PCRT/I/c11d

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

$$	$dQ$	$=$	$+TdS$	$dQ(S,V),T(S,V,N),dN=0$
$$	${\displaystyle }$	$=$	$+T[+{\left(\frac{\partial S}{\partial S}\right)}_{V,N}dS+{\left(\frac{\partial S}{\partial V}\right)}_{S,N}dV]$	$$
$$	${\displaystyle }$	$=$	$+T[+1\cdot dS+0\cdot dV]$	$$
$(01)$	${\displaystyle }$	$=$	$+TdS+0dV$	$$
$$	${\displaystyle }$	$=$	$+d{Q}_{V,N}+0$	$d{Q}_{V,N}(S),d{Q}_{S,N}(V)=0,dN=0$
$(02)$	${\displaystyle }$	$=$	$+d{U}_{V,N}+0$	$d{U}_{V,N}(S),dN=0$
$$	${\left(\frac{\partial Q}{\partial S}\right)}_{V,N}$	$=$	$+T_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)=+T$	${\partial }_{S}Q(S,V,N),T(S,V,N)$
$(03)$	${\displaystyle }$	$=$	${\left(\frac{\partial U}{\partial S}\right)}_{V,N}=+T$	${\partial }_{S}U(S,V,N),T(S,V,N)$
$$	${\left(\frac{\partial Q}{\partial V}\right)}_{S,N}$	$=$	$+0$	${\partial }_{V}Q(S,V,N)$
$(04)$	${\displaystyle }$	$\ne$	${\left(\frac{\partial U}{\partial V}\right)}_{S,N}=-p$	${\partial }_{V}U(S,V,N),p(S,V,N)$
$$	$\frac{\partial }{\partial V}\frac{\partial Q}{\partial S}$	$=$	$-\frac{2}{3}\frac{T_0}{V}{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)=-\frac{2}{3}\frac{T}{V}$	${\partial }_V{\partial }_{S}Q(S,V,N),T(S,V,N)$
$$	${\displaystyle }$	$=$	$\frac{\partial }{\partial V}\frac{\partial U}{\partial S}=-\frac{2}{3}\frac{T}{V}$	${\partial }_V{\partial }_{S}U(S,V,N),T(S,V,N)$
$$	$\frac{\partial }{\partial S}\frac{\partial Q}{\partial V}$	$=$	$+0$	${\partial }_S{\partial }_{V}Q(S,V,N)$
$$	${\displaystyle }$	$\ne$	$\frac{\partial }{\partial S}\frac{\partial U}{\partial V}=-\frac{2}{3}\frac{p}{nR}$	${\partial }_S{\partial }_{V}U(S,V,N),p(S,V,N)$
$(05)$	$\frac{\partial }{\partial V}\frac{\partial Q}{\partial S}$	$\ne$	$\frac{\partial }{\partial S}\frac{\partial Q}{\partial V}$	${\partial }_V{\partial }_{S}Q(S,V,N),{\partial }_S{\partial }_{V}Q(S,V,N)$
$$	$Q_B(S,V,N)$	$=$	$+{\int }_{S,dV=0}TdS$	$dS(S),dV=0,dN=0,T(S,V,N)$
$$	${\displaystyle }$	$=$	$+{\int }_{S,dV=0}{T}_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)dS$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nR{T}_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)+B(V,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nRT(S,V,N)+B(V,N)$	$$
$$	$Q_A(S,V,N)$	$=$	$+{\int }_{V,dS=0}TdS$	$dS(V),dS=0,dN=0,T(S,V,N)$
$$	${\displaystyle }$	$=$	$0+A(S,N)$	$$
$$	$A$	$\ne$	$A(S,N)$	$A=\text{const}$
$$	$B$	$\ne$	$B(V,N)$	$B=\text{const}$
$$	$Q_B(S,V,N)$	$=$	$+\frac{3}{2}nR{T}_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)+B$	$$
$$	$Q_A(S,V,N)$	$=$	$+A$	$Q_A(S,V,N)=\text{const}$
$$	$Q_B(S,V,N)$	$\ne$	$Q_A(S,V,N)$	$$
$$	$Q_{V,N}(S_2,S_1)$	$=$	$Q(S_2,V,N)-Q(S_1,V,N)$	$$
$$	${\displaystyle }$	$=$	$+{\int }_{S_1,V}^{S_2,V}TdS$	$dS(S),dV=0,dN=0,T(S,V,N)$
$(06)$	${\displaystyle }$	$=$	$+\frac{3}{2}nR{T}_0{\left(\frac{V_0}{V}\right)}^{2/3}[\exp \left(\frac{2}{3}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{3}\frac{S_1}{nR}\right)]$	$$
$$	${\displaystyle }$	$=$	$U(S_2,V,N)-U(S_1,V,N)$	$$
$(07)$	${\displaystyle }$	$=$	$U_{V,N}(S_2,S_1)$	$$
$$	$Q_{S,N}(V_2,V_1)$	$=$	$Q(S,V_2,N)-Q(S,V_1,N)$	$$
$$	${\displaystyle }$	$=$	$+{\int }_{V_1,S}^{V_2,S}TdS$	$dS(V),dS=0,dN=0,T(S,V,N)$
$(08)$	${\displaystyle }$	$=$	$0$	$$

