Chemistry Reference
In-Depth Information
Consider a virtual gas whose critical temperature, denoted by
T
c
=
T
tr
, co-
incides with the temperature of the triple point of the given gas. Let us consider
two isotherms: the critical isotherm of the virtual gas and the isotherm of the
given gas for
T
=
T
tr
.
From relation (25), for a fixed
C
(
γ
)
with
T
p
=
T
tr
, we can find
γ
tr
(the
value of
γ
at the triple point). For known values of
Z
tr
and
P
tr
, using formula
(13), we obtain the value of
C
(
γ
tr
)
. Thereby we find on the
(
P, Z
)
diagram the
terminal point of the spinodal (24) at which the derivative of the quarter of the
ellipse vanishes. We have found this by using the experimental values of the
two endpoints of the quarter of the ellipse, which is our gas spinodal. Thus, the
positions of the
critical point
and the
triple point
(well known from standard
tables)
entirely determine
the isotherms of the given gas.
It should be noted that the theory constructed above did not take into account
the interaction of particles anywhere.
As was already pointed out in the author's papers, it is more natural to pass
from momenta
p
to energies
ε
=
p
2
/
2
m
and to generalize the problem of the
transition of the energy
p
2
/
2
m
to the energy
H
=
p
2
/
2
m
+
c/r
k
, i.e., to the
Hamiltonian containing repulsion.
In this case, the following correspondence is used:
p
2
2
m
→
p
2
2
m
1
e
p
2
/
2
m
−
µ/T
−
1
→
1
c
r
k
,
+
e
(
p
2
/
2
m
+
c/r
k
−
µ
)
β
−
1
,
(29)
where
β
= 1
/T
.
Here integration of the distribution corresponding to the number of particles
must be performed with respect to the measure
p
γ
dp r
2
dr
(integration over
angles is omitted). Integration of the pressure distribution must be performed
with respect to the measure
H p
γ
dp r
2
dr
over the three-dimensional volume
V
.
5.
Supercritical State as the Parastatistics of Clusters
When
T
r
>
1
, in order to determine
C
(
γ
)
, we use parastatistics in which
µ
varies from minus infinity to plus infinity. In this case the isotherm issued on the
(
Z.P
)
plane from the point
Z
= 1
, P
= 0
comes to the line
Z
= 1
for certain
positive
µ
.
The value of
K
(
T
r
, γ
)
satisfies the condition
γ
1
1 +
K
N
(
γ
c
=
C
(
γ
)
ζ
(1 +
γ
) =
T
1+
γ
C
(
γ
)
1
−
ζ
(1 +
γ
)
.
(30)
r
