Second law Ficks

Albery and Hadgraft [9,10] obtained solutions to Fick s second law of diffusion to examine the mechanism by which simple nicotinate esters 

If the diffusive flux in a system is J, Section 1.3.5 and Eq. 1.18 are used to write the diffusion equation in the general form 

Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 77 

Accumulation within a volume depends only on the fluxes at its boundary. For example, in one dimension, 

Because Eq. 4.2 has one time and two spatial derivatives, its solution requires three independent conditions an initial condition and two independent boundary conditions. Boundary conditions typically may look like 

In Chapter 3, several different types of diffusivity were introduced for diffusion in a chemically homogeneous system or for interdiffusion in a solution. In each case, Fick s law applies, but the appropriate diffusivity depends on the particular system. The development of the diffusion equation in this chapter depends only on the form of Fick s law, J = -DVc. D is a placeholder for the appropriate diffusivity, just as J and c are placeholders for the type of component that diffuses. 

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The Dunwald—Wagner equation, based on the application of Ficks second law of diffusion into or out of a sphere (radius r) [477], can be written... 

Because of the length and complexity of Pecoras calculation, it will not be reproduced here. Instead, we will perform a somewhat less rigorous, but plausible, evaluation of Cse(r) to reach the same result. In doing this, it must be assumed that the time dependence of Cse(r) is the same as that observed if a large fluctuation in concentration were artificially imposed on the system at time t = 0 and allowed to decay according to Ficks second law of diffusion (58) where... 

If the diffusivity is not a spatially dependent property, Equation 8 can be written as Ficks second law,... 

In this context, the relative terms far, short, small, and large can be defined as follows. Fick s second law of diffusion dictates that the distance, 5, that a species having a diffusion coefficient, D, may diffuse within a period of time, t, is given by (12) ... 

The rate of diffusion of the ion to the electrode surface is given by Fick s Second Law as... 

Hence, the current (at any time) is proportional to the concentration gradient of the electroactive species. As indicated by the above equations, the dififusional flux is time dependent. Such dependence is described by Fick s second law (for linear diffusion) ... 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

The diffusional transport model for systems in which sorbed molecules can be divided in two populations, one formed by completely immobilized molecules and the other by molecules free to diffuse, has been developed by Vieth and Sladek 33) in a modified form of the Fick s second law. However, if linear isotherms are experimentally found, as in the case of the DGEBA-TETA system in Fig. 4, the diffusion of the penetrant may be described by the classical diffusion law with constant value of the effective diffusion coefficient,... 

Equation 10.66 is referred to as Fick s Second Law. This also applies when up is small, corresponding to conditions where C, is always low. This equation can be solved for a number of important boundary conditions, and it should be compared with the corresponding equation for unsteady state heat transfer (equation 9.29). 

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which is in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component, the only condition under which analytical solutions can be produced for the diffusion equations. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick s first and second laws expressed in molar units. 

A crystal is suspended in fresh solvent and 5% of the crystal dissolves in 300 s. How long will it take before 10% of the crystal has dissolved Assume that the solvent can be regarded as infinite in extent, that the mass transfer in the solvent is governed by Fick s second law of diffusion and may be represented as a unidirectional process, and that changes in the surface area of the crystal may be neglected. Start your... 

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

Hence by substitution of the convection term and from Fick s second law of diffusion (eqn. 3.2), we obtain... 

Many extensions have been derived for the Ilkovic equation from the consideration that the dme does not behave as a flat electrode but in fact shows a spherical growth. For instance, Fick s second law of diffusion (cf., eqn. 3.2) becomes12... 

This equation can be derived by means of Laplace transformation of Fick s second law for a planar microelectrode ... 

For the HMDE and for a solution that contains only ox of a reversible redox couple, Reinmuth102, on the basis of Fick s second law for spherical diffusion and its initial and boundary conditions, derived the quantitative relationship (at 25° C)... 

Perhaps the simplest Fick s law permeation model consists of two aqueous compartments, separated by a very thin, pore-free, oily membrane, where the unstirred water layer may be disregarded and the solute is assumed to be negligibly retained in the membrane. At the start (t = 0 s), the sample of concentration CD 0), in mol/cm3 units, is placed into the donor compartment, containing a volume (Vo, in cm3 units) of a buffer solution. The membrane (area A, in cm2 units) separates the donor compartment from the acceptor compartment. The acceptor compartment also contains a volume of buffer (VA, in cm3 units). After a permeation time, t (in seconds), the experiment is stopped. The concentrations in the acceptor and donor compartments, CA(t) and C (t), respectively, are determined. 

It is practical to make the approximation that CM(oo) m Cm (t). This is justified if the membrane is saturated with the sample in a short period of time. This lag (steady-state) time may be approximated from Fick s second law as tlag = h2 / (n2Dm), where h is the membrane thickness in centimeters and Dm is the sample diffusivity inside the membrane, in cm2/s [40,41]. Mathematically, xLAG is the time at which Fick s second law has transformed into the limiting situation of Fick s first law. In the PAMPA approximation, the lag time is taken as the time when solute molecules first appear in the acceptor compartment. This is a tradeoff approximation to achieve high-throughput speed in PAMPA. With h = 125 pm and Dm 10 7 cm2/s, it should take 3 min to saturate the lipid membrane with sample. The observed times are of the order of 20 min (see below), short enough for our purposes. Cools... 

We will first consider the simple case of diffusion of a non-electrolyte. The course of the diffusion (i.e. the dependence of the concentration of the diffusing substance on time and spatial coordinates) cannot be derived directly from Eq. (2.3.18) or Eq. (2.3.19) it is necessary to obtain a differential equation where the dependent variable is the concentration c while the time and the spatial coordinates are independent variables. The derivation is thus based on Eq. (2.2.10) or Eq. (2.2.5), where we set xj> = c and substitute from Eq. (2.3.18) or Eq. (2.3.19) for the fluxes. This yields Fick s second law (in fact, this is only a consequence of Fick s first law respecting the material balance—Eq. 2.2.10), which has the form of a partial differential equation... 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Fick s second law of diffusion can be derived from Fick s first law by using a mass balance approach. Consider the differential fluid element shown in Figure 4. This differential fluid element is simply a small cube of liquid or gas, with volume Ax Ay Az, and will be defined as the system for the mass balance. Assume now that component A enters the cube at position x by diffusion and exits the cube at x + Ax by the same mechanism. For the moment, assume that no diffusion occurs in the y or z directions and that the faces of the cube that are perpendicular to the y and z axes thus are impermeable to the diffusion of A. Under these conditions, the component mass balance for A in this system is... 

Figure 4 Differential fluid element (system) used for the development of Fick s second law. Diffusion occurs only in the x direction, as shown. The front face of the cube is shaded for contrast.

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

Fick s first and second laws of diffusion can be modified to include terms describing fluid convection. As with the discussion of Fick s laws, the equations pre-... 

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