PCRT/I/a7db

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

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

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

A Ausgetauschte Wärme dQ = +TdS

$\qquad \qquad$	$dQ$	$=$	$+\,T\,dS$	$\qquad dQ(S,p),\,T(S,p,N),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,T\,\left[+\,\left({\frac {\partial S}{\partial S}}\right)_{p,N}\,dS+\,\left({\frac {\partial S}{\partial p}}\right)_{S,N}\,dp\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,T\,\left[\,+\,1\cdot dS+\,0\cdot dp\right]$	$\qquad \qquad$
$(01)\qquad$	${\displaystyle }$	$=$	$+\,T\,dS+\,0\,dp$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,dQ_{p,N}+\,0$	$\qquad dQ_{p,N}(S),\,dQ_{S,N}(p)=0,\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$+\,dH_{p,N}+\,0$	$\qquad dH_{p,N}(S),\,dN=0\qquad$
$\qquad \qquad$	$\left({\frac {\partial Q}{\partial S}}\right)_{p,N}$	$=$	$+\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,T$	$\qquad \partial _{S}Q(S,p,N),\,T(S,p,N)\qquad$
$(03)\qquad$	${\displaystyle }$	$=$	$\left({\frac {\partial H}{\partial S}}\right)_{p,N}=+\,T$	$\qquad \partial _{S}H(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial Q}{\partial p}}\right)_{S,N}$	$=$	$+\,0$	$\qquad \partial _{p}Q(S,p,N)\qquad$
$(04)\qquad$	${\displaystyle }$	$\neq$	$\left({\frac {\partial H}{\partial p}}\right)_{S,N}=+\,V$	$\qquad \partial _{p}H(S,p,N),\,V(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial p}}{\frac {\partial Q}{\partial S}}$	$=$	$+\,{\frac {2}{5}}\,{\frac {T_{0}}{p}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {2}{5}}\,{\frac {T}{p}}$	$\qquad \partial _{p}\partial _{S}Q(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {\partial }{\partial p}}{\frac {\partial H}{\partial S}}=+\,{\frac {2}{5}}\,{\frac {T}{p}}$	$\qquad \partial _{p}\partial _{S}H(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial Q}{\partial p}}$	$=$	$+\,0$	$\qquad \partial _{S}\partial _{p}Q(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$\neq$	${\frac {\partial }{\partial S}}{\frac {\partial H}{\partial p}}=+\,{\frac {2}{5}}\,{\frac {V}{n\,R}}$	$\qquad \partial _{S}\partial _{p}H(S,p,N),\,V(S,p,N)\qquad$
$(05)\qquad$	${\frac {\partial }{\partial p}}{\frac {\partial Q}{\partial S}}$	$\neq$	${\frac {\partial }{\partial S}}{\frac {\partial Q}{\partial p}}$	$\qquad \partial _{p}\partial _{S}Q(S,p,N),\,\partial _{S}\partial _{p}Q(S,p,N)\qquad$
$\qquad \qquad$	$Q_{B}(S,p,N)$	$=$	$+\,\int _{S,dp=0}T\,dS$	$\qquad dS(S),\,dp=0,\,dN=0,\,T(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\int _{S,dp=0}T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,dS$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {5}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {5}{2}}\,n\,R\,T(S,p,N)\,+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	$Q_{A}(S,p,N)$	$=$	$+\,\int _{p,dS=0}T\,dS$	$\qquad dS(p),\,dS=0,\,dN=0,\,T(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$0+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	$A$	$\neq$	$A(S,N)$	$\qquad A=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(p,N)$	$\qquad B=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$Q_{B}(S,p,N)$	$=$	$+\,{\frac {5}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,B$	$\qquad \qquad$
$\qquad \qquad$	$Q_{A}(S,p,N)$	$=$	$+\,A$	$\qquad Q_{A}(S,p,N)={\mbox{const}}\qquad$
$\qquad \qquad$	$Q_{B}(S,p,N)$	$\neq$	$\,Q_{A}(S,p,N)$	$\qquad \qquad$
$\qquad \qquad$	$Q_{p,N}(S_{2},S_{1})$	$=$	$\,Q(S_{2},p,N)-\,Q(S_{1},p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\int _{S_{1},p}^{S_{2},p}T\,dS$	$\qquad dS(S),\,dp=0,\,dN=0,\,T(S,p,N)\qquad$
$(06)\qquad$	${\displaystyle }$	$=$	$+\,{\frac {5}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)-\,\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,H(S_{2},p,N)-\,H(S_{1},p,N)$	$\qquad \qquad$
$(07)\qquad$	${\displaystyle }$	$=$	$\,H_{p,N}(S_{2},S_{1})$	$\qquad \qquad$
$\qquad \qquad$	$Q_{S,N}(p_{1},p_{2})$	$=$	$\,Q(S,p_{1},N)-\,Q(S,p_{2},N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,\int _{p_{2},S}^{p_{1},S}T\,dS$	$\qquad dS(p),\,dS=0,\,dN=0,\,T(S,p,N)\qquad$
$(08)\qquad$	${\displaystyle }$	$=$	$0$	$\qquad \qquad$

