PCRT.I.E.05

PCRT.I.E

05 Einmal Rechtsherum im Kreis in der (S,p;N=const)-Ebene

A Wärme hinein, Qa > 0, Arbeit hinaus, Wa < 0: (N,p1,S1) -> (N,p1,S2)

$\qquad \qquad$	$Q_{a}$	$=$	$Q_{p1,N}(S_{2},S_{1})>0$	$\qquad (a:p,\,S;p_{1}S_{1}\to p_{1}S_{2}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {5}{2}}\,n\,R\,T_{0}\,\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}\,\left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (p,S;p_{1}S_{1}\to p_{1}S_{2}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {5}{2}}\,n\,R\,\left[T(N,p_{1},S_{2})-T(N,p_{1},S_{1})\right]$	$\qquad (T;T_{11}\to T_{12}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$H(N,p_{1},S_{2})-H(N,p_{1},S_{1})$	$\qquad (H;H_{11}\to H_{12}),\,dN=0\qquad$
$\qquad \qquad$	$W_{a}$	$=$	$W_{p1,N}(S_{2},S_{1})<0$	$\qquad (a:p,\,S;p_{1}S_{1}\to p_{1}S_{2}),\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$-\,n\,R\,T_{0}\,\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}\,\left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (p,S;p_{1}S_{1}\to p_{1}S_{2}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,n\,R\,\left[T(N,p_{1},S_{2})-T(N,p_{1},S_{1})\right]$	$\qquad (T;T_{11}\to T_{12}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p_{1}\,[V(N,p_{1},S_{2})-V(N,p_{1},S_{1})]$	$\qquad (V;V_{11}\to V_{12}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,[U(N,p_{1},S_{2})-\,H(N,p_{1},S_{2})]-[U(N,p_{1},S_{1})-\,H(N,p_{1},S_{1})]$	$\qquad (U,H;[U-H]_{11}\to [U-H]_{12}),\,dN=0\qquad$

B Arbeit hinaus, Wb < 0: (N,p1,S2) -> (N,p2,S2)

$\qquad \qquad$	$W_{b}$	$=$	$W_{S2,N}(p_{2},p_{1})<0$	$\qquad (b:p,S;p_{1}S_{2}\to p_{2}S_{2}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}-\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}\right]\,\exp \left({\frac {2}{5}}{\frac {S_{2}-S_{0}}{n\,R}}\right)$	$\qquad (p,S;p_{1}S_{2}\rightarrow p_{2}S_{2}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,\left[\,T(N,p_{2},S_{2})-\,T(N,p_{1},S_{2})\right]$	$\qquad (T\,;\,T_{12}\to T_{22}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,p_{2},S_{2})-U(N,p_{1},S_{2})$	$\qquad (U;U_{12}\to U_{22}),\,dN=0\qquad$

C Wärme hinaus, Qc < 0, Arbeit hinein, Wc > 0: (N,p2,S2) -> (N,p2,S1)

$\qquad \qquad$	$Q_{c}$	$=$	$Q_{p2,N}(S_{1},S_{2})<0$	$\qquad (c:p,S;p_{2}S_{2}\to p_{2}S_{1}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {5}{2}}\,n\,R\,T_{0}\,\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\,\left[\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)\right]$	$\qquad (p,S;p_{2}S_{2}\to p_{2}S_{1}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {5}{2}}\,n\,R\,\left[T(N,p_{2},S_{1})-T(N,p_{2},S_{2})\right]$	$\qquad (T;T_{22}\rightarrow T_{21}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$H(N,p_{2},S_{1})-H(N,p_{2},S_{2})$	$\qquad (H;H_{22}\to H_{21}),\,dN=0\qquad$
$\qquad \qquad$	$W_{c}$	$=$	$W_{p2,N}(S_{1},S_{2})>0$	$\qquad (c:p,\,S;p_{2}S_{2}\to p_{2}S_{1}),\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$-\,n\,R\,T_{0}\,\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\,\left[\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)\right]$	$\qquad (p,S;p_{2}S_{2}\to p_{2}S_{1}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,n\,R\,\left[T(N,p_{2},S_{1})-T(N,p_{2},S_{2})\right]$	$\qquad (T;T_{22}\to T_{21}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,p_{2}\,[V(N,p_{2},S_{1})-V(N,p_{2},S_{2})]$	$\qquad (V;V_{22}\to V_{21}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$\,[U(N,p_{2},S_{1})-\,H(N,p_{2},S_{1})]-[U(N,p_{2},S_{2})-\,H(N,p_{2},S_{2})]$	$\qquad (U,H;[U-H]_{22}\to [U-H]_{21}),\,dN=0\qquad$

