< Teil A: Funktionen des Zustands und deren Differentiale in verschiedenen Koordinatensystemen - CPRT/I/4cb4

02 Die Differentiale der Zustandsfunktionen in ihren (eigenen) natürlichen Variablen - CPRT/I/cf51

E, F, G, H

Die Fundamentalgleichungen (differentiale Form, links) bzw. die Zustandsfunktionen (integrale Form, rechts) in Merkform lauten

$(02)\qquad \qquad \qquad \qquad$	$dE(N,S,V)$	$=$	$+$	$\mu \,dN+t\,dS+p\,d(-V)$		$\qquad (0,0,0)$
$(01)\qquad \qquad \qquad \qquad$	$E(N,S,V)$	$=$	$+$	$\mu \,N+t\,S+p\,(-V)$		$\qquad (0,0,0)$
$\qquad \qquad \qquad \qquad \qquad$	$dE(N,S,V)$	$=$	$+$	$\mu \,dN+N\,d\mu +t\,dS+S\,dt+p\,d(-V)+V\,d(-p)$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(03)\qquad \qquad \qquad \qquad$	$F(N,t,V)$	$=$	$+$	$\mu \,N+p\,(-V)$		$\qquad (0,1,0)$
$\qquad \qquad \qquad \qquad \qquad$	$dF(N,t,V)$	$=$	$+$	$\mu \,dN+N\,d\mu +p\,d(-V)+V\,d(-p)$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(04)\qquad \qquad \qquad \qquad$	$dF(N,t,V)$	$=$	$+$	$\mu \,dN-S\,dt+p\,d(-V)$		$\qquad (0,1,0)$
$(05)\qquad \qquad \qquad \qquad$	$G(N,t,p)$	$=$	$+$	$\mu \,N$		$\qquad (0,1,1)$
$\qquad \qquad \qquad \qquad \qquad$	$dG(N,t,p)$	$=$	$+$	$\mu \,dN+N\,d\mu$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(06)\qquad \qquad \qquad \qquad$	$dG(N,t,p)$	$=$	$+$	$\mu \,dN-S\,dt-V\,d(-p)$		$\qquad (0,1,1)$
$(07)\qquad \qquad \qquad \qquad$	$H(N,S,p)$	$=$	$+$	$\mu \,N+t\,S$		$\qquad (0,0,1)$
$\qquad \qquad \qquad \qquad \qquad$	$dH(N,S,p)$	$=$	$+$	$\mu \,dN+N\,d\mu +t\,dS+S\,dt$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(08)\qquad \qquad \qquad \qquad$	$dH(N,S,p)$	$=$	$+$	$\mu \,dN+t\,dS-V\,d(-p)$		$\qquad (0,0,1)$

L, K, J, I

$(15)\qquad \qquad \qquad \qquad$	$L(\mu ,S,p)$	$=$	$+$	$t\,S$		$\qquad (1,0,1)$
$\qquad \qquad \qquad \qquad \qquad$	$dL(\mu ,S,p)$	$=$	$+$	$t\,dS+S\,dt$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(16)\qquad \qquad \qquad \qquad$	$dL(\mu ,S,p)$	$=$	$-$	$N\,d\mu +t\,dS-V\,d(-p)$		$\qquad (1,0,1)$
$(13)\qquad \qquad \qquad \qquad$	$K(\mu ,t,p)$	$=$	$+$	$0$		$\qquad (1,1,1)$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$+$	$0$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(14)\qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt+p\,d(-V)$	$\quad =0$	$\qquad (1,1,1)$
$(11)\qquad \qquad \qquad \qquad$	$J(\mu ,t,V)$	$=$	$+$	$p\,(-V)$		$\qquad (1,1,0)$
$\qquad \qquad \qquad \qquad \qquad$	$dJ(\mu ,t,V)$	$=$	$+$	$p\,d(-V)+V\,d(-p)$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(12)\qquad \qquad \qquad \qquad$	$dJ(\mu ,t,V)$	$=$	$-$	$N\,d\mu -S\,dt+p\,d(-V)$		$\qquad (1,1,0)$
$(09)\qquad \qquad \qquad \qquad$	$I(\mu ,S,V)$	$=$	$+$	$t\,S+p\,(-V)$		$\qquad (1,0,0)$
$\qquad \qquad \qquad \qquad \qquad$	$dI(\mu ,S,V)$	$=$	$+$	$t\,dS+S\,dt+p\,d(-V)+V\,d(-p)$
$\qquad \qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$-$	$N\,d\mu -S\,dt-V\,d(-p)$	$\quad =0$
$(10)\qquad \qquad \qquad \qquad$	$dI(\mu ,S,V)$	$=$	$-$	$N\,d\mu +t\,dS+p\,d(-V)$		$\qquad (1,0,0)$

