A primitive model

Problem 16.1. Explainthe following statement In a symmetrical electron transfer process, where the donor and acceptor species comprise identical ionic centers, for example, Eq. (16.1), the transition state is given by all configurations that are in equilibrium with the donor and acceptor species when both are carrying a charge q = 0.5 ( donor + acceptor). 

Note that Ex is the amount of nuclear energy which should be released following a vertical (i.e. without changing A) electronic transition from state a to b (in fact 

In fact, the degeneracy at Ar is removed by the coupling (the noncrossing rule). This correction 

As a model for a rate process, the surface crossing picture described above can be treated within the Landau-Zener theory (Section 2.4) that yields the probability that a transition between the two electronic states occurs during one dynamical crossing event. Here Atr stands for R of Section 2.4. Using this theory to evaluate the rate of such electron transfer processes involves several assumptions  

The basic assumptions of the Landau-Zener theory need to be satisfied. These involve the applicability of classical mechanics (e.g. the neglect of tunneling) for the nuclear dynamics and the locality of the curve crossing event. 

Thermal relaxation (solvent reorganization) is fast relative to the reaction rate, so that the distribution of nuclear configurations remains thermal throughout the reaction. 

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Figure 5.15. Diagrams representing a primitive model of spin-orbit coupling.

The guiding principle in writing down the self-repelling Gaussian chain model is mathematical simplicity, not microscopic faithfulness. Do we have, any idea why such a primitive model properly can explain the experiments To investigate this question we consider how a realistic microscopic description could be reduced to our model. 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

Figure 4.8 Chemical potential data for a primitive model 1-1 electrolyte from Valleau and Cohen (1980). See Eq. (4.87). The upper results are lny , and the dashed-dot curve is a parabola fitted to those results. The lower results are In / Vc...

Consider a primitive model of a dilute electrolyte solution the system is composed of ions of two types that interact as... 

Kalyuzhnyi, Yu.V., Blum, L., Holovko, M.F., and Protsykevytch, I.A. Primitive model for highly asymmetric electrolytes, associative mean spherical approximation. Physica A, 1997, 236, No. 1-2, p. 85-96. 

In electrochemical proton transfer, such as may occur as a primary step in the hydrogen evolution reaction (h.e.r.) or as a secondary, followup step in organic electrode reactions or O2 reduction, the possibility exists that nonclassical transfer of the H particle may occur by quantum-mechanical tunneling. In homogeneous proton transfer reactions, the consequences of this possibility were investigated quantitatively by Bernal and Fowler and Bell, while Bawn and Ogden examined the H/D kinetic isotope effect that would arise, albeit on the basis of a primitive model, in electrochemical proton discharge and transfer in the h.e.r. 

Substitution of either the exact pair correlation functions or the solutions of the HNC or MSA equations causes all of the A to vanish. The same is true for the Aq, but only for a primitive model symmetrical electrolyte having equal ion diameters. For refined models, the quantity (A > [Eq. (175)] must vanish. Small values of A and (A ) therefore indicate good accuracy in the numerical procedures, whether or not the computed correlation functions accurately represent the assumed Hamiltonian model. 

D. D. Carey, Radial distributions of ions for a primitive model of an electrolyte solution, J. Chem. Phys. 46, 3783 (1967). 

If the L] interaction parameter e) is very small compared to the ELB energy sub), then the first square brackets on the right-hand side of (3.4.15) will be negligibly small (it will be zero for a primitive model in which Ulj is replaced by a hard-sphere potential). In this case, the major contribution to A comes from structural changes induced in the solvent. Here, the... 

Despite the simplifying assumptions in the derivation, such as assuming that the medium, water, is a continuum with no structure, and that the only work is electrostatic, and even more assumptions in calculating the properties of individual ions from the measured properties of electrolytes, as estimated by the Born function comes reasonably close to the measured Gibbs energy of ion solvation, as shown in Figure 6.7. Other thermodynamic properties such as the volume, entropy and enthalpy of solvation can also be obtained by appropriate differentiation of Equation (6.5). As a result, ever since its inception the Born equation has been used as a primitive model for the electrostatic contribution to the properties of an ion in a dielectric solvent. 

Waisman E, Lebowitz JL (1970) Exact solution of an integral equation for structure of a primitive model of electrolytes. J Chem Phys 52 4307-4311... 

Mills P, Anderson CF, Record MX Jr (1986) Grand canonical Monte-Carlo calculations of thermodynamic coefficients for a primitive model of DNA salt-solutions. J Phys Chem 90 6541-6548. doi 10.1021/... 

An application of the B3LYP method also allows explanation of the pecnliarities of accumulations of radicals XIV and XV at 295 K, namely, later appearance of the radical XV signal in the ESR spectrum as compared with radicals XIV. For that the energy variations in a model reacting system in the course of consecutive conversions to iminoxyl radicals XXa, XXb and XXc (structural analogues of radicals XTV and XV) by reactions similar to (Equation 7.43)-(Equation 7.53) were calculated. As a primitive model compound, N-(3-aminophenyl)formamide XXI has been chosen. The results obtained are represented in Figure 7.10. 

Following the success of Chan s simple proposition, and given the accuracy of SAFT in modelling the properties of non-ionic fluids, it is a natural extension to combine a treatment of the solvent and other non-Coulombic terms using SAFT with a contribution to treat charge-charge interactions from a primitive model theory. The Helmholtz energy of the electrolyte solution is usually therefore written as... 

The flat electrode face a primitive model (continuum dielectric) electrolyte. In this case... 

In general, conqiact models are classified into three categories primitive models, macromodels, and behavioral models [1]. A primitive model is the constituent element (e. g., capacitors, inductors, and resistors etc.) in a corrplex system and is typically represented as a single Differential-Algebraic Equation (DAE) derived from basic conservation laws in different domains. The difference between macromodels and behavioral models is subtle, and both describe the dynamic response of a device via a set of equations. Macromodels are constructed via assembly of primitive models or a set of DAEs in a system-level representation, while the behavioral models are more generic and effective forms and built on the underlying domain-physics [1]. 

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