Constant-charge electrostatic model

Figure 5. The critical absolute permeability necessary to sustain the stability of a static foam as a function of liquid saturation. Calculations are for the constant-charge electrostatic model.

Figure 7 reports calculations of the effect of flow velocity on the critical capillary pressure for the constant-charge electrostatic model and for different initial film thicknesses. 

ENORDET AOS 1618) in an 81- fjtm permeability sandpack. Using the parameters listed and the constant- charge electrostatic model for the conjoining/disjoining pressure isotherm, the data are rescaled A ... 

Figure 10. Comparison of the critical-capillary-pressure data of Khatib, Hirasaki and Falls (5) (darkened circles) to the proposed dynamic foam stability theory (solid line). Best fitting parameters for the constant-charge electrostatic model are listed.

The simplest approach to describing the interactions of metal cations dissolved in water with solvent molecules is the Born electrostatic model, which expresses solvation energy as a function of the dielectric constant of the solvent and, through transformation constants, of the ratio between the squared charge of the metal cation and its effective radius. This ratio, which is called the polarizing power of the cation (cf Millero, 1977), defines the strength of the electrostatic interaction in a solvation-hydrolysis process of the type... 

Thus the contribution of the structured ionic cloud to the total potential at the surface of the central ion will not be as it is in the DH theory, and because the electrostatic model requires an equipotential surface to be maintained there, a new model is needed. We therefore approximate an ion to a dielectric sphere of radius a, characterized by the dielectric constant of the solvent D, and having a charge Q, residing on an infinitesimally thin conducting surface. This type of model has been exploited by previous workers (17,18) and may be reconciled with a quantum-mechanical description (18). 

The interphase between an electrolyte solution and an electrode has become known as the electrical double layer. It was recognized early that the interphase behaves like a capacitor in its ability to store charge. Helmholtz therefore proposed a simple electrostatic model of the interphase based on charge separation across a constant distance as illustrated in Figure 2.12. This parallel-plate capacitor model survives principally in the use of the term double layer to describe a situation that is quite obviously far more complex. Helmholtz was unable to account for the experimentally observed potential dependence and ionic strength dependence of the capacitance. For an ideal capacitor, Q = CV, and the capacitance C is not a function of V. 

The crystal lattice energy can be estimated from a simple electrostatic model When this model is applied to an ionic crystal only the electrostatic charges and the shortest anion-cation intermiclear distance need be considered. The summation of all the geometrical interactions be/Kveeti the ions is called the Madelung constant. From this model an equatitWjor the crystal lattice energy is derived ... 

The thermodynamic equilibrium constants shown by Equations 3.15 and 3.16 match the stoichiometric (or concentration based-) constants of stoichiometric models (see Equations 1 and 3 of Reference 1). Since the latter neglect the modulation of the adsorption of a charged species by the surface potential, they are not constant [19] after the addition of the IPR in the mobile phase. Stoichiometric relationships [19] represent only the ratio of equilibrium concentrations and cannot describe equilibrium in the presence of electrostatic interactions. In their stoichiometric approach. 

In this chapter, we discuss two models for the electrostatic interaction between two parallel dissimilar hard plates, that is, the constant surface charge density model and the surface potential model. We start with the low potential case and then we treat with the case of arbitrary potential. 

Born (1) and later Bjerrum (2) developed a theoretical approach to ion-solvent interactions based on a rather simple electrostatic model. Ions are considered as rigid spheres of radius r and charge z in a solvent continuum of dielectric constant e. Changes in enthalpy AH av) and in free energy AG av), respectively, associated with the transfer of the gaseous ions into the solvent are represented by the following equations ... 

There have been many attempts to calculate AH independent of the equilibrium constant. The difficulty of a complete theoretical treatment of the H bond unfortunately requires approximations. The uncertainties thus introduced deprive the calculations of predictive value. Briefly, the usual approximations are based on some sort of electrostatic model, with computation of electrostatic, dispersion, and repulsive contributions by the methods of classical physics. Of course, the calculations require knowledge or estimation of such quantities as molecular arrangement, charge distribution, potential function, etc. Only a few systems have been treated. Reference 1327, for HF dimers 25, for carboxylic acids and 1561b, for ice furnish illustrative examples. Many other references are listed in Section 8.3, where a more complete discussion of the theoretical treatments is given. 

In a major contribution to the understanding of transmission of substituent effects in organic molecules, Westheimer and co-workers (Kirkwood and Westheimer, 1938 Westheimer and Kirkwood, 1938 Westheimer and Shookhoff, 1939 Westheimer et al., 1942) pointed out that the electrostatic field of a charged or dipolar substituent is at least partially transmitted through the molecule itself. Because the dielectric constant of an organic molecule is much smaller than that of a polar solvent such as water, substituent effects are attenuated with increasing distance to a much smaller extent than predicted by Equations (5) and (6). Kirkwood and Westheimer described methods for computation of the effective dielectric constant for a molecule-solvent system. The use of the effective dielectric constant (Dea) in place of the solvent s dielectric constant (D) in Equations (5) and (6) greatly improved the quantitative capabilities of electrostatic models of substituent effects. 

The stability constants of ion pairs (their log /Cassoc values) have been shown to be proportional to the electrostatic function ZMzJd, where z Z/. are the charge of metal cation and ligand, and d rM + ri, the sum of their crystal radii (cf. Fig. 3.5). Mathematical models for predicting ion pair stabilities generally assume this proportionality and include the simple electrostatic model, the Bjerrum model, and the Fuoss model (cf. Langmuir 1979). Such models can predict stabilities in fair agreement with empirical data for monovalent and divalent cation ion pairs. 

Figure 10,18 Schematic plot of surface species and charge (a) and potential ) relationships versus distance from the surface (at the zero plane) used in the constant capacitance (CC) and the diffuse-layer (DL) models. The capacitance, C is held constant in the CC model. The potential is the same at the zero and d planes in the diffuse-layer model i/fj). Reprinted from Adv. Colloid Interface Sci. 12, J. C. Westall and H. Hohl, A comparison of electrostatic models for the oxide/solution interface, pp. 265-294, Copyright 1980 with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.

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