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)

$$	$Q_a$	$=$	$Q_{p1,N}(S_2,S_1)>0$	$(a:p,S;p_1{S}_1\to p_1{S}_2),dN=0$
$(01)$	${\displaystyle }$	$=$	$\frac{5}{2}nR{T}_0{\left(\frac{p_1}{p_0}\right)}^{2/5}[\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)]$	$(p,S;p_1{S}_1\to p_1{S}_2),dN=0$
$$	${\displaystyle }$	$=$	$\frac{5}{2}nR[T(N,p_1,S_2)-T(N,p_1,S_1)]$	$(T;T_{11}\to T_{12}),dN=0$
$$	${\displaystyle }$	$=$	$H(N,p_1,S_2)-H(N,p_1,S_1)$	$(H;H_{11}\to H_{12}),dN=0$
$$	$W_a$	$=$	$W_{p1,N}(S_2,S_1)<0$	$(a:p,S;p_1{S}_1\to p_1{S}_2),dN=0$
$(02)$	${\displaystyle }$	$=$	$-nR{T}_0{\left(\frac{p_1}{p_0}\right)}^{2/5}[\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)]$	$(p,S;p_1{S}_1\to p_1{S}_2),dN=0$
$$	${\displaystyle }$	$=$	$-nR[T(N,p_1,S_2)-T(N,p_1,S_1)]$	$(T;T_{11}\to T_{12}),dN=0$
$$	${\displaystyle }$	$=$	$-p_1[V(N,p_1,S_2)-V(N,p_1,S_1)]$	$(V;V_{11}\to V_{12}),dN=0$
$$	${\displaystyle }$	$=$	$[U(N,p_1,S_2)-H(N,p_1,S_2)]-[U(N,p_1,S_1)-H(N,p_1,S_1)]$	$(U,H;[U-H{]}_{11}\to [U-H{]}_{12}),dN=0$

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

$$	$W_b$	$=$	$W_{S2,N}(p_2,p_1)<0$	$(b:p,S;p_1{S}_2\to p_2{S}_2),dN=0$
$(01)$	${\displaystyle }$	$=$	$\frac{3}{2}nR{T}_0[{\left(\frac{p_2}{p_0}\right)}^{2/5}-{\left(\frac{p_1}{p_0}\right)}^{2/5}]\exp \left(\frac{2}{5}\frac{S_2-S_0}{nR}\right)$	$(p,S;p_1{S}_2\to p_2{S}_2),dN=0$
$$	${\displaystyle }$	$=$	$\frac{3}{2}nR[T(N,p_2,S_2)-T(N,p_1,S_2)]$	$(T;T_{12}\to T_{22}),dN=0$
$$	${\displaystyle }$	$=$	$U(N,p_2,S_2)-U(N,p_1,S_2)$	$(U;U_{12}\to U_{22}),dN=0$

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

$$	$Q_c$	$=$	$Q_{p2,N}(S_1,S_2)<0$	$(c:p,S;p_2{S}_2\to p_2{S}_1),dN=0$
$(01)$	${\displaystyle }$	$=$	$\frac{5}{2}nR{T}_0{\left(\frac{p_2}{p_0}\right)}^{2/5}[\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)]$	$(p,S;p_2{S}_2\to p_2{S}_1),dN=0$
$$	${\displaystyle }$	$=$	$\frac{5}{2}nR[T(N,p_2,S_1)-T(N,p_2,S_2)]$	$(T;T_{22}\to T_{21}),dN=0$
$$	${\displaystyle }$	$=$	$H(N,p_2,S_1)-H(N,p_2,S_2)$	$(H;H_{22}\to H_{21}),dN=0$
$$	$W_c$	$=$	$W_{p2,N}(S_1,S_2)>0$	$(c:p,S;p_2{S}_2\to p_2{S}_1),dN=0$
$(02)$	${\displaystyle }$	$=$	$-nR{T}_0{\left(\frac{p_2}{p_0}\right)}^{2/5}[\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)]$	$(p,S;p_2{S}_2\to p_2{S}_1),dN=0$
$$	${\displaystyle }$	$=$	$-nR[T(N,p_2,S_1)-T(N,p_2,S_2)]$	$(T;T_{22}\to T_{21}),dN=0$
$$	${\displaystyle }$	$=$	$-p_2[V(N,p_2,S_1)-V(N,p_2,S_2)]$	$(V;V_{22}\to V_{21}),dN=0$
$$	${\displaystyle }$	$=$	$[U(N,p_2,S_1)-H(N,p_2,S_1)]-[U(N,p_2,S_2)-H(N,p_2,S_2)]$	$(U,H;[U-H{]}_{22}\to [U-H{]}_{21}),dN=0$

