CPRT/cef7

< Inhaltsverzeichnis - CPRT

Energie E (Bezugssystem)

$(01)$	$dE(N,S,V)$	$=$	$+\mu dN+tdS+pd(-V)$
$(02)$	$E(N,S,V)$	$=$	$+\mu N+tS+p(-V)$
$(02)$	$E(N,S,V)$	$=$	$+dN\frac{\partial E}{\partial N}+dS\frac{\partial E}{\partial S}+dV\frac{\partial E}{\partial V}$
$(03)$	$0(N,S,V)$	$=$	$+N\frac{\partial \mu }{\partial N}+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}$
$(04)$	$0(N,S,V)$	$=$	$+N\frac{\partial t}{\partial N}+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}$
$(05)$	$0(N,S,V)$	$=$	$+N\frac{\partial p}{\partial N}+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}$
$(06)$	$0(N,S,V)$	$=$	$-Nd\mu -Sdt-Vd(-p)$
$(07)$	$0(N,S,V)$	$=$	$+N\frac{\partial dE}{\partial N}+S\frac{\partial dE}{\partial S}+V\frac{\partial dE}{\partial V}\ne dE(N,S,V)$

Herleitungen Energie E

$(01)$	$dE(N,S,V)$	$=$	$+\mu dN+tdS+pd(-V)$
$$	${\displaystyle }$	$=$	$+\frac{\partial E}{\partial N}dN+\frac{\partial E}{\partial S}dS+\frac{\partial E}{\partial V}dV$
$$	${\displaystyle }$	$=$	$+dN\frac{\partial E}{\partial N}+dS\frac{\partial E}{\partial S}+dV\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+dN\mu +dSt+d(-V)p$
$$	$\frac{\partial }{\partial N}\left(dN\frac{\partial E}{\partial N}\right)$	$=$	$dN\frac{\partial }{\partial N}\frac{\partial E}{\partial N},\frac{\partial }{\partial N}\left(dN\right)=d\frac{\partial N}{\partial N}=d(1)=0(N,S,V)$
$$	$d\mu (N,S,V)$	$=$	$+d\frac{\partial E}{\partial N}\ne 0$
$$	${\displaystyle }$	$=$	$+dN\frac{\partial }{\partial N}\frac{\partial E}{\partial N}+dS\frac{\partial }{\partial S}\frac{\partial E}{\partial N}+dV\frac{\partial }{\partial V}\frac{\partial E}{\partial N}$
$$	${\displaystyle }$	$=$	$+dN\frac{\partial }{\partial N}\frac{\partial E}{\partial N}+dS\frac{\partial }{\partial N}\frac{\partial E}{\partial S}+dV\frac{\partial }{\partial N}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial N}\left(dN\frac{\partial E}{\partial N}\right)+\frac{\partial }{\partial N}\left(dS\frac{\partial E}{\partial S}\right)+\frac{\partial }{\partial N}\left(dV\frac{\partial E}{\partial V}\right)$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial N}dE\ne 0$
$$	$\frac{\partial }{\partial S}\left(dS\frac{\partial E}{\partial S}\right)$	$=$	$dS\frac{\partial }{\partial S}\frac{\partial E}{\partial S},\frac{\partial }{\partial S}\left(dS\right)=d\frac{\partial S}{\partial S}=d(1)=0(N,S,V)$
$$	$dt(N,S,V)$	$=$	$+d\frac{\partial E}{\partial S}\ne 0$
$$	${\displaystyle }$	$=$	$+dN\frac{\partial }{\partial N}\frac{\partial E}{\partial S}+dS\frac{\partial }{\partial S}\frac{\partial E}{\partial S}+dV\frac{\partial }{\partial V}\frac{\partial E}{\partial S}$
$$	${\displaystyle }$	$=$	$+dN\frac{\partial }{\partial S}\frac{\partial E}{\partial N}+dS\frac{\partial }{\partial S}\frac{\partial E}{\partial S}+dV\frac{\partial }{\partial S}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial S}\left(dN\frac{\partial E}{\partial N}\right)+\frac{\partial }{\partial S}\left(dS\frac{\partial E}{\partial S}\right)+\frac{\partial }{\partial S}\left(dV\frac{\partial E}{\partial V}\right)$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial S}dE\ne 0$
