Moreover, the errors vary with the kind of phase transition. Nevertheless,
substantial homogeneous nucleation does only occur at very strong
supersaturation, and it depends very steeply on temperature or concentra-
tion, as predicted. As a rule of thumb, the values forrhomare generally
about 1 or 2 nm, and the nucleus then generally contains between 50 and 250
molecules.
Question
Consider air saturated with water vapor at 25 8 C. To what temperature should the air
be cooled before water droplets would spontaneously form? The enthalpy of
evaporation of water equals about 2:45 GJ?m^3 and the surface tension about
75 mN?m^1 at low temperature.
Answer
Assuming homogeneous nucleation of water droplets, we need to findThom. Further
assumingrhomto be equal to the value observed for ice nucleation, i.e., 1.85 nm
(Table 14.2), Eq. (14.6) directly yields Thom¼ 158 C. This would be a small
undercooling.
In practice, however, homogeneous nucleation is observed to occur at about
58 C, at a supersaturation lnb& 1 :43. Inserting the latter value in Eq. (14.9), we derive
rhom¼0.82 nm, which corresponds to a nucleus containing 75 water molecules, and
which is far smaller than an ice nucleus. Equation (14.6) in fact shows that for a
smaller value ofg/DHV,rcrwill be smaller at the same temperature ratio. For water
nucleation to occur, this ratio is therefore much smaller than for ice nucleation (cf.
the values in Table 14.2). It is likely that also the value ofrhomis smaller. Hence it is
not allowed to take a fixed value forrhomand to derive results from it.
14.2.2 Heterogeneous Nucleation
In the previous section is was stated that the (homogeneous) nucleation
temperature of ice in pure water is about 408 C. It is common experience,
however, that ice starts to form at far higher temperatures, sometimes even
at 18 C. How is this to be explained? The following experiment may be
enlightening.
A suitable substance, e.g., a triacylglycerol, is melted, and above the
melting temperature it is emulsified into an aqueous solution with a suitable
surfactant (say, a protein). The emulsion then is brought under a microscope
and slowly cooled. As soon as a droplet crystallizes, which can be seen when