PCRT/I/c11d

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

29 Ausgetauschte Wärme dQ = +TdS und ausgetauschte Arbeit dW = -pdV bei (S,V;N=const)

Wärme und Arbeit sind keine Zustandänderungen, ihre Summe ist eine Zustandsänderung.

A Ausgetauschte Wärme dQ = +TdS

$\qquad \qquad$	$dQ$	$=$	$+\,T\,dS$	$\qquad dQ(S,V),\,T(S,V,N),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,T\,\left[+\,\left({\frac {\partial S}{\partial S}}\right)_{V,N}\,dS+\,\left({\frac {\partial S}{\partial V}}\right)_{S,N}\,dV\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,T\,\left[\,+\,1\cdot dS+\,0\cdot dV\right]$	$\qquad \qquad$
$(01)\qquad$	${\displaystyle }$	$=$	$+\,T\,dS+\,0\,dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,dQ_{V,N}+\,0$	$\qquad dQ_{V,N}(S),\,dQ_{S,N}(V)=0,\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$+\,dU_{V,N}+\,0$	$\qquad dU_{V,N}(S),\,dN=0\qquad$
$\qquad \qquad$	$\left({\frac {\partial Q}{\partial S}}\right)_{V,N}$	$=$	$+\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=+\,T$	$\qquad \partial _{S}Q(S,V,N),\,T(S,V,N)\qquad$
$(03)\qquad$	${\displaystyle }$	$=$	$\left({\frac {\partial U}{\partial S}}\right)_{V,N}=+\,T$	$\qquad \partial _{S}U(S,V,N),\,T(S,V,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial Q}{\partial V}}\right)_{S,N}$	$=$	$+\,0$	$\qquad \partial _{V}Q(S,V,N)\qquad$
$(04)\qquad$	${\displaystyle }$	$\neq$	$\left({\frac {\partial U}{\partial V}}\right)_{S,N}=-\,p$	$\qquad \partial _{V}U(S,V,N),\,p(S,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial V}}{\frac {\partial Q}{\partial S}}$	$=$	$-\,{\frac {2}{3}}\,{\frac {T_{0}}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {2}{3}}\,{\frac {T}{V}}$	$\qquad \partial _{V}\partial _{S}Q(S,V,N),\,T(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {\partial }{\partial V}}{\frac {\partial U}{\partial S}}=-\,{\frac {2}{3}}{\frac {T}{V}}$	$\qquad \partial _{V}\partial _{S}U(S,V,N),\,T(S,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial Q}{\partial V}}$	$=$	$+\,0$	$\qquad \partial _{S}\partial _{V}Q(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$\neq$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial V}}=-\,{\frac {2}{3}}{\frac {p}{n\,R}}$	$\qquad \partial _{S}\partial _{V}U(S,V,N),\,p(S,V,N)\qquad$
$(05)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial Q}{\partial S}}$	$\neq$	${\frac {\partial }{\partial S}}{\frac {\partial Q}{\partial V}}$	$\qquad \partial _{V}\partial _{S}Q(S,V,N),\,\partial _{S}\partial _{V}Q(S,V,N)\qquad$
$\qquad \qquad$	$Q_{B}(S,V,N)$	$=$	$+\,\int _{S,dV=0}T\,dS$	$\qquad dS(S),\,dV=0,\,dN=0,\,T(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\int _{S,dV=0}T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\,dS$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,B(V,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T(S,V,N)\,+\,B(V,N)$	$\qquad \qquad$
$\qquad \qquad$	$Q_{A}(S,V,N)$	$=$	$+\,\int _{V,dS=0}T\,dS$	$\qquad dS(V),\,dS=0,\,dN=0,\,T(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$0+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	$A$	$\neq$	$A(S,N)$	$\qquad A=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(V,N)$	$\qquad B=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$Q_{B}(S,V,N)$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,B$	$\qquad \qquad$
$\qquad \qquad$	$Q_{A}(S,V,N)$	$=$	$+\,A$	$\qquad Q_{A}(S,V,N)={\mbox{const}}\qquad$
$\qquad \qquad$	$Q_{B}(S,V,N)$	$\neq$	$\,Q_{A}(S,V,N)$	$\qquad \qquad$
$\qquad \qquad$	$Q_{V,N}(S_{2},S_{1})$	$=$	$\,Q(S_{2},V,N)-\,Q(S_{1},V,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\int _{S_{1},V}^{S_{2},V}T\,dS$	$\qquad dS(S),\,dV=0,\,dN=0,\,T(S,V,N)\qquad$
$(06)\qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\left[\exp \left({\frac {2}{3}}{\frac {S_{2}}{n\,R}}\right)-\,\exp \left({\frac {2}{3}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,U(S_{2},V,N)-\,U(S_{1},V,N)$	$\qquad \qquad$
$(07)\qquad$	${\displaystyle }$	$=$	$\,U_{V,N}(S_{2},S_{1})$	$\qquad \qquad$
$\qquad \qquad$	$Q_{S,N}(V_{2},V_{1})$	$=$	$\,Q(S,V_{2},N)-\,Q(S,V_{1},N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\int _{V_{1},S}^{V_{2},S}T\,dS$	$\qquad dS(V),\,dS=0,\,dN=0,\,T(S,V,N)\qquad$
$(08)\qquad$	${\displaystyle }$	$=$	$0$	$\qquad \qquad$

