14:440:407 Section 02 Fall 2010 SOLUTION OF HOMEWORK 07
Question 10.2:
(a) Rewrite the expression for the total free energy change for nucleation (Equation 10.1) for
the case of a cubic nucleus of edge length a (instead of a sphere of radius r). Now differentiate this expression
with respect to a (per Equation 10.2) and solve for both the critical cube edge length, a*, and also
Δ
G*.
(b) Is
Δ
G* greater for a cube or a sphere? Why?
Solution:
(a) This problem first asks that we rewrite the expression for the total free energy change for nucleation
(analogous to Equation 10.1) for the case of a cubic nucleus of edge length
a
.
The volume of such a cubic radius is
a
3
, whereas the total surface area is 6
a
2
(since there are six faces each of which has an area of
a
2
).
Thus, the
expression for
Δ
G
is as follows:
Δ
G
=
a
3
Δ
G
v
+
6
a
2
γ
Differentiation of this expression with respect to
a
is as
d
Δ
G
da
=
d
(
a
3
Δ
G
v
)
da
+
d
(
6
a
2
γ
)
da
=
3
a
2
Δ
G
v
+
12
a
γ
If we set this expression equal to zero as
3
a
2
Δ
G
v
+
12
a
γ =
0
and then solve for
a
(=
a
*), we have
a
* =
−
4
γ
Δ
G
v
Substitution of this expression for
a
in the above expression for
Δ
G
yields an equation for
Δ
G
* as
Δ
G
* = (
a
*)
3
Δ
G
v
+
6(
a*
)
2
γ

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