Nernst theory, diffusion

This theory was a development of Nernst s diffusion theory of reactions taking place between substances in different phases, but it is by no means inseparably connected with this. It was employed and developed by Bodenstein, and can usually be made to account more or less satisfactorily for the influence of the various concentrations on the rate of reaction. 

The P s are now permeability coefficients and are related to the mobilities of the ions as in the original Nernst theory. The subscripts in and out refer to the concentrations of the ions inside and outside the membrane and the P s describe diffusion coefficients, mobilities, and the membrane thickness, but, in the Hodgkin-Huxley theory, were used as adjustable parameters. 

The contradiction is only an apparent one, and the difference between the theories lies in the hypotheses. The measurement of the speeds of reaction depends upon the conditions of the experiment. If the reaction between two components of reaction actually takes place instantaneously, we can vary the time of reaction entirely at will by the period of time during which we add one of the components. If the latter is used up with immeasurable rapidity, the measured velocity of reaction must naturally always remain proportional to the added quantity of the reaction components. The Nernst theory is based on relations in which this subsequent delivery is effected only by the diffusion, the reacting agent furnished by the current being... 

Near the end of the last century, Nernst " proposed the basic concept of ionic movement in an electrolyte system. In an electrolyte system, ions are acted upon by two kinds of forces relevant to our discussion the coulombic force and the osmotic force, which is essentially equal to the force of diffusion due to the ionic concentration difference. According to the Nernst theory, the liquid junction potential exhibited at the interface of two electrolyte solution phases with different ionic concentrations arises essentially from the difference in ionic mobilities across the liquid junction. 

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,... 

The diffusion potential theory has its origins in the Nernst theory formulated to describe the liquid junction potential developed at the interface of two electrolytes with different ionic concentrations and ionic mobilities. For two univalent ionic solutions the liquid junction potential is given by... 

The Nernst theory makes the questionable assumption of a constant geometry for diffusion. A more accurate treatment, due to Nielsen ( ), yields the relationship ... 

Nernst-Haskell The theory of dilute diffusion of salts is well developed and has been experimentally verified. For dilute solutions of a single salt, the well-known Nernst-Haskell equation (Reid et al.) is applicable ... 

In an earlier note (p. 9) we mentioned the occurrence of overvoltage in an electrolytic cell (and overpotentials at single electrodes), which means that often the breakthrough of current requires an Uappl = Eiecomp r] V higher than Ehack calculated by the Nernst equation as this phenomenon is connected with activation energy and/or sluggishness of diffusion we shall treat the subject under the kinetic treatment of the theory of electrolysis (Section 3.2). 

In agreement with the theory of electrolysis, treated in Sections 3.1 and 3.2, the parts of the residual current and the limiting current are clearly shown by the nature of the polarographic waves because for the cathodic reduction of Cd2+ and Zn2+ at the dme we have to deal with rapid electron transfer and limited diffusion of the cations from the solution towards the electrode surface and of the metal amalgam formed thereon towards the inside of the Hg drop, we may conclude that the half-wave potential, Eh, is constant [cf., Fig. 3.13 (a ] and agrees with the redox potential of the amalgam, i.e., -0.3521V for Cd2+ + 2e - Cd(Hg) and -0.7628 V for Zn2+ + 2e -> Zn(Hg) (ref. 10). The Nernst equation is... 

Steady diffusion across a thin film is mathematically straightforward but physically subtle. Dissolution film theory, suggested initially by Nernst and Brunner, is essentially based on steady diffusion across a thin film. 

Doi and his coworkers have proposed a semiquantitative theory for the swelling behavior of PAANa gels in electric fields [14]. They have considered the effect of the diffusion of mobile ions due to concentration gradients in the gel. First of all, the changes in ion concentration profiles under an electric field have been calculated using the partial differential Equation 16 (Nernst-Planck equation [21]). 

Barkey, Tobias and Muller formulated the stability analysis for deposition from well-supported solution in the Tafel regime at constant current [48], They used dilute-solution theory to solve the transport equations in a Nernst diffusion layer of thickness S. The concentration and electrostatic potential are given in this approximation... 

