PCRT.I.B.06

PCRT.I.B

06 Das Differential des chemischen Potentials

A Systeme mit Teilchenaustausch

Differenzieren wir die homogenen Formen der Zustandsfunktionen und vergleichen sie mit den Differentialen der Zustandsfunktionen, so erkennen wir, daß es in der Thermodynamik Summen gibt, die sich zu null addieren.

$\qquad \qquad \qquad \qquad \qquad$

$dU$

$=$

$+$

$dT\,S$

$+$

$S\,dT$

$-$

$dp\,V$

$-$

$p\,dV$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dH$

$=$

$+$

$dT\,S$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dF$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$-$

$dp\,V$

$-$

$p\,dV$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dH$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dU$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

$-$

$p\,dV$

${\displaystyle }$

${\displaystyle }$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dH$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$+$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dF$

$=$

$-$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$-$

$p\,dV$

${\displaystyle }$

${\displaystyle }$

$+$

$\mu \,dN$

$\qquad \qquad \qquad \qquad \qquad$

$dG$

$=$

$-$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$\mu \,dN$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

Durch Umformen erhalten wir das Differential des chemischen Potentials

$\qquad \qquad \qquad \qquad \qquad$	$N\,d\mu$	$=$	$-$	$S\,dT$	$-$	$V\,dp$
$(02)\qquad \qquad \qquad \qquad$	$d\mu$	$=$	$-$	${\frac {S}{N}}\,dT$	$-$	${\frac {V}{N}}\,dp$
$(03)\qquad \qquad \qquad \qquad$	$d\mu$	$=$	$-$	$s\,dT$	$-$	$s\,dp$
$(04)\qquad \qquad \qquad \qquad$	$dg$	$=$	$-$	$s\,dT$	$-$	$s\,dp$

Das chemische Potential ist eine intensive Grösse und sein Differential sieht genauso aus wie das Differential der freien Enthalpie. Das chemische Potential ist die Frei Enthalpie pro Stoffmenge. Wir können es auch als die molare freie Enthalpie bezeichnen.

Übung a

Wir leiten das Differential der molaren freien Enthalpie $dg=d\mu$ entsprechend dem Differential des chemischen Potentials $d\mu$ her. Dabei gehen wir im Vergleich zur obigen Herleitung rückwärts vor. Wir leiten zunächst das Differential für die molare freie Enthalpie ausgehend von der freien Enthalpie und ihrem Differential her. Dann gehen wir von der freien Energie und ihrem Differential aus, dann von der Enthalpie und ihrem Differential und schliesslich von der inneren Energie und ihrem Differential aus.

Übung b

Wir starten mit $F=-\,p\,V$. Wir nehmen als $\mu \,n=0$ als Null der Skala der Energiefunktionen $U,H,F,G$. Wie lauten $U,H,G$ ? Wie lauten die Differentiale $dU,dH,dF,dG$ ?

B Systeme ohne Teilchenaustausch

$\qquad \qquad \qquad \qquad \qquad$

$dU$

$=$

$+$

$dT\,S$

$+$

$S\,dT$

$-$

$dp\,V$

$-$

$p\,dV$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dH$

$=$

$+$

$dT\,S$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dF$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$-$

$dp\,V$

$-$

$p\,dV$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dH$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dU$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

$-$

$p\,dV$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dH$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$+$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dF$

$=$

$-$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$-$

$p\,dV$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$\qquad \qquad \qquad \qquad \qquad$

$dG$

$=$

$-$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$(01)\qquad \qquad \qquad \qquad$

$0$

$=$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,V$

${\displaystyle }$

${\displaystyle }$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

Der Ausdruck für das chemische Potential ist für ein System ohne Teilchenaustausch dersselbe wie für ein System mit Teilchenaustausch und er hängt von zwei Variablen ab, hat zwei Freiheitsgrade z.B. T und p, die unabhängig voneinander variiert werden können.

$(02)\qquad \qquad \qquad \qquad$	$d\mu$	$=$	$-$	$s\,dT$	$-$	$s\,dp$
$(03)\qquad \qquad \qquad \qquad$	$dg$	$=$	$-$	$s\,dT$	$-$	$s\,dp$