Diffusion Fick s second law

The simple kinetics for uptake of soluble substrate of the bacteria in a biofilm is traditionally described by a combination of mass transport across the water/biofilm interface, transport in the biofilm itself and the corresponding relevant biotransformations. Transport through the stagnant water layer at the biofilm surface is described by Fick s first law of diffusion. Fick s second law of diffusion and Michaelis-Menten (Monod) kinetics are used for describing the combined transport and transformations in the biofilm itself (Williamson... 

Fick s second law is a partial differential equation that defines the change in concentration within a phase due to the process of molecular diffusion. Fick s second law can be solved numerically, or it can be directly solved to obtain a closed form solution for simplified boundary and initial conditions. 

Descriptions of the absorption of solutes into porous materials can be based on continuum formulations by defining an effective diffusion coefficient, Deff. With an effective diffusivity, Fick s 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,... 

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... 

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)... 

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 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. 

Figure 3 shows a steady diffusion across a membrane. As in the previous case, the membrane separates two well-mixed dilute solutions, and the diffusion coefficient Dm is assumed constant. However, unlike the film, the membrane has different physicochemical characteristics than the solvent. As a result, the diffusing solute molecules may preferentially partition into the membrane or the solvent. As before, applying Fick s second law to diffusion across a membrane, we... 

Figure 5 shows the diffusion of a solute into such an impermeable membrane. The membrane initially contains no solute. At time zero, the concentration of the solute at z = 0 is suddenly increased to c, and maintained at this level. Equilibrium is assumed at the interface of the solution and the membrane. Therefore, the corresponding membrane concentration at z = 0 is Kc1. Since the membrane is impermeable, the concentration on the other side will not be affected by the change at z = 0 and will still be free of solute. This abrupt increase produces a time-dependent concentration profile as the solute penetrates into the membrane. If the solution is assumed to be dilute, Fick s second law Eq. (9) is applicable ... 

In this section we want to discuss unsteady diffusion across a permeable membrane. In other words, we are interested in how concentration and flux change before reaching the steady state discussed in Section IV.B. The membrane is initially free of solute. At time zero, the concentrations on both sides of the membrane are increased, to C and c2. Equilibrium between the solution and the membrane interface is assumed therefore, the corresponding concentrations on the membrane surfaces are Kc, and Kc2. Fick s second law is still applicable ... 

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 ... 

A formal derivation of diffusion in a restricted, high diffusivity path which uses no atomic model of the grain boundary is that due to Fisher, who made a flux balance in unit width of a grain boundary having a thickness of S. There is flux accumulation in the element according to Fick s second law given by... 

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. 

Figure 2a represents the concentration profile of the tin species during the service life of the coating. The diffusion in the polymer matrix is represented by Fick s second law for nonsteady state flow ... 

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