Entropy
edit
S
=
S
(
U
,
V
,
{
N
i
}
)
{\displaystyle S=S(U,V,\{N_{i}\})}
By the definition of a total differential,
d
S
=
∂
S
∂
U
d
U
+
∂
S
∂
V
d
V
+
∑
i
=
1
s
∂
S
∂
N
i
d
N
i
.
{\displaystyle dS={\frac {\partial S}{\partial U}}dU+{\frac {\partial S}{\partial V}}dV+\sum _{i=1}^{s}{\frac {\partial S}{\partial N_{i}}}dN_{i}.}
From the equations of state ,
d
S
=
1
T
d
U
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
.
{\displaystyle dS={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The differentials in the above equation are all of extensive variables , so they may be integrated to yield
S
=
U
T
+
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
.
{\displaystyle S={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right).}
Massieu potential / Helmholtz free entropy
edit
Φ
=
S
−
U
T
{\displaystyle \Phi =S-{\frac {U}{T}}}
Φ
=
U
T
+
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
−
U
T
{\displaystyle \Phi ={\frac {U}{T}}+{\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {U}{T}}}
Φ
=
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
{\displaystyle \Phi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}
Starting over at the definition of
Φ
{\displaystyle \Phi }
and taking the total differential, we have via a Legendre transform (and the chain rule )
d
Φ
=
d
S
−
1
T
d
U
−
U
d
1
T
,
{\displaystyle d\Phi =dS-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}
d
Φ
=
1
T
d
U
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
1
T
d
U
−
U
d
1
T
,
{\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU-Ud{\frac {1}{T}},}
d
Φ
=
−
U
d
1
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
.
{\displaystyle d\Phi =-Ud{\frac {1}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
d
Φ
{\displaystyle d\Phi }
we see that
Φ
=
Φ
(
1
T
,
V
,
{
N
i
}
)
.
{\displaystyle \Phi =\Phi ({\frac {1}{T}},V,\{N_{i}\}).}
If reciprocal variables are not desired,[3 ] : 222
d
Φ
=
d
S
−
T
d
U
−
U
d
T
T
2
,
{\displaystyle d\Phi =dS-{\frac {TdU-UdT}{T^{2}}},}
d
Φ
=
d
S
−
1
T
d
U
+
U
T
2
d
T
,
{\displaystyle d\Phi =dS-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}
d
Φ
=
1
T
d
U
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
1
T
d
U
+
U
T
2
d
T
,
{\displaystyle d\Phi ={\frac {1}{T}}dU+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {1}{T}}dU+{\frac {U}{T^{2}}}dT,}
d
Φ
=
U
T
2
d
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
,
{\displaystyle d\Phi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}
Φ
=
Φ
(
T
,
V
,
{
N
i
}
)
.
{\displaystyle \Phi =\Phi (T,V,\{N_{i}\}).}
Planck potential / Gibbs free entropy
edit
Ξ
=
Φ
−
P
V
T
{\displaystyle \Xi =\Phi -{\frac {PV}{T}}}
Ξ
=
P
V
T
+
∑
i
=
1
s
(
−
μ
i
N
T
)
−
P
V
T
{\displaystyle \Xi ={\frac {PV}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)-{\frac {PV}{T}}}
Ξ
=
∑
i
=
1
s
(
−
μ
i
N
T
)
{\displaystyle \Xi =\sum _{i=1}^{s}\left(-{\frac {\mu _{i}N}{T}}\right)}
Starting over at the definition of
Ξ
{\displaystyle \Xi }
and taking the total differential, we have via a Legendre transform (and the chain rule )
d
Ξ
=
d
Φ
−
P
T
d
V
−
V
d
P
T
{\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-Vd{\frac {P}{T}}}
d
Ξ
=
−
U
d
2
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
P
T
d
V
−
V
d
P
T
{\displaystyle d\Xi =-Ud{\frac {2}{T}}+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-Vd{\frac {P}{T}}}
d
Ξ
=
−
U
d
1
T
−
V
d
P
T
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
.
{\displaystyle d\Xi =-Ud{\frac {1}{T}}-Vd{\frac {P}{T}}+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}.}
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
d
Ξ
{\displaystyle d\Xi }
we see that
Ξ
=
Ξ
(
1
T
,
P
T
,
{
N
i
}
)
.
{\displaystyle \Xi =\Xi \left({\frac {1}{T}},{\frac {P}{T}},\{N_{i}\}\right).}
If reciprocal variables are not desired,[3 ] : 222
d
Ξ
=
d
Φ
−
T
(
P
d
V
+
V
d
P
)
−
P
V
d
T
T
2
,
{\displaystyle d\Xi =d\Phi -{\frac {T(PdV+VdP)-PVdT}{T^{2}}},}
d
Ξ
=
d
Φ
−
P
T
d
V
−
V
T
d
P
+
P
V
T
2
d
T
,
{\displaystyle d\Xi =d\Phi -{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}
d
Ξ
=
U
T
2
d
T
+
P
T
d
V
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
−
P
T
d
V
−
V
T
d
P
+
P
V
T
2
d
T
,
{\displaystyle d\Xi ={\frac {U}{T^{2}}}dT+{\frac {P}{T}}dV+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i}-{\frac {P}{T}}dV-{\frac {V}{T}}dP+{\frac {PV}{T^{2}}}dT,}
d
Ξ
=
U
+
P
V
T
2
d
T
−
V
T
d
P
+
∑
i
=
1
s
(
−
μ
i
T
)
d
N
i
,
{\displaystyle d\Xi ={\frac {U+PV}{T^{2}}}dT-{\frac {V}{T}}dP+\sum _{i=1}^{s}\left(-{\frac {\mu _{i}}{T}}\right)dN_{i},}
Ξ
=
Ξ
(
T
,
P
,
{
N
i
}
)
.
{\displaystyle \Xi =\Xi (T,P,\{N_{i}\}).}