This is illustrated in Figure 14.1b, where tany¼DtrH/Teq. It has been
implicitly assumed thatDHandDSare independent of temperature; for
most systems this is about correct if the temperature range is small (say, a
few times 10K). Values ofDtrHcan generally be obtained by calorimetry.
(Note that in the present case (Fig. 14.1,a!b) the values ofDHandDSare
both negative; for the transitionb!a, they would both be positive, with
TDS>DH.)
It thus appears at first that at temperatures even slightly belowTeq,
phasebwill spontaneously form. This however is not the case, since the
formation of a region of another phase introduces a phase boundary, which
goes along with a specific interfacial free energygab. This introduces a term
DGSthat is positive. For a very small region of the new phase—which is
called an embryo—the surface term will generally be dominant. This is
becauseDtrGis proportional to volume (pd^3 /6) andDGSproportional to
surface area (pd^2 /4). An example may illustrate this.
In the freezing of water,DtrH¼330 MJ?m^3 ; the interfacial tension
gbetween water and ice¼25 mN?m^1 (Table 10.1). Assuming these values
to be correct at 108 C, and considering a spherical embryo of ice of
diameter 5 nm (which would contain about 2000 molecules), we calculate
from Eq. (14.3) forDtrGa value of 8? 10 ^19 J and forDGS 2? 10 ^18 J. This
means that formation of the embryo would cause an increase in free energy
by 12? 10 ^19 J, or about 300 timeskBT, which is extremely unlikely to occur.
Even if such an embryo were present, it would immediately dissolve, since
that will decrease the total free energy. An embryo that can and does grow
to form a new phase is called anucleus. To obtain a nucleus, a much deeper
undercooling (supercooling) would be needed. This is further discussed in
Section 14.2.1.
The reasoning given above applies to most kinds of phase transition,
for instance also forb?ain the same system. An overview (not exhaustive)
is given in Table 14.1; it also specifies the equilibrium temperature and the
transition enthalpy. It should be added that it here concerns so-calledfirst-
order (phase) transitions. In section 16.1 the difference between first- and
second-order transitions will be discussed.
The formation of a new phaseinside a solid phaseis very difficult,
because the transition generally implies a change in density, hence in
volume. This leads to a change in pressure, and thereby to an additional,
generally large and positive, term in free energy for nucleus formation.
Except for some solid!solid transitions where the change in density is
small, this tends to prevent nucleation; any formation of a new phase will
occur at the boundary of the system. Sublimation then does not need
nucleation: the new phase (gas) is already present. The same holds for a
solid phase (crystals) in a solution. When crystals in air tend to melt, a liquid
singke
(singke)
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