PCRT.I.E.02

PCRT.I.E

02 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

$$	$dQ$	$=$	$+TdS$	$dQ(S,p),T(S,p,N),dN=0$
$$	${\displaystyle }$	$=$	$+T[+{\left(\frac{\partial S}{\partial S}\right)}_{p,N}dS+{\left(\frac{\partial S}{\partial p}\right)}_{S,N}dp]$	$$
$$	${\displaystyle }$	$=$	$+T[+1\cdot dS+0\cdot dp]$	$$
$(01)$	${\displaystyle }$	$=$	$+TdS+0dp$	$$
$$	${\displaystyle }$	$=$	$+d{Q}_{p,N}+0$	$d{Q}_{p,N}(S),d{Q}_{S,N}(p)=0,dN=0$
$(02)$	${\displaystyle }$	$=$	$+d{H}_{p,N}+0$	$d{H}_{p,N}(S),dN=0$
$$	${\left(\frac{\partial Q}{\partial S}\right)}_{p,N}$	$=$	$+T_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)=+T$	${\partial }_{S}Q(S,p,N),T(S,p,N)$
$(03)$	${\displaystyle }$	$=$	${\left(\frac{\partial H}{\partial S}\right)}_{p,N}=+T$	${\partial }_{S}H(S,p,N),T(S,p,N)$
$$	${\left(\frac{\partial Q}{\partial p}\right)}_{S,N}$	$=$	$+0$	${\partial }_{p}Q(S,p,N)$
$(04)$	${\displaystyle }$	$\ne$	${\left(\frac{\partial H}{\partial p}\right)}_{S,N}=+V$	${\partial }_{p}H(S,p,N),V(S,p,N)$
$$	$\frac{\partial }{\partial p}\frac{\partial Q}{\partial S}$	$=$	$+\frac{2}{5}\frac{T_0}{p}{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)=+\frac{2}{5}\frac{T}{p}$	${\partial }_p{\partial }_{S}Q(S,p,N),T(S,p,N)$
$$	${\displaystyle }$	$=$	$\frac{\partial }{\partial p}\frac{\partial H}{\partial S}=+\frac{2}{5}\frac{T}{p}$	${\partial }_p{\partial }_{S}H(S,p,N),T(S,p,N)$
$$	$\frac{\partial }{\partial S}\frac{\partial Q}{\partial p}$	$=$	$+0$	${\partial }_S{\partial }_{p}Q(S,p,N)$
$$	${\displaystyle }$	$\ne$	$\frac{\partial }{\partial S}\frac{\partial H}{\partial p}=+\frac{2}{5}\frac{V}{nR}$	${\partial }_S{\partial }_{p}H(S,p,N),V(S,p,N)$
$(05)$	$\frac{\partial }{\partial p}\frac{\partial Q}{\partial S}$	$\ne$	$\frac{\partial }{\partial S}\frac{\partial Q}{\partial p}$	${\partial }_p{\partial }_{S}Q(S,p,N),{\partial }_S{\partial }_{p}Q(S,p,N)$
$$	$Q_B(S,p,N)$	$=$	$+\underset{S,dp=0}{\int }TdS$	$dS(S),dp=0,dN=0,T(S,p,N)$
$$	${\displaystyle }$	$=$	$+\underset{S,dp=0}{\int }{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)dS$	$$
$$	${\displaystyle }$	$=$	$+\frac{5}{2}nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B(p,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{5}{2}nRT(S,p,N)+B(p,N)$	$$
$$	$Q_A(S,p,N)$	$=$	$+\underset{p,dS=0}{\int }TdS$	$dS(p),dS=0,dN=0,T(S,p,N)$
$$	${\displaystyle }$	$=$	$0+A(S,N)$	$$
$$	$A$	$\ne$	$A(S,N)$	$A=\text{const}$
$$	$B$	$\ne$	$B(p,N)$	$B=\text{const}$
$$	$Q_B(S,p,N)$	$=$	$+\frac{5}{2}nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B$	$$
$$	$Q_A(S,p,N)$	$=$	$+A$	$Q_A(S,p,N)=\text{const}$
$$	$Q_B(S,p,N)$	$\ne$	$Q_A(S,p,N)$	$$
$$	$Q_{p,N}(S_2,S_1)$	$=$	$Q(S_2,p,N)-Q(S_1,p,N)$	$$
$$	${\displaystyle }$	$=$	$+\underset{S_1,p}{\overset{S_2,p}{\int }}TdS$	$dS(S),dp=0,dN=0,T(S,p,N)$
$(06)$	${\displaystyle }$	$=$	$+\frac{5}{2}nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}[\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)]$	$$
$$	${\displaystyle }$	$=$	$H(S_2,p,N)-H(S_1,p,N)$	$$
$(07)$	${\displaystyle }$	$=$	$H_{p,N}(S_2,S_1)$	$$
$$	$Q_{S,N}(p_1,p_2)$	$=$	$Q(S,p_1,N)-Q(S,p_2,N)$	$$
$$	${\displaystyle }$	$=$	$+\underset{p_2,S}{\overset{p_1,S}{\int }}TdS$	$dS(p),dS=0,dN=0,T(S,p,N)$
$(08)$	${\displaystyle }$	$=$	$0$	$$

