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1
08 Vom Differential über das Integral zum Zustand der inneren Energie
Unterabschnitt 08 Vom Differential über das Integral zum Zustand der inneren Energie umschalten
1.1
A in (S,V,N)-Koordinaten: dU(S,V) -> U(S,V,N)
1.2
B in (S,p,N)-Koordinaten: dU(S,p) -> U(S,p,N)
1.3
C in (T,V,N)-Koordinaten: dU(T,V) -> U(T,V,N)
Inhaltsverzeichnis umschalten
CPRT.I.C.08
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CPRT.I.C
08 Vom Differential über das Integral zum Zustand der inneren Energie
[
Bearbeiten
]
A in (S,V,N)-Koordinaten: dU(S,V) -> U(S,V,N)
[
Bearbeiten
]
(
3.11.1.1.1
)
{\displaystyle (3.11.1.1.1)\quad }
d
U
{\displaystyle dU}
=
{\displaystyle =}
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
[
+
n
R
T
0
d
S
n
R
−
p
0
V
0
d
V
V
]
{\displaystyle \left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-\,p_{0}\,V_{0}\,{\frac {dV}{V}}\right]}
d
U
(
S
,
V
)
,
d
N
=
0
←
(
3.9.1.1.2
)
{\displaystyle \quad dU(S,V),\,dN=0\quad \leftarrow (3.9.1.1.2)}
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
∫
S
,
d
V
=
0
d
U
{\displaystyle \int _{S,dV=0}\,dU}
U
(
S
,
V
,
N
)
{\displaystyle \quad U(S,V,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
∫
S
,
d
V
=
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
n
R
T
0
d
S
n
R
{\displaystyle \int _{S,dV=0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)n\,R\,T_{0}\,{\frac {dS}{n\,R}}}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
+
B
(
V
,
N
)
{\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)+\,B(V,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
(
∂
U
∂
V
)
S
,
N
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{S,N}}
=
{\displaystyle =}
−
n
R
V
T
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
+
(
∂
B
∂
V
)
N
{\displaystyle -{\frac {n\,R}{V}}\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial B}{\partial V}}\right)_{N}}
∂
V
U
(
S
,
V
,
N
)
,
B
(
V
,
N
)
{\displaystyle \quad \partial _{V}U(S,V,N),\,B(V,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
−
p
+
(
∂
B
∂
V
)
N
{\displaystyle -\,p+\,\left({\frac {\partial B}{\partial V}}\right)_{N}}
p
(
S
,
V
,
N
)
,
B
(
V
,
N
)
{\displaystyle \quad p(S,V,N),\,B(V,N)\qquad }
(
3.11.1.1.2
)
{\displaystyle (3.11.1.1.2)\quad }
(
∂
U
∂
V
)
S
,
N
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{S,N}}
=
{\displaystyle =}
−
p
{\displaystyle -\,p}
∂
V
U
(
S
,
V
,
N
)
,
p
(
S
,
V
,
N
)
{\displaystyle \quad \partial _{V}U(S,V,N),\,p(S,V,N)\qquad }
{\displaystyle \quad }
B
{\displaystyle B}
≠
{\displaystyle \neq }
B
(
V
,
N
)
{\displaystyle B(V,N)}
B
=
const.
{\displaystyle \quad B={\mbox{const.}}\qquad }
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
∫
V
,
d
S
=
0
d
U
{\displaystyle \int _{V,dS=0}\,dU}
U
(
S
,
V
,
N
)
{\displaystyle \quad U(S,V,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
−
∫
S
,
d
V
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
p
0
V
0
d
V
V
{\displaystyle -\,\int _{S,dV}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)p_{0}\,V_{0}\,{\frac {dV}{V}}}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
−
∫
S
,
d
V
p
0
V
0
1
V
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
d
V
{\displaystyle -\,\int _{S,dV}\,p_{0}\,V_{0}\,{\frac {1}{V}}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)dV}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
+
A
(
S
,
N
)
{\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)+\,A(S,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
(
∂
U
∂
S
)
V
,
N
{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{V,N}}
=
{\displaystyle =}
+
T
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
+
(
∂
A
∂
S
)
N
{\displaystyle +\,T_{0}\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial A}{\partial S}}\right)_{N}}
∂
S
U
(
S
,
V
,
N
)
,
A
(
S
,
N
)
{\displaystyle \quad \partial _{S}U(S,V,N),\,A(S,N)\qquad }
(
3.11.1.1.3
)
{\displaystyle (3.11.1.1.3)\quad }
{\displaystyle }
=
{\displaystyle =}
+
T
+
(
∂
A
∂
S
)
N
{\displaystyle +\,T+\,\left({\frac {\partial A}{\partial S}}\right)_{N}}
T
(
S
,
V
,
N
)
,
A
(
S
,
N
)
{\displaystyle \quad T(S,V,N),\,A(S,N)\qquad }
{\displaystyle \quad }
(
∂
U
∂
S
)
V
,
N
{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{V,N}}
=
{\displaystyle =}
+
T
{\displaystyle +\,T}
∂
S
U
(
S
,
V
,
N
)
,
T
(
S
,
V
,
N
)
{\displaystyle \quad \partial _{S}U(S,V,N),\,T(S,V,N)\qquad }
{\displaystyle \quad }
A
{\displaystyle A}
≠
{\displaystyle \neq }
A
(
S
,
N
)
{\displaystyle A(S,N)}
A
=
const.
