Fick’s first and second laws

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. 

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-... 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

Explain the relationship between Fick s first and second law. 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

Since Fick s first and second laws of diffusion are valid independent of whether I) is a function of c, only or not and also of the form of initial and boundary conditions of a particular experiment, it is quite inadequate to specify this particular type of sorption as Fickian. The term Fickian" should be applied more generally to all mass transport phenomena which are governed by Eq. (1), i. e., the Pick diffusion equation. 

Fick s first and second laws describe the diffusion of solute to the surface and the difhision of solvent and other coordination ions away from the surface of a growing crystal. 

Figure 8.5 A diagram of relations among stresses and strain rates. The two full-line arcs are based on the definition of a material s viscosity, and the two dotted arcs are based on mechanical equilibrium or conservation of momentum. From these, two relations are derived that resemble Fick s first and second laws, represented by wiggly lines.

Ordinary molecular diffusion is generally recognized as the primary mechanism for gas transport in the unsaturated zone (19,, . Fundamental theory for ordinary diffusion according to Fick s first and second laws is presented in (22) more extensive theoretical discussions of gas transport in porous media are presented in (2 ... 

Diffusive transfer of solutes in solids is governed by Fick s First and Second Laws, Even though biological solids are not stiucturally homogeneous and diffusion occurs mainly in the fluid occlnded within the solid, Fick s Laws will he expressed in terms of X, that is,... 

Adolf Fick in 1855 developed equations to describe diffusion that are now called Fick s first and second laws of diffusion. In an isotropic material (where the properties do not vary with direction), and when diffusion occurs along only one dimension, which is approximately the case in most packaging systems, Fick s first law can be written ... 

Fick s first and second laws must be modified when describing a system that manifests the Kirkendal effect. Fick s first law becomes (written in terms of species A)... 

The second boundary condition follows from Fick s first and Faraday s laws dc x, t)... 

The quantity of solute B crossing a plane of area A in unit time defines the flux. It is symbolized by J, and is a vector with units of molecules per second. Fick s first law of diffusion states that the flux is directly proportional to the distance gradient of the concentration. The flux is negative because the flow occurs in a direction so as to offset the gradient ... 

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... 

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... 

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. 

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... 

Differential equations of the first order arise with application of the law of mass action under either steady or unsteady conditions, and second order with Fick s or Newton-Fourier laws. A particular problem may be represented by one equation or several that must be solved simultaneously. 

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). 

The flux of particles is in the opposite sense to the direction of the concentration gradient increase. Equation (6) is Fick s first law, which has been experimentally confirmed by many workers. D is the mutual diffusion coefficient (units of m2 s 1), equal to the sum of diffusion coefficients for both reactants, and for mobile solvents D 10 9 m2 s D = DA + jDb. The diffusion coefficient is approximately inversely dependent upon viscosity and is discussed in Sect. 6.9. As spherical symmetry is appropriate for the diffusion of B towards a spherically symmetric A reactant, the flux of B crossing a spherical surface of radius r is given by eqn. (6) where r is the radial coordinate. The total number of reactant B molecules crossing this surface, of area 4jrr2, per second is the particle current I... 

The first term is equivalent to the flux due to concentration gradients (i.e., Fick s first law) and the second term represents the flux due to gradients in the potential. 

Subsequently a well-defined area at the surface is depleted from the adsorbate layer by a focused laser pulse. Since thermal equilibrium at the surface is rapidly recovered, the bare spot can be refilled only by surface diffusion of adsorbates from the surrounding areas [31]. A second laser impulse is applied to desorb the transported adsorbates after a time interval t from the first pulse. The corresponding amount of material can be quantified by mass spectrometry. For the idealized case of a circular depletion region, with a step-like coverage gradient and a concentration-independent diffusivity, the time-dependent refilling from Fick s first law is [32,33] ... 

Fick s second law defines the behavior of a diffusing chemical in space over time. Fick s second law is derived from Fick s first law and the equation of continuity for a solute. For simplicity, we derive Fick s second law in 1-D coordinates. This can readily be extended to multiple dimensions or to spherical coordinates [22]. 

Despite the fact that the skin is a heterogeneous membrane, Fick s laws of diffusion have been successfully used to analyze skin permeation data. Solutions to the second law have been used in mechanistic interpretations (see later) and in considering concentration profiles within the skin. Fick s first law has been used to analyze steady-state diffusion rates and in the development of predictive models for skin permeability. 

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