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PCRT.I.G
04 (N,V,S(V,p))-Koordinatensystem
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{\displaystyle \qquad \qquad }
d
Q
N
{\displaystyle dQ_{N}}
=
{\displaystyle =}
d
Q
V
,
N
+
d
Q
S
,
N
{\displaystyle dQ_{V,N}+dQ_{S,N}}
d
Q
N
(
V
,
S
)
,
d
Q
V
,
N
(
S
)
,
d
Q
S
,
N
(
V
)
{\displaystyle \qquad dQ_{N}(V,S),\,dQ_{V,N}(S),\,dQ_{S,N}(V)\qquad }
{\displaystyle \qquad \qquad }
d
Q
V
,
N
{\displaystyle dQ_{V,N}}
=
{\displaystyle =}
d
U
V
,
N
{\displaystyle dU_{V,N}}
d
U
V
,
N
(
S
)
{\displaystyle \qquad dU_{V,N}(S)\qquad }
{\displaystyle \qquad \qquad }
d
Q
S
,
N
{\displaystyle dQ_{S,N}}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
d
W
N
{\displaystyle dW_{N}}
=
{\displaystyle =}
d
W
V
,
N
+
d
W
S
,
N
{\displaystyle dW_{V,N}+dW_{S,N}}
d
W
N
(
V
,
S
)
,
d
W
V
,
N
(
S
)
,
d
W
S
,
N
(
V
)
{\displaystyle \qquad dW_{N}(V,S),\,dW_{V,N}(S),\,dW_{S,N}(V)\qquad }
{\displaystyle \qquad \qquad }
d
W
V
,
N
{\displaystyle dW_{V,N}}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
d
W
S
,
N
{\displaystyle dW_{S,N}}
=
{\displaystyle =}
d
U
S
,
N
{\displaystyle dU_{S,N}}
d
U
S
,
N
(
V
)
{\displaystyle \qquad dU_{S,N}(V)\qquad }
{\displaystyle \qquad \qquad }
d
U
N
{\displaystyle dU_{N}}
=
{\displaystyle =}
d
U
V
,
N
+
d
U
S
,
N
{\displaystyle dU_{V,N}+dU_{S,N}}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
U
{\displaystyle U}
=
{\displaystyle =}
3
2
n
R
T
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
{\displaystyle {\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)}
U
(
N
,
V
,
S
)
{\displaystyle \qquad U(N,V,S)\qquad }
{\displaystyle \qquad \qquad }
d
Q
N
{\displaystyle dQ_{N}}
=
{\displaystyle =}
d
Q
V
,
N
+
d
Q
p
,
N
{\displaystyle dQ_{V,N}+dQ_{p,N}}
d
Q
N
(
V
,
p
)
,
d
Q
V
,
N
(
p
)
,
d
Q
p
,
N
(
V
)
{\displaystyle \qquad dQ_{N}(V,p),\,dQ_{V,N}(p),\,dQ_{p,N}(V)\qquad }
{\displaystyle \qquad \qquad }
d
W
N
{\displaystyle dW_{N}}
=
{\displaystyle =}
d
W
V
,
N
+
d
W
p
,
N
{\displaystyle dW_{V,N}+dW_{p,N}}
d
W
N
(
V
,
p
)
,
d
W
V
,
N
(
p
)
,
d
W
p
,
N
(
V
)
{\displaystyle \qquad dW_{N}(V,p),\,dW_{V,N}(p),\,dW_{p,N}(V)\qquad }
{\displaystyle \qquad \qquad }
d
Q
V
,
N
+
d
W
V
,
N
{\displaystyle dQ_{V,N}+dW_{V,N}}
=
{\displaystyle =}
d
U
V
,
N
{\displaystyle dU_{V,N}}
d
U
V
,
N
(
p
)
{\displaystyle \qquad dU_{V,N}(p)\qquad }
{\displaystyle \qquad \qquad }
d
Q
p
,
N
+
d
W
p
,
N
{\displaystyle dQ_{p,N}+dW_{p,N}}
=
{\displaystyle =}
d
U
p
,
N
{\displaystyle dU_{p,N}}
d
U
p
,
N
(
V
)
{\displaystyle \qquad dU_{p,N}(V)\qquad }
{\displaystyle \qquad \qquad }
d
U
N
{\displaystyle dU_{N}}
=
{\displaystyle =}
d
U
V
,
N
+
d
U
S
,
N
{\displaystyle dU_{V,N}+dU_{S,N}}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
1
{\displaystyle 1}
=
{\displaystyle =}
(
V
0
V
)
5
/
3
(
p
0
p
)
exp
(
2
3
S
(
V
,
p
)
−
S
0
n
R
)
{\displaystyle \left({\frac {V_{0}}{V}}\right)^{5/3}\left({\frac {p_{0}}{p}}\right)\,\exp \left({\frac {2}{3}}{\frac {S(V,p)-S_{0}}{n\,R}}\right)}