{\displaystyle \quad A={\mbox{const.}}\qquad }
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
+
3
2
n
R
T
0
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
+
A
+
B
{\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)+\,A+\,B}
U
(
S
,
V
,
N
)
{\displaystyle \quad U(S,V,N)\qquad }
(
3.11.1.1.4
)
{\displaystyle (3.11.1.1.4)\quad }
U
(
S
0
,
V
0
,
N
)
{\displaystyle U(S_{0},V_{0},N)}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \quad }
{\displaystyle \quad }
A
+
B
{\displaystyle A+B}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \quad }
(
3.11.1.1.5
)
{\displaystyle (3.11.1.1.5)\quad }
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
(
S
,
V
,
N
)
←
(
3.10.1.1.1
)
{\displaystyle \quad U(S,V,N)\quad \leftarrow (3.10.1.1.1)}
B in (S,p,N)-Koordinaten: dU(S,p) -> U(S,p,N)
[
Bearbeiten
]
(
3.11.1.2.1
)
{\displaystyle (3.11.1.2.1)\quad }
d
U
{\displaystyle dU}
=
{\displaystyle =}
3
5
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
[
+
n
R
T
0
d
S
n
R
+
p
0
V
0
d
p
p
]
{\displaystyle {\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+\,p_{0}\,V_{0}\,{\frac {dp}{p}}\right]}
d
U
(
S
,
p
)
,
d
N
=
0
←
(
3.9.2.2.7
)
{\displaystyle \quad dU(S,p),\,dN=0\quad \leftarrow (3.9.2.2.7)}
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
∫
S
,
d
p
=
0
d
U
{\displaystyle \int _{S,dp=0}\,dU}
U
(
S
,
p
,
N
)
{\displaystyle \quad U(S,p,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
∫
S
,
d
p
=
0
3
5
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
n
R
T
0
d
S
n
R
{\displaystyle \int _{S,dp=0}{\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)n\,R\,T_{0}\,{\frac {dS}{n\,R}}}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
5
2
⋅
3
5
n
R
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
B
(
p
,
N
)
{\displaystyle +\,{\frac {5}{2}}\cdot {\frac {3}{5}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
2
n
R
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
B
(
p
,
N
)
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,B(p,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
(
∂
U
∂
p
)
S
,
N
{\displaystyle \left({\frac {\partial U}{\partial p}}\right)_{S,N}}
=
{\displaystyle =}
+
2
5
⋅
3
2
n
R
p
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
(
∂
B
∂
p
)
N
{\displaystyle +\,{\frac {2}{5}}\cdot {\frac {3}{2}}\,{\frac {n\,R}{p}}\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial B}{\partial p}}\right)_{N}}
∂
p
U
(
S
,
p
,
N
)
,
B
(
p
,
N
)
{\displaystyle \quad \partial _{p}U(S,p,N),\,B(p,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
5
n
R
p
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
(
∂
B
∂
p
)
N
{\displaystyle +\,{\frac {3}{5}}\,{\frac {n\,R}{p}}\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial B}{\partial p}}\right)_{N}}
