Semi-infinite diffusion

In the foregoing, the expressions needed to account for mass transport of O and R, e.g. eqns. (23), (27), (46), and (61c), were introduced as special solutions of the integral equations (22), giving the general relationship between the surface concentrations cG (0, t), cR (0, t) and the faradaic current in the case where mass transport occurs via semi-infinite linear diffusion. It is worth emphasizing that eqns. (22) hold irrespective of the relaxation method applied. Of course, other types of mass transport (e.g. bounded diffusion, semi-infinite spherical diffusion, and convection) may be involved, leading to expressions different from eqns. (22). 

Pick MotECUtAR Diffusion—Semi-Infinite Medium... 

Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

The procedure in use here involves the deposition of a radioactive isotope of the diffusing species on the surface of a rod or bar, the length of which is much longer than tire length of the metal involved in the diffusion process, the so-called semi-infinite sample solution. 

In a continuous steady state reactor, a slightly soluble gas is absorbed into a liquid in which it dissolves and reacts, the reaction being second order with respect to the dissolved gas. Calculate the reaction rate constant on the assumption that the liquid is semi-infinite in extent and that mass transfer resistance in the gas phase is negligible. The diffusivity of the gas in the liquid is 10" 8 m2/s, the gas concentration in the liquid falls to one half of its value in the liquid over a distance of 1 mm, and the rate of absorption at the interface is 4 x 10"6 kmol/m2 s. 

Solute gas is diffusing into a stationary liquid, virtually free of solvent, and of sufficient depth lot it to be regarded as semi-infinite in extent, in what depth of fluid below die surface will 90% of die material which has been transferred across the interface have accumulated in the first minute )... 

FIGURE 26.18 Theoretical temperature rise in the contact area of a pad sliding over a semi-infinite solid for different depths from the surface. Width 2b 2 mm, speed 3 m/s, pressure 2 Mp, p—l, heat conductivity 0.15 W/m/K, heat diffusivity 10 " m /s. 

The time-dependent diffusion equations for Red appropriate to the axisymmetrical geometry, shown in Fig. 10, are identical to Eqs. (9) and (10), given earlier. Although phase 2 is assumed to be semi-infinite in the z-direction, the model can readily be modified for the situation where phase 2 has a finite thickness [61]. 

E. Unsteady Diffusion into a Semi-Infinite Membrane... 

Figure 5 Diffusion into a semi-infinite membrane. The membrane initially contains no solute. At time zero, the concentration of the solution at z = 0 is suddenly increased to and maintained at cx. This abrupt increase produces time-dependent concentration profiles as the solute penetrates into the membrane.

Equation (128) with the initial and boundary conditions of Eqs. (129)—(131) is very similar to the model equations for diffusion in a semi-infinite membrane [Eqs. (64)-(67)]. The concentration profile is... 

In order to simplify the situation, we assume that our porous sample under investigation covers the bottom of an open straight-walled can and fills it to a height d (Figure 1). Such a sample will exhibit the same areal exhalation rate as a free semi-infinite sample of thickness 2d, as long as the walls and the bottom of the can are impermeable and non-absorbant for radon. A one-dimensional analysis of the diffusion of radon from the sample is perfectly adequate under these conditions. To idealize the conditions a bit further we assume that diffusion is the only transport mechanism of radon out from the sample, and that this diffusive transport is governed by Fick s first law. Fick s law applied to a porous medium says that the areal exhalation rate is proportional to the (radon) concentration gradient in the pores at the sample-air interface... 

Experimental measurements of DH in a-Si H using SIMS were first performed by Carlson and Magee (1978). A sample is grown that contains a thin layer in which a small amount (=1-3 at. %) of the bonded hydrogen is replaced with deuterium. When annealed at elevated temperatures, the deuterium diffuses into the top and bottom layers and the deuterium profile is measured using SIMS. The diffusion coefficient is obtained by subtracting the control profile from the annealed profile and fitting the concentration values to the expression, valid for diffusion from a semiinfinite source into a semi-infinite half-plane (Crank, 1956),... 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are /),yi and 2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance <5,- (diffusion layer thickness) from the surface where c, = 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst-Planck equation (10), it can be seen that the flux at any surface is ... 

In the case of semi-infinite diffusion towards a spherical organism of radius r0, with the surface of separation (concentric with the organism s surface) at radius ra, the flux at the membrane surface is [26] ... 

The leading boundary conditions correspond with semi-infinite diffusion with an instantaneous sink of the diffusing species at the plane x = 0 ... 

