## Box 142

KELVIN EFFECT, VAPOR PRESSURE LOWERING, AND THE KÖHLER CURVES

1. Kelvin effect. Recall that the change in free energy of a gas due to a change in pressure (at constant T) from Pt to P2 is given by

Consider the situation in Fig. 14.37 in which a number of moles dn is transferred from a bulk liquid with vapor pressure P{) to a droplet of the same liquid having radius r and over which the vapor pressure is P. Assuming each system is at equilibrium, the free energies of the liquid and gas in each case must be equal. The change in free energy, dG, for transferring dn moles from the bulk liquid to the droplet is therefore the same as the accompanying free energy change for the gas. From Eq. (X), this is given by dG = dnRT\n(P/P{]). (Y)

However, there is also a free energy change due to the increase in surface area, dA, of the droplet caused by the transfer of the dn moles. The surface tension of the liquid, y, is the work required per unit change in surface area to expand a surface against the intermolecular forces that tend to minimize the surface. The area of the initial droplet is 4irr2 and hence the change due to a small change dr caused by transfer of dn moles is 8irr dr. The free energy change in the droplet due to an increase in its surface area is therefore dG = y dA = 87Try dr. (Z)

This transfer of dn moles causes a volume change dV = Airr2 dr. If the molecular weight of the compound is MW and liquid density is p, then p = (MW)dn/(4vr2 dr). This expression can be used to

Po rb 9

FIGURE 14.37 Basis of Kelvin effect for increased vapor pressure over small liquid droplets.

Combining Eqs. (Y) and (AA) gives the Kelvin equation:

This Kelvin equation says that the vapor pressure over a droplet depends exponentially on the inverse of the droplet radius. Thus, as the radius decreases, the vapor pressure over the droplet increases compared to that over the bulk liquid. This equation also holds for water coating an insoluble sphere (Twomey, 1977).

This has important implications for nucleation in the atmosphere. Condensation of a vapor such as water to form a liquid starts when a small number of water molecules form a cluster upon which other gaseous molecules can condense. However, the size of this initial cluster is very small, and from the Kelvin equation, the vapor pressure over the cluster would be so large that it would essentially immediately evaporate at the relatively small supersaturations found in the atmosphere, up to ~2% (Prup-pacher and Klett, 1997). As a result, clouds and fogs would not form unless there was a preexisting particle upon which the water could initially condense. Such particles are known as cloud condensation nuclei, or CCN.

While water is a major component of tropo-spheric particles, and hence largely determines the surface tension (7), organics found in particles may act as surfactants (see Chapter 9.C.2). In this case, their segregation at the air-water interface could potentially lead to a substantial surface tension lowering of such particles, which would lead to a lower equilibrium water vapor pressure over the droplet (Eq. (BB)) and hence activation at smaller supersaturations. This possibility is discussed in more detail in the next section.

2. Vapor pressure lowering. Raoult's law says that the vapor pressure of a solution component, A, whose pure vapor pressure is is proportional to FIGURE 14.37 Basis of Kelvin effect for increased vapor pressure over small liquid droplets.

its mole fraction in solution, xA, i.e., in a two-component solution of A and B, to xA = (1 - xB):

or alternatively