B Ausgetauschte Arbeit dW = -pdV

$\qquad \qquad$	$dW$	$=$	$-\,p\,dV$	$\qquad dW(S,p),dV(S,p),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p\,\left[+\,\left({\frac {\partial V}{\partial S}}\right)_{p,N}\,dS+\,\left({\frac {\partial V}{\partial p}}\right)_{S,N}\,dp\right]$	$\qquad \qquad$
$\qquad \qquad$	$-S\,\left({\frac {\partial V}{\partial S}}\right)_{p,N}$	$=$	$-{\frac {2}{5}}{\frac {S}{n\,R}}\,V$	$\qquad \partial _{S}V(S,p,N)\quad \leftarrow (12.{\mbox{C}}.3)$
$\qquad \qquad$	$-p\,\left({\frac {\partial V}{\partial S}}\right)_{p,N}$	$=$	$-{\frac {2}{5}}\,T$	$\qquad \partial _{S}V(S,p,N),\,T(S,p,N)\quad$
$\qquad \qquad$	$-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}$	$=$	$+\,{\frac {3}{5}}\,V$	$\qquad \partial _{p}V(S,p,N)\quad \leftarrow (12.{\mbox{B}}.3)$
$(01)\qquad$	$dW$	$=$	$-\,{\frac {2}{5}}\,T\,dS+\,{\frac {3}{5}}\,V\,dp$	$\qquad dW(S,p),\,dN=0;\,(S,p,N):\,T,\,V\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,dW_{p,N}+\,dW_{S,N}$	$\qquad dW_{p,N}(S),\,dW_{S,N}(p),\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$+\,[dU_{p,N}-\,dH_{p,N}]+\,dU_{S,N}$	$\qquad dU_{p,N}(S),\,dH_{p,N}(S),\,dU_{S,N}(p),\,dN=0\qquad$
$\qquad \qquad$	$\left({\frac {\partial W}{\partial S}}\right)_{p,N}$	$=$	$-\,{\frac {2}{5}}\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=-\,{\frac {2}{5}}\,T$	$\qquad \partial _{S}W(S,p,N),\,T(S,p,N)\qquad$
$(03)\qquad$	${\displaystyle }$	$\neq$	$\left({\frac {\partial U}{\partial S}}\right)_{p,N}=+\,{\frac {3}{5}}\,T$	$\qquad \partial _{S}U(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	$\left({\frac {\partial W}{\partial p}}\right)_{S,N}$	$=$	$=+\,{\frac {3}{5}}\,V_{0}\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {3}{5}}\,V$	$\qquad \partial _{p}W(S,p,N),\,V(S,p,N)\qquad$
$(04)\qquad$	${\displaystyle }$	$=$	$\left({\frac {\partial U}{\partial p}}\right)_{S,N}=+\,{\frac {3}{5}}\,V$	$\qquad \partial _{p}U(S,p,N),\,V(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial p}}{\frac {\partial W}{\partial S}}$	$=$	$-\,0$	$\qquad \partial _{p}\partial _{S}W(S,p,N)\qquad$
$\qquad$	${\displaystyle }$	$\neq$	${\frac {\partial }{\partial p}}{\frac {\partial U}{\partial S}}=+\,{\frac {6}{25}}\,{\frac {T}{p}}$	$\qquad \partial _{p}\partial _{S}U(S,p,N),\,T(S,p,N)\qquad$
$\qquad \qquad$	${\frac {\partial }{\partial S}}{\frac {\partial W}{\partial p}}$	$=$	$+\,{\frac {6}{25}}\,{\frac {V_{0}}{n\,R}}\left({\frac {p_{0}}{p}}\right)^{3/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)=+\,{\frac {6}{25}}\,{\frac {V}{n\,R}}$	$\qquad \partial _{S}\partial _{p}W(S,p,N),\,V(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {\partial }{\partial S}}{\frac {\partial U}{\partial p}}=+\,{\frac {6}{25}}\,{\frac {V}{n\,R}}$	$\qquad \partial _{S}\partial _{p}U(S,p,N),\,V(S,p,N)\qquad$
$(05)\qquad$	${\frac {\partial }{\partial p}}{\frac {\partial W}{\partial S}}$	$\neq$	${\frac {\partial }{\partial S}}{\frac {\partial W}{\partial p}}$	$\qquad \partial _{p}\partial _{S}W(S,p,N),\,\partial _{S}\partial _{p}W(S,p,N)\qquad$
