PCRT/I/c2ae

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

28 Vom Differential über das Integral zum Zustand der inneren Energie

A (S,V,N)-Koordinaten: dU(S,V) -> U(S,V,N)

$(01)\qquad$	$dU$	$=$	$\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-\,p_{0}\,V_{0}\,{\frac {dV}{V}}\right]$	$\qquad dU(S,V),\,dN=0\quad \leftarrow (11.{\mbox{A}}.02)$
$\qquad \qquad$	$U$	$=$	$\int _{S,dV=0}\,dU$	$\qquad U(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\int _{S,dV=0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)n\,R\,T_{0}\,{\frac {dS}{n\,R}}$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(V,N)$	$\qquad \qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{S,N}$	$=$	$-{\frac {n\,R}{V}}\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial B}{\partial V}}\right)_{N}$	$\qquad \partial _{V}U(S,V,N),\,B(V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p+\,\left({\frac {\partial B}{\partial V}}\right)_{N}$	$\qquad p(S,V,N),\,B(V,N)\qquad$
$(02)\qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{S,N}$	$=$	$-\,p$	$\qquad \partial _{V}U(S,V,N),\,p(S,V,N)\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(V,N)$	$\qquad B={\mbox{const.}}\qquad$
$\qquad \qquad$	$U$	$=$	$\int _{V,dS=0}\,dU$	$\qquad U(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{S,dV}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)p_{0}\,V_{0}\,{\frac {dV}{V}}$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{S,dV}\,p_{0}\,V_{0}\,{\frac {1}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{V,N}$	$=$	$+\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial A}{\partial S}}\right)_{N}$	$\qquad \partial _{S}U(S,V,N),\,A(S,N)\qquad$
$(03)\qquad$	${\displaystyle }$	$=$	$+\,T+\,\left({\frac {\partial A}{\partial S}}\right)_{N}$	$\qquad T(S,V,N),\,A(S,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{V,N}$	$=$	$+\,T$	$\qquad \partial _{S}U(S,V,N),\,T(S,V,N)\qquad$
$\qquad \qquad$	$A$	$\neq$	$A(S,N)$	$\qquad A={\mbox{const.}}\qquad$
$\qquad \qquad$	$U$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,A+\,B$	$\qquad U(S,V,N)\qquad$
$(04)\qquad$	$U(S_{0},V_{0},N)$	$=$	$0$	$\qquad$
$\qquad \qquad$	$A+B$	$=$	$0$	$\qquad$
$(05)\qquad$	$U$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad U(S,V,N)\quad \leftarrow (14.{\mbox{A}}.01)$

B (S,p,N)-Koordinaten: dU(S,p) -> U(S,p,N)

$(01)\qquad$	$dU$	$=$	${\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+\,p_{0}\,V_{0}\,{\frac {dp}{p}}\right]$	$\qquad dU(S,p),\,dN=0\quad \leftarrow (12.{\mbox{B}}.07)$
$\qquad \qquad$	$U$	$=$	$\int _{S,dp=0}\,dU$	$\qquad U(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\int _{S,dp=0}{\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)n\,R\,T_{0}\,{\frac {dS}{n\,R}}$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {5}{2}}\cdot {\frac {3}{5}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial p}}\right)_{S,N}$	$=$	$+\,{\frac {2}{5}}\cdot {\frac {3}{2}}\,{\frac {n\,R}{p}}\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial B}{\partial p}}\right)_{N}$	$\qquad \partial _{p}U(S,p,N),\,B(p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\,{\frac {n\,R}{p}}\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial B}{\partial p}}\right)_{N}$	$\qquad \partial _{p}U(S,p,N),\,B(p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\,V+\,\left({\frac {\partial B}{\partial p}}\right)_{N}$	$\qquad \partial _{p}U(S,p,N),\,V(S,p,N),\,B(p,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial p}}\right)_{S,N}$	$=$	$\left({\frac {\partial (H-\,p\,V)}{\partial p}}\right)_{S,N}=V-\,V\left({\frac {\partial p}{\partial p}}\right)_{S,N}-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}$	$\qquad (S,p,N):\,\partial _{p}U,\,\partial _{p}H,\,\partial _{p}V\qquad$
$\qquad \qquad$	$-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}$	$=$	$+\,{\frac {3}{5}}\,V$	$\qquad (S,p,N):\,\partial _{p}V,\,V\quad \leftarrow (12.{\mbox{B}}.03)$
$(02)\qquad$	$\left({\frac {\partial U}{\partial p}}\right)_{S,N}$	$=$	$+\,{\frac {3}{5}}\,V$	$\qquad (S,p,N):\,\partial _{p}U,\,V\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(p,N)$	$\qquad B={\mbox{const.}}\qquad$
$\qquad \qquad$	$U$	$=$	$\int _{p,dS=0}\,dU$	$\qquad U(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\int _{p,dS=0}\,{\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,p_{0}\,V_{0}\,{\frac {dp}{p}}$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\int _{p,dS=0}\,{\frac {3}{5}}\,{\frac {1}{p}}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,p_{0}\,V_{0}\,dp$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {5}{2}}\cdot {\frac {3}{5}}\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{p,N}$	$=$	$+\,{\frac {2}{5}}\,{\frac {1}{n\,R}}\cdot {\frac {3}{2}}\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial A}{\partial S}}\right)_{N}$	$\qquad \partial _{S}U(S,p,N),\,A(S,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial A}{\partial S}}\right)_{N}$	$\qquad \partial _{S}U(S,p,N),\,A(S,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\,T+\,\left({\frac {\partial A}{\partial S}}\right)_{N}$	$\qquad \partial _{S}U(S,p,N),\,T(S,p,N),\,A(S,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{p,N}$	$=$	$\left({\frac {\partial (H-\,p\,V)}{\partial S}}\right)_{p,N}=T-p\left({\frac {\partial V}{\partial S}}\right)_{p,N}$	$\qquad (S,p,N):\,\partial _{S}U,\,\partial _{S}H,\,\partial _{S}V,\,T\qquad$
$\qquad \qquad$	$-S\left({\frac {\partial V}{\partial S}}\right)_{p,N}$	$=$	$-{\frac {2}{5}}{\frac {S}{n\,R}}\,V$	$\qquad (S,p,N):\partial _{S}V,\,S,\,V\quad \leftarrow (12.{\mbox{C}}.03)$
$(03)\qquad$	$-p\,\left({\frac {\partial V}{\partial S}}\right)_{p,N}$	$=$	$-\,{\frac {2}{5}}\,T$	$\qquad \partial _{S}V(S,p,N),\,T(S,p,N)\qquad$
$(04)\qquad$	$\left({\frac {\partial U}{\partial S}}\right)_{p,N}$	$=$	$+\,{\frac {3}{5}}\,T$	$\qquad \partial _{S}U(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	$A$	$\neq$	$A(S,N)$	$\qquad A={\mbox{const.}}\qquad$
$\qquad \qquad$	$U$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,A+\,B$	$\qquad U(S,p,N)\qquad$
$(05)\qquad$	$U(S_{0},p_{0},N)$	$=$	$0$	$\qquad \qquad$
$\qquad \qquad$	$A+B$	$=$	$0$	$\qquad \qquad$
$(06)\qquad$	$U$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)$	$\qquad U(S,p,N)\qquad \leftarrow (15.{\mbox{B}}.02)$

