CPRT/6971

< Inhaltsverzeichnis - CPRT

Energie G (zwei neue Variable)

$S,V$ kommen, $t,p$ gehen.

$(01)$	$dE(N,S,V)$	$=$	$+\mu dN+tdS+pd(-V)$
$(02)$	$dG(N,t,p)$	$=$	$+\mu dN-Sdt-(-V)dp$
$(02)$	$dG(N,S,V)$	$=$	$+{\mu }_E^{G}dN+t_E^{G}dS+p_E^{G}d(-V)$
$(03)$	$G(N,t,p)$	$=$	$+\mu N$
$(03)$	$G(N,S,V)$	$=$	$+{\mu }_E^{G}N+t_E^{G}S+p_E^G(-V)$

Variante a)

(04a) folgt aus (03) und (03a). (05a) folgt aus (02a) und (03a).

$(01a)$	$-Sdt$	$=$	$+[0-S\frac{\partial t}{\partial N}]dN+[0-S\frac{\partial t}{\partial S}]dS+[0-S\frac{\partial t}{\partial (-V)}]d(-V)(N,S,V)$
$(01a)$	$-(-V)dp$	$=$	$+[0-(-V)\frac{\partial p}{\partial N}]dN+[0-(-V)\frac{\partial p}{\partial S}]dS+[0-(-V)\frac{\partial p}{\partial (-V)}]d(-V)(N,S,V)$
$(02a)$	$dG(N,S,V)$	$=$	$+[\mu -S\frac{\partial t}{\partial N}-(-V)\frac{\partial p}{\partial N}]dN+[0-S\frac{\partial t}{\partial S}-(-V)\frac{\partial p}{\partial S}]dS$
$$	${\displaystyle }$	${\displaystyle }$	$+[0-S\frac{\partial t}{\partial (-V)}-(-V)\frac{\partial p}{\partial (-V)}]d(-V)$
$(03a)$	$G(N,S,V)$	$=$	$+[\mu -S\frac{\partial t}{\partial N}-(-V)\frac{\partial p}{\partial N}]N+[0-S\frac{\partial t}{\partial S}-(-V)\frac{\partial p}{\partial S}]S$
$$	${\displaystyle }$	${\displaystyle }$	$+[0-S\frac{\partial t}{\partial (-V)}-(-V)\frac{\partial p}{\partial (-V)}](-V)$
$(04a)$	$0(N,S,V)$	$=$	$+[0-S\frac{\partial t}{\partial N}-(-V)\frac{\partial p}{\partial N}]N+[0-S\frac{\partial t}{\partial S}-(-V)\frac{\partial p}{\partial S}]S$
$$	${\displaystyle }$	${\displaystyle }$	$+[0-S\frac{\partial t}{\partial (-V)}-(-V)\frac{\partial p}{\partial (-V)}](-V)$
$(05a)$	$0(N,S,V)$	$=$	$-Nd[\mu -S\frac{\partial t}{\partial N}-(-V)\frac{\partial p}{\partial N}]-Sd[0-S\frac{\partial t}{\partial S}-(-V)\frac{\partial p}{\partial S}]$
$$	${\displaystyle }$	${\displaystyle }$	$-(-V)d[0-S\frac{\partial t}{\partial (-V)}-(-V)\frac{\partial p}{\partial (-V)}]$

Variante b) (am einfachsten zu merken)

(04b) folgt aus (03) und (03b). (05b) folgt aus (02b) und (03b).

$(01b)$	$-Sdt$	$=$	$+[0-S\frac{\partial \mu }{\partial S}]dN+[0-S\frac{\partial t}{\partial S}]dS+[0-S\frac{\partial p}{\partial S}]d(-V)(N,S,V)$
$(01b)$	$-(-V)dp$	$=$	$+[0-V\frac{\partial \mu }{\partial V}]dN+[0-V\frac{\partial t}{\partial V}]dS+[0-V\frac{\partial p}{\partial V}]d(-V)(N,S,V)$
$(02b)$	$dG(N,S,V)$	$=$	$+[\mu -S\frac{\partial \mu }{\partial S}-V\frac{\partial \mu }{\partial V}]dN+[0-S\frac{\partial t}{\partial S}-V\frac{\partial t}{\partial V}]dS+[0-S\frac{\partial p}{\partial S}-V\frac{\partial p}{\partial V}]d(-V)$
$(03b)$	$G(N,S,V)$	$=$	$+[\mu -S\frac{\partial \mu }{\partial S}-V\frac{\partial \mu }{\partial V}]N+[0-S\frac{\partial t}{\partial S}-V\frac{\partial t}{\partial V}]S+[0-S\frac{\partial p}{\partial S}-V\frac{\partial p}{\partial V}](-V)$
$(04b)$	$0(N,S,V)$	$=$	$+[0-S\frac{\partial \mu }{\partial S}-V\frac{\partial \mu }{\partial V}]N+[0-S\frac{\partial t}{\partial S}-V\frac{\partial t}{\partial V}]S+[0-S\frac{\partial p}{\partial S}-V\frac{\partial p}{\partial V}](-V)$
$(05b)$	$0(N,S,V)$	$=$	$-Nd[\mu -S\frac{\partial \mu }{\partial S}-V\frac{\partial \mu }{\partial V}]-Sd[0-S\frac{\partial t}{\partial S}-V\frac{\partial t}{\partial V}]-(-V)d[0-S\frac{\partial p}{\partial S}-V\frac{\partial p}{\partial V}]$

Variante c)

(04c) folgt aus (03) und (03c). (05c) folgt aus (02c) und (03c).

$(01c)$	$-Sdt$	$=$	$+[0+N\frac{\partial \mu }{\partial N}+V\frac{\partial \mu }{\partial V}]dN+[0+N\frac{\partial t}{\partial N}+V\frac{\partial t}{\partial V}]dS+[0+N\frac{\partial p}{\partial N}+V\frac{\partial p}{\partial V}]d(-V)$
$(01c)$	$-(-V)dp$	$=$	$+[0+N\frac{\partial \mu }{\partial N}+S\frac{\partial \mu }{\partial S}]dN+[0+N\frac{\partial t}{\partial N}+S\frac{\partial t}{\partial S}]dS+[0+N\frac{\partial p}{\partial N}+S\frac{\partial p}{\partial S}]d(-V)$
$(02c)$	$dG(N,S,V)$	$=$	$+[\mu +N\frac{\partial \mu }{\partial N}]dN+[0+N\frac{\partial t}{\partial N}]dS+[0+N\frac{\partial p}{\partial N}]d(-V)$
$(03c)$	$G(N,S,V)$	$=$	$+[\mu +N\frac{\partial \mu }{\partial N}]N+[0+N\frac{\partial t}{\partial N}]S+[0+N\frac{\partial p}{\partial N}](-V)$
$(04c)$	$0(N,S,V)$	$=$	$+[0+N\frac{\partial \mu }{\partial N}]N+[0+N\frac{\partial t}{\partial N}]S+[0+N\frac{\partial p}{\partial N}](-V)$
$(05c)$	$0(N,S,V)$	$=$	$-Nd[\mu +N\frac{\partial \mu }{\partial N}]-Sd[0+N\frac{\partial t}{\partial N}]-(-V)d[0+N\frac{\partial p}{\partial N}]$