Debye-Hiickel parameter, interactions

A is a Debye-Hiickel parameter (cf. Appendix II) and I is the ionic strength. Pitzer found that binary interaction parameter Xdepends on ionic strength and may conveniently be expressed as ... 

This is the screened image interaction between a point charge and an uncharged plate, both immersed in an electrolyte solution of Debye-Hiickel parameter k. Further, in the absence of electrolytes (k 0), Eq. (14.78) becomes... 

Equation (14.101) is the screened image interaction between a line charge and an uncharged plate, both immersed in an electrolyte solution of Debye-Hiickel parameter K. We see that in the former case (Cpj = 0), the interaction force is repulsion and the latter case (epi = cxd) attraction. Eurther, in the absence of electrolytes (k 0), we can show from Eq. (14.101) that the interaction force —dV IdH per unit length between plate 1 and cylinder 2 with 02 0 is given by... 

The zeta-potential is frequently used to predict the stability of a suspension or the adhesion of suspended particles on macroscopic surfaces (e.g. cellulose fibres, tubing). This is because double layer interaction between particles or between particles and surfaces is governed by the ion distribution in the diffuse layer, which primarily depends on the Debye-Hiickel parameter k and the diffuse layer potential i/ d- The latter, however, is commonly approximated by the zeta potential f (Lyklema 2010, cf. Fig. 3.3). It is quite obvious that repulsion requires high zeta-potential values of equal sign, whereas adhesion occurs in the absence of surface charge or for oppositely charged surfaces. 

The DLVO model (named after its principal creators, Deqaguin, Landau, Verwey and Overbeek) is the most widely used to describe inter-particle surface force potential (1,2). It assumes that the total inter-particle potential is the sum of an attractive van der Waals force and a repulsive double-layer force. The repulsive force due to the double-layer coulombic interaction between equal spheres separated by a distance D generates a positive potential energy Vr. If the radius r of the spheres is large compared to the double-layer thickness 1/k (Kr l, with K the Debye-Hiickel parameter), Fr is described approximately by ... 

Intense ion-ion interactions which are characteristic of salt solutions occur in the concentrated aqueous solutions from which AB cements are prepared. As we have seen, in such solutions the simple Debye-Hiickel limiting law that describes the strength goes up so the repulsive force between the ions becomes increasingly important. This is taken account of in the full Debye-Hiickel equation by the inclusion of a parameter related to ionic size and hence distance of closest approach (Marcus, 1988). 

Table B.l lists all the chemical reactions and their temperature dependence. Table B.2 lists the Debye-Hiickel constants A,p and Av) as a function of temperature and pressure. Table B.3 lists the numerical arrays used for calculating unsymmetrical interactions (Equations 2.62 and 2.66). Table B.4 lists binary Pitzer-equation parameters for cations and anions as a function of temperature. Table B.5 lists ternary Pitzer-equation parameters for cations and anions as a function of temperature. Table B.6 lists binary and ternary Pitzer-equation parameters for soluble gases as a function of temperature. Table B.7 lists equations used to estimate the molar volume of liquid water and water ice as a function of temperature at 1.01 bar pressure and their compressibilities. Table B.8 lists equations for the molar volume and the compressibilities of soluble ions and gases as a function of temperature. Table B.9 lists the molar volumes of solid phases. Table B.10 lists volumetric Pitzer-equation parameters for ion interactions as a function of temperature. Table B.ll lists pressure-dependent coefficients for volumetric Pitzer-equation parameters. Table B.12 lists parameters used to estimate gas fugacities using the Duan et al. (1992b) model.

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

Evidently there are factors at work in an electrolytic solution that have not yet been reckoned with, and the ion size parameter is being asked to include the effects of all these factors simultaneously, even though these other factors probably have little to do with the size of the ions and may vary with concentration. If this were so, the ion size parameter a, calculated back from experiment, would indeed have to vary with concentration. The problem therefore is What factors, forces, and interactions were neglected in the Debye-Hiickel theory of ionic clouds ... 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

Some other kinds of models have shown parameters that seem to follow useful correlation relationships. Among these are the virial coefficient model of Bums (2), the interaction coefficient model of Helgeson, Kirkham, and Flowers (4), and the hydration theory model of Stokes and Robinson (1). The problem shared by all three of these models is that they employ individual ion size parameters in the Debye-Hiickel submodel. This led to restricted applicability to solutions of pure aqueous electrolytes, or thermodynamic inconsistencies in applications to electrolyte mixtures. Wolery and Jackson (in prep.) discuss empirical modification of the Debye-Huckel model to allow ion-size mixing without introducing thermodynamic inconsistencies. It appears worthwhile to examine what might be gained by modifying these other models. This paper looks at the hydration tlieory approach. 

With the increase in salinity it becomes necessary to account for possible minimum distance between ions and their size. In order to account for the interaction between ions, Debye and Huckel introduced additional parameters into Equation (1.69). The obtained equation is considered expanded Debye-Hiickel second approximation equation, which is... 

As explained in section 3.6.1, many modifications have been proposed for the Debye-Hiickel relationship for estimating the mean ionic activity coefficient 7 of an electrolyte in solution and the Davies equation (equation 3.35) was identified as one of the most reliable for concentrations up to about 0.2 molar. More complex modifications of the Debye-Huckel equation (Robinson and Stokes, 1970) can greatly extend the range of 7 estimation, and the Bromley (1973) equation appears to be effective up to about 6 molar. The difficulty with all these extended equations, however, is the need for a large number of interacting parameters to be taken into account for which reliable data are not always available. 

The first two terms are derived from Debye-Hiickel theory, and the third and fourth terms express short-range interactions (e.g., ion-molecular interactions). A( ) can be calculated as a function of temperature using the polynomial equation given by Clegg et al. (1994), which is based on the study of Archer and Wang (1990). Pitzer and Mayorga (1973) determined three parameters mx)... 

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