B Ausgetauschte Arbeit dW = -pdV

$$	$dW$	$=$	$-pdV$	$dW(S,p),dV(S,p),dN=0$
$$	${\displaystyle }$	$=$	$-p[+{\left(\frac{\partial V}{\partial S}\right)}_{p,N}dS+{\left(\frac{\partial V}{\partial p}\right)}_{S,N}dp]$	$$
$$	$-S{\left(\frac{\partial V}{\partial S}\right)}_{p,N}$	$=$	$-\frac{2}{5}\frac{S}{nR}V$	${\partial }_{S}V(S,p,N)\leftarrow (12.\text{C}.3)$
$$	$-p{\left(\frac{\partial V}{\partial S}\right)}_{p,N}$	$=$	$-\frac{2}{5}T$	${\partial }_{S}V(S,p,N),T(S,p,N)$
$$	$-p{\left(\frac{\partial V}{\partial p}\right)}_{S,N}$	$=$	$+\frac{3}{5}V$	${\partial }_{p}V(S,p,N)\leftarrow (12.\text{B}.3)$
$(01)$	$dW$	$=$	$-\frac{2}{5}TdS+\frac{3}{5}Vdp$	$dW(S,p),dN=0;(S,p,N):T,V$
$$	${\displaystyle }$	$=$	$+d{W}_{p,N}+d{W}_{S,N}$	$d{W}_{p,N}(S),d{W}_{S,N}(p),dN=0$
$(02)$	${\displaystyle }$	$=$	$+[d{U}_{p,N}-d{H}_{p,N}]+d{U}_{S,N}$	$d{U}_{p,N}(S),d{H}_{p,N}(S),d{U}_{S,N}(p),dN=0$
$$	${\left(\frac{\partial W}{\partial S}\right)}_{p,N}$	$=$	$-\frac{2}{5}{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)=-\frac{2}{5}T$	${\partial }_{S}W(S,p,N),T(S,p,N)$
$(03)$	${\displaystyle }$	$\ne$	${\left(\frac{\partial U}{\partial S}\right)}_{p,N}=+\frac{3}{5}T$	${\partial }_{S}U(S,p,N),T(S,p,N)$
$$	${\left(\frac{\partial W}{\partial p}\right)}_{S,N}$	$=$	$=+\frac{3}{5}{V}_0{\left(\frac{p_0}{p}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)=+\frac{3}{5}V$	${\partial }_{p}W(S,p,N),V(S,p,N)$
$(04)$	${\displaystyle }$	$=$	${\left(\frac{\partial U}{\partial p}\right)}_{S,N}=+\frac{3}{5}V$	${\partial }_{p}U(S,p,N),V(S,p,N)$
$$	$\frac{\partial }{\partial p}\frac{\partial W}{\partial S}$	$=$	$-0$	${\partial }_p{\partial }_{S}W(S,p,N)$
$$	${\displaystyle }$	$\ne$	$\frac{\partial }{\partial p}\frac{\partial U}{\partial S}=+\frac{6}{25}\frac{T}{p}$	${\partial }_p{\partial }_{S}U(S,p,N),T(S,p,N)$
$$	$\frac{\partial }{\partial S}\frac{\partial W}{\partial p}$	$=$	$+\frac{6}{25}\frac{V_0}{nR}{\left(\frac{p_0}{p}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)=+\frac{6}{25}\frac{V}{nR}$	${\partial }_S{\partial }_{p}W(S,p,N),V(S,p,N)$
$$	${\displaystyle }$	$=$	$\frac{\partial }{\partial S}\frac{\partial U}{\partial p}=+\frac{6}{25}\frac{V}{nR}$	${\partial }_S{\partial }_{p}U(S,p,N),V(S,p,N)$
$(05)$	$\frac{\partial }{\partial p}\frac{\partial W}{\partial S}$	$\ne$	$\frac{\partial }{\partial S}\frac{\partial W}{\partial p}$	${\partial }_p{\partial }_{S}W(S,p,N),{\partial }_S{\partial }_{p}W(S,p,N)$
$$	$W_B(S,p,N)$	$=$	$-\underset{S,dp=0}{\int }pdV$	$dV(S),dp=0,dN=0$
$$	${\displaystyle }$	$=$	$-\frac{2}{5}\underset{S,dp=0}{\int }TdS$	$T(S,p,N)$
$$	${\displaystyle }$	$=$	$-\frac{2}{5}\underset{S,dp=0}{\int }{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)dS$	$$
