(
01
)
{\displaystyle (01)\qquad }
U
{\displaystyle U}
=
{\displaystyle =}
3
2
N
k
T
{\displaystyle {\frac {3}{2}}N\,k\,T}
=
{\displaystyle =}
3
2
n
R
T
{\displaystyle {\frac {3}{2}}n\,R\,T}
U
(
T
)
{\displaystyle \qquad U(T)\qquad }
{\displaystyle \qquad \qquad }
p
V
{\displaystyle p\,V}
=
{\displaystyle =}
N
k
T
{\displaystyle N\,k\,T}
=
{\displaystyle =}
n
R
T
{\displaystyle n\,R\,T}
(
p
V
)
(
T
)
{\displaystyle \qquad (p\,V)(T)\qquad }
{\displaystyle \qquad \qquad }
d
U
{\displaystyle dU}
=
{\displaystyle =}
+
T
d
S
−
p
d
V
{\displaystyle +\,T\,dS-p\,dV}
=
{\displaystyle =}
+
T
d
S
−
n
R
T
V
d
V
{\displaystyle +\,T\,dS-{\frac {n\,R\,T}{V}}\,dV}
d
U
(
S
,
V
)
,
d
N
=
0
{\displaystyle \qquad dU(S,V),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
d
S
{\displaystyle dS}
=
{\displaystyle =}
1
T
d
U
+
n
R
V
d
V
{\displaystyle {\frac {1}{T}}dU+{\frac {n\,R}{V}}\,dV}
=
{\displaystyle =}
+
3
2
n
R
T
d
T
+
n
R
V
d
V
{\displaystyle +\,{\frac {3}{2}}\,{\frac {n\,R}{T}}\,dT+{\frac {n\,R}{V}}\,dV}
d
S
(
T
,
V
)
,
d
N
=
0
{\displaystyle \qquad dS(T,V),\,dN=0\qquad }
{\displaystyle \qquad \qquad }
(
∂
S
∂
T
)
V
,
N
{\displaystyle \left({\frac {\partial S}{\partial T}}\right)_{V,N}}
=
{\displaystyle =}
+
3
2
n
R
T
{\displaystyle +\,{\frac {3}{2}}\,{\frac {n\,R}{T}}}
{\displaystyle }
{\displaystyle }
S
(
T
,
V
,
N
)
{\displaystyle \qquad S(T,V,N)\qquad }
(
02
)
{\displaystyle (02)\qquad }
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}
=
{\displaystyle =}
S
0
−
n
R
{\displaystyle S_{0}-\,n\,R}
S
(
T
,
V
,
N
)
{\displaystyle \qquad S(T,V,N)\qquad }
{\displaystyle \qquad \qquad }
(
∂
S
∂
V
)
T
,
N
{\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
+
n
R
V
{\displaystyle +\,{\frac {n\,R}{V}}}
{\displaystyle }
{\displaystyle }
S
(
T
,
V
,
N
)
{\displaystyle \qquad S(T,V,N)\qquad }
(
03
)
{\displaystyle (03)\qquad }
V
(
∂
S
∂
V
)
T
,
N
{\displaystyle V\,\left({\frac {\partial S}{\partial V}}\right)_{T,N}}
=
{\displaystyle =}
+
n
R
{\displaystyle +\,n\,R}
{\displaystyle }
{\displaystyle }
S
(
T
,
V
,
N
)
{\displaystyle \qquad S(T,V,N)\qquad }
(
04
)
{\displaystyle (04)\qquad }
∂
∂
V
∂
S
∂
T
{\displaystyle {\frac {\partial }{\partial V}}{\frac {\partial S}{\partial T}}}
=
{\displaystyle =}
∂
∂
T
∂
S
∂
V
{\displaystyle {\frac {\partial }{\partial T}}{\frac {\partial S}{\partial V}}}
=
{\displaystyle =}
0
{\displaystyle 0}
S
(
T
,
V
,
N
)
{\displaystyle \qquad S(T,V,N)\qquad }
(
05
)
{\displaystyle (05)\qquad }
d
S
