Davies equation, applications

In this particular example, the Davies equation (15) gives the same value as Equation 1 it seems to be applicable, therefore, to calculating TKs values of oxides and carbonates of bivalent metals from Ks data in 0.2M NaC104. It fails, however, if applied to data which were evaluated in media of higher ionic strengths. 

For soil solutions, the Davies equation, which is a modification of the Debye-Hiickel equation, is commonly used, and is applicable to solutions up to approximately I — 0.7 mol (see also Section 3.2.1) ... 

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

The Davies correction has no theoretical substantiation. Equation (1.77) ignores ion sizes and produces equal values y. for equal charges. It is not recommended for moderate and high ionic strength for calculation of activities coefficient of micro-concentrations. The Davies equation gives sufficiently accurate results at ionic strength of up to 0.5 mole-kg . It is handy for the application for geochemical models imder standard conditions. 

When a stability constant is derived from measurements in an ionic medium, the value will vary when the medium is changed. This is a subject of considerable importance, because often the medium of interest is different from the medium of measurement for example, in biomedical applications, the medium of interest is a biological medium such as blood, but that medium is unsuitable for stability constant measurements. CoiTections for variation of ionic strength can be made by using the Davies equation (Eq. 8) to estimate activity coefficients ... 

The equation of state for ionized monolayers has been discussed by Hachisu [32]. This author has shown by independent derivations using three different approaches that the equation proposed by Davies [33] is applicable in the presence or absence of added electrolyte provided that the Gouy-Chapman electrical double-layer model applies. The Davies equation may be written... 

References Brown, J. W., and R. V. Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw-Hill, New York (2000) Churchill, R. V, Operational Mathematics, 3d ed., McGraw-Hill, New York (1972) Davies, B., Integral Transforms and Their Applications, 3d ed., Springer (2002) Duffy, D. G., Transform Methods for Solving Partial Differential Equations, Chapman Hall/CRC, New York (2004) Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997). 

These parameters can be directly related back to the information contained in the EPM with n components (j) and m species (t). Application of the mass balance constraint equation requires that the concentration of each species must be known. Therefore, activity coefficients are computed if the ionic strength is already known from either the Davis or the extended Debye-Eluckel equation however, if ionic strength is unknown and has to be calculated, equation (5.134) can be converted to a general expression for the concentration of each species by substituting the expression for S to give... 

The Davies and Jones derivation makes some fundamental assumptions concerning the surface concentrations of the lattice ions and the BCF theory is only applicable to very small supersaturations. Thus, both theories have limitations which affect the interpretation of the results of growth experiments. Nielsen [27] has attempted to examine in detail how the parabolic dependence can be explained in terms of the density of kinks on a growth spiral and the adsorption and integration of lattice ions. One of the factors, a = S — 1, comes from the density of kinks on the spiral [eqns. (4) and (68)] and the other factor is proportional to the net flux per kink of ions from the solution into the lattice. Nielsen found it necessary to assume that the adsorption of equivalent amounts of constituent ions occurred and that the surface adsorption layer is in equilibrium with the solution. Rather than eqn. (145), Nielsen expresses the concentration in the adsorption layer in the form of a simple adsorption isotherm equation... 

The models describing hydrolysis and adsorption on oxide surfaces are called surface complexation models in literature. They differ in the assumptions concerning the structure of the double electrical layer, i.e. in the definition of planes situation, where adsorbed ions are located and equations asociating the surface potential with surface charge (t/> = f(5)). The most important models are presented in the papers by Westall and Hohl [102]. Tbe most commonly used is the triple layer model proposed by Davis et al. [103-105] from conceptualization of the electrical double layer discussed by Yates et al. [106] and by Chan et al. [107]. Reviews and representative applications of this model have been given by Davis and Leckie [108] and by Morel et al. [109]. We will base our consideration on this model. 

The method expressed In Equations 18-23 has seen only a few numerical applications, but It represents the ultimate application of the generalized complex variational principle in that only real valued eigenvalue calculations are required. Doubtless the use of this idea coupled with more cleverly chosen trial functions, such as those of Junker (19,20) and Chung and Davis (33) described above will also be possible. 

In this section, applications of the Davies and truncated Guggenheim equations are demonstrated through worked examples from the literature. 

From this latter observation the question immediately arises as to the possibility of applying Equation 22 to channels where there is a sizable gap between the channel and the floor. In addition one might ask to what degree the equation would be applicable to the apparatus of Davies (I), where the channel depth is extremely small since it is formed by the contact of rings with the liquid surface. Table II illustrates the application of Equation 22 to experimental data, where the reported surface viscosity was measured by the method of Dervician and Joly ( ), and the particle times were measured by Davies. In terms of particle time Equation 22 becomes... 

Table II. Application of Equation 29 to Calculation of Surface Viscosity from Davies Measurements (I)...

Equation (85) was verified experimentally by Nicklin et al. (1962) for application to a finite bubble or to a slug rising in a tube. Davies and Taylor (1950) also provided a solution with a slightly different empirical constant. 

Data from UC Davis ChemWiki, "Electrochemistry 5 Applications of the Nemst Equation. chemwiki.ucdavis.edu. 

The value of the new constant B in water at 25°C is 3.291 x 10 m" mol l. For many electrolytes the valne of a is around 300-400 pm, so the product Ba is close to unity for aqueous solutions at room temperature. Taking advantage of this coincidence, and adding an empirical linear term in I (which may be justified by some qualitative reasoning), Davies (1962) has formulated a useful approximate expression. Equation 2.23, which he finds applicable to a great many ionic species in water at room temperature. 

Density functional theory (DFT) as applied to adsorption is a classical statistical mechanic technique. For a discussion of DFT and classical statistical mechanics, with specific applications to surface problems, the text book by Davis [1] is highly recommended. (Here the more commonly used symbol for number density p(r) is used. Davis uses n(r) so one will have to make an adjustment for this text.) The calculations at the moment may be useful for modeling but are questionable for analysis with unknown surfaces. The reason for this is that the specific forces, or input parameters, required for a calculation are dependent upon the atoms assumed to be present on the surface. For unknown surfaces, a reversion to the use of the Brunaver, Emmett and Teller (BET) equation is often employed. 

The anisotropy of the a and P relaxations for drawn poly(ethylene terephthalate) has been studied by several workers.Davies and Ward showed that the a process exhibited little anisotropy but the p process a pronounced anisotropy when measurements were made on sections cut from an oriented rod (extrusion ratio = 3.3 1). In a study of the p relaxation in oriented poly(ethylene terephthalate), Hsu et attempted to modify the Kirkwood equation (equation (42) above) for application to systems of oriented dipoles. They found that the change in the dielectric properties on orientation of a sample could not be explained in terms of the alignment of chains alone, but the effect of conformational changes on dielectric anisotropy needed to be considered. In a further paper these authors showed that satisfactory agreement was obtained between the observed dielectric anisotropy and a theory in which the director order parameter (c/ Section 18.3.4) and the angle between the dipole moment vector of a repeat unit and the chain axis were adjustable parameters. 

---