Equilibrium of a Pure Water Droplet

In Chapter 9 we showed that the vapor pressure over a curved interface always exceeds that of the same substance over a flat surface. The dependence of the water vapor pressure on the droplet diameter is given by the Kelvin equation (9.86) as 

FIGURE 15.2 Ratio of the equilibrium vapor pressure of water over a droplet of diameter Dp, pw(Dp), to that over a flat surface, p°, as a function of droplet diameter for O C and 20 C. 

Droplets in the atmosphere never consist exclusive of water they always contain dissolved compounds. However, understanding the behavior of a pure water droplet is necessary for understanding the behavior of an aqueous droplet solution. 

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Water in the atmosphere exists in the gas phase as water vapor and in the aqueous phase as water droplets and wet aerosol particles. In this section we will investigate the conditions for water equilibrium between the gas and aqueous phases. This equilibrium is complicated by two effects the curvature of the particles and the formation of aqueous solutions. We will start from the simplest case—the equilibrium between a flat pure water surface and the atmosphere. Then the equilibrium of a pure water droplet will be investigated, followed by a flat water solution surface. Finally, these effects will be... 

The steeply rising portion of the Kohler curves represents a region where solute effects dominate. As the droplet diameter increases, the relative importance of the Kelvin effect over the solute effect increases, and finally beyond the critical diameter the domination of the Kelvin effect is evident. In this range all Kohler curves approach the Kelvin equation, represented by the equilibrium of a pure water droplet. Physically, the solute concentration is so small in this range (recall that each Kohler curve refers to fixed solute amount) that the droplet becomes similar to pure water. 

Stability of Atmospheric Droplets We have already seen in the previous section that a pure water droplet cannot be at stable equilibrium with its surroundings. A small perturbation of either the droplet itself or its surroundings causes spontaneous droplet growth or shrinkage. 

This initial stage of droplet formation deserves a careful explanation. Over a flat, pure water surface at 100% relative humidity (saturation with respect to water), water vapor is in equilibrium, which means that the number of water molecules leaving the water surface is balanced by the number arriving at the surface. Molecules at water surfaces are subjected to intermolecular attractive forces exerted by the nearby molecules below. If the water surface area is increased by adding curvature, molecules must be moved from the interior to the surface layer, in which case energy is required to oppose the cohesive forces of the liquid. As a consequence, for a pure water droplet to be at equilibrium, the relative humidity has to exceed the relative humidity at equilibrium over a flat, pure water surface, or be supersaturated. The flux of molecules to and from a surface produces what is known as vapor pressure. The equilibrium vapor pressure is less over a salt solution than it is over pure water at the same temperature. This effect balances to some extent the increase in equilibrium vapor pressure caused by the surface curvature of small droplets. Droplets with high concentrations of solute can then be at equilibrium at subsaturation. 

FIGURE 2 Equilibrium saturation ratio versus droplet radius for different masses of sodium chloride nuclei (solid lines). Asterisks show droplet radius where solution drop containing indicated mass of sodium chloride will continue to grow without further increase in the saturation ratio. Dashed curve is for a pure water droplet. RH, Relative humidity. [Adapted from Byers, H. R. (1965). Elements of Cloud Physics, courtesy of the University of Chicago Press and the author.]... 

The term ysv is the interfacial tension of the solid material in equilibrium with a fluid vapor yLV is the surface tension of the fluid material in equilibrium with its vapor and ySL is the interfacial tension between the solid and liquid materials. Complete, spontaneous wetting occurs when 9 = 0° or when the material spreads uniformly over a substrate to form a thin sheet. A contact angle of 0° occurs with pure water droplet on a clean, glass shde. Therefore, for complete spontaneous wetting, cos 9 > 1.0 or when... 

Equations (17.21), (17.24), (17.26), and (17.27) are different forms of the Kohler equations (Kohler 1921, 1926). These equations express the two effects that determine the vapor pressure over an aqueous solution droplet—the Kelvin effect that tends to increase vapor pressure and the solute effect that tends to decrease vapor pressure. For a pure water drop there is no solute effect and the Kelvin effect results in higher vapor pressures compared to a flat interface. By contrast, the vapor pressure of an aqueous solution drop can be larger or smaller than the vapor pressure over a pure water surface depending on the magnitude of the solute effect term B/Dp relative to the curvature term A/Dp. Note that both effects increase with decreasing droplet size but the solute effect increases much faster. One should also note that a droplet may be in equilibrium in a subsaturated environment if DpA < B. 