{\displaystyle dS}
=
{\displaystyle =}
+
3
2
n
R
d
T
T
+
n
R
d
V
V
{\displaystyle +\,{\frac {3}{2}}\,n\,R\,{\frac {dT}{T}}+n\,R\,{\frac {dV}{V}}}
=
{\displaystyle =}
+
3
2
n
R
d
(
T
/
T
0
)
T
/
T
0
+
n
R
d
(
V
/
V
0
)
V
/
V
0
{\displaystyle +\,{\frac {3}{2}}n\,R\,{\frac {d(T/T_{0})}{T/T_{0}}}+n\,R{\frac {d(V/V_{0})}{V/V_{0}}}}
d
S
(
T
,
V
)
,
d
N
=
0
{\displaystyle \qquad dS(T,V),\,dN=0\qquad }
(
06
)
{\displaystyle (06)\qquad }
S
{\displaystyle S}
=
{\displaystyle =}
+
3
2
n
R
ln
T
T
0
+
n
R
ln
V
V
0
+
S
0
{\displaystyle +\,{\frac {3}{2}}n\,R\,\ln {\frac {T}{T_{0}}}+n\,R\,\ln {\frac {V}{V_{0}}}+\,S_{0}}
=
{\displaystyle =}
+
n
R
ln
(
T
T
0
)
3
/
2
(
V
V
0
)
+
S
0
{\displaystyle +\,n\,R\,\ln \left({\frac {T}{T_{0}}}\right)^{3/2}\left({\frac {V}{V_{0}}}\right)+\,S_{0}}
S
(
T
,
V
,
N
)
,
S
0
(
T
0
,
V
0
,
N
)
{\displaystyle \qquad S(T,V,N),\,S_{0}(T_{0},V_{0},N)\qquad }
{\displaystyle \qquad \qquad }
G
{\displaystyle G}
=
{\displaystyle =}
U
−
S
T
+
V
p
{\displaystyle U-\,S\,T+\,V\,p}
{\displaystyle }
{\displaystyle }
(
S
,
V
,
N
)
:
G
,
U
,
T
,
p
{\displaystyle \qquad (S,V,N):\,G,\,U,\,T,\,p\qquad }
{\displaystyle \qquad \qquad }
G
{\displaystyle G}
=
{\displaystyle =}
U
−
T
S
+
V
p
{\displaystyle U-\,T\,S+\,V\,p}
{\displaystyle }
{\displaystyle }
(
T
,
V
,
N
)
:
G
,
U
,
T
,
p
{\displaystyle \qquad (T,V,N):\,G,\,U,\,T,\,p\qquad }
{\displaystyle \qquad \qquad }
0
{\displaystyle 0}
=
{\displaystyle =}
U
−
T
S
+
V
p
{\displaystyle U-\,T\,S+\,V\,p}
{\displaystyle }
{\displaystyle }
(
T
0
,
V
0
,
N
)
:
U
,
S
,
p
;
G
=
0
{\displaystyle (T_{0},V_{0},N)\,:\,U,\,S,\,p\,;\,G=0\qquad }
(
07
)
{\displaystyle (07)\qquad }
S
0
{\displaystyle S_{0}}
=
{\displaystyle =}
U
0
+
V
0
p
0
T
0
{\displaystyle {\frac {U_{0}+\,V_{0}\,p_{0}}{T_{0}}}}
=
{\displaystyle =}
+
5
2
n
R
{\displaystyle +\,{\frac {5}{2}}\,n\,R}
(
T
0
,
V
0
,
N
)
:
S
,
U
,
p
;
G
=
0
{\displaystyle (T_{0},V_{0},N)\,:\,S,\,U,\,p\,;\,G=0\qquad }
(
08
)
{\displaystyle (08)\qquad }
1
{\displaystyle 1}
=
{\displaystyle =}
(
T
0
T
)
3
/
2
(
V
0
V
)
exp
(
S
−
S
0
n
R
)
{\displaystyle \left({\frac {T_{0}}{T}}\right)^{3/2}\left({\frac {V_{0}}{V}}\right)\exp \left({\frac {S-S_{0}}{n\,R}}\right)}
=
{\displaystyle =}
(
T
0
T
)
(
V
0
V
)
2
/
3
exp
(
2
3
S
−
S
0
n
R
)
{\displaystyle \left({\frac {T_{0}}{T}}\right)\left({\frac {V_{0}}{V}}\right)^{2/3}\exp \left({\frac {2}{3}}{\frac {S-S_{0}}{n\,R}}\right)}
V
(
S
,
T
,
N
)
,
T
(
S
,
V
,
N
)
{\displaystyle \qquad V(S,T,N),T(S,V,N)\qquad }