∂
p
U
(
S
,
p
,
N
)
,
B
(
p
,
N
)
{\displaystyle \quad \partial _{p}U(S,p,N),\,B(p,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
5
V
+
(
∂
B
∂
p
)
N
{\displaystyle +\,{\frac {3}{5}}\,V+\,\left({\frac {\partial B}{\partial p}}\right)_{N}}
∂
p
U
(
S
,
p
,
N
)
,
V
(
S
,
p
,
N
)
,
B
(
p
,
N
)
{\displaystyle \quad \partial _{p}U(S,p,N),\,V(S,p,N),\,B(p,N)\qquad }
{\displaystyle \quad }
(
∂
U
∂
p
)
S
,
N
{\displaystyle \left({\frac {\partial U}{\partial p}}\right)_{S,N}}
=
{\displaystyle =}
(
∂
(
H
−
p
V
)
∂
p
)
S
,
N
=
V
−
V
(
∂
p
∂
p
)
S
,
N
−
p
(
∂
V
∂
p
)
S
,
N
{\displaystyle \left({\frac {\partial (H-\,p\,V)}{\partial p}}\right)_{S,N}=V-\,V\left({\frac {\partial p}{\partial p}}\right)_{S,N}-p\left({\frac {\partial V}{\partial p}}\right)_{S,N}}
(
S
,
p
,
N
)
:
∂
p
U
,
∂
p
H
,
∂
p
V
{\displaystyle \quad (S,p,N):\,\partial _{p}U,\,\partial _{p}H,\,\partial _{p}V\qquad }
{\displaystyle \quad }
−
p
(
∂
V
∂
p
)
S
,
N
{\displaystyle -p\left({\frac {\partial V}{\partial p}}\right)_{S,N}}
=
{\displaystyle =}
+
3
5
V
{\displaystyle +\,{\frac {3}{5}}\,V}
(
S
,
p
,
N
)
:
∂
p
V
,
V
←
(
3.9.2.2.3
)
{\displaystyle \quad (S,p,N):\,\partial _{p}V,\,V\quad \leftarrow (3.9.2.2.3)}
(
3.11.1.2.2
)
{\displaystyle (3.11.1.2.2)\quad }
(
∂
U
∂
p
)
S
,
N
{\displaystyle \left({\frac {\partial U}{\partial p}}\right)_{S,N}}
=
{\displaystyle =}
+
3
5
V
{\displaystyle +\,{\frac {3}{5}}\,V}
(
S
,
p
,
N
)
:
∂
p
U
,
V
{\displaystyle \quad (S,p,N):\,\partial _{p}U,\,V\qquad }
{\displaystyle \quad }
B
{\displaystyle B}
≠
{\displaystyle \neq }
B
(
p
,
N
)
{\displaystyle B(p,N)}
B
=
const.
{\displaystyle \quad B={\mbox{const.}}\qquad }
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
∫
p
,
d
S
=
0
d
U
{\displaystyle \int _{p,dS=0}\,dU}
U
(
S
,
V
,
N
)
{\displaystyle \quad U(S,V,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
∫
p
,
d
S
=
0
3
5
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
p
0
V
0
d
p
p
{\displaystyle \int _{p,dS=0}\,{\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,p_{0}\,V_{0}\,{\frac {dp}{p}}}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
∫
p
,
d
S
=
0
3
5
1
p
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
p
0
V
0
d
p
{\displaystyle \int _{p,dS=0}\,{\frac {3}{5}}\,{\frac {1}{p}}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\,p_{0}\,V_{0}\,dp}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
5
2
⋅
3
5
p
0
V
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
A
(
S
,
N
)
{\displaystyle +\,{\frac {5}{2}}\cdot {\frac {3}{5}}\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,A(S,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
2
p
0
V
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
