Equations Fick s second law

The reaction scheme at and near the phase boundary during the phase transformation is depicted in Figure 10-14. The width of the defect relaxation zone around the moving boundary is AifR, it designates the region in which the relaxation processes take place. The boundary moves with velocity ub(f) and establishes the boundary conditions for diffusion in the adjacent phases a and p. The conservation of mass couples the various processes. This is shown schematically in Figure 10-14b where the thermodynamic conditions illustrated in Figure 10-12 are also taken into account. The transport equations (Fick s second laws) have to be solved in both the a and p... 

If we attach a new reference frame (z) to the moving (stable planar) boundary, z = ,-V t. The transport equation (Fick s second law) reads in the z-system... 

Finally, let us briefly point out some essential features of the stability analysis for a more general transport problem. It can be exemplified by the moving a//9 phase boundary in the ternary system of Figure 11-12. Referring to Figure 11-7 and Eqn. (11.10), it was a single independent (vacancy) flux that caused the motion of the boundary. In the case of two or more independent components, we have to formulate the transport equation (Fick s second law) for each component, both in the a- and /9-phase. Each of the fluxes jf couples at the boundary b with jf, i = A,B,... (see, for example, Eqn. (11.2)). Furthermore, in the bulk, the fluxes are also coupled (e.g., by electroneutrality or site conservation). 

Diffusion Equations. Fick s second law can be solved for the special case of a layer of substance of constant (with time) concentration diffusing into a semi-infinite medium which initially contains no solute. Using the method of the Laplace transform (8) the solution yields an expression for the concentration in the medium ... 

Minimum mass transport occurs even in the absence of water filtration. We will review uni dimensional (linear) mass transport. If V = 0 and q, = 0, advective-dispersive equation acquires the format of a linear equation Fick s second law (equation 3.12). 

In the aforementioned step 1, the solvent diffuses into polymer networks (case-I diffusion). At this time, the solvent absorption behavior of the gel can be obtained by solving the diffusion equation (Fick s second law). In... 

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... 

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). 

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] ... 

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 ... 

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. 

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 law is a concise mathematical statement however, it is not directly applicable to solutions of most pharmaceutical problems. Fick s second law presents a more general and useful equation in resolving most diffusion problems. Fick s second law can be derived from Fick s first law. 

Equation (9) is Fick s second law of diffusion, derived on the assumption that D is constant. Fick s second law essentially states that the rate of change in concentration in a volume within the diffusional field is proportional to the rate of change in the spatial concentration gradient at that point in the field, the proportionality constant being the diffusion coefficient. 

The choice of vx is a matter of convenience for the system of interest. Table 1 summarizes the various definitions of vx and corresponding, /Y, commonly in use [3], The various diffusion coefficients listed in Table 1 are interconvertible, and formulas have been derived. For polymer-solvent systems, the volume average velocity, vv, is generally used, resulting in the simplest form of Jx,i- Assuming that this vv = 0, implying that the volume of the system does not change, the equation of continuity reduces to the common form of Fick s second law. In one dimension, this is... 

The general law describing the diffusion process of a fluid is the Fick s second law, Equation (3), which expresses the dependence of the molecular concentration with time57 58 ... 

This is known as Fick s second law of diffusion or more commonly as the diffusion equation. In these equations, J is called the flux of the diffusing species, with units of [amount of substance (atoms or equivalent units) m2 s-1], c is the concentration of the diffusing species, with units of [amount of substance (atoms or equivalent units) m-3] at position x (m) after time t (s) D is the diffusion coefficient, units (m2 s 1). 

This is Fick s second law of diffusion, the diffusion equation. 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

In the framework of Scheme 2.1, we start with the case where the electron transfer does not interfere kinetically. As compared to the simple Nemstian electron transfer case (Section 6.1.2), the main change occurs in die partial derivative equation pertaining to B, where a kinetic term is introduced in Fick s second law. A corresponding equation for C should also be taken into account, leading to the following system of partial derivative equations, accompanied by a series of initial and boundary conditions (assuming that the diffusion coefficients of A, B, and C are the same) ... 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

Guy et al. [5] derived an equation for the diffusion-controlled release of a drug from a sphere, radius r, by applying the three-dimensional form of Fick s second law of diffusion after transformation to spherical coordinates. This equation can be rearranged as ... 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

It is clear that since the mercury drop approximates a sphere, the theory of spherical, and not linear, diffusion might have to be used. However, detailed considerations accessible in monographs show that if the electrodic reaction is driven for a sufficiently short time (/ < a few seconds) and if the mercury-drop radius is not too small (r > 0.05 cm), then the equations of linear diffusion can be used with validity. Thus, the partial differential equations for the diffusion of A and D are [see Fick s second law cf. Eq. (4.32)]... 

The treatment of nonsteady-state diffusion is a question of solving Fick s second law of diffusion. In many cases, however, the equations can be taken from the treatments of the analogous problems in heat flow in solids. The point is that heat flow and diffusion are described by mathematically similar methods. 

Since P(x,t) is normalized, its integral over all values of x equals unity. Likewise, integrating Equation (57) over all values of x gives c0 The same quantity of solute is present at all times whether it is concentrated at the origin or is spread out from —x to +x. The function given by Equation (57) is a solution to Fick s second law, shown in Equation (26), for a onedimensional problem in which all the material is present initially at x = 0 and at a concentration c0. 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

As a first step we have to formulate the diffusion/reaction equation of A and D in the water-film. For this purpose we combine Fick s second law (Eq. 18-14) with the forward/backward reaction of the aldehyde (Eq. 12-16) ... 

This approach of subdividing space into an increasing number of discrete pieces provides the basis for many numerical computer models (e.g., the so-called finite difference models) an example will be discussed in Chapter 23. Although these models are extremely powerful and convenient for the analysis of field data, they often conceal the basic principles which are responsible for a given result. Therefore, in the next chapter we will discuss models which are not only continuous in time, but also continuous along one or several space axes. In this context continuous in space means that the concentrations are given not only as steadily varying functions in time [QY)], but also as functions in space [C,(r,x) or C,(t,x,y,z)]. Such models lead to partial differential equations. A prominent example is Fick s second law (Eq. 18-14). 

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). 

The right-hand side of eqn. (9), which is the diffusion equation or Fick s second law, involves two spherically symmetric derivatives of p(r, t). In the general case of three-dimensional space, lacking any symmetry, it can be shown that the Laplacian operator... 

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