The cleaning procedure of PTFE and PE wafers was as follows the surfaces were rinsed with alcohol and water, then the substrates were soaked in a sulfochromic acid from 30 to 60 minutes at the temperature 50°C. The surfaces then were washed with distilled water and dried with a strong jet of nitrogen. The equilibrium macroscopic contact angles obtained were 105° and 90° for PTFE and PE substrates, respectively (for pure water droplets). 

Equations have been developed to express the equilibrium supersaturation of a droplet of a given radius in terms of the composition and size of the CCNs. A family of curves showing the fractional equilibrium relative humidity (relative humidity/100%) of water droplets containing differing masses of sodium chloride salt is shown in Fig. 2. A curve for pure water droplets is also shown. Asterisks at the peak of each curve show the critical radius and supersaturation of the droplets and indicate that the higher the mass of the salt, the lower the critical supersaturation. A family of curves can also be generated for other salts found in the atmosphere, such as ammonium sulfate. 

Hi) Failure of adsorption equilibration. Origins (i) and (ii) apply when the three tensions involved have their equilibrium values, i.e. when all adsorption processes are relaxed. However, Incomplete adsorption at any of the three interfaces also gives rise to differences between a(adv) and a(rec). This phenomenon is not a real type of hysteresis but rather the result of lack of patience if we wait long enough the Deborah number De = r(ads)/t(obs) becomes 1. Here T(ads) is the characteristic time for the establishment of adsorption equilibrium and t(obs) the measuring time. However, as these phenomena are often observed, we shall include them in the present discussion. A typical illustration, already referred to in connection with [3.2.1] is that of a benzene droplet placed on top of pure water. First it spreads, but later it retracts to form a droplet. The reason is that it takes some time to equilibrate benzene adsorption at the water-air interface. 

The chemistry that occurs in cloud and fog droplets in the atmosphere has been shown, in the last decade or so, to be highly complex. Most atmospheric species are soluble to some extent, and the liquid-phase reactions that are possible lead to a diverse spectrum of products. The aspect of atmospheric aqueous-phase chemistry that has received the most attention is that involving dissolved S02. Sulfur dioxide is not particularly soluble in pure water, but the presence of other dissolved species such as H202 or 03 displaces the dissolution equilibrium for S02, effectively... 

The adsorption of a surfactant at an interface between CO2 and a second fluid, such as water, may be determined directly from measurement of the interfacial tension (change in Gibbs free energy with surface area), y, versus surfactant concentration. A novel tandem variable-volume pendant drop tensiometer has been developed to measure equilibrium and dynamic values of y as a function ofT.p and time (Figure 2.4-1) [21]. An organic [21] or aqueous phase [18] is preequilibrated with CO2 in the first variable-volume cell (drop-phase cell). A droplet of this liquid is injected into the second variable-volume cell, with two windows at 180° mounted on a diameter, containing either pure CO2 or CO2 and surfactant. 

Winsor s type II diagram and phase behavior, which is noted 2 for a similar reason as exposed previously, embodies the opposite situation, in which the poly-phasic equilibrium consists of an inverse micellar oil solution Si that eventually solubilizes enough water to become a W/O microemulsion with separated or percolated water droplets ora bicontinuous one, in equilibrium with an es.sentially pure aqueous phase. In this case the tie line slope is slanted the other way and the critical point is located at the extreme left of the binodai curve. 

Systems of chemical interest often are not chemically pure substances but are mixtures of two or more chemically distinct species. The mixture may be a solution if it is uniform at the molecular level or a dispersion if it is uniform on a much larger scale, such as that of small droplets of oil dispersed in liquid water. A system of one or more chemically distinct species may exist in different phases, and the number of components of a system is the number of independent species needed to specify the composition of all the phases in a system. For instance, a closed system of water vapor over a dilute water solution of copper sulfate is a two-component system. This is because the liquid and vapor of water are in equilibrium and only the total amount of water is adjustable, while the amount of... 

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