Classical Nucleation Theory - Aerosols: Measurement, Dosimetry, and Health Effects - page 523

where ψ and T 0 are parameters determined by itting Equation 19.67 to the dependencies of surface
tension on temperature available from reference topics (see, e.g., [49,50]). The temperature depen-
dence of saturated vapor pressure is given by the formula:

D
T

log 10 ( P

Sat Torr =

( )) C − (19.68)

where C and D are parameters determined by itting Equation 19.68 to the reference topics data on
saturated vapor pressure. The sound velocity was evaluated using Equation 19.63. The values of
σ S /σ ∞ as determined from the solution of Equations 19.61 through 19.64 are plotted in Figure 19.6
for different temperatures. To avoid confusion we presented three points for each metal, correspond-
ing to the ends and middle of the experimental temperature range. The points in Figure 19.6 are
not dependences σ S /σ ∞ versus R S , but there are three independent values σ S /σ ∞ for each metal cor-
responding to three different temperatures and three values of R S . One can see from Figures 19.4
and 19.6 that all the metals considered can be divided into two groups Li, Na, Cs, Ag (monovalent
metals) and Mg, Zn, Hg (bivalent metals). The monovalent metals are characterized by the ratio
σ S /σ ∞ > 1. The bivalent metals Zn, Hg, Mg show σ S /σ ∞ < 1. It is important to note that in all the cases
the surface tension for drops of radius about 1 nm differs essentially from that of the lat surface,
that is, the surface tension is a strong function of size.
Table 19.2 summarizes important parameters which were determined from the experimental
nucleation rate, supersaturation, and temperature by solution Equations 19.60/19.61 and 19.64. One
can see that normally N S is about a few tens of atoms, but for Ag it is 4 ≤ N S ≤ 7. Nevertheless, the
Gibbs theory is rigorous even for such a small value N S (because it is rigorous for the radius R S
whatever small). Nevertheless, Equations 19.60 and 19.61 can be considered as of less accuracy in
the case of Ag with respect to those in the case of other metals, because Equations 19.3 and 19.4 for
the Zeldovitch factor were derived under the assumption that the critical nucleus contains not less
than 10-30 molecules [1]. However, even if the accuracy of Equations 19.60 and 19.61 in the case of
Ag would be 50% it gives the error in the drop surface tension as low as 0.4%.
As seen from Table 19.2 the translational correction factor Φ R / N 1 is about 10 6 -10 7 for the major-
ity of metals presented in Table 19.2 and only for Ag it is as high as 10 11 . The reason in such a high
value of this factor for Ag is in low concentration n 1 for the supersaturated vapor which is three to
ive orders of magnitude less than that for the other metals. The rotational correction factor Q ro K is

2.0

875

971

Li

T = 588 K

1033

Na

1.5

689

Ag

757

646

606

556

1.0

Hg

Mg

268

734

794

875

340

Zn

370

0.5

600 661

725

0.2 0.3 0.4 0.5 0.6

0.7 0.8

R s (nm)

FIGURE 19.6 σ S /σ ∞ and R S as calculated by solution of Equations 19.61 and 19.64 using the experimental
measurements as shown in Figure 19.5. Three points were selected for each metal corresponding to the ends
and middle of the experimental temperature range. Temperature is shown for each point in the plot.