Diffusion potential derivation

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... 

When both solutions are binary and identical in nature and differ only by their concentration and the component E of the held strength is given by Eq. (4.18), the diffusion potential 9 can be expressed by Eq. (4.19). An equation of this type was derived by Walther Nemst in 1888. Like other equations resting on Eick s law (4.1), this equation, is approximate and becomes less exact with increasing concentration. For the more general case of multicomponent solutions, the Henderson equation (1907),... 

A semi-permeable membrane, which is unequally permeable to different components and thus may show a potential difference across the membrane. In case (1), a diffusion potential occurs only if there is a difference in mobility between cation and anion. In case (2), we have to deal with the biologically important Donnan equilibrium e.g., a cell membrane may be permeable to small inorganic ions but impermeable to ions derived from high-molecular-weight proteins, so that across the membrane an osmotic pressure occurs in addition to a Donnan potential. The values concerned can be approximately calculated from the equations derived by Donnan35. In case (3), an intermediate situation, there is a combined effect of diffusion and the Donnan potential, so that its calculation becomes uncertain. 

The remaining concentration of electron holes and, therefore, the electrical conductivity are functions of the water vapor pressure. This function can be derived by applying the formula (15a) of the exhaustion boundary layer. There we have, however, to substitute for Using the law of mass action of (21), we obtain for the diffusion potential... 

The discussion up to this point has been concerned essentially with metal alloys in which the atoms are necessarily electrically neutral. In ionic systems, an electric diffusion potential builds up during the spinodal decomposition process. The local gradient of this potential provides an additional driving force, which acts upon the diffusing species and this has to be taken into account in the derivation of the equivalents of Eqns. (12.28) and (12.30). The formal treatment of this situation has not yet been carried out satisfactorily [A.V. Virkar, M. R. Plichta (1983)]. We can expect that the spinodal process is governed by the slower cation, for example, in a ternary AX-BX crystal. The electrical part of the driving force is generally nonlinear so that linearized kinetic equations cannot immediately be applied. 

The potential is an aggregate of all reversible work terms that can be transported with the species i. Using Lagrange multipliers, Cahn and Larche derive a potential that is a sum of the diffusant s elastic energy and its chemical potential [4]. Cahn and Larche coined the term diffusion potential to describe this sum. Our use of the term is consistent with theirs. 

The expression for this diffusion potential is derived in Exercise 13.3. 

Vapor transport differs from surface diffusional transport, where the flux is always in the surface plane. For both surface diffusion and vapor transport, the diffusion potential at the surface is proportional to the local value of 7sk if the surface free energy is isotropic. For surface diffusion, the interface normal velocity is related to a derivative (i.e., the divergence of the flux). Also, the total volume is conserved during surface diffusion. For vapor transport, the interface normal velocity is directly proportional to the vapor flux, and the total number of atoms is not necessarily conserved. 

The particular characteristics of morphological evolution are determined by the dominant transport mechanism their analyses derive from the diffusion potential, which depends on the local curvature. For a surface of revolution about the z-axis, the curvature is given by Eq. C.16 that is,... 

With all of the preliminaries out of the way, let us now derive the expression for the diffusion potential across a membrane for the case in which most of the net passive ionic flux density is due to K+, Na+, and Cl movements. Using the permeability coefficients of the three ions and substituting in the net flux density of each species as defined by Equation 3.16, Equation 3.17 becomes... 

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 important step in the derivation of the diffusion potential is the statement that under conditions of steady state, the electroneutrality field sees to it that the quantity of positive charge flowing into a volume element is equal in magnitude but opposite in sign to the quantity of negative charge flowing in (Fig. 4.82). That is. 

This is the basic equation for the diffusion potential. It has been derived here on the basis of a realistic point of view, namely, that the diffusion potential arises from the nonequilibrium process of diffusion. 

