Ficks Laws

A mixture of two gases A and B is considered whose concentrations c and Cq are functions of the space variables, all other variables, especially the pressure and the temperature, being constant. FICK s law states that the flux density of A is proportional to the concentration gradient of A  

The negative sign indicates that the matter is transported from regions of high concentration towards those of lower concentration in A. 

FICK s law takes the following more general form  

The superscript c indicates that the transport of matter results from a concentration gradient. 

the following relationships have been established in Chapter III  

The diffusional flux of component A is proportional to the concentration gradient grad cA. For the time being we will limit ourselves to a mixture of two components A and B. We will also assume that diffusion only takes place along one coordinate axis, for example the y-axis. The diffusional flux can be described by an empirical statement corresponding to Fourier s law 

In a multicomponent mixture consisting of N components the diffusional flux j A of component A is given by [1.23] 

From this equation (1.161) is obtained for the special case of N = 2. 

14Adolph Fick (1829-1901), Professor of Physiology in Zurich and Wurzburg, discovered the fundamental laws of diffusion. 

By exchanging the indices A and B in (1.161) the diffusional flux for component B in the binary mixture can be found. As the sum of the two diffusional fluxes disappears, according to (1.156), it follows that 

It is a striking symptom of the common ignorance in this field that not one of the phenomenological schemes which are fit to describe the general case of diffusion is widely known. 

---

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... 

Each film is in stagnant or laminar flow, such that mass transfer across the films is by a process of molecular diffusion and can therefore be described by Ficks Law. 

Ficks Law states that the flux j (mol/s m2) for molecular diffusion, for any given component is given by... 

If the equilibrium at the crystal face is established rapidly so as not to be rate-limiting, the crystallization process would be controlled by solute transport, which is proportional to the solute concentration gradient across the stagnant lay r-(Cs). The rate of crystallization could be modeled using Ficks law of diffusion and is given by... 

Here the second term is the diffusion flux. We write the latter in the commonly used weak irreversibility approximation, namely, as the Fick law... 

In general the diffusion of a solute like Xe is not driven by the concentration gradients but by the chemical potential. In polymer blends with incompatible components, as we investigate here, in the equilibrium situation it may occur that different domains have different concentrations of Xe. Then the usual Fick laws do not hold. 

Most chemical compounds are characterised by diffusion of the components across the bulks of their growing layers in the form of atoms or ions. The process of bulk diffusion is described by Fick s laws. The first Fick law relates the flux of atoms of a given component to its diffusion coefficient and concentration gradient in the direction of diffusion at constant surface area of contacting phases ... 

The second Fick law describes a change in concentration of diffusing particles at a given point of space with passing time ... 

Note that equation (5.1) follows immediately from Fick s laws on the assumption of a quasi-stationary distribution of the concentration of components within the diffusion boundary layer. Indeed, if in this layer cAl t 0, then the second Fick law yields cA x const. It means that the distribution of the concentration of component A within this layer is close to linear (Fig. 5.1). Anywhere outside of this layer, the concentration of A is assumed to be the same and equal to an instantaneous value, c. This implies sufficiently intensive agitation of the liquid. In such a case, the flow of A atoms across the diffusion boundary layer under the condition of constancy of the surface area of the dissolving solid is... 

Use of Ficks law to describe the diffusion process requires the solute particle to be small compared with the diffusion boundary layer. The analysis presented above suggests that, for Peelet numbers greater than 100, the ratio 8o/0p is proportional to (Pe)ua/R. The solid curves in Figure 3 are truncated at the value of the Peelet number corresponding to Pe/R3 10 "2, where an inspection of the radial concentration profile revealed that the ratio 8d/Op is about ten. 

Wagner enhancement factor — describes usually the relationships between the classical - diffusion coefficient (- self-diffusion coefficient) of charged species i and the ambipolar - diffusion coefficient. The latter quantity is the proportionality coefficient between the - concentration gradient and the - steady-state flux of these species under zero-current conditions, when the - charge transfer is compensated by the fluxes of other species (- electrons or other sort(s) of -> ions). The enhancement factors show an increasing diffusion rate with respect to that expected from a mechanistic use of -> Ficks laws, due to an internal -> electrical field accelerating transfer of less mobile species [i, ii]. 

In performing the analysis, the apparatus registers the total hydrogen flux q(t) through the surface of sample. According to the Fick law, this flux is ... 

The particle was divided into 30 to 100 shells for the calculation and Ficks Law was assumed to hold. Heat generated was taken as rate of adsorption times the heat of adsorption, and the latter quantity was assumed to be independent of amount adsorbed. With increasing tern-... 

For nitrogen on 4A, Figure 1, the nonisothermal rate data despite the heating could be represented reasonably by usual isothermal Ficks law equations (I), if D/R is taken as 7.5 X 10" min" Thus, the value of D/R calculated from the isothermal equation was 50% larger than that used to derive the nonisothermal curve in Figure 1. Here the temperature maximum occurs at low amounts adsorbed, and the increased rate owing to increased diffusivities is nearly compensated by the decreased equilibrium adsorption at the observed temperatures. Propane... 

The transportation of heat and a substance in the air flow is governed by the Fourier and Fick laws (3.55). The thermal conductivity coefficient Dh/(P Cp) and the mass transfer coefficient DE, both possessing the dimension m2/s, determine the intensity... 

According to (562), the distinctive feature of the interval t, tf) is the deviation from the Fick law, that is, the Maxwell-Cattaneo law is correct rather than the Fick law (487)... 

The first term on the right is the usual Fick law or diffusive flux and A is the hydro-dynamic diffusion coefficient of the ion... 

The diffusing fluxes of biogenic elements across sediment-seawater interfaces mainly depend on the diffusion of the concentration difference caused by the concentration gradient near the interfaces. The net diffusing fluxes of biogenic elements across the sediment-water interface in some of China s sea regions can be estimated by the First Fick Law, and the results are listed in Table 1.10. 

---