$$	$\frac{\partial }{\partial V}\left(dV\frac{\partial E}{\partial V}\right)$	$=$	$dV\frac{\partial }{\partial V}\frac{\partial E}{\partial V},\frac{\partial }{\partial V}\left(dV\right)=d\frac{\partial V}{\partial V}=d(1)=0(N,S,V)$
$$	$dp(N,S,V)$	$=$	$+d\frac{\partial E}{\partial (-V)}\ne 0$
$$	${\displaystyle }$	$=$	$+dN\frac{\partial }{\partial N}\frac{\partial E}{\partial (-V)}+dS\frac{\partial }{\partial S}\frac{\partial E}{\partial (-V)}+dV\frac{\partial }{\partial V}\frac{\partial E}{\partial (-V)}$
$$	${\displaystyle }$	$=$	$-dN\frac{\partial }{\partial N}\frac{\partial E}{\partial V}-dS\frac{\partial }{\partial S}\frac{\partial E}{\partial V}-dV\frac{\partial }{\partial V}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$-dN\frac{\partial }{\partial V}\frac{\partial E}{\partial N}-dS\frac{\partial }{\partial V}\frac{\partial E}{\partial S}-dV\frac{\partial }{\partial V}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$-\frac{\partial }{\partial V}\left(dN\frac{\partial E}{\partial N}\right)-\frac{\partial }{\partial V}\left(dS\frac{\partial E}{\partial S}\right)-\frac{\partial }{\partial V}\left(dV\frac{\partial E}{\partial V}\right)$
$$	${\displaystyle }$	$=$	$-\frac{\partial }{\partial V}dE$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial (-V)}dE\ne 0$
$(02)$	$E(N,S,V)$	$=$	$+\mu N+tS+p(-V)$
$$	${\displaystyle }$	$=$	$+\frac{\partial E}{\partial N}N+\frac{\partial E}{\partial S}S+\frac{\partial E}{\partial V}V$
$$	${\displaystyle }$	$=$	$+N\frac{\partial E}{\partial N}+S\frac{\partial E}{\partial S}+V\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+N\mu +St+(-V)p$
$$	$\frac{\partial }{\partial N}\left(N\frac{\partial E}{\partial N}\right)$	$=$	$\frac{\partial E}{\partial N}+N\frac{\partial }{\partial N}\frac{\partial E}{\partial N}(N,S,V)$
$(03)$	$0(N,S,V)$	$=$	$+N\frac{\partial \mu }{\partial N}+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}$
$$	${\displaystyle }$	$=$	$+N\frac{\partial }{\partial N}\frac{\partial E}{\partial N}+S\frac{\partial }{\partial S}\frac{\partial E}{\partial N}+V\frac{\partial }{\partial V}\frac{\partial E}{\partial N}$
$$	${\displaystyle }$	$=$	$+N\frac{\partial }{\partial N}\frac{\partial E}{\partial N}+S\frac{\partial }{\partial N}\frac{\partial E}{\partial S}+V\frac{\partial }{\partial N}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial N}\left(N\frac{\partial E}{\partial N}\right)+\frac{\partial }{\partial N}\left(S\frac{\partial E}{\partial S}\right)+\frac{\partial }{\partial N}\left(V\frac{\partial E}{\partial V}\right)-\frac{\partial E}{\partial N}$
$$	${\displaystyle }$	$=$	$+\frac{\partial E}{\partial N}-\frac{\partial E}{\partial N}$
$$	${\displaystyle }$	$=$	$+\mu -\mu =0$
$$	$\frac{\partial }{\partial S}\left(S\frac{\partial E}{\partial S}\right)$	$=$	$\frac{\partial E}{\partial S}+S\frac{\partial }{\partial S}\frac{\partial E}{\partial S}(N,S,V)$
$(04)$	$0(N,S,V)$	$=$	$+N\frac{\partial t}{\partial N}+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}$