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}}{n\,R}}\right)+\,A(S,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
(
∂
U
∂
S
)
p
,
N
{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{p,N}}
=
{\displaystyle =}
+
2
5
1
n
R
⋅
3
2
p
0
V
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
(
∂
A
∂
S
)
N
{\displaystyle +\,{\frac {2}{5}}\,{\frac {1}{n\,R}}\cdot {\frac {3}{2}}\,p_{0}\,V_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial A}{\partial S}}\right)_{N}}
∂
S
U
(
S
,
p
,
N
)
,
A
(
S
,
N
)
{\displaystyle \quad \partial _{S}U(S,p,N),\,A(S,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
5
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
(
∂
A
∂
S
)
N
{\displaystyle +\,{\frac {3}{5}}\,T_{0}\,\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,\left({\frac {\partial A}{\partial S}}\right)_{N}}
∂
S
U
(
S
,
p
,
N
)
,
A
(
S
,
N
)
{\displaystyle \quad \partial _{S}U(S,p,N),\,A(S,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
5
T
+
(
∂
A
∂
S
)
N
{\displaystyle +\,{\frac {3}{5}}\,T+\,\left({\frac {\partial A}{\partial S}}\right)_{N}}
∂
S
U
(
S
,
p
,
N
)
,
T
(
S
,
p
,
N
)
,
A
(
S
,
N
)
{\displaystyle \quad \partial _{S}U(S,p,N),\,T(S,p,N),\,A(S,N)\qquad }
{\displaystyle \quad }
(
∂
U
∂
S
)
p
,
N
{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{p,N}}
=
{\displaystyle =}
(
∂
(
H
−
p
V
)
∂
S
)
p
,
N
=
T
−
p
(
∂
V
∂
S
)
p
,
N
{\displaystyle \left({\frac {\partial (H-\,p\,V)}{\partial S}}\right)_{p,N}=T-p\left({\frac {\partial V}{\partial S}}\right)_{p,N}}
(
S
,
p
,
N
)
:
∂
S
U
,
∂
S
H
,
∂
S
V
,
T
{\displaystyle \quad (S,p,N):\,\partial _{S}U,\,\partial _{S}H,\,\partial _{S}V,\,T\qquad }
{\displaystyle \quad }
−
S
(
∂
V
∂
S
)
p
,
N
{\displaystyle -S\left({\frac {\partial V}{\partial S}}\right)_{p,N}}
=
{\displaystyle =}
−
2
5
S
n
R
V
{\displaystyle -{\frac {2}{5}}{\frac {S}{n\,R}}\,V}
(
S
,
p
,
N
)
:
∂
S
V
,
S
,
V
←
(
3.9.2.3.3
)
{\displaystyle (S,p,N):\partial _{S}V,\,S,\,V\quad \leftarrow (3.9.2.3.3)}
(
3.11.1.2.3
)
{\displaystyle (3.11.1.2.3)\quad }
−
p
(
∂
V
∂
S
)
p
,
N
{\displaystyle -p\,\left({\frac {\partial V}{\partial S}}\right)_{p,N}}
=
{\displaystyle =}
−
2
5
T
{\displaystyle -\,{\frac {2}{5}}\,T}
∂
S
V
(
S
,
p
,
N
)
,
T
(
S
,
p
,
N
)
{\displaystyle \quad \partial _{S}V(S,p,N),\,T(S,p,N)\qquad }
(
3.11.1.2.4
)
{\displaystyle (3.11.1.2.4)\quad }
(
∂
U
∂
S
)
p
,
N
{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{p,N}}
=
{\displaystyle =}
+
3
5
T
{\displaystyle +\,{\frac {3}{5}}\,T}
∂
S
U
(
S
,
p
,
N
)
,
T
(
S
,
p
,
N
)
{\displaystyle \quad \partial _{S}U(S,p,N),\,T(S,p,N)\qquad }
{\displaystyle \quad }
A
{\displaystyle A}
≠
{\displaystyle \neq }
A
(
S
,
N
)
{\displaystyle A(S,N)}
A
=
const.