D Arbeit hinein, Wd > 0: (p2,S1) -> (p1,S1)

$\qquad \qquad$	$W_{d}$	$=$	$W_{S1,N}(p_{1},p_{2})>0$	$\qquad (d:p,S;\,p_{2}S_{1}\to p_{1}S_{1}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}-\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\right]\,\exp \left({\frac {2}{5}}{\frac {S_{1}-S_{0}}{n\,R}}\right)$	$\qquad (p,S;\,p_{2}S_{1}\to p_{1}S_{1}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\left[T(N,p_{1},S_{1})-T(N,p_{2},S_{1})\right]$	$\qquad (T;\,T_{21}\to T_{11}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,p_{1},S_{1})-U(N,p_{2},S_{1})$	$\qquad (U;\,U_{21}\rightarrow U_{11}),\,dN=0\qquad$

E Wärme gesamt, Qa + Qc > 0: (N,p1,S1) -> (N,p1,S2), (N,p2,S2) -> (N,p2,S1)

$\qquad \qquad$	$Q_{a}+Q_{c}$	$=$	$Q_{p2}(S_{1},S_{2})+Q_{p1}(S_{2},S_{1})>0$	$\qquad (a+c:S,p;11\to 12,22\to 21),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {5}{2}}\,N\,k\,T_{0}\,\left[\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}-\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\right]$	$\qquad \qquad$
$(01)\qquad$	${\displaystyle }$	${\displaystyle }$	$\times \left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{N\,k}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{1}}{N\,k}}\right)\right]$	$\qquad (S,p;11\to 12,22\to 21),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {5}{2}}\,N\,k\,[T(N,p_{2},S_{1})-T(N,p_{2},S_{2})]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,{\frac {5}{2}}\,N\,k\,[T(N,p_{1},S_{2})-T(N,p_{1},S_{1})]$	$\qquad (T;T_{11}\to T_{12}\to T_{22}\to T_{21}\to T_{11})\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$H(N,p_{2},S_{1})-H(N,p_{2},S_{2})$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,H(N,p_{1},S_{2})-H(N,p_{1},S_{1})$	$\qquad (H;H_{11}\to H_{12}\to H_{22}\to H_{21}\to H_{11})\qquad$
$\qquad \qquad$	$\oint _{p,S}T\,dS$	$=$	$\int _{p_{1},S_{1}}^{p_{1},S_{2}}T\,dS+\int _{p_{1},S_{2}}^{p_{2},S_{2}}T\,dS+\int _{p_{2},S_{2}}^{p_{2},S_{1}}T\,dS+\int _{p_{2},S_{1}}^{p_{1},S_{1}}T\,dS$	$\qquad (S,p;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$
$\qquad \qquad$	$0$	$=$	$\int _{p_{1},S_{2}}^{p_{2},S_{2}}T\,dS+\int _{p_{2},S_{1}}^{p_{1},S_{1}}T\,dS$	$\qquad (S,p;12\to 22,21\to 11),\,dN=0\qquad$
$(02)\qquad$	$\oint _{p,S}T\,dS$	$=$	$\int _{S,p_{1},p_{2}}T\,dS$	$\qquad (p,S;11\to 12,\,22\to 21),\,dN=0\qquad$

F Arbeit gesamt, Wa + Wb + Wc + Wd < 0: (p1,S1) -> (p1,S2) -> (p2,S2) -> (p2,S1) -> (p1,S1)