U
(
N
,
V
,
S
(
V
,
p
)
)
{\displaystyle \qquad U(N,V,S(V,p))\qquad }
{\displaystyle \qquad \qquad }
(
V
V
0
)
(
p
p
0
)
{\displaystyle \left({\frac {V}{V_{0}}}\right)\left({\frac {p}{p_{0}}}\right)}
=
{\displaystyle =}
(
V
0
V
)
2
/
3
exp
(
2
3
S
(
V
,
p
)
−
S
0
n
R
)
{\displaystyle \left({\frac {V_{0}}{V}}\right)^{2/3}\,\exp \left({\frac {2}{3}}{\frac {S(V,p)-S_{0}}{n\,R}}\right)}
U
(
N
,
V
,
S
(
V
,
p
)
)
{\displaystyle \qquad U(N,V,S(V,p))\qquad }
{\displaystyle \qquad \qquad }
U
{\displaystyle U}
=
{\displaystyle =}
3
2
n
R
T
0
(
V
V
0
)
(
p
p
0
)
{\displaystyle {\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V}{V_{0}}}\right)\left({\frac {p}{p_{0}}}\right)}
U
(
N
,
V
,
p
)
{\displaystyle \qquad U(N,V,p)\qquad }
{\displaystyle \qquad \qquad }
Q
V
1
,
N
(
S
12
,
S
11
)
{\displaystyle Q_{V1,N}(S_{12},S_{11})}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
1
)
2
/
3
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
{\displaystyle }
×
[
exp
(
2
3
S
12
−
S
0
n
R
)
−
exp
(
2
3
S
11
−
S
0
n
R
)
]
{\displaystyle \times \left[\exp \left({\frac {2}{3}}{\frac {S_{12}-S_{0}}{n\,R}}\right)-\,\exp \left({\frac {2}{3}}{\frac {S_{11}-S_{0}}{n\,R}}\right)\right]}
(
V
,
p
,
S
:
V
1
p
1
S
11
→
V
1
p
2
S
12
)
,
d
N
=
0
{\displaystyle \qquad (V,p,S:V_{1}p_{1}S_{11}\to V_{1}p_{2}S_{12}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
Q
V
1
,
N
(
p
2
,
p
1
)
{\displaystyle Q_{V1,N}(p_{2},p_{1})}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
1
V
0
)
[
(
p
2
p
0
)
−
(
p
1
p
0
)
]
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{1}}{V_{0}}}\right)\left[\left({\frac {p_{2}}{p_{0}}}\right)-\,\left({\frac {p_{1}}{p_{0}}}\right)\right]}
(
V
,
p
:
V
1
p
1
→
V
1
p
2
)
,
d
N
=
0
{\displaystyle \qquad (V,p:V_{1}p_{1}\to V_{1}p_{2}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
W
N
(
V
2
,
S
22
,
V
1
,
S
12
)
{\displaystyle W_{N}(V_{2},S_{22},V_{1},S_{12})}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
2
)
2
/
3
exp
(
2
3
S
22
−
S
0
n
R
)
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S_{22}-S_{0}}{n\,R}}\right)}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
{\displaystyle }
−
3
2
n
R
T
0
(
V
0
V
1
)
2
/
3
exp
(
2
3
S
12
−
S
0
n
R
)
{\displaystyle -\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S_{12}-S_{0}}{n\,R}}\right)}
(
V
,
p
,
S
:
V
1
p
2
S
12
→
V
2
p
2
S
22
)
,
d
N
=
0
{\displaystyle \qquad (V,p,S:V_{1}p_{2}S_{12}\to V_{2}p_{2}S_{22}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
W
p
2
,
N
(
V
2
,
V
1
)
{\displaystyle W_{p2,N}(V_{2},V_{1})}
=
{\displaystyle =}
+
3
2
n
R
T
0
[
(
V
2
V
0
)
−
(
V
1
V
0
)
]
(
p
2
p
0
)
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{2}}{V_{0}}}\right)-\,\left({\frac {V_{1}}{V_{0}}}\right)\right]\left({\frac {p_{2}}{p_{0}}}\right)}
(
V
,
p
:
V
1
p
2
→
V
2
p
2
)
,
d
N
=
0
