Chemistry Reference
In-Depth Information
where ∆
G
B
is the volume free energy per molecule,
m
is the number of molecules
in the nucleus,
r
is the nucleus radius, and
˃
is what Jacobs and Tompkins called the
strain energy per unit area. This is the energy associated with the local deformation
of the lattice due to the difference in the molecular volumes of A and B. Instead of
using the radius as a variable, Jacobs and Tompkins employ the number of mol-
ecules, which is linked to the radius as:
3
=
4
3
π
r
(4.66)
m
,
V
m
where
V
m
is the volume of a molecule. Then Eq. 4.65 turns into Eq. 4.67:
1
3
2
3
(4.67)
2
∆
GmG
=
∆
+
(
36
π
V
)
m
σ
.
B
m
From the condition of ∆
G
maximum, the critical nucleus size is:
3
2
σ
*
2
m
m
=−
36
π
V
.
(4.68)
3
G
∆
B
Substitution of
m
*
into Eq. 4.67 yields the height of the free energy barrier:
2
=
16
3
π
m
V
G
*
∆
G
.
(4.69)
2
(
∆
)
B
Jacobs and Tompkins did not introduce temperature dependence into ∆
G
*
. How-
ever, this can be easily done by assuming that at some temperature
T
0
the product
phase B can be at equilibrium with reactant phase A. Then one can follow the usual
reasoning (Eqs. 3.37 and 3.38) to arrive at an equation that establishes a temperature
dependence of the free energy barrier height:
2
2
16
VT
HTT
π
m
Z
T
*
0
∆
G
=
=
,
(4.70)
2
2
2
3
(
∆
) (
−
) ()
∆
B
0
where
Z
is a constant that collects all parameters that can be considered temperature
independent and ∆
T
=
T
−
T
0
is superheating. The temperature dependence of the
nucleation rate for the product phase B is obtained by substituting the resulting ∆
G*
into the Fisher-Turnbull [
99
] equation (Eq. 3.42) that yields:
−
Z
−
E
(4.71)
wT
(
)
=
w
exp
exp
D
.
0
(
)
2
RT
RT
∆
T
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