Aus Gleichungen (05) und (16) folgt:

(05)\qquad \qquad \qquad \qquad

G(N,t,p)

=

+

\mu \,N

\qquad \qquad

\mu (t,p)

=

+

{\frac {G}{N}}

=

g(t,p)

(14)\qquad \qquad \qquad \qquad

dK(\mu ,t,p)

=

-

N\,d\mu

-

V\,d(-p)

-

S\,dt

=

0

\qquad \qquad

d\mu (t,p)

=

-

{\frac {S}{N}}\,dt

+

{\frac {V}{N}}\,dp

=

dg(t,p)

E, F, G, H, L, K, J, I

Die Fundamentalgleichungen (differentiale Form, links) bzw. die Zustandsfunktionen (integrale Form, rechts) in Merkform lauten

(17)\qquad \qquad \qquad \qquad

E(N,S,V)

=

+

\mu \,N

+

t\,S

+

p\,(-V)

\qquad (0,0,0)

\qquad \qquad

E

=

+

G

+

I

(18)\qquad \qquad \qquad \qquad

F(N,t,V)

=

+

\mu \,N

+

p\,(-V)

\qquad (0,1,0)

\qquad \qquad

F

=

+

G

+

J

(19)\qquad \qquad \qquad \qquad

G(N,t,p)

=

+

\mu \,N

\qquad (0,1,1)

\qquad \qquad

G

=

+

G

+

K

(20)\qquad \qquad \qquad \qquad

H(N,S,p)

=

+

\mu \,N

+

t\,S

\qquad (0,0,1)

\qquad \qquad

H

=

+

G

+

L

(24)\qquad \qquad \qquad \qquad

L(\mu ,S,p)

=

+

t\,S

\qquad (1,0,1)

\qquad \qquad

L

=

-

G

+

H

(23)\qquad \qquad \qquad \qquad

K(\mu ,t,p)

=

\quad =0

\qquad (1,1,1)

\qquad \qquad

K

=

-

G

+

G\quad =0

(22)\qquad \qquad \qquad \qquad

J(\mu ,t,V)

=

+

p\,(-V)

\qquad (1,1,0)

\qquad \qquad

J

=

-

G

+

F

(21)\qquad \qquad \qquad \qquad

I(\mu ,S,V)

=

+

t\,S

+

p\,(-V)

\qquad (1,0,0)

\qquad \qquad

I

=

-

G

+

E

s, v, e, f, g, h, l, k, j, i

Zur Stoffmenge proportionale Grössen wie $S,V,U,H,F,G$ lassen sich immer als Produkt von Stoffmenge mal Grösse pro Stoffmenge schreiben.

(25)\qquad \qquad \qquad \qquad

S

=

N\,s

\qquad \qquad

dS

=

d(N\,s)

=

+

N\,ds

+

s\,dN

(26)\qquad \qquad \qquad \qquad

V

=

N\,v

\qquad \qquad

dV

=

d(N\,v)

=

+

N\,dv

+

v\,dN

(27)\qquad \qquad \qquad \qquad

E

=

N\,e

=

N\,(

+

\mu \cdot 1

+

t\,s

+

p\,(-v)

)

\qquad \qquad

dE

=

d(N\,e)

=

+

N\,de

+

e\,dN

(28)\qquad \qquad \qquad \qquad

F

=

N\,f

=

N\,(

+

\mu \cdot 1

+

p\,(-v)

)

\qquad \qquad

dF

=

d(N\,f)

=

+

N\,df

+

f\,dN

(29)\qquad \qquad \qquad \qquad

G

=

N\,g

=

N\,(

+

\mu \cdot 1

)

\qquad \qquad

dG

=

d(N\,g)

