Fick’s law diffusion

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

Numerous fluxes obey the empirical relationship in Eq. 1, such as the flux of electric charges in a conductor (Ohm s law), molecular diffusion (Fick s law), conduction of heat (Fourier s law), or flow of water in an aquifer (Darcy s law). For two or more simultaneous transport fluxes, the driving forces of one may affect the other, resulting in mutually dependent fluxes in a coupled process ... 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

Fick s law of diffusion A law relating the rate of diffusion of a substance in a given direction to the gradient of its concentration. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Equimolar Counterdiffusion in Binary Cases. If the flux of A is balanced by an equal flux of B in the opposite direction (frequently encountered in binary distillation columns), there is no net flow through the film and like is directly given by Fick s law. In an ideal gas, where the diffusivity can be shown to be independent of concentration, integration of Fick s law leads to a linear concentration profile through the film and to the following expression where (P/RT)y is substituted for... 

The rate of diffusion of the carbon atoms is given by Fick s laws of diffusion. In one dimension,... 

When a relatively slow catalytic reaction takes place in a stirred solution, the reactants are suppHed to the catalyst from the immediately neighboring solution so readily that virtually no concentration gradients exist. The intrinsic chemical kinetics determines the rate of the reaction. However, when the intrinsic rate of the reaction is very high and/or the transport of the reactant slow, as in a viscous polymer solution, the concentration gradients become significant, and the transport of reactants to the catalyst cannot keep the catalyst suppHed sufficientiy for the rate of the reaction to be that corresponding to the intrinsic chemical kinetics. Assume that the transport of the reactant in solution is described by Fick s law of diffusion with a diffusion coefficient D, and the intrinsic chemical kinetics is of the foUowing form... 

For an ion to move through the lattice, there must be an empty equivalent vacancy or interstitial site available, and it must possess sufficient energy to overcome the potential barrier between the two sites. Ionic conductivity, or the transport of charge by mobile ions, is a diffusion and activated process. From Fick s Law, J = —D dn/dx), for diffusion of a species in a concentration gradient, the diffusion coefficient D is given by... 

The dislocation cannot glide upwards by the shearing of atom planes - the atomic geometry is wrong - but the dislocation can move upwards if atoms at the bottom of the half-plane are able to diffuse away (Fig. 19.2). We have come across Fick s Law in which diffusion is driven by differences in concentration. A mechanical force can do exactly the same thing, and this is what leads to the diffusion of atoms away from the... 

Diffusion plays an important part in peak dispersion. It not only contributes to dispersion directly (i.e., longitudinal diffusion), but also plays a part in the dispersion that results from solute transfer between the two phases. Consider the situation depicted in Figure 4, where a sample of solute is introduced in plane (A), plane (A) having unit cros-sectional area. Solute will diffuse according to Fick s law in both directions ( x) and, at a point (x) from the sample point, according to Ficks law, the mass of solute transported across unit area in unit time (mx) will be given by... 

If the rate of a reaction is governed by the encounter frequency, it is said to be diffusion-controlled. This frequency imposes an upper limit on the rate of reaction that can be evaluated by the use of Fick s laws of diffusion. The mathematical expression of this phenomenon was first presented by von Smoluchowski.2 We shall adopt a simple approach,3,4 although more rigorous derivations have been given.5... 

The kinetics of transport depends on the nature and concentration of the penetrant and on whether the plastic is in the glassy or rubbery state. The simplest situation is found when the penetrant is a gas and the polymer is above its glass transition. Under these conditions Fick s law, with a concentration independent diffusion coefficient, D, and Henry s law are obeyed. Differences in concentration, C, are related to the flux of matter passing through the unit area in unit time, Jx, and to the concentration gradient by,... 

Fick s Law of diffusion is normally expressed in molar units or ... 

In a steady-state process, a gas is absorbed in a liquid with which it undergoes an irreversible reaction. The mass transfer process is governed by Fick s law, and the liquid is sufficiently deep for it to be regarded as effectively infinite in depth. On increasing the temperature, the concentration of reactant at the liquid surface CAi falls to 0.8 times its original value. The diffusivity is unchanged, but the reaction constant increases by a factor of 1.35. It is found that the mass transfer rate at the liquid surface falls to 0.83 times its original value. What is the order of the chemical reaction ... 

It is shown in Chapter 10, from Fick s Law of diffusion, that the rate of diffusion of a constituent A in a mixture is proportional to its concentration gradient. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

The charge consumed by oxidation swelling under diffusion control, once the structure is relaxed, depends on the anodic potentials applied at each moment. The process can be quantified by Fick s law ... 

Diffusion rates for liquids in an elastomer are easily measured by absorption (immersion) testing, a simple process as indicated in Figure 23.6. An initially weighed immersed sheet sample of elastomer is removed from the liquid periodically, rapidly dabbed with tissue paper, reweighed, and replaced. A plot of mass increase versus root time is drawn (also see Figure 23.6), root time being chosen due to the form of appropriate solutions of Fick s laws. 

Confined flows typically exhibit laminar-flow regimes, i.e. rely on a diffusion mixing mechanism, and consequently are only slowly mixed when the diffusion distance is set too large. For this reason, in view of the potential of microfabrication, many authors pointed to the enhancement of mass transfer that can be achieved on further decreasing the diffusional length scales. By simple correlations based on Fick s law, it is evident that short liquid mixing times in the order of milliseconds should result on decreasing the diffusion distance to a few micrometers. 

As shown in Fig. 1.12, diffusional flow contributions in engineering situations are usually expressed by Fick s Law for molecular diffusion... 

Fick s law, as modified to describe the diffusion of available (diffusible) solutes in soil and written in Cartesian coordinates is therefore... 

Figure 4 Schematic representation of a small section of a diffusion profile illustrating the application of Fick s law to determine the concentration change in the central volume element as a result of the fluxes (F) across the two planes at L and R (see text for details).

To inject a general note it may be pointed out that two very important laws, called Fick s laws, form the basis of diffusion theory. The first law can be expressed in the following form ... 

Diffusion is not straightforward inside the resin phase, and this is due to the restrictive influences of the polymer network and because of the charge distribution connected with the fixed ions of the functional groups. The resin phase is consequently related to a porous solid. The effectual diffusivities of metal ions in the resin phase may differ but are largely less than those in the aqueous phase external to the resin phase. If Fick s law is applied to diffusion in a resin bead of radius, r, it may be represented as... 

The book is organized into eight chapters. Chapter 1 describes the physicochemical needs of pharmaceutical research and development. Chapter 2 defines the flux model, based on Fick s laws of diffusion, in terms of solubility, permeability, and charge state (pH), and lays the foundation for the rest of the book. Chapter 3 covers the topic of ionization constants—how to measure pKa values accurately and quickly, and which methods to use. Bjerrum analysis is revealed as the secret weapon behind the most effective approaches. Chapter 4 discusses experimental... 

As mentioned, the gradient of the diffusion electric potential is suppressed in the case of diffusion of ions present in a low concentration in an excess of indifferent electrolyte ( base electrolyte ). Under these conditions, the simple form of Fick s law (2.3.18) holds for the diffusion of the given ion. The... 

A similar situation occurs in tracer diffusion. This type of diffusion occurs for different abundances of an isotope in a component of the electrolyte at various sites in the solution, although the overall concentration of the electrolyte is identical at all points. Since the labelled and the original ions have the same diffusion coefficient, diffusion of the individual isotopes proceeds without formation of the diffusion potential gradient, so that the diffusion can again be described by the simple form of Fick s law. 

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