Multicomponent Equilibrium Thermodynamics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 63

3.3.3 V APOR P RESSURE OVER C URVED S URFACES
Consider the formation of fog droplets in the atmosphere, and condensation (nucle-
ation) of small clusters leading to the formation of aerosols, cloud droplets, and
raindrops. All of these involve highly curved interfaces. The development of equi-
librium thermodynamic quantities (free energy and chemical potential) for these
systems involves modifications to the vapor pressure relationships for solutes dis-
tributed between a liquid and vapor phase. The vapor pressure over a curved surface
is dependent on its radius of curvature. Let us consider the curved surface of a liquid in
contact with its vapor. From Chapter 2 we have at constant T , the following expression
for the molar free energy change:

1
r 1 +

,

1
r 2

Δ G = V m Δ P = V m · σ (3.42)

where V m is the molar volume of the liquid and r 1 and r 2 are the principal radii of cur-
vature of the surface. Over a plane surface we can use theYoung-Laplace equation for
Δ

P from Chapter 2. Further the chemical potential between the curved surface (with
vapor pressure P ∗ c
i

) and a plane surface (with saturation vapor pressure P i ) is given by

RT ln P ∗ c

.

c i

P i

Δ G = μ

i − μ i =

(3.43)

Equating the two free energy differences, we have the following equation:

exp σ

1
r 1 +

.

P ∗ c
i

V m
RT

1
r 2 P i =

(3.44)

In environmental engineering, we are particularly interested in spherical surfaces
(e.g., fog, rain, cloud, and mist) for which r 1 = r 2 = r . Hence we have

exp 2

.

P ∗ c
i

r · σ V m P i =

(3.45)

RT

This is the Kelvinequation , which gives the vapor pressure over a curved surface, P ∗ c
i
relative to that over a plane surface, P i given the surface tension of the liquid, radius
of the drop, and temperature. For a solid crystal in equilibrium with a liquid also the
Kelvin equation applies if the vapor pressures are replaced with the activity of the
solute in the solvent.
Consider the case of water, the most ubiquitous of phases encountered in envi-
ronmental engineering. The surface tension of water at 298 K is 72 mN/m and its
molar volume is 18

10 − 6 m 3 /mol. Figure 3.4 shows the value of P ∗ c
i

/P i for vari-

×

ous size water drops. When r ≥ 1000 nm the normal vapor pressure is not affected,

100 nm there is an appreciable increase in vapor pressure. For liquids
of large molar volume and surface tension, the effect becomes even more significant.
An example is mercury, which vaporizes rapidly when comminuted. The conclusion

whereas for r ≤