PCRT/I/ac7e

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

05 Partielle Ableitungen

Halten wir die Entropie $S={\mbox{const}}$, das Volumen $V={\mbox{const}}$, den Druck $p={\mbox{const}}$ oder die Stoffmenge $N={\mbox{const}}$, so ist ihre Änderung null (z.B. $d(V={\mbox{const}})\equiv dV=0$). Wir erhalten dann, für die Änderung, die partiellen ersten Ableitungen und die gemischt-partiellen zweiten Ableitungen der Inneren Energie

A Innere Energie U(N,S,V)

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

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

$=$

$+$

$\mu \,dN$

$,$

$\quad \left({\frac {\partial U}{\partial N}}\right)_{S,V}$

$=$

$+$

$\mu =$

$+$

$g_{s,v}$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(02)\quad \qquad \qquad \qquad$

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

$=$

$+$

$T\,dS$

$,$

$\quad \left({\frac {\partial U}{\partial S}}\right)_{V,N}$

$=$

$+$

$T$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(03)\quad \qquad \qquad \qquad$

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

$=$

$+$

$p\,d(-V)$

$,$

$\quad \left({\frac {\partial U}{\partial V}}\right)_{S,N}$

$=$

$-$

$p$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(04)\quad \qquad \qquad \qquad$

$+$

$\left({\frac {\partial \mu }{\partial S}}\right)_{V,N}$

$=$

$+$

$\left({\frac {\partial T}{\partial N}}\right)_{S,V}$

$,\quad$

$+$

$\left({\frac {\partial \mu }{\partial V}}\right)_{S,N}$

$=$

$-$

$\left({\frac {\partial p}{\partial N}}\right)_{S,V}$

$,\quad$

$+$

$\left({\frac {\partial T}{\partial V}}\right)_{S,N}$

$=$

$-$

$\left({\frac {\partial p}{\partial S}}\right)_{V,N}$

${\displaystyle }$

für die Änderung, die ersten partiellen Ableitungen und die zweiten gemischt-partiellen Ableitungen der Enthalpie

B Enthalpie H(N,S,p)

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

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

$=$

$+$

$\mu \,dN$

$,$

$\quad \left({\frac {\partial H}{\partial N}}\right)_{S,p}$

$=$

$+$

$\mu$

$=$

$+$

$g_{s,p}$

$(02)\quad \qquad \qquad \qquad$

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

$=$

$+$

$T\,dS$

$,$

$\quad \left({\frac {\partial H}{\partial S}}\right)_{p,N}$

$=$

$+$

$T$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(03)\quad \qquad \qquad \qquad$

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

$=$

$-$

$V\,d(-p)$

$,$

$\quad \left({\frac {\partial H}{\partial p}}\right)_{S,N}$

$=$

$+$

$V$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(04)\quad \qquad \qquad \qquad$

$+$

$\left({\frac {\partial \mu }{\partial S}}\right)_{p,N}$

$=$

$+$

$\left({\frac {\partial T}{\partial N}}\right)_{S,p}$

$,\quad$

$+$

$\left({\frac {\partial \mu }{\partial p}}\right)_{S,N}$

$=$

$+$

$\left({\frac {\partial V}{\partial N}}\right)_{S,p}$

$,\quad$

$+$

$\left({\frac {\partial T}{\partial p}}\right)_{S,N}$

$=$

$+$

$\left({\frac {\partial V}{\partial S}}\right)_{p,N}$

${\displaystyle }$

für die Änderung, die erstesn partiellen Ableitungen und die gemischten zweiten partiellen Ableitungen der Freien Energie

C Freie Energie F(N,T,V)

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

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

$=$

$+$

$\mu \,dN$

$,$

$\quad \left({\frac {\partial F}{\partial N}}\right)_{T,V}$

$=$

$+$

$\mu$

$=$

$+$

$g_{T,v}$

$(02)\quad \qquad \qquad \qquad$

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

$=$

$-$

$S\,dT$

$,$

$\quad \left({\frac {\partial F}{\partial T}}\right)_{V,N}$

$=$

$-$

$S$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(03)\quad \qquad \qquad \qquad$

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

$=$

$+$

$p\,d(-V)$

$,$

$\quad \left({\frac {\partial F}{\partial V}}\right)_{T,N}$

$=$

$-$

$p$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(04)\quad \qquad \qquad \qquad$

$+$

$\left({\frac {\partial \mu }{\partial T}}\right)_{V,N}$

$=$

$-$

$\left({\frac {\partial S}{\partial N}}\right)_{T,V}$

$,\quad$

$+$

$\left({\frac {\partial \mu }{\partial V}}\right)_{T,N}$

$=$

$-$

$\left({\frac {\partial p}{\partial N}}\right)_{T,V}$

$,\quad$

$-$

$\left({\frac {\partial S}{\partial V}}\right)_{T,N}$

$=$

$-$

$\left({\frac {\partial p}{\partial T}}\right)_{V,N}$

${\displaystyle }$

und für die Änderung, die ersten partiellen Ableitungen und die zweiten gemischt-partiellen Ableitungen der Freien Enthalpie

D Freie Enthalpie G(N,T,p)

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

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

$=$

$+$

$\mu \,dN$

$,$

$\quad \left({\frac {\partial G}{\partial N}}\right)_{T,p}$

$=$

$+$

$\mu$

$=$

$+$

$g_{T,p}$

$(02)\quad \qquad \qquad \qquad$

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

$=$

$-$

$S\,dT$

$,$

$\quad \left({\frac {\partial G}{\partial T}}\right)_{p,N}$

$=$

$-$

$S$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(03)\quad \qquad \qquad \qquad$

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

$=$

$-$

$V\,d(-p)$

$,$

$\quad \left({\frac {\partial G}{\partial p}}\right)_{T,N}$

$=$

$+$

$V$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

$(04)\quad \qquad \qquad \qquad$

$+$

$\left({\frac {\partial \mu }{\partial T}}\right)_{p,N}$

$=$

$-$

$\left({\frac {\partial S}{\partial N}}\right)_{T,p}$

$,\quad$

$+$

$\left({\frac {\partial \mu }{\partial p}}\right)_{T,N}$

$=$

$+$

$\left({\frac {\partial V}{\partial N}}\right)_{T,p}$

$,\quad$

$-$

$\left({\frac {\partial S}{\partial p}}\right)_{T,N}$

$=$

$+$

$\left({\frac {\partial V}{\partial T}}\right)_{p,N}$

${\displaystyle }$