{\displaystyle \quad A={\mbox{const.}}\qquad }
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
3
2
n
R
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
+
A
+
B
{\displaystyle {\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)+\,A+\,B}
U
(
S
,
p
,
N
)
{\displaystyle \quad U(S,p,N)\qquad }
(
3.11.1.2.5
)
{\displaystyle (3.11.1.2.5)\quad }
U
(
S
0
,
p
0
,
N
)
{\displaystyle U(S_{0},p_{0},N)}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \qquad \qquad }
{\displaystyle \quad }
A
+
B
{\displaystyle A+B}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \quad \qquad }
(
3.11.1.2.6
)
{\displaystyle (3.11.1.2.6)\quad }
U
{\displaystyle U}
=
{\displaystyle =}
3
2
n
R
T
0
(
p
p
0
)
2
/
5
exp
(
2
5
S
−
S
0
n
R
)
{\displaystyle {\frac {3}{2}}\,n\,R\,T_{0}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)}
U
(
S
,
p
,
N
)
←
(
3.10.2.2.2
)
{\displaystyle \quad U(S,p,N)\qquad \leftarrow (3.10.2.2.2)}
C in (T,V,N)-Koordinaten: dU(T,V) -> U(T,V,N)
[
Bearbeiten
]
(
3.11.1.3.1
)
{\displaystyle (3.11.1.3.1)\quad }
d
U
{\displaystyle dU}
=
{\displaystyle =}
3
2
n
R
d
T
{\displaystyle {\frac {3}{2}}\,n\,R\,dT}
d
U
(
T
,
V
)
,
d
N
=
0
←
(
3.9.3.3.3
)
{\displaystyle \quad dU(T,V),\,dN=0\quad \leftarrow (3.9.3.3.3)}
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
∫
T
,
d
V
=
0
d
U
{\displaystyle \int _{T,dV=0}\,dU}
U
(
T
,
V
,
N
)
{\displaystyle \quad U(T,V,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
∫
S
,
d
V
=
0
3
2
n
R
d
T
{\displaystyle \int _{S,dV=0}{\frac {3}{2}}\,n\,R\,dT}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
3
2
n
R
T
+
B
(
V
,
N
)
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T+\,B(V,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
(
∂
U
∂
V
)
T
,
N
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
+
(
∂
B
∂
V
)
N
{\displaystyle +\,\left({\frac {\partial B}{\partial V}}\right)_{N}}
∂
V
U
(
T
,
V
,
N
)
,
B
(
V
,
N
)
{\displaystyle \quad \partial _{V}U(T,V,N),\,B(V,N)\qquad }
{\displaystyle \quad }
(
∂
U
∂
V
)
T
,
N
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
(
∂
(
F
+
T
S
)
∂
V
)
T
,
N
=
−
p
+
S
(
∂
T
∂
V
)
T
,
N
+
T
(
∂
S
∂
V
)
T
,
N
{\displaystyle \left({\frac {\partial (F+\,T\,S)}{\partial V}}\right)_{T,N}=-\,p+\,S\left({\frac {\partial T}{\partial V}}\right)_{T,N}+\,T\left({\frac {\partial S}{\partial V}}\right)_{T,N}}
(
T
,
V
,
N
)
:
∂
V
U
,
∂
V
H
,
∂
V
S
{\displaystyle \quad (T,V,N):\,\partial _{V}U,\,\partial _{V}H,\,\partial _{V}S\qquad }
{\displaystyle \quad }
−
V
(
∂
S
∂
V
)
T
,
N
{\displaystyle -V\,\left({\frac {\partial S}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
−
n
R
{\displaystyle -\,n\,R}
∂
V
S
(
T
,
V
,
N
)
←
(
3.9.3.2.1
)
{\displaystyle \quad \partial _{V}S(T,V,N)\quad \leftarrow (3.9.3.2.1)}
{\displaystyle \quad }
+
T
(
∂
S
∂
V
)
T
,
N
{\displaystyle +T\left({\frac {\partial S}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
+
n
R
T
V
=
+
p
{\displaystyle +\,{\frac {n\,R\,T}{V}}=+\,p}
(
T
,
V
,
N
)
:
∂
V
S
,
p
{\displaystyle \quad (T,V,N):\,\partial _{V}S,\,p\quad }
(
3.11.1.3.2
)
{\displaystyle (3.11.1.3.2)\quad }
(
∂
U
∂
V
)
T
,
N
{\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
0
{\displaystyle 0}
(
T
,
V
,
N
)
:
∂
V
U
{\displaystyle \quad (T,V,N):\,\partial _{V}U\qquad }
{\displaystyle \quad }
B
{\displaystyle B}
≠
{\displaystyle \neq }
B
(
V
,
N
)
{\displaystyle B(V,N)}
B
=
const.