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

$$	$W_d$	$=$	$W_{S1,N}(p_1,p_2)>0$	$(d:p,S;p_2{S}_1\to p_1{S}_1),dN=0$
$(01)$	${\displaystyle }$	$=$	$\frac{3}{2}nR{T}_0[{\left(\frac{p_1}{p_0}\right)}^{2/5}-{\left(\frac{p_2}{p_0}\right)}^{2/5}]\exp \left(\frac{2}{5}\frac{S_1-S_0}{nR}\right)$	$(p,S;p_2{S}_1\to p_1{S}_1),dN=0$
$$	${\displaystyle }$	$=$	$\frac{3}{2}nR[T(N,p_1,S_1)-T(N,p_2,S_1)]$	$(T;T_{21}\to T_{11}),dN=0$
$$	${\displaystyle }$	$=$	$U(N,p_1,S_1)-U(N,p_2,S_1)$	$(U;U_{21}\to U_{11}),dN=0$

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

$$	$Q_a+Q_c$	$=$	$Q_{p2}(S_1,S_2)+Q_{p1}(S_2,S_1)>0$	$(a+c:S,p;11\to 12,22\to 21),dN=0$
$$	${\displaystyle }$	$=$	$\frac{5}{2}Nk{T}_0[{\left(\frac{p_1}{p_0}\right)}^{2/5}-{\left(\frac{p_2}{p_0}\right)}^{2/5}]$	$$
$(01)$	${\displaystyle }$	${\displaystyle }$	$\times [\exp \left(\frac{2}{5}\frac{S_2}{Nk}\right)-\exp \left(\frac{2}{5}\frac{S_1}{Nk}\right)]$	$(S,p;11\to 12,22\to 21),dN=0$
$$	${\displaystyle }$	$=$	$\frac{5}{2}Nk[T(N,p_2,S_1)-T(N,p_2,S_2)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$+\frac{5}{2}Nk[T(N,p_1,S_2)-T(N,p_1,S_1)]$	$(T;T_{11}\to T_{12}\to T_{22}\to T_{21}\to T_{11})$
$$	${\displaystyle }$	$=$	$H(N,p_2,S_1)-H(N,p_2,S_2)$	$$
$$	${\displaystyle }$	${\displaystyle }$	$+H(N,p_1,S_2)-H(N,p_1,S_1)$	$(H;H_{11}\to H_{12}\to H_{22}\to H_{21}\to H_{11})$
$$	${\oint }_{p,S}TdS$	$=$	$\underset{p_1,S_1}{\overset{p_1,S_2}{\int }}TdS+\underset{p_1,S_2}{\overset{p_2,S_2}{\int }}TdS+\underset{p_2,S_2}{\overset{p_2,S_1}{\int }}TdS+\underset{p_2,S_1}{\overset{p_1,S_1}{\int }}TdS$	$(S,p;11\to 12\to 22\to 21\to 11),dN=0$
$$	$0$	$=$	$\underset{p_1,S_2}{\overset{p_2,S_2}{\int }}TdS+\underset{p_2,S_1}{\overset{p_1,S_1}{\int }}TdS$	$(S,p;12\to 22,21\to 11),dN=0$
$(02)$	${\oint }_{p,S}TdS$	$=$	$\underset{S,p_1,p_2}{\int }TdS$	$(p,S;11\to 12,22\to 21),dN=0$

