PCRT/I/ec9f

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

13 Differential Null-Funktion, Differential Chemisches Potential: System ohne Teilchenaustausch

Diesmal differenzieren wir das Produkt zweier Grössen in einer eigentlich natürlichen, aber ungewohnten Weise: $d(a\,b)=da\,b+a\,db$. Die Differentiale mit den intensiven Variablen rücken jetzt von der zweiten an die ersten Stelle, umgekehrt die Differentiale mit den extensiven Variablen von der ersten an die zweite Stelle.

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

$dU_{N,S,V}$

$=$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

$+$

$dT\,S$

$+$

$S\,dT$

$+$

$dp\,(-V)$

$+$

$p\,d(-V)$

${\displaystyle }$

${\displaystyle }$

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

$(dU_{S,V})_{N}$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$+$

$dT\,S$

$+$

$S\,dT$

$+$

$dp\,(-V)$

$+$

$p\,d(-V)$

${\displaystyle }$

${\displaystyle }$

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

$dU(N;S,V)$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

$+$

$p\,d(-V)$

${\displaystyle }$

${\displaystyle }$

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

$dL(\mu ,T,p)$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$+$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,(-V)$

${\displaystyle }$

${\displaystyle }$

$\qquad =0$

${\displaystyle }$

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

$dF_{N,T,V}$

$=$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,(-V)$

$+$

$p\,d(-V)$

${\displaystyle }$

${\displaystyle }$

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

$(dF_{T,V})_{N}$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,(-V)$

$+$

$p\,d(-V)$

${\displaystyle }$

${\displaystyle }$

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

$dF(N;T,V)$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$-$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$p\,d(-V)$

${\displaystyle }$

${\displaystyle }$

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

$dL(\mu ,T,p)$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$+$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,(-V)$

${\displaystyle }$

${\displaystyle }$

$\qquad =0$

${\displaystyle }$

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

$dG_{N,T,p}$

$=$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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

$(dG_{T,p})_{N}$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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

$dG(N;T,p)$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$-$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$-$

$dp\,(-V)$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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

$dL(\mu ,T,p)$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$+$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,(-V)$

${\displaystyle }$

${\displaystyle }$

$\qquad =0$

${\displaystyle }$

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

$dH_{N,S,p}$

$=$

$+$

$d\mu \,N$

$+$

$\mu \,dN$

$+$

$dT\,S$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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

$(dH_{S,p})_{N}$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$+$

$dT\,S$

$+$

$S\,dT$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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

$dH(N;S,p)$

$=$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$+$

$S\,dT$

$-$

$dp\,(-V)$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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

$dL(\mu ,T,p)$

$=$

$+$

$d\mu \,N$

${\displaystyle }$

${\displaystyle }$

$+$

$dT\,S$

${\displaystyle }$

${\displaystyle }$

$+$

$dp\,(-V)$

${\displaystyle }$

${\displaystyle }$

$\qquad =0$

${\displaystyle }$

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.

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