Chemical separation equations with diffusion

Approximate Solutions of Chemical Separation Equations with Diffusion... 

The relationships corresponding to equations (l)-(5) above are as follows The obse ed rate constants (kjfobs) and activation parameters (AH xfobs AS -pfobs) for disappearance of R by reaction with T depend on the second order rate constant for diffusive collision to form the cage pair (k pj), equation 12) and the fractional cage efficiency (Fjc equation 11). The latter is determined by the first order rate constants for the chemical reaction (kjc) and diffusive (re)separation (kj j) of the T/R collisional cage pair. The associated activation parameters are given by equations (13) and (14). 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

With so many uncertainties, it is hardly surprising that the difficulties inherent in a successful application of the diffusion equation (or molecular pair analysis) to recombination probability experiments are very considerable. Chemically induced dynamic polarisation (Sect. 4) is a fairly new technique which may assist in the study of recombination of radicals following their diffusive separation from the solvent cage. 

The Donnan potential can also be regarded as a special case of a diffusion potential. We can assume that the mobile ions are initially in the same region as the immobile ones. In time, some of the mobile ions will tend to diffuse away. This tendency, based on thermal motion, causes a slight charge separation, which sets up an electrical potential difference between the Donnan phase and the bulk of the adjacent solution. For the case of a single species of mobile cations with the anions fixed in the membrane (both assumed to be monovalent), the diffusion potential across that part of the aqueous phase next to the membrane can be described by Equation 3.11 n — El = (u- — u+)/(u + w+)](i 77F)ln (c11/ 1) that we derived for diffusion toward regions of lower chemical potential in a solution. Fixed anions have zero mobility (u = 0) hence (u — u+)/(u — u+) here is —uJu+> or —1. Equation 3.11 then becomes En — El = — (RT/F) In (cll/cl)> which is the same as the Nernst potential (Eq. 3.6) for monovalent cations [—In = In (cVc11)]. 

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). 

Equation (11-86) expresses the combined effect of external and internal mass-transport resistance. Note that the reduction in rate due to internal diffusion (through rj) is combined with the rate constant k for the chemical step, while the external effect is separate. If the external resistance is negligible, then k a r]k. Ifinternal transport is insignificant, then rj 1. If both conditions are satisfied, the rate is determined solely by the chemical step that is, Eq. (11-85) reduces to Fp = k Cf,. 

To describe chemical reactions upon reencoimter, which in general will occur with different probabilities for the singlet and triplet states, one has to separate the variables qss TqTo again. However, because electron-spin polarization does not arise during a diffusive excursion and is randomized in the approach of the radicals to their contact distance, it suffices to retain the component 2Im(ps7 u) in the density matrix the component 2Re(ps7 u) is not needed. In this new basis, the mixing matrix M of Equation (17) becomes... 

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