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)

$$	$dQ$	$=$	$+TdS$	$dQ(T,V),dS(T,V),dN=0$
$$	${\displaystyle }$	$=$	$+T[+{\left(\frac{\partial S}{\partial T}\right)}_{V,N}dT+{\left(\frac{\partial S}{\partial V}\right)}_{T,N}dV]$	$$
$$	$-T{\left(\frac{\partial S}{\partial T}\right)}_{V,N}$	$=$	$-\frac{3}{2}nR$	${\partial }_{T}S(T,V,N)\leftarrow (3.9.3.3.1)$
$$	$+T{\left(\frac{\partial S}{\partial T}\right)}_{V,N}$	$=$	$+\frac{3}{2}nR$	${\partial }_{T}S(T,V,N)$
$$	$-V{\left(\frac{\partial S}{\partial V}\right)}_{T,N}$	$=$	$-nR$	${\partial }_{V}S(T,V,N)\leftarrow (3.9.3.2.1)$
$$	$+T{\left(\frac{\partial S}{\partial V}\right)}_{T,N}$	$=$	$+\frac{nRT}{V}$	${\partial }_{V}S(T,V,N)$
$(3.11.2.5.1)$	$dQ$	$=$	$+\frac{3}{2}nRdT+\frac{nRT}{V}dV$	$dQ(T,V),dN=0$
$$	${\displaystyle }$	$=$	$+d{Q}_{V,N}+d{Q}_{T,N}$	$d{Q}_{V,N}(T),d{Q}_{T,N}(V),dN=0$
$(3.11.2.5.2)$	${\displaystyle }$	$=$	$+d{U}_{V,N}-d{F}_{T,N}$	$d{U}_{V,N}(T),d{F}_{T,N}(V),dN=0$
$$	${\left(\frac{\partial Q}{\partial T}\right)}_{V,N}$	$=$	$+\frac{3}{2}nR$	${\partial }_{T}Q(T,V,N)$
$(3.11.2.5.3)$	${\displaystyle }$	$=$	${\left(\frac{\partial U}{\partial T}\right)}_{V,N}=+\frac{3}{2}nR$	${\partial }_{T}U(T,V,N)$
$$	${\left(\frac{\partial Q}{\partial V}\right)}_{T,N}$	$=$	$+\frac{nRT}{V}$	${\partial }_{V}Q(T,V,N)$
$(3.11.2.5.4)$	${\displaystyle }$	$\ne$	${\left(\frac{\partial U}{\partial V}\right)}_{T,N}=+0$	${\partial }_{V}U(T,V,N)$
$$	$\frac{\partial }{\partial V}\frac{\partial Q}{\partial T}$	$=$	$+0$	${\partial }_V{\partial }_{T}Q(T,V,N)$
$$	${\displaystyle }$	$=$	$\frac{\partial }{\partial V}\frac{\partial U}{\partial T}=+0$	${\partial }_V{\partial }_{T}U(T,V,N)$
$$	$\frac{\partial }{\partial T}\frac{\partial Q}{\partial V}$	$=$	$+\frac{nR}{V}$	${\partial }_T{\partial }_{V}Q(T,V,N)$
$$	${\displaystyle }$	$\ne$	$\frac{\partial }{\partial T}\frac{\partial U}{\partial V}=+0$	${\partial }_T{\partial }_{V}U(T,V,N)$
$(3.11.2.5.5)$	$\frac{\partial }{\partial V}\frac{\partial Q}{\partial T}$	$\ne$	$\frac{\partial }{\partial T}\frac{\partial Q}{\partial V}$	${\partial }_V{\partial }_{T}Q(T,V,N),{\partial }_T{\partial }_{V}Q(T,V,N)$