There is, however, another method of deriving the diffusion potential. One takes note of the fact that when a steady-state electroneutrality field has developed, the system relevant to a study of the diffusion potential hangs together in a delicate balance. The diffusion flux is exactly balanced by the electric flux the concentrations and the electrostatic potential throughout the interphase region do not vary with time. (Remember the derivation of the Einstein relation in Section 4.4.) In fact, one may turn a blind eye to the drift and pretend that the whole system is in equilibrium. 

An equation has been derived for the diffusion potential [cf. Eq. (4.283)], but it is a dijferential equation relating the infinitesimal potential difference dyj developed across an infinitesimally thick lamina dx in the interphase region. What one measures experimentally, however, is the total potential difference h.y/ = ij/ -y/ across a transition region extending fromx = 0 to x = Z (Fig. 4.85). Hence, to theorize about the... 

When two electrolyte solutions at different concentrations are separated by an ion--permeable membrane, a potential difference is generally established between the two solutions. This potential difference, known as membrane potential, plays an important role in electrochemical phenomena observed in various biomembrane systems. In the stationary state, the membrane potential arises from both the diffusion potential [1,2] and the membrane boundary potential [3-6]. To calculate the membrane potential, one must simultaneously solve the Nernst-Planck equation and the Poisson equation. Analytic formulas for the membrane potential can be derived only if the electric held within the membrane is assumed to be constant [1,2]. In this chapter, we remove this constant held assumption and numerically solve the above-mentioned nonlinear equations to calculate the membrane potential [7]. 

In order to explain the Na stimulation of ATP synthesis driven by a diffusion potential the presence of a Na /H antiporter was proposed [175]. In this artificial system the acidification of the cytoplasm, which occurs in response to electrogenic potassium efflux, could be prevented by the antiporter. Subsequently, Na /H antiporter activity has been demonstrated in both Methanobacterium thermoautotrophicum [176] and in Methanosarcina harden [108]. An important result of these studies was that the Na /H antiporter could be inhibited by amiloride and harmaline, which have been described as inhibitors of eucaryotic Na" /H" antiporters [177]. Using these inhibitors it has been shown that an active antiporter is essential for methanogenesis from H2/CO2 [176,178]. The antiporter also accepts Li instead of Na, since Li stimulates CH4 formation from H2/CO2 in the absence of Na [176]. In subsequent studies the use of amiloride and the more potent derivative ethyl-isopropylamiloride permitted the discrimination of primary and secondary Na potentials generated in partial reactions of the CO2 reduction pathway. 

The continued availability of adequate trophic support appears to be crucial not only for the development of nerve cells and their interconnecting circuitry, but also for the maintenance of neurons and their synapses in the adult (31). There is considerable evidence that diffusible, target-derived trophic factors play important roles in the development of specific retinal cell types. In particular, the neurotrophins [nerve growth factor (NGF), brain-derived neurotrophic factor (BDNF), neurotrophin (NT)-3 and NT-4/5] have received considerable attention for their potential roles in both developing and adult nervous systems (66,67). 

One can use a thermodynamic approach to derive the equation describing the condition of equilibrium between sedimentation and diffusion. The derivation involves the assumption of a constant gravity-chemical potential (i.e. the generalized chemical potential which includes the action of the external gravity field) and the assumption that the laws established for ideal systems can be applied to dilute dispersions, i.e. 

Oberst FW and Andrews HL, The electrolytic dissociation of morphine derivatives and certain synthetic analgetic compounds, JPET, 71, 38-41 (1941). Cited in Perrin Bases 2879 ref Ol. NB Results were reported as K, values. For codeine, Rb = 1-bl x 10 giving pJCb = 5.79. The potentiometric study used an asymmetric cell with diffusion potentials. 

The equations for the fluxes of and and the expression for the diffusion potential will be derived now for a filament on the basis of the assumptions discussed in Section 2.2. Across the length / of the proton-con-ducting filament a chemical potential difference FKh= —i Tln Ch(/)/ch(0) is assumed to be maintained by two synchronized chemical reactions which inject and take out H at z = 0 and z = 1 respectively. The K layer has the radial extension from r = ator = a-ft/ with the thickness d corresponding roughly to the Debye length. The concentration of in the K layer will... 

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