Nernst-Planck flux equation

Barry, P. H. (1998). Derivation of unstirred-layer transport number equations from the Nernst-Planck flux equations, Biophys. J., 74, 2903-2905. 

Based on the Nernst-Planck flux equation and Eyring s rate theory, a simple theoretical model was evolved for the description of the transport of ions through thick carrier membranes5 (see also Ref. 15). The primary... 

Application of the Nernst-Planck Flux Equations Combined with the... 

Thus, essentially, the Nernst-Planck flux equation is found. In principle, however, the theory enables the determination of the absolute values of the diffusion coefficients. 

In general it can be said that the experimental material is not extensive. The experimental material concerns the application of irreversible thermodynamics, the application of refined Nernst-Planck flux equations and the application of quasi-thermodynamics. The latter is used to derive equations for membrane potentials. 

Assuming an ideal solution in which the activity of a component is identical to its concentration and no kinetic coupling occurs between individual fluxes, Equation 5.8 becomes identical with the Nernst-Planck flux equation [18], which is given by ... 

To provide a quantitative expression for the diffusion flux 7 one cannot use the Nernst-Planck flux equation (4.231) because the latter describes the independent flow of one ionic species and in the case under discussion it has been shown that the migration current of the H ions is profoundly affected by the concentration of the K" ions. A simple modification of the Nernst-Planck equation can be argued as follows. 

From these modified forms of the Nernst-Planck flux equation (4.231), one can see that even if f,. - 0, it is still possible to have a flux of a species provided there is a concentration gradient, which is often brought into existence by interfacial charge-transfer reactions at the electrode-electrolyte interfaces consuming or generating the species. 

Nernst-Planck flux equations has to be made. The modification that will now be described is a more detailed version which will lead us back to transport numbers. 

Electroosmotic effects also influence current efficiency, not only in terms of coupling effects on the fluxes of various species but also in terms of their impact on steady-state membrane water levels and polymer structure. The effects of electroosmosis on membrane permselectivity have recently been treated through the classical Nernst-Planck flux equations, and water transport numbers in chlor-alkali cell environments have been reported by several workers.Even with classical approaches, the relationship between electroosmosis and permselectivity is seen to be quite complicated. Treatments which include molecular transport of water can also affect membrane permselectivity, as seen in Fig. 17. The different results for the two types of experiments here can be attributed largely to the effects of osmosis. A slight improvement in current efficiency results when osmosis occurs from anolyte to catholyte. Another frequently observed consequence of water transport is higher membrane conductance, " " which is an important factor in the overall energy efficiency of an operating cell. 

Aguilella VM, Maf6 S, PeUicer J (1987) On the nature of the diffusion potential derived from Nernst-Planck flux equations using the electroneutrality assumption. Electrochim Acta 32 483 88... 

The model upon which this equation is based neglects potential differences that may occur due to IR inside the membrane and assumes that the entire membrane potential consists of the difference of the two interfacial potential differences. In some cases, this may be a good approximation.4 In other cases, the potential difference through the membrane (determined by fluxes considered in the Nernst-Planck-type equations) may dominate. A comparison of Eq. (14.5) with (14.6) shows that both models lead to equations that have the same form. 

Donnan dialysis The BAHLM systems with ion-exchange membranes, based on Donnan dialysis [18,19], will be considered below. Donnan dialysis is a continuously operating ion-exchange process. There are many theoretical models describing transport mechanisms and kinetics of DL) [18-26]. All transport kinetics models are based on Fick s or Nernst-Planck s equations for ion fluxes. In both cases, the authors introduce many assumptions and simplifications. 

Planck derived a more general theory concerning ionic movement in solution by using the so-called Nernst-Planck diffusion equation. The following is the most simple expression of ionic fluxes for the one-dimensional case,... 

Ion transport across membranes can be evaluated by using mucosal and serosal electrodes to read transepithelial current (I) and potential difference OP). With these parameters, equivalent circuit analysis can be utilized to account for the relative contributions of transcellular and paracellular pathways. Ionic flux (J) is defined by the Nernst-Planck equation,... 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are /),yi and 2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance <5,- (diffusion layer thickness) from the surface where c, = 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst-Planck equation (10), it can be seen that the flux at any surface is ... 

If there is a net transport of charge across the membrane, the membrane potential will influence the solute transfer and also be affected by it, complicating the data treatment. The starting point for most descriptions of the internalisation flux of permeant ions, i, is the one-dimensional Nernst-Planck equation (cf. equation (10)) that combines a concentration gradient with the corresponding electric potential gradient [270] ... 

