Electrostatic arguments

A semi-infinite crystal cut along a polar direction has an infinite electrostatic energy, because the electrostatic field has a non-zero mean value in the material. Yet, under some specific conditions that we will now consider, the macroscopic field may be cancelled out and the surface stabilized. The following arguments rely upon a macroscopic analysis, which means that  

Macroscopic electric field Let us consider a slab of equi-distant layers, bearing positive and negative charge densities equal to + cr (Fig. 3.7a). The repeat unit has a dipole moment density equal to aR. As for an association of capacitors. Gauss theorem allows an estimation of the electric field S perpendicular to the layers and of the electrostatic potential V. The field S turns out to be equal to zero between two double-layers and equal to 47rcr inside a double-layer. Its mean value in the slab is, therefore, non-zero  

S = 2na. The electrostatic potential, on the other hand, increases from the left to the right of the slab in Fig. 3.7a, by an amount dV = AnaR per double layer. is large, typically of the order of several tens eV. Unlike non-polar surfaces, it is no longer possible to use the concept of a surface Madelung constant, because the electrostatic potential is different from that of the bulk in every layer. The total electrostatic energy is proportional to the slab thickness. It is infinite for macroscopic systems 

In the following, we will consider two ways of cancelling the macroscopic field on the basis of purely electrostatic arguments. We will temporarily forget that the system has electronic degrees of freedom allowing spontaneous charge redistributions in response to the electrostatic potentials. 

Addition of a surface dipole An additional modification of the surface charges may lower the total energy. Let us consider that a dipole density a R is added on the two faces of the system (Fig. 3.7c). Compared with the preceding estimation, the electrostatic field is changed only inside the double-layer on which the dipole has been placed. A value of a = a 12 = a jA yields oscillations of V around zero, so that the macroscopic electrostatic energy of the system vanishes. A similar result could be obtained by a modification of the outer double-layer spacing 6R, such that the dipole moment a R — SR)/2 = a R. This corresponds to an inward relaxation equal to 5R = R/2. 

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From simple electrostatic arguments it is clear that the force can be written in the following form [33] ... 

If we consider that classical electrostatic arguments may be relevant, we would predict that the net charge on each metal end and the distance be-... 

The electrostatic argument describes a situation that could occur in some bilayer systems. However, there are two problems with this argument as an... 

The assumption of linear response played a prominent role in the derivation (given above) of the SCRF equations, and one aspect of the physics implied by this assumption is worthy of special emphasis. This aspect is the partitioning of Gp into a solute-solvent interaction part Gss and a intrasolvent part Gss The partitioning is quite general since it follows entirely from the assumption of linear response. Since classical electrostatics with a constant permittivity is a special case of linear response, it can be derived by any number of classical electrostatic arguments. The result is [114, 116-119]... 

Given the success of the electrostatic argument in explaining the known abzyme selectivity, a further prediction was made1101 for the homologous 6-exo versus 7-endo competition for substrate 1 (n = 2) because the theozyme had no recognition elements... 

The influence of substituents on the stability of singlet m-benzynes is weak, and small variations have been rationalized on the basis of electrostatic arguments. The triplet states are even less affected by substimtion. Note that high structural flexibility is obviously characteristic for all m-benzynes. It can therefore not be taken for granted that calculated tendencies (and in particular those obtained with DFT methods) represent actual physical trends instead of just reflecting the deficiencies of the respective level of theory. 

Trend lines are shown which emphasize that the values for AhydFF (M +, g)conv and AhydH (X, g)conv vary with ionic radius. This is to be expected from simple electrostatic arguments. The smaller ions would be expected to have greater interactions with water than the larger ones. 

With the ionic cloud on the electrode, the resemblance of the Gouy—Chapman model to that of the theoiy of ion-ion interactions in solution reviewed in Chapter 3 is evident. There, it was necessary to arbitrarily choose one ion and spotlight it as the central ion, or source, of the field. Here, the discussion resolves on ion-electrode interactions with the electrode as the source of the field. The response of an ion, however, does not depend on how the electric field is produced (i.e., whether the source is a central ion or a charged electrode). It depends only on the value of the field at the location of the ion. Hence, the electrostatic arguments in the problems of ion-ion interactions and ion-electrode interactions must be similar. 

The other limiting condition occurs when electrons are pumped into the electrode to make it very negatively charged. What will the dipoles do On the basis of a simple electrostatic argument, the flipped-up dipoles will turn around and flop down, fn this new state, the hydrogens are facing the electrode and the oxygen atom is toward the solution [Fig. 6.76(b)]. 

Expressions similar to equation (1) can also be derived from simple electrostatic arguments for reaction between two ions of charge ZA and ZB, the rate constant k (presumed to be measured at zero ionic strength) in a medium of dielectric constant eR is given by equation (2) ... 

The equilibria considered up to now have all involved inner sphere complexes. There is the possibility that an inner sphere complex may react with free ligands in solution this includes the solvent itself, to give an outer sphere complex where the ligand enters the secondary solvation shell of the inner sphere complex. If the two species involved in this type of interaction are of opposite sign, which is the situation where this type of complex formation is expected to be most effective, the outer sphere complex is called an ion pair. Fuoss65 has derived an expression (equation 38) for the ion pair formation constant, XIP, from electrostatic arguments ... 

This functional is also physically motivated as it expresses the balance of two terms a favorable (negative) solute-solvent interaction energy and an unfavorable (positive) solvent-solvent interaction. At equilibrium the second term is equal to half of the first as expected also from basic electrostatic arguments. 

The geometries of the octahedral and tetrahedral coordination sites shown in figs 2.3 and 2.6a suggest that the value of the tetrahedral crystal field splitting parameter, A, will be smaller than the octahedral parameter, A0, for each transition metal ion. It may be shown by simple electrostatic arguments and by group theory that... 

In 1954 Weiss32 used Bernal and Fowler s simplified solvation model,16 with an Inner Sphere of ionic coordination, i.e., a small spherical double layer around the ion of charge ze, followed by a sharp discontinuity at radius q, the edge of the Outer Sphere or Dielectric Continuum. He used a simple electrostatic argument to determine the energy to remove an electron at optical frequency from the Inner Sphere ... 

No single method for protein study gives much information by itself and spectrophotometric titrations also have been best exploited in conjunction with auxiliary studies. For example, Tanford has routinely coupled his studies with results obtained from electrometric H+-titration studies, to show with certainty that only part of the anomalous phenolic titration of proteins can be explained by a general electrostatic argument. The review by Tanford in this volume discusses these problems in detail. 

When the proton acceptor molecule contains an atom like O or S, a second lone pair is pre-,sent that makes prediction of the equilibrium geometry less obvious. If one assumes an sp hybridization, the two lone pairs are disposed as in Fig. 2.3A, which would yield an angle P, between the HYH bisector and the X--Y axis, in the vicinity of 125°. The alternate type of hybridization, sp, leaves one of the lone pairs in a p-orbital, oriented 90° from the other lone pair. This arrangement would lead to a 180° p angle. Geometry B is also favored by certain electrostatic arguments. Specifically, it would permit the dipole moment of YH2, collinear with the HYH bisector, to align itself with the dipole of the X—H molecule. 

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