$\qquad \qquad$	$W_{B}(S,p,N)$	$=$	$-\,\int _{S,dp=0}p\,dV$	$\qquad dV(S),\,dp=0,\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,{\frac {2}{5}}\,\int _{S,dp=0}T\,dS$	$\qquad T(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,{\frac {2}{5}}\,\int _{S,dp=0}T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,dS$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,{\frac {2}{5}}\cdot {\frac {5}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,n\,R\,T(S,p,N)\,+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{3/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p\,V(S,p,N)\,+\,B(p,N)$	$\qquad \qquad$
$\qquad \qquad$	$W_{A}(S,p,N)$	$=$	$-\,\int _{p,dS=0}p\,dV$	$\qquad dV(p),\,dS=0,\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\int _{p,dS=0}V\,dp$	$\qquad V(S,p,N)\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\int _{p,dS=0}V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,dp$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{5}}\cdot {\frac {5}{2}}\,p\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,p\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,p\,V(S,p,N)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T(S,p,N)\,+\,A(S,N)$	$\qquad \qquad$
$\qquad \qquad$	$A$	$\neq$	$A(S,N)$	$\qquad A=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$B$	$\neq$	$B(p,N)$	$\qquad B=\,{\mbox{const}}\qquad$
$\qquad \qquad$	$W_{B}(S,p,N)$	$=$	$-\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B$	$\qquad \qquad$
$\qquad \qquad$	$W_{A}(S,p,N)$	$=$	$+\,{\frac {3}{2}}\,p\,V_{0}\,\left({\frac {p_{0}}{p}}\right)^{3/5}\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)\,+\,A$	$\qquad \qquad$
$\qquad \qquad$	$W_{B}(S,p,N)$	$\neq$	$W_{A}(S,p,N)$	$\qquad \qquad$
$\qquad \qquad$	$W_{p,N}(S_{2},S_{1})$	$=$	$W(S_{2},p,N)-\,W(S_{1},p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{S_{1},p}^{S_{2},p}p\,dV$	$\qquad dV(S),\,dp=0,\,dN=0\qquad$
$(06)\qquad$	${\displaystyle }$	$=$	$-\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\,\left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,n\,R\,[T(S_{2},p,N)-\,T(S_{1},p,N)]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p\,V(S_{2},p,N)-\,p\,V(S_{1},p,N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,[U(S_{2},p,N)-\,H(S_{2},p,N)]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,[U(S_{1},p,N)-\,H(S_{1},p,N)]$	$\qquad \qquad$
$(07)\qquad$	${\displaystyle }$	$=$	$\,U_{p,N}(S_{2},S_{1})-\,H_{p,N}(S_{2},S_{1})$	$\qquad \qquad$
$\qquad \qquad$	$W_{S,N}(p_{1},p_{2})$	$=$	$W(S,p_{1},N)-\,W(S,p_{2},N)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,\int _{p_{2},S}^{p_{1},S}p\,dV$	$\qquad dV(p),\,dS=0,\,dN=0\qquad$
$(08)\qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}-\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\right]\,\exp \left({\frac {2}{5}}{\frac {S-\,S_{0}}{n\,R}}\right)$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,U(S,p_{1},N)-\,U(S,p_{2},N)$	$\qquad \qquad$
$(09)\qquad$	${\displaystyle }$	$=$	$\,U_{S,N}(p_{1},p_{2})$	$\qquad \qquad$