CPRT.I.D.03

CPRT.I.D

03 Wärme dQ = +TdS und Arbeit dW = -pdV bei (T,V;N=const)

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

A Ausgetauschte Wärme dQ = +TdS bei (T,V;N=const)

$\quad$	$dQ$	$=$	$+\,T\,dS$	$\quad dQ(T,V),\,dS(T,V),\,dN=0\qquad$
$\quad$	${\displaystyle }$	$=$	$+\,T\,\left[+\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}\,dT+\,\left({\frac {\partial S}{\partial V}}\right)_{T,N}\,dV\right]$	$\quad \qquad$
$\quad$	$-T\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}$	$=$	$-{\frac {3}{2}}\,n\,R$	$\quad \partial _{T}S(T,V,N)\quad \leftarrow (3.9.3.3.1)$
$\quad$	$+T\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}$	$=$	$+{\frac {3}{2}}\,n\,R$	$\quad \partial _{T}S(T,V,N)\quad$
$\quad$	$-V\left({\frac {\partial S}{\partial V}}\right)_{T,N}$	$=$	$-\,n\,R$	$\quad \partial _{V}S(T,V,N)\quad \leftarrow (3.9.3.2.1)$
$\quad$	$+T\left({\frac {\partial S}{\partial V}}\right)_{T,N}$	$=$	$+\,{\frac {n\,R\,T}{V}}$	$\quad \partial _{V}S(T,V,N)\quad$
$(3.11.2.5.1)\quad$	$dQ$	$=$	$+\,{\frac {3}{2}}\,n\,R\,dT+\,{\frac {n\,R\,T}{V}}\,dV$	$\quad dQ(T,V),\,dN=0\qquad$
$\quad$	${\displaystyle }$	$=$	$+\,dQ_{V,N}+\,dQ_{T,N}$	$\quad dQ_{V,N}(T),\,dQ_{T,N}(V),\,dN=0\qquad$
$(3.11.2.5.2)\quad$	${\displaystyle }$	$=$	$+\,dU_{V,N}-\,dF_{T,N}$	$\quad dU_{V,N}(T),\,dF_{T,N}(V),\,dN=0\qquad$
$\quad$	$\left({\frac {\partial Q}{\partial T}}\right)_{V,N}$	$=$	$+\,{\frac {3}{2}}\,n\,R$	$\quad \partial _{T}Q(T,V,N)\qquad$
$(3.11.2.5.3)\quad$	${\displaystyle }$	$=$	$\left({\frac {\partial U}{\partial T}}\right)_{V,N}=+\,{\frac {3}{2}}\,n\,R$	$\quad \partial _{T}U(T,V,N)\qquad$
$\quad$	$\left({\frac {\partial Q}{\partial V}}\right)_{T,N}$	$=$	$+\,{\frac {n\,R\,T}{V}}$	$\quad \partial _{V}Q(T,V,N)\qquad$
$(3.11.2.5.4)\quad$	${\displaystyle }$	$\neq$	$\left({\frac {\partial U}{\partial V}}\right)_{T,N}=+\,0$	$\quad \partial _{V}U(T,V,N)\qquad$
$\quad$	${\frac {\partial }{\partial V}}{\frac {\partial Q}{\partial T}}$	$=$	$+\,0$	$\quad \partial _{V}\partial _{T}Q(T,V,N)\qquad$
$\quad$	${\displaystyle }$	$=$	${\frac {\partial }{\partial V}}{\frac {\partial U}{\partial T}}=+\,0$	$\quad \partial _{V}\partial _{T}U(T,V,N)\qquad$
$\quad$	${\frac {\partial }{\partial T}}{\frac {\partial Q}{\partial V}}$	$=$	$+\,{\frac {n\,R}{V}}$	$\quad \partial _{T}\partial _{V}Q(T,V,N)\qquad$