=

+

N\,dg

+

g\,dN

(30)\qquad \qquad \qquad \qquad

H

=

N\,h

=

N\,(

+

\mu \cdot 1

+

t\,s

)

\qquad \qquad

dH

=

d(N\,h)

=

+

N\,dh

+

h\,dN

(34)\qquad \qquad \qquad \qquad

L

=

N\,l

=

N\,(

+

t\,s

)

\qquad \qquad

dL

=

d(N\,l)

=

+

N\,dl

+

l\,dN

(33)\qquad \qquad \qquad \qquad

K

=

N\,k

=

N\,(

)\quad =0

\qquad \qquad

dK

=

d(N\,k)

=

+

N\,dk

+

k\,dN\quad =0

(32)\qquad \qquad \qquad \qquad

J

=

N\,j

=

N\,(

+

p\,(-v)

)

\qquad \qquad

dJ

=

d(N\,j)

=

+

N\,dj

+

j\,dN

(31)\qquad \qquad \qquad \qquad

I

=

N\,i

=

N\,(

+

t\,s

+

p\,(-v)

)

\qquad \qquad

dI

=

d(N\,i)

=

+

N\,di

+

i\,dN

de, df, dg, dh

$\qquad \qquad \qquad \qquad \qquad$	$N\,de$	$=N\,d(+\,\mu \,{\it {1}}+t\,s+p\,(-v))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,de$	$=N\,(\,+\,{\it {1}}\,d\mu +t\,ds+s\,dt+p\,d(-v)+v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,de$	$=+N\,{\it {1}}\,d\mu +\,t\,N\,ds+N\,s\,dt+p\,N\,d(-v)+N\,v\,d(-p)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,de$	$=+\,t\,N\,ds+p\,N\,d(-v)-(-N\,{\it {1}}\,d\mu -N\,s\,dt-N\,v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,de$	$=+\,t\,N\,ds+p\,N\,d(-v)-(-N\,d\mu -S\,dt-V\,d(-p))$
$(35)\qquad \qquad \qquad \qquad$	$N\,de$	$=+\,t\,N\,ds+p\,N\,d(-v)$
$(36)\qquad \qquad \qquad \qquad$	$N\,de$	$=+\,dQ+dW$
$\qquad \qquad \qquad \qquad \qquad$	$N\,df$	$=N\,d(+\,\mu \,1+p\,(-v))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,df$	$=N\,(+\,{\it {1}}\,d\mu +p\,d(-v)+v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,df$	$=+p\,N\,d(-v)+\,N\,{\it {1}}\,d\mu +N\,v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,df$	$=+\,p\,N\,d(-v)-N\,s\,dt-(-\,N\,{\it {1}}\,d\mu -N\,s\,dt-N\,v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,df$	$=+\,p\,N\,d(-v)-N\,s\,dt-(-\,N\,d\mu -S\,dt-V\,d(-p))$
$(37)\qquad \qquad \qquad \qquad$	$N\,df$	$=-\,p\,N\,dv-N\,s\,dt$
$(38)\qquad \qquad \qquad \qquad$	$N\,df$	$=+\,d{\overline {Q}}+dW$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dg$	$=N\,d(+\,\mu \,{\it {1}})$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dg$	$=+\,N\,{\it {1}}\,d\mu$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dg$	$=-\,N\,s\,dt-N\,v\,d(-p)-(-\,N\,{\it {1}}\,d\mu -N\,s\,dt-N\,v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dg$	$=-\,N\,s\,dt-N\,v\,d(-p)-(-\,N\,d\mu -S\,dt-V\,d(-p))$
$(39)\qquad \qquad \qquad \qquad$	$N\,dg$	$=-\,N\,s\,dt-N\,v\,d(-p)$
$(40)\qquad \qquad \qquad \qquad$	$N\,dg$	$=+\,d{\overline {Q}}+d{\overline {W}}$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dh$	$=N\,d(+\,\mu \,{\it {1}}+t\,s)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dh$	$=N\,(+\,{\it {1}}\,d\mu +t\,ds+s\,dt)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dh$	$=+N\,{\it {1}}\,d\mu +\,t\,N\,ds+N\,s\,dt$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dh$	$=+\,t\,N\,ds-N\,v\,d(-p)-(-N\,{\it {1}}\,d\mu -N\,s\,dt-N\,v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dh$	$=+\,t\,N\,ds-N\,v\,d(-p)-(-N\,d\mu -S\,dt-V\,d(-p))$
$(41)\qquad \qquad \qquad \qquad$	$N\,dh$	$=+\,t\,N\,ds-N\,v\,d(-p)$
$(42)\qquad \qquad \qquad \qquad$	$N\,dh$	$=+\,dQ+d{\overline {W}}$

