Langmuir 1993,9, 3722-3723
3722
A Vapor Pressure Equation for Droplets V. A. Kuzf Imtituto de Fisica de Ltquidos y Sistemas Biolbgicos, IFLYSIB (UNLP-CONICET-CIC), C.C. 565, 1900 La Plata, Argentina Received October 13,1992. I n Final Form: September 9 , 1 9 9 9
A thermodynamicderivation of the vapor pressure equation for curved surfaces (droplets) is presented here. It is shown that the vapor pressure dependsnot only on the usual variables (surfacetension, droplet radius, temperature, and molar liquid volume) but also on the number of moles of the different phases. The present equation is valid for droplets having the same number of moles in the liquid and at the interfacialphase (limit of very smalldroplets). For dropletsof standard sizeit leadsto the Kelvin equation. Introduction A careful search has been devoted to the vapor pressure equation of a pure liquid in the case of a plane surface (see refs 1-3). Less or no PVT or theoretical research has been done on systems having strongly curved surfaces which is the case for small d r ~ p l e t s The . ~ vapor pressure, in these divariantb systems,changes with temperature and droplet radius. The oldest and fundamental relationship governing this phenomenon was given by Kelvin in 1871. Questions such as the influence of curvature on surface tension or on the triplet point, compressibility of the droplets, or the effect of curvature on the heat of evaporationhave also been carefully~ t u d i e d The . ~ Kelvin equation is used to understand processes such as evaporation-condensation (homogeneousnucleation theory), growth of crystals, capillary condensation, formation of clouds, e?. There are doubts about the range of its applicability. Here we shall try to establish this range. Our approach will be a thermodynamic one. By taking the variation of the total thermodynamic Gibbs potential of the system of noninteracting droplets in equilibrium with the saturated vapor and by considering the equilibrium conditions, we find a vapor pressure equation which reduces to that of Kelvin when the molar interfacialvolume is neglected in relation with the molar liquid volume of the droplet.
Vapor Pressure of Droplets Let us consider a droplet which is able to exchange matter with the surroundingvapor. The Gibbs free energy is a homogeneous function of first degree in nl, ne, and ne; that is
+
G = plnl+ peng pane (1) where n and p are the molar number and the chemical potential of each respective phase. The superscripts 1, g, and a indicate the liquid, the gas, and the interface, respectively. The total differential of G is
+
dG = d(plnl + peng) pc dn" + nu dp" (2) The coexisting phases will be governed by the corret Member of the Consejo Nacional de Inveatigacionee CientIficas y TBcnicas (CONICET) of Argentina a Abstract publiehedin Advance ACSAbetracte,October 15,1993. (1) Ambroee, D.Vapor Preesure in Chemical Thermodynamics;The Chemical Society: London, 1973; Vol. 1. (2) Rowlineon, J. S.; Swinton, F. L. Liquid and Liquid Mixtures; Butterworth Scientific: London, 1981. (3)Kuz, V. A. Fluid Phase Equilib. 1991,66, 113. (4) Beysens, D.; Knobler, C. M. Phye. Rev. Lett. 1986,57,1433. (5) Defay,R.;Prigogine,I. Surface Tensionand Adsorption;Long", Green k Co.: London,1966.
sponding Gibbs-Duhem equation; in particular for the surface phase, when the system is in thermal equilibrium, the following equation results: p e dn" = -a dA (3) Here a is the surface tension while A is the area. By substitution of eq 3 into eq 2 and considering the system near equilibrium, we have
d(pln' + peng)+ n" dp" = a dA (4) The physicochemical equilibrium of the system requires that
= p g = P" (5) Then, introducing this condition into eq 4, it becomes d(pgnl+ pgn3 + n" dpe = u dA (6) By integrating this equation, the following results: 1 4*3uv1 (7) (1 + ng/nl + ne/nl) (4/3)?rr3 where v1 = V/nl is the partial molar volume of the liquid drop, A = 4m-2is the area, and V = (4/3)urS is the volume. In the above derivation it has been assumed that fne dpe = n'$dpg. By substitution of p g for a perfect gas, eq 7 becomes pg
z
where pais the vapor pressure corresponding to a plane surface, R is the gas constant, and Tis the temperature. This equation is similar to the Kelvin equation h@/p,) = 2av1/rRT (9) which is obtained via the condition of mechanical equilibrium (Laplace equation). In both eqs 8 and 9,the In@/p&depends on the standard variables u, V I , r, and T. The Kelvin equation is valid under standard conditions, that is, for any droplet having a size such that the interfacial volume Vu
