Diffusion, theoretical treatment

To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

Theoretically, dispersion can take place by diffusion in the stationary phase but, as will be seen, in practice, is much less in magnitude than that in the mobile phase. The theoretical treatment is similar to that for dispersion in the mobile phase using equation (10). 

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

The theoretical treatment which has been developed in Sections 10.2-10.4 relates to mass transfer within a single phase in which no discontinuities exist. In many important applications of mass transfer, however, material is transferred across a phase boundary. Thus, in distillation a vapour and liquid are brought into contact in the fractionating column and the more volatile material is transferred from the liquid to the vapour while the less volatile constituent is transferred in the opposite direction this is an example of equimolecular counterdiffusion. In gas absorption, the soluble gas diffuses to the surface, dissolves in the liquid, and then passes into the bulk of the liquid, and the carrier gas is not transferred. In both of these examples, one phase is a liquid and the other a gas. In liquid -liquid extraction however, a solute is transferred from one liquid solvent to another across a phase boundary, and in the dissolution of a crystal the solute is transferred from a solid to a liquid. 

The treatment of the two-phase SECM problem applicable to immiscible liquid-liquid systems, requires a consideration of mass transfer in both liquid phases, unless conditions are selected so that the phase that does not contain the tip (denoted as phase 2 throughout this chapter) can be assumed to be maintained at a constant composition. Many SECM experiments on liquid-liquid interfaces have therefore employed much higher concentrations of the reactant of interest in phase 2 compared to the phase containing the tip (phase 1), so that depletion and diffusional effects in phase 2 can be eliminated [18,47,48]. This has the advantage that simpler theoretical treatments can be used, but places obvious limitations on the range of conditions under which reactions can be studied. In this section we review SECM theory appropriate to liquid-liquid interfaces at the full level where there are no restrictions on either the concentrations or diffusion coefficients of the reactants in the two phases. Specific attention is given to SECM feedback [49] and SECMIT [9], which represent the most widely used modes of operation. The extension of the models described to other techniques, such as DPSC, is relatively straightforward. 

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

The transport of electro active species from the bulk of the solution to the electrode may be governed not only by diffusion but also by adsorption of the species on the electrode surface. When both the mechanisms are operative, the overall electrochemical process may give considerably complicated results. The theoretical treatment is complex and of limited interest to inorganic chemists, therefore, a qualitative approach will be adopted to identify the presence of adsorption phenomena. 

Finally, the result of a theoretical treatment of a similar system with almost complete association in the membrane will be given without calculations [66,67]. The diffusion potential in the membrane depends not only on electrodiffusion of J, and A but also on diffusion of associates JA and KA. The resultant formula for the membrane potential is... 

The initial theoretical treatment of these mechanisms of deposition was given by Lorenz (31-34). The initial experimental studies on surface diffusion were published by Mehl and Bockris (35, 38). Conway and Bockris (36, 40) calculated activation energies for the ion-transfer process at various surface sites. The simulation of crystal growth with surface diffusion was discussed by Gilmer and Bennema (43). 

The theoretical treatment of the selectivities of various types of liquid membranes is mainly due to the efforts of Eisenman and his school (18—25) and others (26,27). The membrane potential E is subdivided into two components (Eq. (4)), namely a boundary potential Eb, produced by the exchange equilibria at the phase boundaries between the membrane and the outside solutions, and a diffusion potential Ed, produced by the diffusion of ions in the membrane itself ... 

Lee et al. evolved a comprehensive analytical-theoretical treatment, based on the solution of the reorientational isotropic diffusion equation, for an ensemble of high-spin systems under motion. These authors developed an analytical expression for the slow-tumbling motional region that relates the orientational-motion correlation time t (in s), or the corresponding tumbling rate t, with the step separation bB, of the ESR fine structure of a quartet by Eq. 8,... 

Analytical treatment of the diffusion-reaction problem in a many-body system composed of Coulombically interacting particles poses a very complex problem. Except for some approximate treatments, most theoretical treatments of the multipair effects have been performed by computer simulations. In the most direct approach, random trajectories and reactions of several ion pairs were followed by a Monte Carlo simulation [18]. In another approach [19], the approximate Independent Reaction Times (IRT) technique was used, in which an actual reaction time in a cluster of ions was assumed to be the smallest one selected from the set of reaction times associated with each independent ion pair. 

There have been many other theoretical analyses of geminate radical recombination probabilities, some of which are considered further below. They can be divided into three types (a) diffusion equation treatments, (b) first passage time methods, and (c) kinetic theory applications. 

The theoretical treatment of mass transfer in LSV and CV assumes that only diffusion is operative. Supporting electrolyte concentrations of the order of 0.1 M are generally used at substrate concentrations of the order of 10-3 M, which should preclude the necessity of considering mass transfer by migration. Here, it is assumed that planar stationary electrodes are used under circumstances where diffusion can be considered to be semi-infinite linear diffusion. Other types of electrode may give rise to spherical, cyclindrical or rectangular diffusion and these cases have been treated. 

The theoretical treatment of ionic conductivity in solids is very similar to that of diffusion, the main difference is the superimposition of the potential field upon the potential barrier to migration (Fig. 3). 

As with the familiar theoretical treatment of ideal gases, the logical starting point for understanding the process of diffusion is consideration of the motion of a... 

Traditionally, experimental values of Zeff have been derived from measurements of the lifetime spectra of positrons that are diffusing, and eventually annihilating, in a gas. The lifetime of each positron is measured separately, and these individual pieces of data are accumulated to form the lifetime spectrum. (The positron-trap technique, to be described in subsection 6.2.2, uses a different approach.) An alternative but equivalent procedure, which is adopted in electron diffusion studies and also in the theoretical treatment of positron diffusion, is to consider the injection of a swarm of positrons into the gas at a given time and then to investigate the time dependence of the speed distribution, as the positrons thermalize and annihilate, by solving the appropriate diffusion equation. The experimentally measured Zeg, termed Z ), is the average over the speed distribution of the positrons, y(v,t), where y(v,t) dv is the number density of positrons with speeds in the interval v to v + dv at time t after the swarm is injected into the gas. The time-dependent speed-averaged Zef[ is therefore... 

The advantages derived from the use of microscopic liquid-liquid interfaces have been highlighted in Sect. 5.5.3, and different approaches to support such small liquidlliquid interfaces in pores, pipettes, and capillaries have been addressed. The theoretical treatment of ion transfer through these interfaces needs to consider the asymmetry of the diffusion fields inside and outside the pore or pipette (i.e., diffusion can be approximated as linear in the inner phase, whereas radial diffusion is significant in the outer phase, especially for small sizes) [36, 40, 42-44]. 

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