B Ausgetauschte Arbeit dW = -pdV

$\qquad \qquad$	$dW$	$=$	$-\,p\,dV$	$\qquad dW(S,V),\,p(S,V,N),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p\,\left[+\,\left({\frac {\partial V}{\partial S}}\right)_{V,N}\,dS+\,\left({\frac {\partial V}{\partial V}}\right)_{S,N}\,dV\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p\,\left[\,+\,0\cdot dS+\,1\cdot dV\right]$	$\qquad \qquad \qquad$
$(01)\qquad$	${\displaystyle }$	$=$	$-\,0\,dS-\,p\,dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,0+\,dW_{S,N}$	$\qquad dW_{V,N}(S)=0,\,dW_{S,N}(V),\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$+\,0+\,dU_{S,N}$	$\qquad dU_{S,N}(V),\,dN=0\qquad$
$\qquad \qquad$	$\left({\frac {\partial W}{\partial S}}\right)_{V,N}$	$=$	$-\,0$	$\qquad \partial _{S}W(S,V,N)\qquad$
$(03)\qquad$	${\displaystyle }$	$\neq$	$\left({\frac {\partial U}{\partial S}}\right)_{V,N}=+\,T$	$\qquad \partial _{S}U(S,V,N),\,T(S,V,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial W}{\partial V}}\right)_{S,N}$	$=$	$-\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,p$	$\qquad \partial _{V}W(S,V,N),\,p(S,V,N)\qquad$
$(04)\qquad$	${\displaystyle }$	$=$	$\left({\frac {\partial U}{\partial V}}\right)_{S,N}=-\,p$	$\qquad \partial _{V}U(S,V,N),\,p(S,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial V}}{\frac {\partial W}{\partial S}}$	$=$	$-\,0$	$\qquad \partial _{V}\partial _{S}W(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$\neq$	${\frac {\partial }{\partial V}}{\frac {\partial U}{\partial S}}=-{\frac {2}{3}}{\frac {T}{V}}$	$\qquad \partial _{V}\partial _{S}U(S,V,N),\,T(S,V,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial W}{\partial V}}$	$=$	$-\,{\frac {2}{3}}\,{\frac {p_{0}}{n\,R}}\left({\frac {V_{0}}{V}}\right)^{5/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {2}{3}}\,{\frac {p}{n\,R}}$	$\qquad \partial _{S}\partial _{V}W(S,V,N),\,p(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial V}}=-\,{\frac {2}{3}}\,{\frac {p}{n\,R}}$	$\qquad \partial _{S}\partial _{V}U(S,V,N),\,p(S,V,N)\qquad$
$(05)\qquad$	${\frac {\partial }{\partial V}}{\frac {\partial W}{\partial S}}$	$\neq$	${\frac {\partial }{\partial S}}{\frac {\partial W}{\partial V}}$	$\qquad \partial _{V}\partial _{S}W(S,V,N),\,\partial _{S}\partial _{V}W(S,V,N)\qquad$
$\qquad \qquad$	$W_{B}(S,V,N)$	$=$	$-\,\int _{S,dV=0}p\,dV$	$\qquad dV(S),\,dV=0,\,dN=0,\,p(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,0+\,B(V,N)$	$\qquad \qquad$
$\qquad \qquad$	$W_{A}(S,V,N)$	$=$	$-\,\int _{V,dS=0}p\,dV$	$\qquad dV(V),\,dS=0,\,dN=0,\,p(S,V,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{V,dS=0}p_{0}\,\left({\frac {V_{0}}{V}}\right)^{5/3}\,\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\,dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,V\,p_{0}\left({\frac {V_{0}}{V}}\right)^{5/3}\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,V\,p(S,V,N)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,p_{0}\,V_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T(S,V,N)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	$A$	$\neq$	$A(S,N)$	$\qquad A=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(V,N)$	$\qquad B=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$W_{B}(S,V,N)$	$=$	$+\,B$	$\qquad W_{B}(S,V,N)=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$W_{A}(S,V,N)$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A$	$\qquad \qquad$
$\qquad \qquad$	$W_{B}(S,V,N)$	$\neq$	$\,W_{A}(S,V,N)$	$\qquad \qquad$
$\qquad \qquad$	$W_{V,N}(S_{2},S_{1})$	$=$	$\,W(S_{2},V,N)-\,W(S_{1},V,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{S_{1},V}^{S_{2},V}p\,dV$	$\qquad dV(S),\,dV=0,\,dN=0,\,p(S,V,N)\qquad$
$(06)\qquad$	${\displaystyle }$	$=$	$0$	$\qquad \qquad$
$\qquad \qquad$	$W_{S,N}(V_{2},V_{1})$	$=$	$\,W(S,V_{2},N)-\,W(S,V_{1},N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{V_{1},S}^{V_{2},S}p\,dV$	$\qquad dV(V),\,dS=0,\,dN=0,\,p(S,V,N)\qquad$
$(07)\qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}-\,\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}\right]\,\exp \left({\frac {2}{3}}{\frac {S-\,S_{0}}{n\,R}}\right)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,U(S,V_{2},N)-\,U(S,V_{1},N)$	$\qquad \qquad$
$(08)\qquad$	${\displaystyle }$	$=$	$\,U_{S,N}(V_{2},V_{1})$	$\qquad \qquad$