First law of Fick

consider a crystalline medium, limited by two plane interfaces with O and X as X-coordinates, between which particles Sdiffuse. This diffusion is carried out in an anisotropic way according to Ox direction only, perpendicular to the two interfaces. If we consider an area X of a surface perpendicular to the direction of diffusion, it comprises files, along which the particles diffuse. Each file is located by its index j (0 j 0). The diffusion is carried out in each file by n jumps with 

It is assumed that the sueeessive jumps of a particle are independent of each other and that the equilibrium positions of the particle are on the same energy level. 

The flux of diffusion relating to the j fde throngh the plane with jc, as X- 

The flux of diffusion is the difference between a flux from left to right and a flux from right to left. From left to right, applying the relation showing the elementary speed of a jump between the planes i and i + 1 as X-coordinates (section 4.2.2), it becomes 

F(i +l,j) is the probability so that the position i + 1 is free to accommodate the particle in the filey. This relation involves the probability rthat has a particle at the top of the barrier to fall down on the opposite side at its arrival. All the equilibrium positions being equivalent, the barriers are symmetrical and thus we can choose the valne Vi for n. In the place of the c(/,yj surface concentrations, we generally prefer to nse the volumetric concentrations, associated with the preceding ones by 

---

Exactly the same mechanism determines diffusion processes. Using the first law of Fick we have a diffusion current. 

This constant diffusion flow according to the first law of Fick can be considered to be analogous to the irreversible flow given by Eq. (41). In Eq. (52) z is a function of time alone. In the case that z depends on space in the sense that the velocity of the... 

According to the first law of Fick about a uni-dimentional flow, the mass dm transferred through an area of cross section A for time dr is proportional to the concentration gradient of the substance across film thickness dC/dh... 

For the first law of Fick, the application of relation [5.8] gives, by clarifying the gradient and the components of flux, and by supposing the coefficient of diffusion independent of the direction (isotropic diffusion)... 

Derive the Cottrell equation by combining Fick s first law of diffusion with the tune-dependent change of the concentration gradient during a potential-step experiment. 

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 ... 

Diffusion is the movement of mass due to a spatial gradient in chemical potential and as a result of the random thermal motion of molecules. While the thermodynamic basis for diffusion is best apprehended in terms of chemical potential, the theories describing the rate of diffusion are based instead on a simpler and more experimentally accessible variable, concentration. The most fundamental of these theories of diffusion are Fick s laws. Fick s first law of diffusion states that in the presence of a concentration gradient, the observed rate of mass transfer is proportional to the spatial gradient in concentration. In one dimension (x), the mathematical form of Fick s first law is... 

We can obtain an additional expression for the diffusion current by considering Fick s first law of diffusion, first introduced in chapter 1, equation (1.34). If J is the flux of species to the electrode, it will be related to the observed current, /, by ... 

The original and simplest form of the diffusion layer theory was developed by Nemst [105] and Brunner [106], who assumed that the mass flux is given by Fick s first law of diffusion. In that case,... 

Transport across the direction of flow can be determined using Fick s first law of diffusion for the flux of material in particles/cm2 sec-1 in a steady, time-independent state... 

Diffusion is quantified by measuring the concentration of the diffusing species at different distances from the release point after a given time has elapsed at a precise temperature. Raw experimental data thus consists of concentration and distance values. The degree of diffusion is represented by a diffusion coefficient, which is extracted from the concentration-distance results by solution of one of two diffusion equations. For one-dimensional diffusion, along x, they are Fick s first law of diffusion ... 

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... 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

In the absence of field effects, equations (4.22), (4.23), (4.25), and (4.26) amount to Fick s second and first laws of diffusion with an apparent diffusion coefficient given by equation (4.24). 

The rate at which they diffuse depends on the concentration gradient, dN dx the larger the gradient, the faster the rate of diffusion. This is the basis of the well-known Fick s first law of diffusion ... 

This is the correct expression for use in the analysis of closed diffusion-cell experiments for the measurement of diffusion coefficients. Equation (48) is known as Fick s First Law of Diffusion. Note that Na = — NB corresponds to saying that w = 0. 

We saw above, from Fick s first law of diffusion, that the flux at a distance r from the central particle is given by... 

In Chap. 2 and 3, the motion of two reactants was considered and a diffusion equation was derived based upon the equation of continuity and Fick s first law of diffusion (see, for instance, Chap. 2 and Chap. 3, Sect. 1.1). When one reactant (say D) can transfer energy or an electron to the other reactant (say A) over distances greater than the encounter separation, an additional term must be considered in the equation of continuity. The two-body density n (rj, r2, t) decays with a rate coefficient l(r, — r2) due to long-range transfer. Furthermore, if energy is being transferred from an excited donor to an acceptor, the donor molecular excited state will decay, even in the absence of acceptor molecules with a natural lifetime r0. Hence, the equation of continuity (42) becomes extended to include two such terms and is... 

Fick s first law of diffusion relates the diffusive flux of species k to its mass fraction or mole fraction gradient. For a binary mixture of species j and k, the mass flux of species k relative to the mass average velocity is, related to the mass fraction gradient of k as... 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

Both the current and the concentration profiles of O and R are determined by diffusion to and from the electrode. These quantities are obtained from calculations using Fick s laws of diffusion. Fick s first law of diffusion... 

Diffusion processes occur in all systems where concentration differences exist. Diffusion is the main mechanism which aids Ln the elimination of concentration gradients. Fick s first law of diffusion defines this phenomenon by correlating mass flow and concentration gradient. This law may be shown as... 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

The rate of diffusion is proportional to the concentration gradient, and the proportionality constant is defined as the diffusion coefficient (D) in Fick s first law of diffusion. Experimental determination of D is commonly performed ex vivo due to the difficulty of measuring concentration gradients in the interstitium. In vivo measurement can be performed in specific tissues, using transparent chamber preparations in combination with the FRAP technique (Berk et al., 1997 Jain et al., 1997 Pluen et al, 2001). However, the in vivo approach is limited only to fluorescent molecules or solutes whose D is not affected by labeling with fluorescent markers. 

---