Curvature solubility

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

Only two possibilities exist for explaining the existence of cloud formation in the atmosphere. If there were no particles to act as cloud condensation nuclei (CCN), water would condense into clouds at relative humidities (RH) of around 300%. That is, air can remain supersaturated below 300% with water vapor for long periods of fime. If this were to occur, condensation would occur on surface objects and the hydrologic cycle would be very different from what is observed. Thus, a second possibility must be the case particles are present in the air and act as CCN at much lower RH. These particles must be small enough to have small settling velocity, stay in the air for long periods of time and be lofted to the top of the troposphere by ordinary updrafts of cm/s velocity. Two further possibilities exist - the particles can either be water soluble or insoluble. In order to understand why it is likely that CCN are soluble, we examine the consequences of the effect of curvature on the saturation water pressure of water. 

Thomson-Freundlich equation relates the solubility of a particle to its curvature radius (e.g., Swalin, 1962). Using this equation, Kirkaldy and Young (1987) show how periodic precipitation may result from the capillary resistance of the matrix to grow small precipitates. Formally, the conditions read... 

Equation (69) is incorrect because, as stressed above, it is the curvature which affects properties such as vapor pressure and solubility, not the actual extents of... 

Note how the Gran plot in Figure 4.9 is curved at intermediate and small addition levels, thus making adjustment quite difficult. Such curvature is best taken to indicate that a complication is present, e.g. that the precipitate has an appreciable solubility (when determining Kf) or that dissociation of a complex has occurred (when trying to measure Kcompiexation)-... 

Evidently the lamp black is more soluble in the benzene than in the water phase. On increasing the concentration of lamp black in the mixture the curvature of the particles increases and the mean diameter of the emulsion decreases as noted by Moore J.AXj.S). xli. 944, 1919) who obtained the following figures. The... 

The effect of curvature or surface energy on the osmotic pressure and thus on the solubility is thus removed when... 

With short chain derivatives, the forces of repulsion are higher than the ones of attraction the curvature is high and spherical micelles are formed at a concentration called the critical micellar concentration (cmc). This concentration can be detected by a change in the physico-chemical properties of the solution (e.g. surface tension, Fig. 3 a). Above a characteristic temperature (referred as Krafft temperature), the tensio-active molecules are infinitely soluble in the form of micelles (Fig. 3 b). 

It is well known that the aqueous phase behavior of surfactants is influenced by, for example, the presence of short-chain alcohols [66,78]. These co-surfactants increase the effective value of the packing parameter [67,79] due to a decrease in the area per head group and therefore favor the formation of structures with a lower curvature. It was found that organic dyes such as thymol blue, dimidiiunbromide and methyl orange that are not soluble in pure supercritical CO2, could be conveniently solubihzed in AOT water-in-C02 reverse microemulsions with 2,2,3,3,4,4,5,5-octafluoro-l-pentanol as a co-surfactant [80]. In a recent report [81] the solubilization capacity of water in a Tx-lOO/cyclohexane/water system was foimd to be influenced by the compressed gases, which worked as a co-surfactant. 

Different surfactants are usually characterised by the solubility behaviour of their hydrophilic and hydrophobic molecule fraction in polar solvents, expressed by the HLB-value (hydrophilic-lipophilic-balance) of the surfactant. The HLB-value of a specific surfactant is often listed by the producer or can be easily calculated from listed increments [67]. If the water in a microemulsion contains electrolytes, the solubility of the surfactant in the water changes. It can be increased or decreased, depending on the kind of electrolyte [68,69]. The effect of electrolytes is explained by the HSAB principle (hard-soft-acid-base). For example, salts of hard acids and hard bases reduce the solubility of the surfactant in water. The solubility is increased by salts of soft acids and hard bases or by salts of hard acids and soft bases. Correspondingly, the solubility of the surfactant in water is increased by sodium alkyl sulfonates and decreased by sodium chloride or sodium sulfate. In the meantime, the physical interactions of the surfactant molecules and other components in microemulsions is well understood and the HSAB-principle was verified. The salts in water mainly influence the curvature of the surfactant film in a microemulsion. The curvature of the surfactant film can be expressed, analogous to the HLB-value, by the packing parameter Sp. The packing parameter is the ratio between the hydrophilic and lipophilic surfactant molecule part [70] ... 

Equation (42) provides a thermodynamically valid way to determine y for an interface involving a solid. The thermodynamic approach makes it clear that curvature has an effect on activity for any curved surface. The surface free energy interpretation of y is more plausible for solids than the surface tension interpretation, which is so useful for liquid surfaces. Either interpretation is valid in both cases, and there are situations in which both are useful. From solubility studies on a particle of known size, y5 can be determined by the method of Example 6.2. 

The radius of curvature of sharp points or protuberances on the particles has a larger effect on the solubility of irregular particles than the equivalent radius of the particles themselves. 

Suppose that the capillarity effect of curvature on solubility is included in Exercise 22.3. Describe qualitatively what happens to the tip growth rate as the tip radius decreases without limit. 

The effect of curvature on vapour pressure (and, similarly, on solubility) provides a ready explanation for the ability of vapours (and solutions) to supersaturate. If condensation has to take place via droplets containing only a few molecules, the high vapour pressures involved will present an energy barrier to the process, whereas in the presence of foreign matter this barrier can be by-passed. 

The fluid phase that fills the voids between particles can be multiphase, such as oil-and-water or water-and-air. Molecules at the interface between the two fluids experience asymmetric time-average van der Waals forces. This results in a curved interface that tends to decrease in surface area of the interface. The pressure difference between the two fluids A/j = v, — 11,2 depends on the curvature of the interface characterized by radii r and r-2, and the surface tension, If (Table 2). In fluid-air interfaces, the vapor pressure is affected by the curvature of the air-water interface as expressed in Kelvin s equation. Curvature affects solubility in liquid-liquid interfaces. Unique force equilibrium conditions also develop near the tripartite point where the interface between the two fluids approaches the solid surface of a particle. The resulting contact angle 0 captures this interaction. 

The ionic strength effect is not limited to ksv variation as described by eq. (22). The addition of large amounts of electrolytes may also modify the quencher solubility and thus its efficiency. This effect has been used by some authors, in systems very different from those examined in this work, in order to determine the association constant of the inhibitor salt (Mac, 1997 Mac and Tokarczyk, 1999) as the electrolyte concentration is increased, the quencher ion associates, so that the effective concentration of the inhibitor ion decreases, leading to a downward curvature of the Stern-Volmer plot. Such a curvature can be quantitatively related... 

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