PCRT/I/c63a

< Lektionen: Physikalische Chemie/Reversible Thermodynamik - PCRT/I

08 Differentiale in den natürlichen Variablen der Enthalpie (N;S,p)

A Enthalpie H(N;S,p)

$\qquad \qquad$	$dH(N;S,p)$	$=$	$+$	$T\,dS$	$-$	$V\,d(-p)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dH(N;S,p)$	$=$	$+$	$T\,dS$	$+$	$V\,dp$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$dH(N;S,p)$	$=$	$+$	$(T-0)\,dS$	$+$	$(V-0)\,dp$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$d{\big (}+\,Q+{\overline {W}}{\big )}$	$=$	${\displaystyle }$	$dH(N;S,p)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$

B Innere Energie U(N;S,p)

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

B Freie Enthalpie G(N;S,p)

$\qquad \qquad$	$(dG_{T,p})_{N}$	$=$	$-$	$S\,dT$	$-$	$V\,d(-p)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dG_{T,p})_{N}$	$=$	$-$	$S\,dT$	$+$	$V\,dp$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG_{N,T,p}$	$=$	${\displaystyle }$	$d(H-(+\,T\,S))$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG_{N,T,p}$	$=$	${\displaystyle }$	$dH-T\,dS-S\,dT$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dG_{T,p})_{N}$	$=$	$-$	$S\,dT+V\,dp$	$+$	$T\,dS-T\,dS$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG(N;T,p)$	$=$	$-$	$S\,dT+V\,dp$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG(N;T,p)$	$=$	$-$	$S\,dT-V\,d(-p)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dG(N;T,p)$	$=$	$-$	$d{\overline {Q}}-d{\overline {W}}$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$-\,S\,dT(s,p)$	$=$	$-$	$S\,{\big (}\partial T/\partial s{\big )}_{p}\,ds$	$-$	$S\,{\big (}\partial T/\partial p{\big )}_{s}\,dp$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$-\,S\,dT(s,p)$	$=$	$-$	$S\,{\big (}\partial T/\partial s{\big )}_{p}\,ds$	$-$	$S\,{\big (}\partial V/\partial S{\big )}_{p,N}\,dp\qquad \leftarrow {\mbox{Gl.}}(05.{\mbox{B}}.04)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$dG(N;S,p)$	$=$	$+$	$\left[\,0-s\left({\frac {\partial T}{\partial s}}\right)_{p}\right]\,dS$	$+$	$\left[\,V-S\left({\frac {\partial V}{\partial S}}\right)_{p,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$
$(02)\qquad$	$dG(N;S,p)+(+\,T\,dS)$	$=$	$+$	$\left[\,T-s\left({\frac {\partial T}{\partial s}}\right)_{p}\right]\,dS$	$+$	$\left[\,V-S\left({\frac {\partial V}{\partial S}}\right)_{p,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$
$(03)\qquad$	$dG(N;S,p)+{\big (}+\,V\,d(-p){\big )}$	$=$	$+$	$\left[\,0-s\left({\frac {\partial T}{\partial s}}\right)_{p}\right]\,dS$	$+$	$\left[\,0-S\left({\frac {\partial V}{\partial S}}\right)_{p,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$
$(03)\qquad$	$+\,d{\overline {Q}}$	$=$	$+$	$\left[\,0-s\left({\frac {\partial T}{\partial s}}\right)_{p}\right]\,dS$	$+$	$\left[\,0-S\left({\frac {\partial V}{\partial S}}\right)_{p,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$

C Freie Energie F(N;S,p)

$\qquad \qquad$	$(dF_{T,V})_{N}$	$=$	$-$	$S\,dT$	$+$	$p\,d(-V)$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dF_{T,V})_{N}$	$=$	$-$	$S\,dT$	$-$	$p\,dV$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dF_{N,T,V}$	$=$	${\displaystyle }$	$d(H-(+\,T\,S)+(-\,p\,V))$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dF_{N,T,V}$	$=$	${\displaystyle }$	$dH-\,T\,dS-\,S\,dT$	$-$	$p\,dV-\,V\,dp$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$(dF_{T,V})_{N}$	$=$	${\displaystyle }$	$-\,S\,dT-p\,dV+T\,dS-T\,dS$	$+$	$V\,dp-\,V\,dp$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dF(N;T,V)$	$=$	$-$	$S\,dT-p\,dV$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dF(N;T,V)$	$=$	$-$	$S\,dT+p\,d(-V)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$dF(N;T,V)$	$=$	$-$	$d{\overline {Q}}+dW$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$-\,S\,dT(s,p)$	$=$	$-$	$S\,{\big (}\partial T/\partial s{\big )}_{p}\,ds$	$-$	$S\,{\big (}\partial V/\partial S{\big )}_{p,N}\,dp$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\qquad \qquad$	$-\,p\,dV_{N}(S,p)$	$=$	$-$	$p\,{\big (}\partial T/\partial p{\big )}_{s}\,ds$	$-$	$p\,{\big (}\partial V/\partial p{\big )}_{S,N}\,dp\qquad \leftarrow {\mbox{Gl.}}(05.{\mbox{B}}.04)$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$dF(N;S,p)$	$=$	$+$	$\left[\,0-s\left({\frac {\partial T}{\partial s}}\right)_{p}-p\left({\frac {\partial T}{\partial p}}\right)_{s}\right]\,dS$	$+$	$\left[\,0-s\left({\frac {\partial V}{\partial S}}\right)_{p,N}-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$
$(01)\qquad$	$+\,d{\overline {Q}}+dW$	$=$	$+$	$\left[\,0-s\left({\frac {\partial T}{\partial s}}\right)_{p}-p\left({\frac {\partial T}{\partial p}}\right)_{s}\right]\,dS$	$+$	$\left[\,0-s\left({\frac {\partial V}{\partial S}}\right)_{p,N}-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$
$(02)\qquad$	$(dF_{S,p})_{N}+{\big (}+\,T\,dS-\,V\,d(-p){\big )}$	$=$	$+$	$\left[\,T-s\left({\frac {\partial T}{\partial s}}\right)_{p}-p\left({\frac {\partial T}{\partial p}}\right)_{s}\right]\,dS$	$+$	$\left[\,V-S\left({\frac {\partial V}{\partial S}}\right)_{p,N}-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}\right]\,dp$	${\displaystyle }$	${\displaystyle }$

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 $H=U-(-\,p\,V)=U+p\,V$ : $dH=dU+p\,dV+V\,dp$, $+\partial V(S,p)/\partial V$, $-\partial p(S,V)/\partial S$ 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 $H=G+(+\,T\,S)=G+T\,S$ : $dH=dG+T\,dS+S\,dT$, $+\partial T(S,p)/\partial p$, $-\partial S(T,p)/\partial p$ 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 $H=F+(+\,T\,S)-(-\,p\,V)=F+\,T\,S+\,p\,V$ : $dH=dF+T\,dS+S\,dT+p\,dV+V\,dp$, $+\partial T(S,p)/\partial p$, $-\partial S(T,V)/\partial V$, $+\partial V(S,p)/\partial S$, $-\partial p(S,V)/\partial S$ 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 dH, dU, dG und dF im (S,p,N)-Koordinatensystem trotzdem richtig?