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.
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