Chemistry Reference
In-Depth Information
where n denotes the number of molecules of gas A at pressure P involved in the
process, then in the absence of surface tension effects, the free-energy change of
the process will be given by
nkT ln
P
P
0
G
¼
ð
8
:
11
Þ
where P is the pressure or activity of A in the vapor phase and P
0
is that in the liquid
phase. The ratio P
=
P
0
is often referred to as the degree of supersaturation of the
system. A liquid drop of radius r will have a surface energy equal to 4pr
2
s, so that
the actual free-energy change on drop formation will be
nkT ln
P
4pr
2
s
¼
P
0
þ
ð
:
Þ
G
8
12
Both elements to the right in Eq. (8.12) can be written in terms of the drop radius r.
If r is the density of the liquid and M its molecular weight, the equation becomes
4
3
pr
3
r
M
RT ln
P
4pr
2
s
G
¼
P
0
þ
ð
8
:
13
Þ
where the two terms are of opposite sign and have a different dependence on r.A
plot of
G versus r exhibits a maximum as illustrated in Figure 8.8 for a hypothe-
tical material with a density of one, molar volume of 20, and pressure or activity
ratio of 4 at a given temperature. The radius at which the plot is a maximum may be
defined as the critical radius, r
c
, which can be determined from Eq. (8.13) by setting
25
20
15
10
5
0
0.2.4 .6
.8
1.0
1.2
1.0
1.6
Figure 8.8.
In nucleation processes there will be a critical particle radius r
c
below which
free-energy considerations will cause the incipient aerosol particle to evaporate; above r
c
,
particle growth will occur.