di, dj, dk, dl

$\qquad \qquad \qquad \qquad \qquad$	$N\,di$	$=N\,d(+\,t\,s+p\,(-v))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,di$	$=N\,(+\,t\,ds+s\,dt+p\,d(-v)+v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,di$	$=N\,(+\,t\,ds+p\,d(-v))+N\,(+\,s\,dt+v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,di$	$=N\,(-\,{\it {1}}\,d\mu +t\,ds+p\,d(-v))-N\,(-\,{\it {1}}\,d\mu -s\,dt-v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,di$	$=N\,(-\,{\it {1}}\,d\mu +t\,ds+p\,d(-v))$
$(43)\qquad \qquad \qquad \qquad$	$N\,di$	$=-\,N\,{\it {1}}\,d\mu +t\,N\,ds+p\,N\,d(-v)$
$(44)\qquad \qquad \qquad \qquad$	$N\,di$	$=+\,d{\overline {\Xi }}+dW+dQ$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dj$	$=N\,d(+\,p\,(-v))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dj$	$=N\,(+\,p\,d(-v)+v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dj$	$=N\,(+\,p\,d(-v))+N\,(+\,v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dj$	$=N\,(-\,{\it {1}}\,d\mu -s\,dt+p\,d(-v))-N\,(-\,{\it {1}}\,d\mu -s\,dt-v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dj$	$=N\,(-\,{\it {1}}\,d\mu -s\,dt+p\,d(-v))$
$(45)\qquad \qquad \qquad \qquad$	$N\,dj$	$=-\,N\,{\it {1}}\,d\mu -N\,s\,dt+p\,N\,d(-v)$
$(46)\qquad \qquad \qquad \qquad$	$N\,dj$	$=+\,d{\overline {\Xi }}+d{\overline {Q}}+dW$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dk$	$=N\,d(\,0\,)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dk$	$=N\,(\,0\,)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dk$	$=N\,(-\,{\it {1}}\,d\mu -s\,dt-v\,d(-p))$
$(47)\qquad \qquad \qquad \qquad$	$N\,dk$	$=-\,N\,{\it {1}}\,d\mu -N\,s\,dt-N\,v\,d(-p)$
$(48)\qquad \qquad \qquad \qquad$	$N\,dk$	$=+\,d{\overline {\Xi }}+d{\overline {Q}}+d{\overline {W}}$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dl$	$=N\,d(+\,t\,s)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dl$	$=N\,(+\,t\,ds+s\,dt\,)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dl$	$=N\,(+\,t\,ds)+N\,(s\,dt\,)$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dl$	$=N\,(-\,{\it {1}}\,d\mu +t\,ds-v\,d(-p))-N\,(-\,{\it {1}}\,d\mu -s\,dt-v\,d(-p))$
$\qquad \qquad \qquad \qquad \qquad$	$N\,dl$	$=N\,(-\,{\it {1}}\,d\mu +t\,ds-v\,d(-p))$
$(49)\qquad \qquad \qquad \qquad$	$N\,dl$	$=-\,N\,{\it {1}}\,d\mu +t\,N\,ds-\,N\,v\,d(-p)$
$(50)\qquad \qquad \qquad \qquad$	$N\,dl$	$=+\,d{\overline {\Xi }}+dQ+d{\overline {W}}$