Therefore we expect Df, identified as the fast diffusion coefficient measured in dynamic light-scattering experiments, in infinitely dilute polyelectrolyte solutions to be very high at low salt concentrations and to decrease to self-diffusion coefficient D KRg 1) as the salt concentration is increased. The above result for KRg 1 limit is analogous to the Nernst-Hartley equation reported in Ref. 33. The theory described here accounts for stmctural correlations inside poly electrolyte chains. 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

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. 

The film model referred to in Chapters 2 and 5 provides, in fact, an oversimplified picture of what happens in the vicinity of interface. On the basis of the film model proposed by Nernst in 1904, Whitman [2] proposed in 1923 the two-film theory of gas absorption. Although this is a very useful concept, it is impossible to predict the individual (film) coefficient of mass transfer, unless the thickness of the laminar sublayer is known. According to this theory, the mass transfer rate should be proportional to the diffusivity, and inversely proportional to the thickness of the laminar film. However, as we usually do not know the thickness of the laminar film, a convenient concept of the effective film thickness has been assumed (as... 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

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. 

Also, in the late 1950s and 1960s some particularly seminal papers on ion exchange kinetics appeared by Helfferich (1962b, 1963, 1965) that are classics in the field. In this research it was definitively shown that the rate-limiting steps in ion exchange phenomena were film diffusion (FD) and/ or particle diffusion (PD). Additionally, the Nernst-Planck theories were explored and applied to an array of adsorbents (Chapter 5). 

C is the concentration of a cross-section, i.e. the reaction takes place directly at the electrode surface. If we suppose that the adjustment of the equilibrium takes place there with extreme rapidity, according to Nernst, then the reaction velocity will have to be based solely on the diffusion velocity, and will, therefore, be directly proportional to the concentration of the depolarizer in the electrolyte. The theory hence demands the formula... 

The two theories of Nernst and Haber above mentioned seem to contradict one another in important points. The electrical speed of reaction in the diffusion theory (Nernst) is directly a speed of diffusion Haber s formula holds good only in case the diffusion is excluded as much as possible. 

Modern opinion views the Nernst-Plank theory as a special case of applying the thermodynamics of irreversible processes to ion exchange. It may also be argued theoretically and experimentally that the observed characteristics of ion exchange rate behaviour can only be fully explained by including chemical reaction as a flux-coupling mechanism as well as the diffusion potential. From a research standpoint it is most probable that future theoretical advances in ion exchange kinetics will result from the further application of non-equilibrium thermodynamics. 

The expression a = nDI2/3 for the ionic conductivity a follows directly from the comparison with the phenomenological theories [2, 8] and indeed gives the well-known Nernst-Einstein relation connecting the conductivity with the mutual diffusion coefficient. Note that D = lirri, , 0 D(k, z) is the hy-... 

The mechanism of dissolution was proposed by Nernst (1904) using a film-model theory. Under the influence of non-reactive chemical forces, a solid particle immersed in a liquid experiences two consecutive processes. The first of these is solvation of the solid at the solid-liquid interface, which causes the formation of a thin stagnant layer of saturated solution around the particle. The second step in the dissolution process consists of diffusion of dissolved molecules from this boundary layer into the bulk fluid. In principle, one may control the dissolution through manipulation of the saturated solution at the surface. For example, one might generate a thin layer of saturated solution at the solid surface by a surface reaction with a high energy barrier (Mooney et al., 1981), but this application is not commonly employed in pharmaceutical applications. 

Nernst made numerous other important contributions to physical chemistry. For example, his distribution law described the concentration distribution of a solute in two immiscible liquids and allowed the calculation of extraction processes. He also formulated several significant theories, such as those on the electrostriction of ions, the diffusion layer at electrodes, and the solubility product. In addition, he established new methods to measure dielectric constants and to synthesize ammonia, on which the German chemist Fritz Haber later successfully followed up. see also Electrochemistry Haber, Fritz Ostwald, Friedrich Wilhelm Physical Chemistry. 

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