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

$$	$W_a+W_c$	$=$	$W_{p2}(S1,S2)+W_{p1}(S2,S1)<0$	$(p,S;11\to 12,22\to 21),dN=0$
$(01)$	${\displaystyle }$	$=$	$-nR{T}_0[{\left(\frac{p_1}{p_0}\right)}^{2/5}-{\left(\frac{p_2}{p_0}\right)}^{2/5}]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$\times [\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)]$	$(p,S;11\to 12,22\to 21),dN=0$
$$	${\displaystyle }$	$=$	$-nR[T(N,p_2,S_1)-T(N,p_2,S_2)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$-nR[T(N,p_1,S_2)-T(N,p_1,S_1)]$	$(T;T_{11}\to T_{12},T_{22}\to T_{21}),dN=0$
$(02)$	${\displaystyle }$	$=$	$+[U(N,p_2,S_1)-H(N,p_2,S_1)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$-[U(N,p_2,S_2)-H(N,p_2,S_2)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$+[U(N,p_1,S_2)-H(N,p_1,S_2)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$-[U(N,p_1,S_1)-H(N,p_1,S_1)]$	$(U,H;[U-H{]}_{11}\to [U-H{]}_{12},[U-H{]}_{22}\to [U-H{]}_{21}])$
$$	$W_b+W_d$	$=$	$W_{S1}(p1,p2)+W_{S2}(p2,p1)<0$	$(p,S;12\to 22,21\to 11),dN=0$
$(03)$	${\displaystyle }$	$=$	$-\frac{3}{2}nR{T}_0[{\left(\frac{p_1}{p_0}\right)}^{2/5}-{\left(\frac{p_2}{p_0}\right)}^{2/5}]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$\times [\exp \left(\frac{2}{5}\frac{S_2}{nR}\right)-\exp \left(\frac{2}{5}\frac{S_1}{nR}\right)]$	$(p,S;12\to 22,21\to 11),dN=0$
$$	${\displaystyle }$	$=$	$+\frac{3}{2}nR[T(N,p_1,S_1)-T(N,p_2,S_1)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$+\frac{3}{2}nR[T(N,p_2,S_2)-T(N,p_1,S_2)]$	$(T;T_{12}\to T_{22},T_{21}\to T_{11}),dN=0$
$(04)$	${\displaystyle }$	$=$	$+[U(N,p_1,S_1)-U(N,p_2,S_1)]$	$$
$$	${\displaystyle }$	${\displaystyle }$	$+[U(N,p_2,S_2)-U(N,p_1,S_2)]$	$(U;U_{12}\to U_{22},U_{21}\to U_{11}),dN=0$
$$	$-{\oint }_{p,S}pdV$	$=$	$-\underset{p_1,S_1}{\overset{p_1,S_2}{\int }}pdV-\underset{p_1,S_2}{\overset{p_2,S_2}{\int }}pdV$	$$
$$	${\displaystyle }$	${\displaystyle }$	$-\underset{p_2,S_2}{\overset{p_2,S_1}{\int }}pdV-\underset{p_2,S_1}{\overset{p_1,S_1}{\int }}pdV$	$(p,S;11\to 12\to 22\to 21\to 11),dN=0$

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

$(01)$	$-\underset{p,S_2,S_1}{\int }pdV$	$/$	$\underset{S,p_1,p_2}{\int }TdS$	$=$	$\frac{W_a+W_b+W_c+W_d}{Q_a+Q_c}$	$=$	$-100\mathrm{\%}$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$(p,S;11\to 12\to 22\to 21\to 11)$
$(02)$	$-{\oint }_{p,S}pdV$	$/$	${\oint }_{p,S}TdS$	$=$	$\frac{W_a+W_b+W_c+W_d}{Q_a+Q_c}$	$=$	$-100\mathrm{\%}$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$(p,S;11\to 12\to 22\to 21\to 11)$
$(03)$	$-\underset{p,S_2,S_1}{\int }pdV$	$+$	$\underset{S,p_1,p_2}{\int }TdS$	$=$	$Q_a+W_a+W_b+Q_c+W_c+W_d$	$=$	$0$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$(p,S;11\to 12\to 22\to 21\to 11)$
$(04)$	$-{\oint }_{p,S}pdV$	$+$	${\oint }_{p,S}TdS$	$=$	$Q_a+W_a+W_b+Q_c+W_c+W_d$	$=$	$0$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$(p,S;11\to 12\to 22\to 21\to 11)$

Übung a

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

Übung b

Wir stellen uns vor, wir haben die $U(S,p,N=\text{const})$- und die $H(S,p,N=\text{const})$-Fläche im $(S,p,N=\text{const})$-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?