$\quad$	${\displaystyle }$	$\neq$	${\frac {\partial }{\partial T}}{\frac {\partial U}{\partial V}}=+\,0$	$\quad \partial _{T}\partial _{V}U(T,V,N)\qquad$
$(3.11.2.5.5)\quad$	${\frac {\partial }{\partial V}}{\frac {\partial Q}{\partial T}}$	$\neq$	${\frac {\partial }{\partial T}}{\frac {\partial Q}{\partial V}}$	$\quad \partial _{V}\partial _{T}Q(T,V,N),\,\partial _{T}\partial _{V}Q(T,V,N)\qquad$
$\quad$	$Q_{B}(T,V,N)$	$=$	$+\,\int _{T,dV=0}T\,dS$	$\quad dS(T),\,dV=0,\,dN=0\qquad$
$\quad$	${\displaystyle }$	$=$	$+\,\int _{T,dV=0}{\frac {3}{2}}\,n\,R\,dT$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T\,+\,B(V,N)$	$\quad \qquad$
$\quad$	$Q_{A}(T,V,N)$	$=$	$+\,\int _{V,dT=0}T\,dS$	$\quad dS(V),\,dT=0,\,dN=0\qquad$
$\quad$	${\displaystyle }$	$=$	$+\,\int _{V,dT=0}{\frac {n\,R\,T}{V}}\,dV$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$+\,n\,R\,T\,\ln V+\,A(T,N)$	$\quad \qquad$
$\quad$	$A$	$=$	$A(T,N)$	$\quad \qquad$
$\quad$	$B$	$\neq$	$B(V,N)$	$\quad B=\,{\mbox{const}}\qquad$
$\quad$	$Q_{B}(T,V,N)$	$=$	$+\,{\frac {3}{2}}\,n\,R\,T\,+\,B$	$\quad \qquad$
$\quad$	$Q_{A}(T,V,N)$	$=$	$+\,n\,R\,T\,\ln V+\,A(T,N)$	$\quad A(T,N)=-\,n\,R\,T\,\ln V_{0}\qquad$
$\quad$	$Q_{B}(T,V,N)$	$\neq$	$Q_{A}(T,V,N)$	$\quad \qquad$
$\quad$	$Q_{V,N}(T_{2},T_{1})$	$=$	$\,Q(T_{2},V,N)-\,Q(T_{1},V,N)$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$+\,\int _{T_{1},V}^{T_{2},V}T\,dS$	$\quad dS(T),\,dV=0,\,dN=0\qquad$
$(3.11.2.5.6)\quad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,(T_{2}-T_{1})$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$\,U(T,V_{2},N)-\,U(T,V_{1},N)$	$\quad \qquad$
$(3.11.2.5.7)\quad$	${\displaystyle }$	$=$	$\,U_{T,N}(V_{2},V_{1})$	$\quad \qquad$
$\quad$	$Q_{T,N}(V_{2},V_{1})$	$=$	$\,Q(T,V_{2},N)-\,Q(T,V_{1},N)$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$+\,\int _{V_{1},T}^{V_{2},T}T\,dS$	$\quad dS(V),\,dT=0,\,dN=0\qquad$
$(3.11.2.5.8)\quad$	${\displaystyle }$	$=$	$+\,n\,R\,T\,\ln {\frac {V_{2}}{V_{1}}}$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$+\,n\,R\,T\,\left[\ln {\frac {V_{2}}{V_{0}}}-\,\ln {\frac {V_{1}}{V_{0}}}\right]$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$-\,[F(T,V_{2},N)-\,F(T,V_{1},N)]$	$\quad \qquad$
$(3.11.2.5.9)\quad$	${\displaystyle }$	$=$	$-\,F_{T,N}(V_{2},V_{1})$	$\quad \qquad$

