,dV))$, $-\partial p(T,V)/\partial T$, $+\partial V(T,p)/\partial T$ wie Übung A.d.

Übung e

Mit Gl.(05) wie Übung A.a.

Übung f

Mit Gl.(05) wie Übung A.b.

Übung g

Mit Gl.(06) wie Übung A.c.

Übung h

Mit $F=U-(+\,T\,S)=U-\,T\,S$ : $d(+\,T\,S)=+\,T\,dS+\,S\,dT$ ,$dF=d(U-\,T\,S)=dU-T\,dS-S\,dT$, $-\partial S(T,V)/\partial V$, $+\partial T(S,V)/\partial V$ wie Übung A.d.

Übung i

Mit Gl.(08) wie Übung A.a.

Übung j

Mit Gl.(08) wie Übung A.b.

Übung k

Mit Gl.(09) wie Übung A.c.

Übung l

Mit $F=H-(+\,T\,S)+(-\,p\,V)=H-T\,S-\,p\,V$ : $dF=dH-T\,dS-S\,dT-p\,dV-V\,dp$, $-\partial S(T,V)/\partial V$, $+\partial T(S,p)/\partial p$, $-\partial p(T,V)/\partial T$, $+\partial V(S,p)/\partial S$ wie Übung A.d.

Übung m

Wie viele Differentiale können wir in der Gleichung für das totale Differential der freien Enthalpie $dG=-\,S\,dT+\,V\,dp=0$ unabhängig voneinander verändern. Wieviel unabhängige Variablen, wieviel Freiheitsgrade, hat ein ideales Gas? Welcher Term fehlt in der Gleichung für das totale Differential der freien Enthalpie und warum sind die Formeln für dF, dG, dU und dH im (T,V,N)-Koordinatensystem trotzdem richtig?