$$	${\displaystyle }$	$=$	$+N\frac{\partial }{\partial N}\frac{\partial E}{\partial S}+S\frac{\partial }{\partial S}\frac{\partial E}{\partial S}+V\frac{\partial }{\partial V}\frac{\partial E}{\partial S}$
$$	${\displaystyle }$	$=$	$+N\frac{\partial }{\partial S}\frac{\partial E}{\partial N}+S\frac{\partial }{\partial S}\frac{\partial E}{\partial S}+V\frac{\partial }{\partial S}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+\frac{\partial }{\partial S}\left(N\frac{\partial E}{\partial N}\right)+\frac{\partial }{\partial S}\left(S\frac{\partial E}{\partial S}\right)+\frac{\partial }{\partial S}\left(V\frac{\partial E}{\partial V}\right)-\frac{\partial E}{\partial S}$
$$	${\displaystyle }$	$=$	$+\frac{\partial E}{\partial S}-\frac{\partial E}{\partial S}$
$$	${\displaystyle }$	$=$	$+t-t=0$
$$	$\frac{\partial }{\partial V}\left(V\frac{\partial E}{\partial V}\right)$	$=$	$\frac{\partial E}{\partial V}+V\frac{\partial }{\partial V}\frac{\partial E}{\partial V}(N,S,V)$
$(05)$	$0(N,S,V)$	$=$	$+N\frac{\partial p}{\partial N}+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}$
$$	${\displaystyle }$	$=$	$+N\frac{\partial }{\partial N}\frac{\partial E}{\partial (-V)}+S\frac{\partial }{\partial S}\frac{\partial E}{\partial (-V)}+V\frac{\partial }{\partial V}\frac{\partial E}{\partial (-V)}$
$$	${\displaystyle }$	$=$	$-N\frac{\partial }{\partial N}\frac{\partial E}{\partial V}-S\frac{\partial }{\partial S}\frac{\partial E}{\partial V}-V\frac{\partial }{\partial V}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$-N\frac{\partial }{\partial V}\frac{\partial E}{\partial N}-S\frac{\partial }{\partial V}\frac{\partial E}{\partial S}-V\frac{\partial }{\partial V}\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$-\frac{\partial }{\partial V}\left(N\frac{\partial E}{\partial N}\right)-\frac{\partial }{\partial V}\left(S\frac{\partial E}{\partial S}\right)-\frac{\partial }{\partial V}\left(V\frac{\partial E}{\partial V}\right)+\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$-\frac{\partial E}{\partial V}+\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+\frac{\partial E}{\partial (-V)}-\frac{\partial E}{\partial (-V)}$
$$	${\displaystyle }$	$=$	$+p-p=0$
$(06)$	$0(N,S,V)$	$=$	$-Nd\mu -Sdt-Vd(-p)$
$$	${\displaystyle }$	$=$	$-Nd\frac{\partial E}{\partial N}-Sd\frac{\partial E}{\partial S}-Vd\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$-d\left(N\frac{\partial E}{\partial N}\right)-d\left(S\frac{\partial E}{\partial S}\right)-d\left(V\frac{\partial E}{\partial V}\right)$
$$	${\displaystyle }$	${\displaystyle }$	$+\frac{\partial E}{\partial N}dN+\frac{\partial E}{\partial S}dS+\frac{\partial E}{\partial V}dV$
$$	${\displaystyle }$	$=$	$-dE+dE=0$
$(07)$	$0(N,S,V)$	$=$	$+N\frac{\partial dE}{\partial N}+S\frac{\partial dE}{\partial S}+V\frac{\partial dE}{\partial V}\ne dE(N,S,V)$
$$	${\displaystyle }$	$=$	$+Nd\frac{\partial E}{\partial N}+Sd\frac{\partial E}{\partial S}+Vd\frac{\partial E}{\partial V}$
$$	${\displaystyle }$	$=$	$+Nd\mu +Sdt+Vd(-p)=0$