Diffusion layer equation

The conductivity of liquid and glass membranes is determined by ion-migration (absence of an excess of supporting electrolyte is assumed) in the diffusion layer. Equation (25) should then be written as ... 

Now we can write the foregoing Levich eqn. 3.89 as a steady-state diffusion layer equation (cf., eqn. 3.4) ... 

The negative sign in Eq. (11) means that a negative value of (or i/ s) leads to a positive charge in the diffuse layer. Equation (11) can also be reversed to express i/ s as a function of era ... 

In Equation (2.4), is the differential capacitance of the Helmholtz layer [Equation (2.5)] and is the differential capacitance of the diffuse layer [Equation (2.6)] ... 

In order to obtain the total net charge within the diffuse layer. Equation (2.9) can be integrated from x = dtox = °°, resulting in ... 

The most important parameter in equations (7.37) to (7.39) is k, which has the dimension of the reciprocal of length. In water at 25 C, = 0.329 /7A". Distance is the Debye-Huckel length and represents the thickness of the diffuse layer. This happens to be a misnomer because, over distance /c , the potential decreases only by j/exp(l) = tpd/2.1, but, in the weak potential approximation, the diffuse layer [equation (7.38)] can be treated as a parallel-plate capacitor Q = 6K with plates separated by distance /c . The variation in the potential in the solution, as a function of the distance from the surface, depends on the concentration and the charge of the ions present in the electrolyte (Figure 7.6). 

The rate of dissolving of a solid is determined by the rate of diffusion through a boundary layer of solution. Derive the equation for the net rate of dissolving. Take Co to be the saturation concentration and rf to be the effective thickness of the diffusion layer denote diffusion coefficient by . 

The diffusion layer widtli is very much dependent on tire degree of agitation of tire electrolyte. Thus, via tire parameter 5, tire hydrodynamics of tire solution can be considered. Experimentally, defined hydrodynamic conditions are achieved by a rotating cylinder, disc or ring-disc electrodes, for which analytical solutions for tire diffusion equation are available [37, 4T, 42 and 43]. 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

Volt mmetiy. Diffusional effects, as embodied in equation 1, can be avoided by simply stirring the solution or rotating the electrode, eg, using the rotating disk electrode (RDE) at high rpm (3,7). The resultant concentration profiles then appear as shown in Figure 5. A time-independent Nernst diffusion layer having a thickness dictated by the laws of hydrodynamics is estabUshed. For the RDE,... 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

In a boundary layer equation the mass center is considered with the help of the velocity (u, Uy, u ) and therefore a distribution of the velocity of the mass center is desirable. The diffusion velocity and diffusion factor are determined with regard to velocity v, giving a formula for Vax /x, but not for /ax - x useful approach is offered by Eq. (4.268c), using the artificial multiplication factor (v - ax /... 

Figure 1.3 la to c shows how an increase in the concentration of dissolved oxygen or an increase in velocity increases /Y and thereby increases. It has been shown in equation 1.73 that /Y increases with the concentration of oxygen and temperature, and with decrease in thickness of the diffusion layer, and similar considerations apply to. Thus Uhlig, Triadis and Stern found that the corrosion rate of mild steel in slowly moving water at... 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

The outer layer (beyond the compact layer), referred to as the diffuse layer (or Gouy layer), is a three-dimensional region of scattered ions, which extends from the OHP into the bulk solution. Such an ionic distribution reflects the counterbalance between ordering forces of the electrical field and the disorder caused by a random thermal motion. Based on the equilibrium between these two opposing effects, the concentration of ionic species at a given distance from the surface, C(x), decays exponentially with the ratio between the electro static energy (zF) and the thermal energy (R 7). in accordance with the Boltzmann equation ... 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

We can also calculate the individual values of (with the equations reported in Section 5.2) and of by determining the concentration distribution in the diffusion layer, and from this, the distribution of solution conductivity. The resulting combined value of q> coincides with the value determined from Eq. (6.32). 

It follows from this equation that the effective diffusion-layer thickness can be described as... 

A second approach is to assume that the drop surface approaching the electrode is a moving plane. This is appropriate, since the diffusion layer is almost always considerably smaller than the size of the drop, for most of its lifetime under practical conditions. To a good approximation, the convective effect close to the moving front is then calculated based on velocities which are twice those determined from Eq. (23), in order to account for the moving center of the drop. The convective-diffusion equation which describes this case is given by... 

The description of the ion transfer process is closely related to the structure of the electrical double layer at the ITIES [50]. The most widely used approach is the combination of the BV equation and the modified Verwey-Niessen (MVN) model. In the MVN model, the electrical double layer at the ITIES is composed of two diffuse layers and one ion-free or inner layer (Fig. 8). The positions delimiting the inner layer are denoted by X2 and X2, and represent the positions of closest approach of the transferring ion to the ITIES from the organic and aqueous side, respectively. The total Galvani potential drop across the interfacial region, AgCp = cj) — [Pg.545]

Alternatively, the flux density /, can be expressed in terms of the ion concentrations, c" and c° at the positions Xq and Xq just outside the diffuse layers, and the total Galvani potential difference A [Pg.545]

When the ITIES is polarized with a potential difference 0, there is a separation of electrical charge across it. According to the Gouy-Chapman theory, the charges in the aqueous and organic diffuse layers are related to the potential drops and A0 in the respective layers by the equations... 

Although it is possible to control the dissolution rate of a drug by controlling its particle size and solubility, the pharmaceutical manufacturer has very little, if any, control over the D/h term in the Nernst-Brunner equation, Eq. (1). In deriving the equation it was assumed that h, the thickness of the stationary diffusion layer, was independent of particle size. In fact, this is not necessarily true. The diffusion layer probably increases as particle size increases. Furthermore, h decreases as the stirring rate increases. In vivo, as GI motility increases or decreases, h would be expected to decrease or increase. In deriving the Nernst-Brunner equation, it was also assumed that all the particles were... 

This dissolution process can be considered to be diffusion-layer controlled. This is best explained by considering the rate of diffusion from the solid surface to the bulk solution through an unstirred liquid film as the rate-determining step. This dissolution process at steady state is described by the Noyes-Whitney equation ... 

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