C (T,V,N)-Koordinaten: dU(T,V) -> U(T,V,N)

$(01)\qquad$	$dU$	$=$	${\frac {3}{2}}\,n\,R\,dT$	$\qquad dU(T,V),\,dN=0\quad \leftarrow (13.{\mbox{C}}.03)$
$\qquad \qquad$	$U$	$=$	$\int _{T,dV=0}\,dU$	$\qquad U(T,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\int _{S,dV=0}{\frac {3}{2}}\,n\,R\,dT$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T+\,B(V,N)$	$\qquad \qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{T,N}$	$=$	$+\,\left({\frac {\partial B}{\partial V}}\right)_{N}$	$\qquad \partial _{V}U(T,V,N),\,B(V,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{T,N}$	$=$	$\left({\frac {\partial (F+\,T\,S)}{\partial V}}\right)_{T,N}=-\,p+\,S\left({\frac {\partial T}{\partial V}}\right)_{T,N}+\,T\left({\frac {\partial S}{\partial V}}\right)_{T,N}$	$\qquad (T,V,N):\,\partial _{V}U,\,\partial _{V}H,\,\partial _{V}S\qquad$
$\qquad \qquad$	$-V\,\left({\frac {\partial S}{\partial V}}\right)_{T,N}$	$=$	$-\,n\,R$	$\qquad \partial _{V}S(T,V,N)\quad \leftarrow (13.{\mbox{B}}.01)$
$\qquad \qquad$	$+T\left({\frac {\partial S}{\partial V}}\right)_{T,N}$	$=$	$+\,{\frac {n\,R\,T}{V}}=+\,p$	$\qquad (T,V,N):\,\partial _{V}S,\,p\quad$
$(02)\qquad$	$\left({\frac {\partial U}{\partial V}}\right)_{T,N}$	$=$	$0$	$\qquad (T,V,N):\,\partial _{V}U\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(V,N)$	$\qquad B={\mbox{const.}}\qquad$
$\qquad \qquad$	$U$	$=$	$\int _{V,dT=0}\,dU$	$\qquad U(T,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\int _{V,dT=0}\,0\,\,dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,A(T,N)$	$\qquad \qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial T}}\right)_{V,N}$	$=$	$+\,\left({\frac {\partial A}{\partial T}}\right)_{N}$	$\qquad \partial _{T}U(T,V,N),\,A(T,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial U}{\partial T}}\right)_{V,N}$	$=$	$\left({\frac {\partial (F+\,T\,S)}{\partial T}}\right)_{V,N}=-\,S+\,S\left({\frac {\partial T}{\partial T}}\right)_{V,N}+\,T\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}$	$\qquad (T,V,N):\,\partial _{T}U,\,\partial _{T}F,\,\partial _{T}S\qquad$
$\qquad \qquad$	$-T\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}$	$=$	$-\,{\frac {3}{2}}n\,R$	$\qquad \partial _{T}S(T,V,N)\quad \leftarrow (13.{\mbox{C}}.01)$
$(03)\qquad$	$\left({\frac {\partial U}{\partial T}}\right)_{V,N}$	$=$	$+\,{\frac {3}{2}}n\,R$	$\qquad \partial _{T}U(T,V,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial A}{\partial T}}\right)_{N}$	$=$	$+\,{\frac {3}{2}}n\,R$	$\qquad A(T,N)\qquad$
$\qquad \qquad$	$A(T,N)$	$=$	$+\,{\frac {3}{2}}n\,R\,T$	$\qquad \qquad$
$\qquad \qquad$	$U$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T+\,B$	$\qquad U(T,V,N)\qquad$
$(04)\qquad$	$U(T_{0},V_{0},N)$	$=$	$0$	$\qquad \qquad$
$\qquad \qquad$	$A+B$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T$	$\qquad \qquad$
$(05)\qquad$	$U$	$=$	${\frac {3}{2}}\,n\,R\,T$	$\qquad U(T,V,N)\qquad \leftarrow (16.{\mbox{B}}.01)$