Figure 17. Comparison of Ju versus t plots predicted by different submodels for a system with one type of site (adsorbing and internalising) linear isotherm with dSS approximation (O) applying equation (43) with A)i = 5.2 x 10 6 m Langmuirian isotherm with dSS approximation (continuous line) applying equation (46) Langmuirian isotherm with semi-infinite diffusion (dotted line) by numerically solving integral equation (7)). Other parameters c(, = 5x 10-4 mol m-3, Dm = 8 x 10 10 m2 s-1, Kn = 2 x 10-5 m, k — 5 x 10 4 s 1, ro = 1.8 x 10 6 m, r0 + <5M = 10 5 m, KM — 2.88x 10 3mol m 3, Tmax = 1.5 x 10-8 mol m-2...

As a first example, the transient case with Henry isotherm can be considered. Expressions developed in Section 2.3 apply with D replacing Dm,ct m replacing cM (including the substitution of c v M by < M and cfsM by c f ) and Ku (defined as r/cM(r0,t) in both cases, i.e. with or without the presence of L) by AT i / (1 + Kc ). Other cases with analytical solutions arise from the steady-state situation. The supply flux under semi-infinite steady-state diffusion is [57] ... 

If instead of semi-infinite diffusion, some distance (5m acts as an effective diffusion layer thickness (Nernst layer approximation), then a modified expression of equation (63) applies where ro is substituted by 1 / (1 /Vo + 1 /<5m ) (see equation (38) above). For some hydrodynamic regimes, which for simplicity, are not dealt with here, the diffusion coefficient might need to be powered to some exponent [57,58],... 

Let us assume parallel flux in a semi-infinite medium bound by the plane x=0. Diffusion of a given element takes place from the plane x=0 kept at concentration Cint. Introducing a Boltzmann variable u with constant diffusion coefficient such as... 

Consider an inert metal with a fractal electrode immersed into an electrolyte containing a redox couple. We assume semi-infinite diffusion of the oxidized species Ox and the reduced species Red in the electrolyte. For the sake of simplicity we assume that the solution contains initially (at the time, t = 0) only the oxidized form and the bulk and surface concentrations are identical, i.e., cox=cL- The electrode is initially subjected to an electrode potential where no reaction takes place. The only reaction occurring when the potential is lowered, is the reversible reduction of Ox to Red, i.e., Ox + ze = Red. In addition, it is assumed that the overall reaction is limited by diffusion of Ox in electrolyte. 

In cyclic voltammetry, simple relationships similar to equations (1.15) may also be derived from the current-potential curves thanks to convolutive manipulations of the raw data using the function 1 /s/nt, which is characteristic of transient linear and semi-infinite diffusion.24,25 Indeed, as... 

FIGURE 1.11. Convolution of the cyclic voltammetric current with the function I j Jnt, characteristic of transient linear and semi-infinite diffusion. Application to the correction of ohmic drop, a —, Nernstian voltammogram distorted by ohmic drop , ideal Nernstian voltammogram. b Convoluted current vs. the applied potential, E. c Correction of the potential scale, d Logarithmic analysis. 

The convolution treatment of the linear and semi-infinite diffusion reactant transport (Section 1.3.2) leads to the following relationship between the concentrations at the electrode surface and the current ... 

In this section we establish the equation of the forward scan current potential curve in dimensionless form (equation 1.3), justify the construction of the reverse trace depicted in Figure 1.4, and derive the charge-potential forward and reverse curves, also in dimensionless form. Linear and semi-infinite diffusion is described by means of the one-dimensional first and second Fick s laws applied to the reactant concentrations. This does not imply necessarily that their activity coefficients are unity but merely that they are constant within the diffusion layer. In this case, the activity coefficient is integrated in the diffusion coefficient. The latter is assumed to be the same for A and B (D). 

For many adsorbates, especially organic substances, the concept of semi-infinite linear diffusion can give us some ideas on the time necessary for an adsorbate to be adsorbed. The number of mols adsorbate, n, diffusing to a unit area of a surface per second, is proportional to the bulk concentration of adsorbate, c ... 

The diffusion coefficients for Rb, Cs and Sr in obsidian can be calculated from the aqueous rate data in Table 1 as well as from the XPS depth profiles. A simple single-component diffusion model (9j characterizes onedimensional transport into a semi-infinite solid where the diffusion coefficient (cm2-s 1) is defined by ... 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

Diffusion in Matrix. The transport equation for a semi-infinite medium of uniform initial concentration of mobile species, with the surface concentration equal to zero for time greater than zero, is given by Crank (13). The rate of mass transfer at the surface for this model is ... 

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