If water movement in the membrane is also to be considered, then one way to do this is to again use the Nernst—Planck equation. Because water has a zero valence, eq 29 reduces to Pick s law, eq 17. However, it is also well documented that, as the protons move across the membrane, they induce a flow of water in the same direction. Technically, this electroosmotic flow is a result of the proton—water interaction and is not a dilute solution effect, since the membrane is taken to be the solvent. As shown in the next section, the electroosmotic flux is proportional to the current density and can be added to the diffusive flux to get the overall flux of water... 

The flnely-porous membrane model (, ) assumes that a substantial amount of salt is transported by convective flow through the narrow pores of the membrane. Integrating the Nernst-Planck equation for salt transport O) and using the appropriate boundary conditions, the following relationship is obtained between the salt rejection and the volume flux ... 

Since the fluxes of electrolyte ions are zero at the electrode surface (rc), the Nernst-Planck equation for the anion becomes... 

In the above it is already said that due to drastic-simplification of the flux equations, the Nernst-Planck equations arise. To show this it is necessary to first write equation (1) in a somewhat different form. For the purpose, the velocities of the several components are split up in a velocity of the particles relative to one another and a velocity of trans-... 

I.I. Concentration gradient Across a Membrane. In the instance that a membrane separates two solutions of the same electrolyte, but with different concentrations F. Helfferich (ref. 55, page 319) calculated the ion-fluxes and the profiles of the internal concentrations, starting from the Nernst-Planck equations. Gradients of activity coefficients could be involved. However, convection (osmosis) had to be neglected. 

R. Schlogl (144) obtained, through his general integration of the Nernst-Planck equations, also values for the diffusion potential. The approximations in the calculations are the same as those used for the fluxes (cf. 3.4). 

The Nernst-Planck equation constitutes the starting point for the electrotransport models [55-57], The overall flux of the ionic species i (/,) comprises the diffusion term driven by the chemical potential gradient (dc,/dx) and the electric transference term due to the electrical potential gradient (d /dx) ... 

As the skin is relatively thick compared to the space-charge layers at its boundaries, the bulk of the membrane may be expected to be electroneutral [56,57], The Nernst-Planck equation can be solved, therefore, by imposing the electroneutrality condition C,/C= C. /C, where the subscripts j and k refer to positive and negative ions, respectively, and C is the average total ion concentration in the membrane. In the case of a homogenous and uncharged membrane bathed by a 1 1 electrolyte, the total ion concentration profile across the membrane is linear and the resulting steady-state flux is described by... 

The flux of the species O, /0(x, t), is described in terms of the three components that constitute the Nernst-Planck equation (Equation 6.14). The parameters are defined below. 

This is a three-component (A, B, C) system, and the transport or diffusion of each component can be described using the Nernst-Planck equation. The relation of the diffusion flux of the i component (i, j, k = A, B, C) can be obtained by combining three Nernst-Planck equations, for the respective components, under the conditions of zero electric current inside the resin particle ... 

Species flux can be described by the Nernst-Planck equation,... 

The flux (J), a common measure of the rate of mass transport at a fixed point, is defined as the number of molecules penetrating a unit area of an imaginary plane in a unit of time and is expressed in units of mol cm V1. The flux to the electrode is described mathematically by a differential equation, known as the Nernst-Planck equation, given here for one dimension... 

In this chapter, we present most of the equations that apply to the systems and processes to be dealt with later. Most of these are expressed as equations of concentration dynamics, that is, concentration of one or more solution species as a function of time, as well as other variables, in the form of differential equations. Fundamentally, these are transport (diffusion-, convection-and migration-) equations but may be complicated by chemical processes occurring heterogeneously (i.e. at the electrode surface - electrochemical reaction) or homogeneously (in the solution bulk chemical reaction). The transport components are all included in the general Nernst-Planck equation (see also Bard and Faulkner 2001) for the flux Jj of species j... 

Nernst-Planck equation — This equation describes the flux of charged particles by diffusion and electrostatic forces. When the ion with charge ze is distributed at concentration c in the potential, cp, it has a one-dimensional flux of the ion, / = -Ddc/dx - (zF/RT) Dcdcp/dx [i]. This can be derived from the concept that the force caused by the gradient of the electrochemical potential is balanced with frictional force by viscosity, t], of the medium. When a spherical ion with radius ro is in the inner potential, cp, the gradient of the electrochemical potential per ion is given by... 

Equation 3.8, which is often called the Nernst-Planck equation, is a general expression for the one-dimensional flux density of species j either across a membrane or in a solution in terms of two components of the driving force — the gradients in activity and in electrical potential. 

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