Nernst-Planck’s equation

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. 

Equation 774 is referred to as the generalized form of the Nernst-Planck s equation. 

Nernst-Planck s equation for water transport in polymer membrane ... 

Resolution of the Nernst-Planck s Equation in the Case of a Localized Galvanic Cell on a Real Microstnicture... 

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... 

The bulk transport of ions in electrochemical systems without the contribution of advection is described by Poisson-Nernst-Planck (PNP) equations (Rubinstein, 1990).The well-known Nernst-Planck equation describes the processes of the process that drives the ions from regions of higher concentration to regions of lower concentration, and electromigration (also referred to as migration), the process that launches the ions in the direction of the electric field (Bard and Faulkner, 1980). Since the ions themselves contribute to the local electric potential, Poisson s equation that relates the electrostatic potential to local ion concentrations is solved simultaneously to describe this effect. The electroneutrality assumption simplifies the mathematical treatise of bulk transport in most electrochemical systems. Nevertheless, this no charge density accumulation assumption does not hold true at the interphase regions of the electric double layer between the solid and the Uquid, hence the cause of most electrokinetic phenomena in clay-electrolyte systems. 

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 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... 

Nernst-Einstein equatioon, 5 587 Nernst equation, 9 571 12 206 19 206 Nernst-Planck equation, 9 612-613 Nerol, 3 233 24 479, 501, 503-506 grades of, 24 505 hydrogenation of, 24 506 price of, 24 505 Nerolidol, 24 546-547 Neroliodyl acetate, 24 547 Nerve agents, 5 815, 818-821 Neryl, 24 479 Neryl esters, 24 505 Nesmeyanov reaction, 3 75 Nested fullerenes, 12 231 Nested situations, amount of coverage in experimental design texts compared, S 395t... 

Equation (11.4) provides a convenient value for that constant. Planck s statement asserts that 5qk is zero only for pure solids and pure liquids, whereas Nernst assumed that his theorem was applicable to all condensed phases, including solutions. According to Planck, solutions at 0 K have a positive entropy equal to the entropy of mixing. (The entropy of mixing is discussed in Chapters 10 and 14). 

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 first model to describe the membrane in the above fashion was that of Bernardi and Verbrugge, "° which was based on earlier work by Verbrugge and Hill. " 214 model utilized a dilute solution approach that used the Nernst— Planck equation (eq 29) to describe the movement of protons, except that now v is not equal to zero. The reason is that, because there are two phases, the protons are in the water and the velocity of the water is give by Schlogl s equation ... 

Similar statements can be made about holes. They, too, have to be transported to the interface to be available for the receipt of electrons there. These matters all come under the influence of the Nernst-Planck equation, which is dealt with in (Section 4.4.15). There it is shown that a charged particle can move under two influences. The one is the concentration gradient, so here one is back with Fick s law (Section 4.2.2). On the other hand, as the particles are changed, they will be influenced by the electric field, the gradient of the potential-distance relation inside the semiconductor. Electrons that feel a concentration gradient near the interface, encouraging them to move from the interior of the semiconductor to the surface, get seized by the electric field inside the semiconductor and accelerated further to the interface. 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

T. Tsuru, S. Nakao and S. Kimura, Calculation of ion rejection by extended Nernst-Planck equation with charged reverse osmosis membranes for single and mixed electrolytes. J. Chem. Eng. Japan 24 (1991) 511-517. 

As long as the system remains close to equilibrium and the fluxes are independent, the fluxes are treated as proportional to the driving forces. Experience (Table 4.17) commends this view for diffusion [Pick s law, Eq. (4.16)], conduction [Ohm s law, Eq. (4.130)], and heat flow (Fourier s law). Thus, the independent flux of an ionic species 1 given by the Nernst-Planck equation (4.231) is written... 

The third form of mass transport is convection driven by pressure. When forced circulation exists in electrolyte, convection may be the dominant form of mass transport. Thus, in general, a flux Jj (mol/s cm) of species j may occur due to the above three types of mass transport mechanisms. The flux can be described by the Nernst-Planck equation [5]... 

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

The flux, Jo(x, t), is defined as the transport of O per unit area (mol s cm ). It can be divided into three components, diffusion, migration, and convection, as originally expressed in the Nernst-Planck equation, written for one-dimensional mass transport along the x-axis in Eq. 18. 

However, in interdiffusion of ions of different mobilities, Fick s law fluxes would be unequal and disturb electroneutrality. Here, the first, minute deviation from local electroneutrality generates an electric potential gradient (diffusion potential) that produces electric transference of ions superimposed on diffusion. This is the mechanism by which the system manages to balance the fluxes so as to maintain electroneutrality (Schlogl and Helfferich, 1957 Helfferich, 1962a Helfferich and Hwang, 1988). The flux now obeys the Nernst-Planck equation (Nernst, 1888 1889 Planck, 1890)... 

Einstein-Schmolukowski, 378, 405 Gibbs-Duhem, 262 LaPlace, 392 Leonard-Jones, 45 Nernst-Einstein, 456 Nernst Planck, 476 Onsager, 494 Planck-Henderson, 500 Poisson, 235, 344 Poisson-Boltzmann, 239 Sackur-Tetrode equation, 128 Setchenow s, 172 Tafel, 2... 

---