{\displaystyle \qquad (V,p:V_{1}p_{2}\to V_{2}p_{2}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
Q
V
2
,
N
(
S
21
,
S
22
)
{\displaystyle Q_{V2,N}(S_{21},S_{22})}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
2
)
2
/
3
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
{\displaystyle }
×
[
exp
(
2
3
S
21
−
S
0
n
R
)
−
exp
(
2
3
S
22
−
S
0
n
R
)
]
{\displaystyle \times \left[\exp \left({\frac {2}{3}}{\frac {S_{21}-S_{0}}{n\,R}}\right)-\,\exp \left({\frac {2}{3}}{\frac {S_{22}-S_{0}}{n\,R}}\right)\right]}
(
V
,
p
,
S
:
V
2
p
2
S
22
→
V
2
p
1
S
21
)
,
d
N
=
0
{\displaystyle \qquad (V,p,S:V_{2}p_{2}S_{22}\to V_{2}p_{1}S_{21}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
Q
V
2
,
N
(
p
1
,
p
2
)
{\displaystyle Q_{V2,N}(p_{1},p_{2})}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
2
V
0
)
[
(
p
1
p
0
)
−
(
p
2
p
0
)
]
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{2}}{V_{0}}}\right)\left[\left({\frac {p_{1}}{p_{0}}}\right)-\,\left({\frac {p_{2}}{p_{0}}}\right)\right]}
(
V
,
p
:
V
2
p
2
→
V
2
p
1
)
,
d
N
=
0
{\displaystyle \qquad (V,p:V_{2}p_{2}\to V_{2}p_{1}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
W
N
(
V
1
,
S
11
,
V
2
,
S
21
)
{\displaystyle W_{N}(V_{1},S_{11},V_{2},S_{21})}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
1
)
2
/
3
exp
(
2
3
S
11
−
S
0
n
R
)
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S_{11}-S_{0}}{n\,R}}\right)}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
{\displaystyle }
−
3
2
n
R
T
0
(
V
0
V
2
)
2
/
3
exp
(
2
3
S
21
−
S
0
n
R
)
{\displaystyle -\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S_{21}-S_{0}}{n\,R}}\right)}
(
V
,
p
,
S
:
V
2
p
1
S
21
→
V
1
p
1
S
11
)
,
d
N
=
0
{\displaystyle \qquad (V,p,S:V_{2}p_{1}S_{21}\to V_{1}p_{1}S_{11}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
W
p
1
,
N
(
V
1
,
V
2
)
{\displaystyle W_{p1,N}(V_{1},V_{2})}
=
{\displaystyle =}
+
3
2
n
R
T
0
[
(
V
1
V
0
)
−
(
V
2
V
0
)
]
(
p
1
p
0
)
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{1}}{V_{0}}}\right)-\,\left({\frac {V_{2}}{V_{0}}}\right)\right]\left({\frac {p_{1}}{p_{0}}}\right)}
(
V
,
p
:
V
2
p
1
→
V
1
p
1
)
,
d
N
=
0
{\displaystyle \qquad (V,p:V_{2}p_{1}\to V_{1}p_{1}),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
0
{\displaystyle 0}
=
{\displaystyle =}
Q
V
1
,
N
(
S
12
,
S
11
)
+
W
N
(
V
2
,
S
22
,
V
1
,
S
12
)
{\displaystyle Q_{V1,N}(S_{12},S_{11})+W_{N}(V_{2},S_{22},V_{1},S_{12})}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
{\displaystyle }
+
Q
V
2
,
N
(
S
21
,
S
22
)
+
W
N
(
V
1
,
S
11
,
V
2
,
S
21
)
{\displaystyle +\,Q_{V2,N}(S_{21},S_{22})+W_{N}(V_{1},S_{11},V_{2},S_{21})}
(
V
S
:
111
→
112
→
222
→
221
→
111
)
{\displaystyle \qquad (VS:111\to 112\to 222\to 221\to 111)\qquad }
{\displaystyle \qquad \qquad }
0
{\displaystyle 0}
=
{\displaystyle =}
Q
V
1
,
N
(
p
2
,
p
1
)
+
W
p
2
,
N
(
V
2
,
V
1
)
{\displaystyle Q_{V1,N}(p_{2},p_{1})+W_{p2,N}(V_{2},V_{1})}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