$$	$Q_B(T,V,N)$	$=$	$+\underset{T,dV=0}{\int }TdS$	$dS(T),dV=0,dN=0$
$$	${\displaystyle }$	$=$	$+\underset{T,dV=0}{\int }\frac{3}{2}nRdT$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nRT+B(V,N)$	$$
$$	$Q_A(T,V,N)$	$=$	$+\underset{V,dT=0}{\int }TdS$	$dS(V),dT=0,dN=0$
$$	${\displaystyle }$	$=$	$+\underset{V,dT=0}{\int }\frac{nRT}{V}dV$	$$
$$	${\displaystyle }$	$=$	$+nRT\ln V+A(T,N)$	$$
$$	$A$	$=$	$A(T,N)$	$$
$$	$B$	$\ne$	$B(V,N)$	$B=\text{const}$
$$	$Q_B(T,V,N)$	$=$	$+\frac{3}{2}nRT+B$	$$
$$	$Q_A(T,V,N)$	$=$	$+nRT\ln V+A(T,N)$	$A(T,N)=-nRT\ln V_0$
$$	$Q_B(T,V,N)$	$\ne$	$Q_A(T,V,N)$	$$
$$	$Q_{V,N}(T_2,T_1)$	$=$	$Q(T_2,V,N)-Q(T_1,V,N)$	$$
$$	${\displaystyle }$	$=$	$+\underset{T_1,V}{\overset{T_2,V}{\int }}TdS$	$dS(T),dV=0,dN=0$
$(3.11.2.5.6)$	${\displaystyle }$	$=$	$+\frac{3}{2}nR(T_2-T_1)$	$$
$$	${\displaystyle }$	$=$	$U(T,V_2,N)-U(T,V_1,N)$	$$
$(3.11.2.5.7)$	${\displaystyle }$	$=$	$U_{T,N}(V_2,V_1)$	$$
$$	$Q_{T,N}(V_2,V_1)$	$=$	$Q(T,V_2,N)-Q(T,V_1,N)$	$$
$$	${\displaystyle }$	$=$	$+\underset{V_1,T}{\overset{V_2,T}{\int }}TdS$	$dS(V),dT=0,dN=0$
$(3.11.2.5.8)$	${\displaystyle }$	$=$	$+nRT\ln \frac{V_2}{V_1}$	$$
$$	${\displaystyle }$	$=$	$+nRT[\ln \frac{V_2}{V_0}-\ln \frac{V_1}{V_0}]$	$$
$$	${\displaystyle }$	$=$	$-[F(T,V_2,N)-F(T,V_1,N)]$	$$
$(3.11.2.5.9)$	${\displaystyle }$	$=$	$-F_{T,N}(V_2,V_1)$	$$

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

$$	$dW$	$=$	$-pdV$	$dW(T,V),p(T,V,N),dN=0$
$$	${\displaystyle }$	$=$	$-p[+{\left(\frac{\partial V}{\partial T}\right)}_{V,N}dT+{\left(\frac{\partial V}{\partial V}\right)}_{T,N}dV]$	$$
$$	${\displaystyle }$	$=$	$-p[+0\cdot dT+1\cdot dV]$	$$
$(3.11.2.6.1)$	${\displaystyle }$	$=$	$-0dT-pdV$	$$
$$	${\displaystyle }$	$=$	$+0+d{W}_{T,N}$	$d{W}_{V,N}(T)=0,d{W}_{T,N}(V),dN=0$
$(3.11.2.6.2)$	${\displaystyle }$	$=$	$+0+d{F}_{T,N}$	$d{F}_{T,N}(V),dN=0$
$$	${\left(\frac{\partial W}{\partial T}\right)}_{V,N}$	$=$	$-0$	${\partial }_{T}W(T,V,N)$
$(3.11.2.6.3)$	${\displaystyle }$	$\ne$	${\left(\frac{\partial F}{\partial T}\right)}_{V,N}=-S$	${\partial }_{T}F(T,V,N),S(T,V,N)$
