Page 154 - Science at the nanoscale

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June 5, 2009
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Formation and Self-Assembly at the Nanoscale
At equilibrium:
Solute A + Solvent B
A(Solvated)
A(Solid)
Solid A
(Solute)
ˇ µ
(Solute)
µ
(Solid)
(Solid)= µ ˇ A
A
µ
A
A
Figure 7.2.
Schematic of liquid solute/solid equilibrium.
a pure substance is normally taken as unity. The activity of the
solvent is also often assumed to be unity in dilute solution.
Equation 7.5 thus provides a relationship between the
Gibbs energy and the concentration for a species in solution.
Alternatively, we can also express this relationship in terms of the
equilibrium constant K eq :
a A (l)
(7.6)
= a A (l)
K eq =
a A (s)
o
(7.7)
∆G = ∆G + RT ln K eq
o
∆G is defined as the Gibb energy at standard states (T = 298.15 K;
−3
P = 101,325 Pa) for concentration at 1 mol dm
.
Gibbs Energy at the Nanoscale
7.1.4
In Chapter 5, we have learnt that particles at the nanoscale pos-
sess a high surface energy (γ) due to their large surface-to-volume
ratios. This substantial amount of surface energy is expected to
contribute significantly to the Gibbs energy of the nanoparticles. ch07
The effect is most evidently observed in the reduction of melting
temperatures, which has been found to reduce with the decreas-
2
ing radius of the nanoparticles. While a quantitative derivation
of the contribution of γ to ∆G remains a topic under extensive
research, we can at least provide some qualitative considerations
as described below.
Let us begin by examining the effect of size on the shift of
a chemical equilibrium between reagent particles A to prod-
j
uct particles B . The equilibrium constant at constant P and T
k
2 P. Buffat, J.-P. Borel, Phys. Rev. A, 13, (1976) 2287.