Environmental Engineering Reference
In-Depth Information
σ is to be evaluated by Equation 19.48. To estimate the Debye frequency one may use the following
formula [1]: ν
max
≈
u
/2
r
, 2
r
is the molecule diameter and
u
is the sound velocity in the bulk liquid
reference phase.
Let us compare our correction factor Φ/
N
1
as estimated by Equation 19.59 for
Ar
with that as
determined by Kusaka (Φ
K
/
N
1
) numerically by Equation 19.44 [11].
Figure 19.1 shows the correction factor Φ/
N
1
as calculated [38] by Equation 19.59 for
R
S
= 2.0
(in Lennard-Jones units) and supersaturation ratio
S
= 2.0 as well as the contributions to this factor
from Φ
R
/
N
1
,
Q
ro
K
, and (
Q
v
)
−3
. The important observation is that the contribution from the transla-
tional degrees of freedom Φ
R
/
N
1
is in the range 3 × 10
2
− 10
4
, while the main contribution comes from
the rotational degrees of freedom
Q
ro
K
which is about 3 × 10
7
in the temperature range 80-110 K.
The vibrational contribution (
Q
v
)
−3
is small enough being in the range 1-8, which corresponds to the
range of
Q
v
from 1 to 0.5. Such a small magnitude of the vibrational partition function corresponds
to the condition
k
B
T < h
ν
max
, that is, to the low-temperature limit where the quantum nature of the
oscillator is to be accounted; that is why the Equation 19.53 was written in the quantum form.
Figure 19.2 compares the correction factor Φ/
N
1
and that of Kusaka Φ
K
/
N
1
for
R
S
= 1.8, 2.0, and
2.5. One can see that these two factors are in a good agreement with each other. The temperature
10
12
10
11
10
10
10
9
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
Φ/
N
1
K
Q
rot
Φ
R
/
N
1
(
Q
v
)
-3
80
90
100
110
T
(K)
FIGURE 19.1
Correction factor Φ/
N
1
(Equation 19.59) and contributions from Φ
R
/
N
1
(Equations 19.35 and
19.48),
Q
ro
K
(Equation 19.50), and (
Q
v
)
−3
(Equation 19.53) versus temperature for
R
S
= 2.0 (in Lennard-Jones
units),
S
= 2.0.
10
13
10
12
R
s
=
2.5
10
11
R
s
=
2.0
10
10
R
s
=
1.8
10
9
80
90
100
110
T
(K)
FIGURE 19.2
Free energy correction factor for different values of the drop radius
R
S
versus temperature.
Symbols, Φ
K
/
N
1
(Numerical calculations by Kusaka, I.,
Phys. Rev. E
., 73, 031607, 2006.); lines, Φ/
N
1
(as evalu-
ated by Equation 19.59). The magnitudes of
R
S
are shown in Lennard-Jones units.
Search WWH ::
Custom Search