$$	${\left(\frac{\partial W}{\partial V}\right)}_{T,N}$	$=$	$-p$	${\partial }_{V}W(T,V,N),p(T,V,N)$
$(3.11.2.6.4)$	${\displaystyle }$	$=$	${\left(\frac{\partial F}{\partial V}\right)}_{T,N}=-p$	${\partial }_{V}F(T,V,N),p(T,V,N)$
$$	$\frac{\partial }{\partial V}\frac{\partial W}{\partial T}$	$=$	$-0$	${\partial }_V{\partial }_{T}W(T,V,N)$
$$	${\displaystyle }$	$\ne$	$\frac{\partial }{\partial V}\frac{\partial F}{\partial T}=-\frac{nR}{V}$	${\partial }_V{\partial }_{T}F(T,V,N)$
$$	$\frac{\partial }{\partial T}\frac{\partial W}{\partial V}$	$=$	$-\frac{nR}{V}$	${\partial }_T{\partial }_{V}W(T,V,N)$
$$	${\displaystyle }$	$=$	$\frac{\partial }{\partial T}\frac{\partial F}{\partial V}=-\frac{nR}{V}$	${\partial }_T{\partial }_{V}F(T,V,N)$
$(3.11.2.6.5)$	$\frac{\partial }{\partial V}\frac{\partial W}{\partial T}$	$\ne$	$\frac{\partial }{\partial T}\frac{\partial W}{\partial V}$	${\partial }_V{\partial }_{T}W(T,V,N),{\partial }_T{\partial }_{V}W(T,V,N)$
$$	$W_B(T,V,N)$	$=$	$-\underset{T,dV=0}{\int }pdV$	$dV(T),dV=0,dN=0,p(T,V,N)$
$$	${\displaystyle }$	$=$	$-0+B(V,N)$	$$
$$	$W_A(T,V,N)$	$=$	$-\underset{V,dT=0}{\int }pdV$	$dV(V),dT=0,dN=0,p(T,V,N)$
$$	${\displaystyle }$	$=$	$-\underset{V,dT=0}{\int }\frac{nRT}{V}dV$	$$
$$	${\displaystyle }$	$=$	$-nRT\ln V+A(T,N)$	$$
$$	$A$	$=$	$A(T,N)$	$$
$$	$B$	$\ne$	$B(V,N)$	$B=\text{const}$
$$	$W_B(T,V,N)$	$=$	$+B$	$W_{V,N}=\text{const}$
$$	$W_A(T,V,N)$	$=$	$-nRT\ln V+A(T,N)$	$A(T,N)=+nRT\ln V_0$
$$	$W_B(T,V,N)$	$\ne$	$W_A(T,V,N)$	$$
$$	$W_{V,N}(T_2,T_1)$	$=$	$W(T_2,V,N)-W(T_1,V,N)$	$$
$$	${\displaystyle }$	$=$	$-\underset{T_1,V}{\overset{T_2,V}{\int }}pdV$	$dV(T),dV=0,dN=0,p(T,V,N)$
$(3.11.2.6.6)$	${\displaystyle }$	$=$	$0$	$$
$$	$W_{T,N}(V_2,V_1)$	$=$	$W(T,V_2,N)-W(T,V_1,N)$	$$
$$	${\displaystyle }$	$=$	$-\underset{V_1,T}{\overset{V_2,T}{\int }}pdV$	$dV(V),dT=0,dN=0,p(T,V,N)$
$(3.11.2.6.7)$	${\displaystyle }$	$=$	$-nRT\ln \frac{V_2}{V_1}$	$$
$$	${\displaystyle }$	$=$	$-nRT[\ln \frac{V_2}{V_0}-\ln \frac{V_1}{V_0}]$	$$
$$	${\displaystyle }$	$=$	$F(T,V_2,N)-F(T,V_1,N)$	$$
$(3.11.2.6.8)$	${\displaystyle }$	$=$	$F_{T,N}(V_2,V_1)$	$$