68 Liquid-gas and liquid-liquid interfaces
pressure p 0 to the droplet with a vapour pressure pr, the free energy
increase is also equal to dnRT In pr/po, assuming ideal gaseous
behaviour. Equating these free energy increases,dnRT In pr I po = 8 TTJ r drand sincedn = 4irr^2 dr p I Mthen(^2) 2^ (4.4)
pr r
where p is the density of the liquid, Vm is the molar volume of the
liquid and M is the molar mass. For example, for water droplets
(assuming y to be constant),
r = l(T^7 m pJpQ-l.Ol
l(T^8 m 1.1
l(T^9 m 3.0
This expression, known as the Kelvin equation, has been verified
experimentally. It can also be applied to a concave capillary
meniscus; in this case the curvature is negative and a vapour pressure
lowering is predicted (see page 125).
The effect of curvature on vapour pressure (and, similarly, on
solubility) provides a ready explanation for the ability of vapours
(and solutions) to supersaturate. If condensation has to take place via
droplets containing only a few molecules, the high vapour pressures
involved will present an energy barrier to the process, whereas in the
presence of foreign matter this barrier can be by-passed.
An important example of this phenomenon is to be found in the
ageing of colloidal dispersions (often referred to as Ostwald ripening).
In any dispersion there exists a dynamic equilibrium whereby the rates
of dissolution and deposition of the dispersed phase balance in order
that saturation solubility of the dispersed material in the dispersion
medium be maintained. In a polydispersed sol the smaller particles will
have a greater solubility than the larger particles and so will tend to
dissolve, while the larger particles will tend to grow at their expense. In