Electrostatic model

The protonation and deprotonation of a surface site in contact with solution 

Combination of Equations 2.15 and 2.16 gives pH- and ionic-strength-independent =SH / =S . When the activities of surface species are proportional to their concentrations, pll- and ionic-strength-independent 

The purely electrostatic diffuse layer model often underestimates the affinity of the counterions to the surface. In the Stem model, the surface charge is partially balanced by chemisorbed counterions (the Stem layer), and the rest of the surface charge is balanced by a diffuse layer. In the Stern model, the interface is modeled as two capacitors in series. One capacitor has a constant capacitance (independent of pH and ionic strength), which represents the affinity of the surface to chemisorbed counterions, and which is an adjustable parameter the relationship between a, and Vd in the other capacitor (the diffuse layer) is expressed by Equation 2.18. A version of the Stern model with two different values of C (below and above pHg) has also been used. The capacitance of the Stem layer reflects the size of the hydrated counterion and varies from one salt to another. The correlation between cation size and Stern layer thickness was studied for a silica-alkali chloride system in [733]. Ion specificity of adsorption on titania was discussed in terms of differential capacity as a function of pH in [545]. The Stern model with the shear plane set at the end of the diffuse layer overestimated the absolute values of the potential of titania [734]. A better fit was obtained with the location of the shear plane as an additional adjustable parameter (fitted separately for each ionic strength). Chemisorption of counterions can also be quantified within the chemical model in terms of expressions similar to the mass law (Section 2.9.3.3). 

The electrostatic models discussed in Sections 2.9.2.1 through 2.9.2.5 apply to a simple chemical model involving one reaction (Reaction 2.14) of the transfer of one species (proton) from the solution to the surface. More complex chemical models involving the transfer of two or more species from the solution to the surface and/or allowing various distances of the adsorbed solution species from the surface require more complex electrostatic models. Usually, three or more capacitors in series are considered. All the capacitors but one have constant capacitances, and the relationship between and Vd in one capacitor (diffuse layer) is expressed by Equation 2.18. 

---

Fowler P W and A D Buckingham 1991. Central or Distributed Multipole Moments Electrostatic Models of Aromatic Dimers. Chemical Physics Letters 176 11-18. 

Price S L, R J Harrison and M F Guest 1989. An Ab Initio Distributed Multipole Study of the Electrostatic Potential Around an Undecapeptide Cyclosporin Derivative and a Comparison with Point Charge Electrostatic Models. Journal of Computational Chemistry 10 552-567. 

Hurley, A. C., Proc. Roy. Soc. London) A226, 170, 179, The electrostatic calculation of molecular energies. I. Methods of calculating molecular energies. II. Approximate wave functions and the electrostatic model/ ... 

Since the dyes contain hydrophobic moieties, the fading reactions, which should be decelerated by polyelectrolytes according to the electrostatic model, can even be accelerated by hydrophobic polyelectrolytes. As shown in Fig. 17, the addition of... 

The von Smoluchowski equation must be corrected when the partners are ions to account for attractive or repulsive forces. They can be approximated by an electrostatic model. The quantity by which Eq. (9-10) or (9-13) is to be multiplied is... 

The concept of electrostatic complimentarity is somewhat meaningless without the ability to estimate its contribution to AAg. Thus, it is quite significant that the electrostatic contribution to AAthat should be evaluated by rigorous FEP methods can be estimated with a given enzyme-substrate structure by rather simple electrostatic models (e.g., the PDLD model). It is also significant that calculated electrostatic contributions to A A g seem to account for its observed value (at least for the enzymes studied in this book). This indicates that simple calculations of electrostatic free energy can provide the correlation between structure and catalytic activity (Ref. 10). 

But why linearly and why with a slope of-1, or something thereabout, the reader may righteously ask. In anticipation of the quantum mechanical treatment in Chapter 5 we can briefly discuss here a simple electrostatic model which fully accounts for the observed behaviour. After all, as the detailed quantum mechanical treatment has shown, direct electrostatic... 

