Chemistry Reference
In-Depth Information
the difference between the change in free energy in forming a crystal with a surface (
∆
g
)
and that of forming an infinite crystal (
∆
g
B
b
B
). In fact, starting with Equations (8a) and (8b),
we have
r
P
2
P
α
= (
∆
g
−∆
g
B
b
B
)/4
π
(15a)
r
P
2
P
=
∆
g
B
s
B
/4
π
(15b)
The free energy terms,
∆
g
B
i
B
, are themselves comprised of two terms: the change in
enthalpy,
s
B
i
B
). Now let's examine
each of the terms. The statement that a phase is less stable is equivalent to one stating that
∆
∆
h
B
i
B
, and the change in entropy,
∆
s
B
i
B
, i.e.
,
∆
g
B
i
B
=
∆
h
B
i
B
-
T
∆
g
B
b
B
is smaller. That this should be so makes sense because the poorer bonding of the
more disordered phase ensures that
h
B
b
B
will be smaller for its formation, and the higher
level of disorder has the same impact on
∆
∆
s
B
b
B
, as illustrated in Figure 9. Therefore, the
only way that
α
can become smaller is for the total change of free energy to decrease by
even more, i.e.,
g
B
s
B
must decrease as well. Equation (15b) reflects this. As with the bulk,
since the surface of the less stable phase is likely to be more disordered than the surface
of the most stable phase, both
∆
s
B
s
B
are likely to be smaller. Nonetheless, this is not
a proof, and the Ostwald-Lussac law of phases should only be viewed as a guiding
principle rather than a statement of fact.
The nucleus shape
Equations (8)
∆
h
B
s
B
and
∆
(14) above were derived under the assumption of a spherical shape of
the nucleus. If the nucleus has any other shape, obviously, the coefficients included in the
parameters A and B in Equation (14) will take different values, but the overall
conclusions on nucleation kinetics, it appears, will not change significantly. There is an
important caveat in the latter statement, and it is in the assumption that we have a means
to predict the nucleus shape in advance. In some cases, this is indeed so. Thus, thinking
of a fluid phase nucleating within another fluid, Gibbs suggested that the shape of the
nuclei is the one ensuring the lowest free energy of the nuclei, i.e., the sphere (Gibbs
1876, 1878). Accounting for crystal anisotropy and applying the free energy
minimization selection criterion, one comes up with cubic or other faceted shapes and
only slight modifications to Equations (8)
−
−
(14).
Figure 9.
Schematic illustrating the
relative contributions of enthalpy and
entropy changes to the free energy
change upon crystallization into the
most stable and a less stable phase.