B Ausgetauschte Arbeit dW = -pdV

$$	$dW$	$=$	$-pdV$	$dW(S,V),p(S,V,N),dN=0$
$$	${\displaystyle }$	$=$	$-p[+{\left(\frac{\partial V}{\partial S}\right)}_{V,N}dS+{\left(\frac{\partial V}{\partial V}\right)}_{S,N}dV]$	$$
$$	${\displaystyle }$	$=$	$-p[+0\cdot dS+1\cdot dV]$	$$
$(01)$	${\displaystyle }$	$=$	$-0dS-pdV$	$$
$$	${\displaystyle }$	$=$	$+0+d{W}_{S,N}$	$d{W}_{V,N}(S)=0,d{W}_{S,N}(V),dN=0$
$(02)$	${\displaystyle }$	$=$	$+0+d{U}_{S,N}$	$d{U}_{S,N}(V),dN=0$
$$	${\left(\frac{\partial W}{\partial S}\right)}_{V,N}$	$=$	$-0$	${\partial }_{S}W(S,V,N)$
$(03)$	${\displaystyle }$	$\ne$	${\left(\frac{\partial U}{\partial S}\right)}_{V,N}=+T$	${\partial }_{S}U(S,V,N),T(S,V,N)$
$$	${\left(\frac{\partial W}{\partial V}\right)}_{S,N}$	$=$	$-p_0{\left(\frac{V_0}{V}\right)}^{5/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)=-p$	${\partial }_{V}W(S,V,N),p(S,V,N)$
$(04)$	${\displaystyle }$	$=$	${\left(\frac{\partial U}{\partial V}\right)}_{S,N}=-p$	${\partial }_{V}U(S,V,N),p(S,V,N)$
$$	$\frac{\partial }{\partial V}\frac{\partial W}{\partial S}$	$=$	$-0$	${\partial }_V{\partial }_{S}W(S,V,N)$
$$	${\displaystyle }$	$\ne$	$\frac{\partial }{\partial V}\frac{\partial U}{\partial S}=-\frac{2}{3}\frac{T}{V}$	${\partial }_V{\partial }_{S}U(S,V,N),T(S,V,N)$
$$	$\frac{\partial }{\partial S}\frac{\partial W}{\partial V}$	$=$	$-\frac{2}{3}\frac{p_0}{nR}{\left(\frac{V_0}{V}\right)}^{5/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)=-\frac{2}{3}\frac{p}{nR}$	${\partial }_S{\partial }_{V}W(S,V,N),p(S,V,N)$
$$	${\displaystyle }$	$=$	$\frac{\partial }{\partial S}\frac{\partial U}{\partial V}=-\frac{2}{3}\frac{p}{nR}$	${\partial }_S{\partial }_{V}U(S,V,N),p(S,V,N)$
$(05)$	$\frac{\partial }{\partial V}\frac{\partial W}{\partial S}$	$\ne$	$\frac{\partial }{\partial S}\frac{\partial W}{\partial V}$	${\partial }_V{\partial }_{S}W(S,V,N),{\partial }_S{\partial }_{V}W(S,V,N)$
$$	$W_B(S,V,N)$	$=$	$-{\int }_{S,dV=0}pdV$	$dV(S),dV=0,dN=0,p(S,V,N)$
$$	${\displaystyle }$	$=$	$-0+B(V,N)$	$$
$$	$W_A(S,V,N)$	$=$	$-{\int }_{V,dS=0}pdV$	$dV(V),dS=0,dN=0,p(S,V,N)$
$$	${\displaystyle }$	$=$	$-{\int }_{V,dS=0}{p}_0{\left(\frac{V_0}{V}\right)}^{5/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)dV$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}V{p}_0{\left(\frac{V_0}{V}\right)}^{5/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}Vp(S,V,N)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}{p}_0{V}_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nR{T}_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nRT(S,V,N)+A(S,N)$	$$
$$	$A$	$\ne$	$A(S,N)$	$A=\text{const}$
$$	$B$	$\ne$	$B(V,N)$	$B=\text{const}$
$$	$W_B(S,V,N)$	$=$	$+B$	$W_B(S,V,N)=\text{const}$
$$	$W_A(S,V,N)$	$=$	$+\frac{3}{2}nR{T}_0{\left(\frac{V_0}{V}\right)}^{2/3}\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)+A$	$$
$$	$W_B(S,V,N)$	$\ne$	$W_A(S,V,N)$	$$
$$	$W_{V,N}(S_2,S_1)$	$=$	$W(S_2,V,N)-W(S_1,V,N)$	$$
$$	${\displaystyle }$	$=$	$-{\int }_{S_1,V}^{S_2,V}pdV$	$dV(S),dV=0,dN=0,p(S,V,N)$
$(06)$	${\displaystyle }$	$=$	$0$	$$
$$	$W_{S,N}(V_2,V_1)$	$=$	$W(S,V_2,N)-W(S,V_1,N)$	$$
$$	${\displaystyle }$	$=$	$-{\int }_{V_1,S}^{V_2,S}pdV$	$dV(V),dS=0,dN=0,p(S,V,N)$
$(07)$	${\displaystyle }$	$=$	$+\frac{3}{2}nR{T}_0[{\left(\frac{V_0}{V_2}\right)}^{2/3}-{\left(\frac{V_0}{V_1}\right)}^{2/3}]\exp \left(\frac{2}{3}\frac{S-S_0}{nR}\right)$	$$
$$	${\displaystyle }$	$=$	$U(S,V_2,N)-U(S,V_1,N)$	$$
$(08)$	${\displaystyle }$	$=$	$U_{S,N}(V_2,V_1)$	$$