dE, dF, dG, dH, dI, dJ, dK, dL

$(51)\qquad \qquad \qquad \qquad$	$dE(N,S,V)$	$=$			$+$	$dQ$	$+$	$dW$	$+$	$e\,dN$
$(52)\qquad \qquad \qquad \qquad$	$dF(N,t,V)$	$=$			$+$	$d{\overline {Q}}$	$+$	$dW$	$+$	$f\,dN$
$(53)\qquad \qquad \qquad \qquad$	$dG(N,t,p)$	$=$			$+$	$d{\overline {Q}}$	$+$	$d{\overline {W}}$	$+$	$g\,dN$
$(54)\qquad \qquad \qquad \qquad$	$dH(N,S,p)$	$=$			$+$	$dQ$	$+$	$d{\overline {W}}$	$+$	$h\,dN$
$(55)\qquad \qquad \qquad \qquad$	$dI(\mu ,S,V)$	$=$	$+$	$d{\overline {\Xi }}$	$+$	$dQ$	$+$	$dW$	$+$	$i\,dN$
$(56)\qquad \qquad \qquad \qquad$	$dJ(\mu ,t,V)$	$=$	$+$	$d{\overline {\Xi }}$	$+$	$d{\overline {Q}}$	$+$	$dW$	$+$	$j\,dN$
$(57)\qquad \qquad \qquad \qquad$	$dK(\mu ,t,p)$	$=$	$+$	$d{\overline {\Xi }}$	$+$	$d{\overline {Q}}$	$+$	$d{\overline {W}}$	$+$	$k\,dN$
$(58)\qquad \qquad \qquad \qquad$	$dL(\mu ,S,p)$	$=$	$+$	$d{\overline {\Xi }}$	$+$	$dQ$	$+$	$d{\overline {W}}$	$+$	$l\,dN$

$(59)\qquad \qquad \qquad \qquad$			$+$	$dQ$	$+$	$dW$	$+$	$e\,dN$	$=$	$dE(N,V;S)$
$(60)\qquad \qquad \qquad \qquad$			$+$	$d{\overline {Q}}$	$+$	$dW$	$+$	$f\,dN$	$=$	$dF(N,t,V)$
$(61)\qquad \qquad \qquad \qquad$			$+$	$d{\overline {Q}}$	$+$	$d{\overline {W}}$	$+$	$g\,dN$	$=$	$dG(N,t,p)$
$(62)\qquad \qquad \qquad \qquad$			$+$	$dQ$	$+$	$d{\overline {W}}$	$+$	$h\,dN$	$=$	$dH(N,S,p)$
$(63)\qquad \qquad \qquad \qquad$	$+$	$d{\overline {\Xi }}$	$+$	$dQ$	$+$	$dW$	$+$	$i\,dN$	$=$	$dI(\mu ,S,V)$
$(64)\qquad \qquad \qquad \qquad$	$+$	$d{\overline {\Xi }}$	$+$	$d{\overline {Q}}$	$+$	$dW$	$+$	$j\,dN$	$=$	$dJ(\mu ,t,V)$
$(65)\qquad \qquad \qquad \qquad$	$+$	$d{\overline {\Xi }}$	$+$	$d{\overline {Q}}$	$+$	$d{\overline {W}}$	$+$	$k\,dN$	$=$	$dK(\mu ,t,p)$
$(66)\qquad \qquad \qquad \qquad$	$+$	$d{\overline {\Xi }}$	$+$	$dQ$	$+$	$d{\overline {W}}$	$+$	$l\,dN$	$=$	$dL(\mu ,S,p)$

dQ, dW

$(67)\qquad \qquad \qquad \qquad$	$+$	$dQ$	$=$	$dE(N,V;S)$
$(68)\qquad \qquad \qquad \qquad$	$+$	$dW$	$=$	$dE(N,S;V)$
$(69)\qquad \qquad \qquad \qquad$	$+$	$dW$	$=$	$dF(N,t;V)$
$(70)\qquad \qquad \qquad \qquad$	$+$	$dQ$	$=$	$dH(N,p;S)$
$(71)\qquad \qquad \qquad \qquad$	$+$	$dQ$	$=$	$dI(\mu ,V;S)\qquad ,\,dN(\mu ,V;S)=0$
$(72)\qquad \qquad \qquad \qquad$	$+$	$dW$	$=$	$dI(\mu ,S;V)\qquad ,\,dN(\mu ,S;V)=0$
$(73)\qquad \qquad \qquad \qquad$	$+$	$dW$	$=$	$dJ(\mu ,t;V)\qquad ,\,dN(\mu ,t;V)=0$
$(74)\qquad \qquad \qquad \qquad$	$+$	$dQ$	$=$	$dL(\mu ,p;S)\qquad ,\,dN(\mu ,S;p)=0$