Chemistry Reference
In-Depth Information
The relation
N
=
C
(
γ
)
T
γ
+
r
ζ
(
γ
+ 1)
, where
γ
=
γ
(
T
r
)
, according to (27)
meets the linear relation
N
=
A
(
γ
)
T
r
, where
A
(
γ
) = (Λ
γ
c
−
γ
z
(
γ
))
1
/
(1+
γ
)
.
We can make the normalization of activity
a
at the point
T
c
, and we can find
a
0
by matching the liquid and gaseous branches at
T
c
for the pressure, in order
to prevent the phase transition on the critical isotherm at
T
r
= 1
.
In what follows, we normalize the activity for
T
r
<
1
with respect to the
value of
a
0
computed below. Then the chemical potentials (in thermodynam-
ics, the thermodynamic Gibbs potentials for the liquid and gaseous branches)
coincide, and therefore there can be no phase transition “gas-liquid” at
T
r
= 1
.
Now, for the isochore-isotherm of the “incompressible liquid” to take place,
we must construct it with regard to the relation
N
c
=
ζ
(
γ
c
+ 1)
, namely,
N
γ
=
C
(
γ
)
T
γ
+1
ζ
(
γ
+ 1)
,
(40)
r
where
γ
=
γ
(
T
r
)
according to (27), remains constant on the liquid isotherm.
We obtain the value
γ
=
γ
(
T
r
)
from the implicit equation
A
(
γ
) =
C
(
γ
(
T
r
))
T
γ
(
T
r
)
ζ
(
γ
(
T
r
) + 1)
.
r
Thus, for each
T
r
<
1
, we find the spinodal curve in the domain of negative
γ
[14],
Λ
(
γ
−
γ
c
)
/
(1+
γ
)
z
(
γ
)
1
/
(1+
γ
)
=
C
(
γ
(
T
r
))
T
γ
(
T
r
)
ζ
(
γ
(
T
r
) + 1)
,
(41)
r
We choose the least value of
γ
(it has the largest absolute value) which is
one of the two solutions of equation (41) and denote it by
γ
(
T
r
)
. In particular,
for
T
r
= 1
, we write
γ
0
=
γ
(1)
.
Let
a
g
=
e
−
µ/T
the activity of the
liquid. We present the condition for the coincidence of
M
and of the activities
at the point of the phase transition,
−
µ/T
be the activity of the gas and
a
l
=
e
a
l
a
0
C
(
γ
(
T
r
))
T
γ
(
T
r
)
|
γ
(
T
r
)|+
γ
c
T
−|
γ
(
T
r
)|
r
Li
2+
γ
(
T
r
)
(
a
g
) = Λ
Li
2−|
γ
(
T
r
)|
,
(42)
r
Λ
γ
c
−
γ
0
ζ
(2 +
γ
c
)
Li
2+
γ
0
(
a
0
) = 1
,
a
l
a
0
.
a
g
=
(43)
These two equations determine the value of the chemical potential
µ
=
µ
=
T
ln
a
g
at which the phase transition of the “ gas” into the “liquid” occurs.