{\displaystyle \quad B={\mbox{const.}}\qquad }
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
∫
V
,
d
T
=
0
d
U
{\displaystyle \int _{V,dT=0}\,dU}
U
(
T
,
V
,
N
)
{\displaystyle \quad U(T,V,N)\qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
∫
V
,
d
T
=
0
0
d
V
{\displaystyle \int _{V,dT=0}\,0\,\,dV}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
{\displaystyle }
=
{\displaystyle =}
+
A
(
T
,
N
)
{\displaystyle +\,A(T,N)}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
(
∂
U
∂
T
)
V
,
N
{\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{V,N}}
=
{\displaystyle =}
+
(
∂
A
∂
T
)
N
{\displaystyle +\,\left({\frac {\partial A}{\partial T}}\right)_{N}}
∂
T
U
(
T
,
V
,
N
)
,
A
(
T
,
N
)
{\displaystyle \quad \partial _{T}U(T,V,N),\,A(T,N)\qquad }
{\displaystyle \quad }
(
∂
U
∂
T
)
V
,
N
{\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{V,N}}
=
{\displaystyle =}
(
∂
(
F
+
T
S
)
∂
T
)
V
,
N
=
−
S
+
S
(
∂
T
∂
T
)
V
,
N
+
T
(
∂
S
∂
T
)
V
,
N
{\displaystyle \left({\frac {\partial (F+\,T\,S)}{\partial T}}\right)_{V,N}=-\,S+\,S\left({\frac {\partial T}{\partial T}}\right)_{V,N}+\,T\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}}
(
T
,
V
,
N
)
:
∂
T
U
,
∂
T
F
,
∂
T
S
{\displaystyle \quad (T,V,N):\,\partial _{T}U,\,\partial _{T}F,\,\partial _{T}S\qquad }
{\displaystyle \quad }
−
T
(
∂
S
∂
T
)
V
,
N
{\displaystyle -T\,\left({\frac {\partial S}{\partial T}}\right)_{V,N}}
=
{\displaystyle =}
−
3
2
n
R
{\displaystyle -\,{\frac {3}{2}}n\,R}
∂
T
S
(
T
,
V
,
N
)
←
(
3.9.3.3.1
)
{\displaystyle \quad \partial _{T}S(T,V,N)\quad \leftarrow (3.9.3.3.1)}
(
3.11.1.3.3
)
{\displaystyle (3.11.1.3.3)\quad }
(
∂
U
∂
T
)
V
,
N
{\displaystyle \left({\frac {\partial U}{\partial T}}\right)_{V,N}}
=
{\displaystyle =}
+
3
2
n
R
{\displaystyle +\,{\frac {3}{2}}n\,R}
∂
T
U
(
T
,
V
,
N
)
{\displaystyle \quad \partial _{T}U(T,V,N)\qquad }
{\displaystyle \quad }
(
∂
A
∂
T
)
N
{\displaystyle \left({\frac {\partial A}{\partial T}}\right)_{N}}
=
{\displaystyle =}
+
3
2
n
R
{\displaystyle +\,{\frac {3}{2}}n\,R}
A
(
T
,
N
)
{\displaystyle \quad A(T,N)\qquad }
{\displaystyle \quad }
A
(
T
,
N
)
{\displaystyle A(T,N)}
=
{\displaystyle =}
+
3
2
n
R
T
{\displaystyle +\,{\frac {3}{2}}n\,R\,T}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
U
{\displaystyle U}
=
{\displaystyle =}
+
3
2
n
R
T
+
B
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T+\,B}
U
(
T
,
V
,
N
)
{\displaystyle \quad U(T,V,N)\qquad }
(
3.11.1.3.4
)
{\displaystyle (3.11.1.3.4)\quad }
U
(
T
0
,
V
0
,
N
)
{\displaystyle U(T_{0},V_{0},N)}
=
{\displaystyle =}
0
{\displaystyle 0}
{\displaystyle \quad \qquad }
{\displaystyle \quad }
A
+
B
{\displaystyle A+B}
=
{\displaystyle =}
+
3
2
n
R
T
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,T}
{\displaystyle \quad \qquad }
(
3.11.1.3.5
)
{\displaystyle (3.11.1.3.5)\quad }
U
{\displaystyle U}
=
{\displaystyle =}
3
2
n
R
T
{\displaystyle {\frac {3}{2}}\,n\,R\,T}
U
(
T
,
V
,
N
)
←
(
3.10.3.2.1
)
{\displaystyle \quad U(T,V,N)\qquad \leftarrow (3.10.3.2.1)}
![{\displaystyle {\frac {3}{5}}\left({\frac {p}{p_{0}}}\right)^{2/5}\exp \left({\frac {2}{5}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}+\,p_{0}\,V_{0}\,{\frac {dp}{p}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d05dc25d7b1d72c1a3e56ebf52e238f8555b79fb)
![{\displaystyle \left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)\left[+\,n\,R\,T_{0}\,{\frac {dS}{n\,R}}-\,p_{0}\,V_{0}\,{\frac {dV}{V}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6df2ca2772a5c2abafcc9b34573636951e4fa08)
