Nernst-Planck expression

This expression is sometimes referred to as the Nernst-Planck expression, and given the many simplifying assumptions made in deriving it, should be used with care. 

Application of elementary conservation laws leads to formulation of a general expression for which is often denoted as 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... 

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. 

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. 

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. 

UF and RO models may all apply to some extent to NF. Charge, however, appears to play a more important role than for other pressure driven membrane processes. The Extended-Nemst Planck Equation (equation (3.28)) is a means of describing NF behaviour. The extended Nernst Planck equation, proposed by Deen et al. (1980), includes the Donnan expression, which describes the partitioning of solutes between solution and membrane. The model can be used to calculate an effective pore size (which does not necessarily mean that pores exist), and to determine thickness and effective charge of the membrane. This information can then be used to predict the separation of mixtures (Bowen and Mukhtar (1996)). No assumptions regarding membrane morphology ate required (Peeters (1997)). The terms represent transport due to diffusion, electric field gradient and convection respectively. Jsi is the flux of an ion i, Di,i> is the ion diffusivity in the membane, R the gas constant, F the Faraday constant, y the electrical potential and Ki,c the convective hindrance factor in the membrane. 

The above expression can be derived from the Nernst-Planck equations with the assumption that concentration does not vary with time. In addition the space dimension of Equation 22.4 is reduced to a two-compartment — cell interior and exterior — model. Thus the steady-state of Equation 22.4 teUs us that when there is no net current flow, the transmembrane potential will equal a quantity called the Nernst potential E = (RT/ZiP) ln([C]o/[C],) z, is the valence of the ion, [C,]o is the concentration of the ion on the outside, and [C,] is the concentration on the inside. Quantities such as ion mobility are effectively lumped into a nonlinear time-varying conductance (G,m hf). 

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,... 

Under steady-state conditions the partial currents persist, so that // = const., and Eq. (88) is then a first-order nonlinear differential equation containing unknown functions q and E and unknown constant U This fundamental electrodiffusion equation, conventionally referred to as the Nernst-Planck equation, may be obtained by direct differentiation of the expression for the electrochemical potential of ion species / present in the dilute solution... 

Recently, Marcus has attempted to treat the ion transfer as a combined activation-controlled and transport-controlled process [68]. In the Marcus theory for ion transfer between two immiscible solutions, the ion transport is described using a Nernst-Planck type equation [57]. This equation leads to an expression of similar to that obtained by Kakiuchi [60],... 

The probability distribution functions in Eq. [59] applied to the trajectories of particles flowing into and out of a system provides a justification for using the Nernst-Planck equation (Eq. [54]) The net ionic directional fluxes can be expressed in terms of differences between the probability fluxes, normalized to the concentration at the sides of the region of interest. That ionic fluxes and differences in probability fluxes are related thus supplies a connection between the solution of the Nernst-Planck equation (Eq. [54]) and the Smoluchowski equation (Eq. [59]), and it provides a direct justification for using Eq. [54] for the study of ions subjected to Brownian dynamics in solution. 

When the flux expression (2) is substituted into (9), the general equation ( Nernst-Planck ) for the concentration and potential flelds is obtained ... 

Planck (loc. cit. 276) has observed that the point on which the whole matter turns is the establishment of a characteristic equation for each substance, which shall agree with Nernst s theorem. For if this is known we can calculate the pressure of the saturated vapour by means of Maxwell s theorem ( 90). He further remarks that, although a very large number of characteristic equations (van der Waals, Clausius s, etc.) are in existence, none of them leads to an expression for the pressure of the saturated vapour which passes over into (9) 210, at very low temperatures. Another condition which must be satisfied is... 

This is an expression of Nernst s postulate which may be stated as the entropy change in a reaction at absolute zero is zero. The above relationships were established on the basis of measurements on reactions involving completely ordered crystalline substances only. Extending Nernst s result, Planck stated that the entropy, S0, of any perfectly ordered crystalline substance at absolute zero should be zero. This is the statement of the third law of thermodynamics. The third law, therefore, provides a means of calculating the absolute value of the entropy of a substance at any temperature. The statement of the third law is confined to pure crystalline solids simply because it has been observed that entropies of solutions and supercooled liquids do not approach a value of zero on being cooled. 

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