In order to attempt a more quantitative description one may start from the early theoretical considerations of Boudart9 who was first to tackle the problem of predicting the change in heats of adsorption with changing work function O. According to his early semiempirical electrostatic model when the work function of a surface changes by AO then the heat of adsorption, -AHad, of covalently bonded adsorbed species should change by ... 

As shown in Chapter 4 (section 4.5.9.2), Equation (8.14) can also be derived via a rigorous electrostatic model which takes into account the presence of the effective double layer on the catalyst surface and gives in general ... 

This might be compatible with the electrostatic model in that the radial extensions of Ad and 5d orbitals are greater than that of 3d but then the diffuseness of these orbitals increases along the series in Eq. (6.10) and that would tend to decrease the Zioct values. 

Godt, R.E. (1981). A simple electrostatic model can explain the effect of pH upon the force-pCa relation of skinned muscle fibers. Biophys. J. 35, 385-392. 

The effective pressure produced by the electrodes on the elastomeric film is a function of applied voltage and can be derived by using an electrostatic model proposed by Pelrine et al. This pressure P is given by... 

Wood and Blundy (2001) developed an electrostatic model to describe this process. In essence this is a continuum approach, analogous to the lattice strain model, wherein the crystal lattice is viewed as an isotropic dielectric medium. For a series of ions with the optimum ionic radius at site M, (A(m))> partitioning is then controlled by the charge on the substituent (Z ) relative to the optimum charge at the site of interest, (Fig. 10) ... 

Experimental clinopyroxene-melt (Brooker et al. 2003) and wollastonite-melt (Law et al. 2000) partitioning data can be fitted to the electrostatic model (see Fig. 9). Only species entering the large cation site (e.g., M2 in clinopyroxene) are considered. For each isovalent group (actinides, lanthanides etc.) derived as shown in Figure 4, is... 

Mineral-Melt Partitioning ofU-Series Nuclides Electrostatic model... 

By analogy with clinopyroxene it is likely that Pa enters the orthopyroxene M2 site. In light of the fact that Du and Dxh in orthopyroxene are approximately ten times lower than in clinopyroxene, it is likely that Z)pa is also lower in orthopyroxene. However, this effect is offset to some extent by the smaller M2 site in orthopyroxene, which will tend to be more favourable to Pa than the M2 site in clinopyroxene. We have used the electrostatic model, applied to the two orthopyroxene-melt partitioning experiments of McDade et al. (2003a,b) to derive (Fig. 13). Both datasets, at 1.5 and 3 GPa, are... 

Figure 13. Electrostatic model fitted to partition coefficients for cations entering the M2-site in orthopyroxene, based on the experiments of McDade et al. (2003a,b). The curves are fits to Equation (7) and can be used to estimate and Do(m2) > from which D-p ui) can be calculated via the lattice strain model. The fit parameters are given in the legend.

Table 5. Electrostatic model fit parameters for garnet and estimates of DpJD ...

Figure 17. Electrostatic model fitted to partition coefficients for cations entering the M4-site in amphibole, based on the experiments of Brenan et al. (1995) and La Tourrette et al. (1995). A single M4-site is assnmed, rather than M4 and M4, as proposed by Botlazzi et al. (1999). The curves are fits to Eqnation (7) and can be nsed to estimate Do(M4), from which Z)pa(M4) can be calculated via the lattice strain model. Becanse of the mnltiplicity of sites in amphibole, it is unlikely that extrapolation of the curves to zero charge gives a reliable estimate for Dr . The fit parameters are Zo(M2> = 1-87 and ps = 38.1 A (La Tonrrette et al. 1995), and 2.31, 36.1 A (Brenan et al. 1995).

Figure 18. Electrostatic model fitted to partition coefficients for cations entering the large M-site in plagioclase, based on the experimental results of Bindeman and Davis (2000) and Blundy and Brooker (2003). The solid curve is a fit of the and Dll data of Blundy and Brooker (2003) to...

---