B Ausgetauschte Arbeit dW = -pdV bei (T,V;N=const)

$\quad$	$dW$	$=$	$-\,p\,dV$	$\quad dW(T,V),\,p(T,V,N),\,dN=0\qquad$
$\quad$	${\displaystyle }$	$=$	$-\,p\,\left[+\,\left({\frac {\partial V}{\partial T}}\right)_{V,N}\,dT+\,\left({\frac {\partial V}{\partial V}}\right)_{T,N}\,dV\right]$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$-\,p\,\left[\,+\,0\cdot dT+\,1\cdot dV\right]$	$\quad \qquad$
$(3.11.2.6.1)\quad$	${\displaystyle }$	$=$	$-\,0\,dT-\,p\,dV$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$+\,0+\,dW_{T,N}$	$\quad dW_{V,N}(T)=0,\,dW_{T,N}(V),\,dN=0\qquad$
$(3.11.2.6.2)\quad$	${\displaystyle }$	$=$	$+\,0+\,dF_{T,N}$	$\quad dF_{T,N}(V),\,dN=0\qquad$
$\quad$	$\left({\frac {\partial W}{\partial T}}\right)_{V,N}$	$=$	$-\,0$	$\quad \partial _{T}W(T,V,N)\qquad$
$(3.11.2.6.3)\quad$	${\displaystyle }$	$\neq$	$\left({\frac {\partial F}{\partial T}}\right)_{V,N}=-\,S$	$\quad \partial _{T}F(T,V,N),\,S(T,V,N)\qquad$
$\quad$	$\left({\frac {\partial W}{\partial V}}\right)_{T,N}$	$=$	$-\,p$	$\quad \partial _{V}W(T,V,N),\,p(T,V,N)\qquad$
$(3.11.2.6.4)\quad$	${\displaystyle }$	$=$	$\left({\frac {\partial F}{\partial V}}\right)_{T,N}=-\,p$	$\quad \partial _{V}F(T,V,N),\,p(T,V,N)\qquad$
$\quad$	${\frac {\partial }{\partial V}}{\frac {\partial W}{\partial T}}$	$=$	$-\,0$	$\quad \partial _{V}\partial _{T}W(T,V,N)\qquad$
$\quad$	${\displaystyle }$	$\neq$	${\frac {\partial }{\partial V}}{\frac {\partial F}{\partial T}}=-{\frac {n\,R}{V}}$	$\quad \partial _{V}\partial _{T}F(T,V,N)\qquad$
$\quad$	${\frac {\partial }{\partial T}}{\frac {\partial W}{\partial V}}$	$=$	$-\,{\frac {n\,R}{V}}$	$\quad \partial _{T}\partial _{V}W(T,V,N)\qquad$
$\quad$	${\displaystyle }$	$=$	${\frac {\partial }{\partial T}}{\frac {\partial F}{\partial V}}=-\,{\frac {n\,R}{V}}$	$\quad \partial _{T}\partial _{V}F(T,V,N)\qquad$
$(3.11.2.6.5)\quad$	${\frac {\partial }{\partial V}}{\frac {\partial W}{\partial T}}$	$\neq$	${\frac {\partial }{\partial T}}{\frac {\partial W}{\partial V}}$	$\quad \partial _{V}\partial _{T}W(T,V,N),\,\partial _{T}\partial _{V}W(T,V,N)\qquad$
$\quad$	$W_{B}(T,V,N)$	$=$	$-\,\int _{T,dV=0}p\,dV$	$\quad dV(T),\,dV=0,\,dN=0,\,p(T,V,N)\qquad$
$\quad$	${\displaystyle }$	$=$	$-\,0+\,B(V,N)$	$\quad \qquad$
$\quad$	$W_{A}(T,V,N)$	$=$	$-\,\int _{V,dT=0}p\,dV$	$\quad dV(V),\,dT=0,\,dN=0,\,p(T,V,N)\qquad$
$\quad$	${\displaystyle }$	$=$	$-\,\int _{V,dT=0}{\frac {n\,R\,T}{V}}\,dV$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$-\,n\,R\,T\,\ln V\,+\,A(T,N)$	$\quad \qquad$
$\quad$	$A$	$=$	$A(T,N)$	$\quad \qquad$
$\quad$	$B$	$\neq$	$B(V,N)$	$\quad B=\,{\mbox{const}}\qquad$
$\quad$	$W_{B}(T,V,N)$	$=$	$+\,B$	$\quad W_{V,N}=\,{\mbox{const}}\qquad$
$\quad$	$W_{A}(T,V,N)$	$=$	$-\,n\,R\,T\,\ln V+\,A(T,N)$	$\quad A(T,N)=+\,\,n\,R\,T\,\ln V_{0}\qquad$
$\quad$	$W_{B}(T,V,N)$	$\neq$	$W_{A}(T,V,N)$	$\quad \qquad$
$\quad$	$W_{V,N}(T_{2},T_{1})$	$=$	$W(T_{2},V,N)-\,W(T_{1},V,N)$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$-\,\int _{T_{1},V}^{T_{2},V}p\,dV$	$\quad dV(T),\,dV=0,\,dN=0,\,p(T,V,N)\qquad$
$(3.11.2.6.6)\quad$	${\displaystyle }$	$=$	$0$	$\quad \qquad$
$\quad$	$W_{T,N}(V_{2},V_{1})$	$=$	$W(T,V_{2},N)-\,W(T,V_{1},N)$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$-\,\int _{V_{1},T}^{V_{2},T}p\,dV$	$\quad dV(V),\,dT=0,\,dN=0,\,p(T,V,N)\qquad$
$(3.11.2.6.7)\quad$	${\displaystyle }$	$=$	$-\,n\,R\,T\,\ln {\frac {V_{2}}{V_{1}}}$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$-\,n\,R\,T\,\left[\ln {\frac {V_{2}}{V_{0}}}-\,\ln {\frac {V_{1}}{V_{0}}}\right]$	$\quad \qquad$
$\quad$	${\displaystyle }$	$=$	$\,F(T,V_{2},N)-\,F(T,V_{1},N)$	$\quad \qquad$
$(3.11.2.6.8)\quad$	${\displaystyle }$	$=$	$\,F_{T,N}(V_{2},V_{1})$	$\quad \qquad$