$$	${\displaystyle }$	$=$	$-\frac{2}{5}\cdot \frac{5}{2}nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B(p,N)$	$$
$$	${\displaystyle }$	$=$	$-nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B(p,N)$	$$
$$	${\displaystyle }$	$=$	$-nRT(S,p,N)+B(p,N)$	$$
$$	${\displaystyle }$	$=$	$-p_0{V}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B(p,N)$	$$
$$	${\displaystyle }$	$=$	$-p{V}_0{\left(\frac{p}{p_0}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B(p,N)$	$$
$$	${\displaystyle }$	$=$	$-pV(S,p,N)+B(p,N)$	$$
$$	$W_A(S,p,N)$	$=$	$-\underset{p,dS=0}{\int }pdV$	$dV(p),dS=0,dN=0$
$$	${\displaystyle }$	$=$	$+\frac{3}{5}\underset{p,dS=0}{\int }Vdp$	$V(S,p,N)$
$$	${\displaystyle }$	$=$	$+\frac{3}{5}\underset{p,dS=0}{\int }{V}_0{\left(\frac{p_0}{p}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)dp$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{5}\cdot \frac{5}{2}p{V}_0{\left(\frac{p_0}{p}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}p{V}_0{\left(\frac{p_0}{p}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}pV(S,p,N)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}{p}_0{V}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+A(S,N)$	$$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nRT(S,p,N)+A(S,N)$	$$
$$	$A$	$\ne$	$A(S,N)$	$A=\text{const}$
$$	$B$	$\ne$	$B(p,N)$	$B=\text{const}$
$$	$W_B(S,p,N)$	$=$	$-nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+B$	$$
$$	$W_A(S,p,N)$	$=$	$+\frac{3}{2}p{V}_0{\left(\frac{p_0}{p}\right)}^{3/5}\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)+A$	$$
$$	$W_B(S,p,N)$	$\ne$	$W_A(S,p,N)$	$$
$$	$W_{p,N}(S_2,S_1)$	$=$	$W(S_2,p,N)-W(S_1,p,N)$	$$
$$	${\displaystyle }$	$=$	$-\underset{S_1,p}{\overset{S_2,p}{\int }}pdV$	$dV(S),dp=0,dN=0$
$(06)$	${\displaystyle }$	$=$	$-nR{T}_0{\left(\frac{p}{p_0}\right)}^{2/5}[\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)]$	$$
$$	${\displaystyle }$	$=$	$-nR[T(S_2,p,N)-T(S_1,p,N)]$	$$
$$	${\displaystyle }$	$=$	$-pV(S_2,p,N)-pV(S_1,p,N)$	$$
$$	${\displaystyle }$	$=$	$[U(S_2,p,N)-H(S_2,p,N)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$+[U(S_1,p,N)-H(S_1,p,N)]$	$$
$(07)$	${\displaystyle }$	$=$	$U_{p,N}(S_2,S_1)-H_{p,N}(S_2,S_1)$	$$
$$	$W_{S,N}(p_1,p_2)$	$=$	$W(S,p_1,N)-W(S,p_2,N)$	$$
$$	${\displaystyle }$	$=$	$-\underset{p_2,S}{\overset{p_1,S}{\int }}pdV$	$dV(p),dS=0,dN=0$
$(08)$	${\displaystyle }$	$=$	$+\frac{3}{2}nR{T}_0[{\left(\frac{p_1}{p_0}\right)}^{2/5}-{\left(\frac{p_2}{p_0}\right)}^{2/5}]\exp \left(\frac{2}{5}\frac{S-S_0}{nR}\right)$	$$
$$	${\displaystyle }$	$=$	$U(S,p_1,N)-U(S,p_2,N)$	$$
$(09)$	${\displaystyle }$	$=$	$U_{S,N}(p_1,p_2)$	$$