{\displaystyle }
+
Q
V
2
,
N
(
p
1
,
p
2
)
+
W
p
1
,
N
(
V
1
,
V
2
)
{\displaystyle +\,Q_{V2,N}(p_{1},p_{2})+W_{p1,N}(V_{1},V_{2})}
(
V
p
:
11
→
12
→
22
→
21
→
11
)
,
d
N
=
0
{\displaystyle \qquad (Vp:11\to 12\to 22\to 21\to 11),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
Q
N
{\displaystyle Q_{N}}
=
{\displaystyle =}
Q
N
(
→
)
+
Q
N
(
←
)
{\displaystyle Q_{N}(\rightarrow )+Q_{N}(\leftarrow )}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
=
{\displaystyle =}
Q
V
1
,
N
(
S
12
,
S
11
)
+
Q
V
2
,
N
(
S
21
,
S
22
)
{\displaystyle Q_{V1,N}(S_{12},S_{11})+Q_{V2,N}(S_{21},S_{22})}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
=
{\displaystyle =}
Q
V
1
,
N
(
p
2
,
p
1
)
+
Q
V
2
,
N
(
p
1
,
p
2
)
{\displaystyle Q_{V1,N}(p_{2},p_{1})+Q_{V2,N}(p_{1},p_{2})}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
W
N
{\displaystyle W_{N}}
=
{\displaystyle =}
W
N
(
←
)
+
W
N
(
→
)
{\displaystyle W_{N}(\leftarrow )+W_{N}(\rightarrow )}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
=
{\displaystyle =}
W
N
(
V
2
,
S
22
,
V
1
,
S
12
)
+
W
N
(
V
1
,
S
11
,
V
2
,
S
21
)
{\displaystyle W_{N}(V_{2},S_{22},V_{1},S_{12})+W_{N}(V_{1},S_{11},V_{2},S_{21})}
{\displaystyle \qquad \qquad }
{\displaystyle \qquad \qquad }
{\displaystyle }
=
{\displaystyle =}
W
p
2
,
N
(
V
2
,
V
1
)
+
W
p
1
,
N
(
V
1
,
V
2
)
{\displaystyle W_{p2,N}(V_{2},V_{1})+W_{p1,N}(V_{1},V_{2})}
{\displaystyle \qquad \qquad }
![{\displaystyle \times \left[\exp \left({\frac {2}{3}}{\frac {S_{12}-S_{0}}{n\,R}}\right)-\,\exp \left({\frac {2}{3}}{\frac {S_{11}-S_{0}}{n\,R}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5229252a115e90859934181642fcc21e52c9d726)
![{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{1}}{V_{0}}}\right)\left[\left({\frac {p_{2}}{p_{0}}}\right)-\,\left({\frac {p_{1}}{p_{0}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2291ad650359bf71b44d6cfa59e04eb9fadc048)
![{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{2}}{V_{0}}}\right)-\,\left({\frac {V_{1}}{V_{0}}}\right)\right]\left({\frac {p_{2}}{p_{0}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03b9c5d9eb73b576477147515b55b1161a754344)
![{\displaystyle \times \left[\exp \left({\frac {2}{3}}{\frac {S_{21}-S_{0}}{n\,R}}\right)-\,\exp \left({\frac {2}{3}}{\frac {S_{22}-S_{0}}{n\,R}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd09e04e04026ab959fb63207e60ac5f04ddfaf)
![{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{2}}{V_{0}}}\right)\left[\left({\frac {p_{1}}{p_{0}}}\right)-\,\left({\frac {p_{2}}{p_{0}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e436bf9d478823bf74c490c34c22890679ca7b1e)
![{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{1}}{V_{0}}}\right)-\,\left({\frac {V_{2}}{V_{0}}}\right)\right]\left({\frac {p_{1}}{p_{0}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bef9fdca767bc026fd0e84b6ba1a40598fec7a5d)