$\qquad \qquad$	$W_{a}+\,W_{c}$	$=$	$W_{p2}(S1,S2)+W_{p1}(S2,S1)<0$	$\qquad (p,S;11\to 12,22\to 21),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	$-\,n\,R\,T_{0}\,\left[\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}-\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\times \left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (p,S;11\to 12,\,22\to 21),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$-\,n\,R\,\left[\,T(N,p_{2},S_{1})-T(N,p_{2},S_{2})\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$-\,n\,R\,\left[\,T(N,p_{1},S_{2})-T(N,p_{1},S_{1})\right]$	$\qquad (T\,;T_{11}\to T_{12},\,T_{22}\to T_{21}),\,dN=0\qquad$
$(02)\qquad$	${\displaystyle }$	$=$	$+\,[U(N,p_{2},S_{1})-\,H(N,p_{2},S_{1})]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$-\,[U(N,p_{2},S_{2})-\,H(N,p_{2},S_{2})]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,[U(N,p_{1},S_{2})-\,H(N,p_{1},S_{2})]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$-\,[U(N,p_{1},S_{1})-\,H(N,p_{1},S_{1})]$	$\qquad (U,H\,;[U-H]_{11}\to [U-H]_{12},\,[U-H]_{22}\to [U-H]_{21}])\qquad$
$\qquad \qquad$	$W_{b}+\,W_{d}$	$=$	$W_{S1}(p1,p2)+W_{S2}(p2,p1)<0$	$\qquad (p,S;12\to 22,21\to 11),\,dN=0\qquad$
$(03)\qquad$	${\displaystyle }$	$=$	$-\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {p_{1}}{p_{0}}}\right)^{2/5}-\left({\frac {p_{2}}{p_{0}}}\right)^{2/5}\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\times \left[\exp \left({\frac {2}{5}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{5}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (p,S;12\to 22,\,21\to 11),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$+\,{\frac {3}{2}}\,n\,R\,\left[\,T(N,p_{1},S_{1})-T(N,p_{2},S_{1})\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,{\frac {3}{2}}\,n\,R\,\left[\,T(N,p_{2},S_{2})-T(N,p_{1},S_{2})\right]$	$\qquad (T\,;T_{12}\to T_{22},\,T_{21}\to T_{11}),\,dN=0\qquad$
$(04)\qquad$	${\displaystyle }$	$=$	$+\,[U(N,p_{1},S_{1})-\,U(N,p_{2},S_{1})]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,[U(N,p_{2},S_{2})-\,U(N,p_{1},S_{2})]$	$\qquad (U\,;U_{12}\to U_{22},\,U_{21}\to U_{11}),\,dN=0\qquad$
$\qquad \qquad$	$-\oint _{p,S}p\,dV$	$=$	$-\int _{p_{1},S_{1}}^{p_{1},S_{2}}p\,dV-\int _{p_{1},S_{2}}^{p_{2},S_{2}}p\,dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$-\int _{p_{2},S_{2}}^{p_{2},S_{1}}p\,dV-\int _{p_{2},S_{1}}^{p_{1},S_{1}}p\,dV$	$\qquad (p,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$

G Verhältnis und Summe der Integrale von dW = -pdV und dQ = +TdS

$(01)\qquad$	$-\int _{p,S_{2},S_{1}}p\,dV$	${\Big /}$	$\int _{S,p_{1},p_{2}}T\,dS$	$=$	${\frac {W_{a}+W_{b}+W_{c}+W_{d}}{Q_{a}+Q_{c}}}$	$=$	$-100\%$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\qquad (p,S;11\to 12\to 22\to 21\to 11)\qquad$
$(02)\qquad$	$-\oint _{p,S}p\,dV$	${\Big /}$	$\oint _{p,S}T\,dS$	$=$	${\frac {W_{a}+W_{b}+W_{c}+W_{d}}{Q_{a}+Q_{c}}}$	$=$	$-100\%$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\qquad (p,S;11\to 12\to 22\to 21\to 11)\qquad$
$(03)\qquad$	$-\int _{p,S_{2},S_{1}}p\,dV$	$+$	$\int _{S,p_{1},p_{2}}T\,dS$	$=$	$Q_{a}+W_{a}+W_{b}+Q_{c}+W_{c}+W_{d}$	$=$	$0$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\qquad (p,S;11\to 12\to 22\to 21\to 11)\qquad$
$(04)\quad$	$-\oint _{p,S}p\,dV$	$+$	$\oint _{p,S}T\,dS$	$=$	$Q_{a}+W_{a}+W_{b}+Q_{c}+W_{c}+W_{d}$	$=$	$0$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\qquad (p,S;11\to 12\to 22\to 21\to 11)\qquad$

Übung a

Wie Übung 21.a für die (S,p;N=const)-Ebene.

Übung b

Wir stellen uns vor, wir haben die $U(S,p,N={\mbox{const}})$- und die $H(S,p,N={\mbox{const}})$-Fläche im $(S,p,N={\mbox{const}})$-Koordinatensystem gegeben. Wie berechnen wir $W_{a},\,W_{b},\,W_{c},\,W_{d},\,Q_{a},\,Q_{c}$ bzw. welchen Flächen entsprechen diese Arbeits- und Wärmeströme? Welche Anforderungen müssen wir an die $U$- und $H$-Flächen stellen, damit die Flächen der Arbeits- und Wärmeströme die gleichen Vorzeichen haben, wie es die Vorstellung aus der Betrachtung der Formeln vorgibt?