potentials will be equal. Thus
μ
L
particle = μ
S
particle
(2.117)
or
μ
L
bulk +
2g
L
V
L
r
= μ
S
bulk +
2g
S
V
S
r
(2.118)
Rearranging
μ
L
bulk − μ
S
bulk =
2
r
g
L
V
L
− g
S
V
S
À
Á
(2.119)
Now the change in the chemical potential μ
L
bulk − μ
S
bulk is the Gibbs energy
corresponding to the phase change (Equation 2.120):
ΔG trans = μ
L
bulk − μ
S
bulk
(2.120)
From earlier sections we know that ΔG trans = ΔH trans − TΔS trans and so
ΔS trans = ΔH trans /T trans , in which we can write
ΔG trans = ΔH trans −
TΔH trans
T trans
(2.121)
In terms of our transition, we have
ΔG trans = ΔH trans −
TΔH trans
T mpt
= ΔH trans 1 −
T
T mpt
(2.122)
Therefore, from Equation 2.119, we have
ΔH trans 1 −
T
T mpt
=
2
r
g
L
V
L
− g
S
V
S
À
Á
(2.123)
Rearranging the above equation gives Equation 2.124:
T
T mpt
= 1 −
1
r
2
ΔH trans
g
L
V
L
− g
S
V
S
À
Á
(2.124)
It is should be noted that in the above equation T refers to the melting
temperature of the nanoparticle of some radius r, where T mpt refers to the
bulk phase melting point. We see that this equation has the exact same
functional form as Equation 2.7
The total Gibbs energy change for nanoparticle melting transition can be
written according to Equation 2.125:
CHAPTER 2: Thermodynamics and Nanoscience
58