Schematic variation of the radius (r) of a soluble particle as a function of relative humidity of the air. r0
dry radius droplet. The phase change takes place at the relative humidity at which a saturated solution of the substance considered is in vapour equilibrium with its environment.
It is known from physical chemistry that the equilibrium vapour pressure is smaller over solutions than over pure water. In the case of ideal solutions this vapour pressure decrease is proportional to x0, the mole fraction of the solvent (Raoult's law). If the solution is real, the interaction of solvent and solute molecules cannot be neglected. For this reason a correction factor has to be applied to calculate the vapour pressure lowering. This correction factor is the so-called osmotic coefficient of water (gj. We also have to take into account that the soluble substance dissociates into ions, forming an electrolyte.
If we further raise the relative humidity after the phase change (see Fig. 37); the radius of the droplet increases and the solution becomes weaker and weaker. This means that at higher relative humidity a more dilute solution is in dynamic equilibrium with the vapour environment. It should be mentioned that the equilibrium radius is governed also by the curvature of the droplet. Since the relation between the curvature and droplet radius is given by the well-known Thomson equation, we may write (Dufour and Defay, 1963; E. Meszaros, 1969):
p kTr where p and p ' are the real vapour pressure and the equilibrium vapour pressure relative to pure water, respectively, a is the surface tension of the solution, vm is the reciprocal of the number of water molecules in the solution, while k and Tgive the Boltzman constant and the absolute temperature, respectively.
After a simple mathematical transformation equation [4.13] yields
P 2 avm
p kTr since x0 -1 = - vx, where x is the mole fraction of the solute, while v the number of ions formed by the dissociation of one solute molecule. If we multiply [4.14] by 100, we obtain a relation between relative humidity and particle radius. The solid line of Fig. 37 represents schematically the results of calculations made using these equations. This curve was experimentally verified first by Dessens (1949) who used a microscope to study the change of radius of atmospheric particles (droplets) captured on spider web as the relative humidity was changed. He discovered that the phase change takes place at a lower relative humidity with decreasing than with increasing humidity (dotted line). This so-called hysteresis phenomenon was later confirmed by several other investigators (see Junge, 1963). It goes without saying that in the case of sulfuric acid solution droplets a sudden change in the particle radius is not observed. It was also more recently shown (Winkler and Junge, 1972)
9 Meszaros that the curve is also smoothed if the particle is composed of a mixture of different salts.
Equations [4.13] and [4.14] are also supported by some atmospheric measurements. Thus, E. Meszaros (1970) measured the size distribution of the mass of atmospheric sulfate particles by means of a cascade impactor backed up by membrane filters. He found that the geometric mean radius of the distribution averaged by humidity intervals varies as a function of the relative humidity as shown by the points in Fig. 38. The solid line of this figure gives the theoretical relation calculated from equation [4.14] for an ammonium sulfate particle with a dry radius of 0.14 /¿m, the value found for the geometric mean radius at low relative humidity. The line shows that the particle radius increases by a factor of two at a relative humidity of 80 %. Near 100 % the droplet radius is several times larger than the dry particle size. Comparison of the curve with the experimental points indicates that the behaviour of the atmospheric particle population is well approximated by the theory outlined. It cannot be excluded, however, that the real phase change is less sudden than the theory predicts.
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