Concentration profile, linear

Equimolar Counterdiffusion in Binary Cases. If the flux of A is balanced by an equal flux of B in the opposite direction (frequently encountered in binary distillation columns), there is no net flow through the film and like is directly given by Fick s law. In an ideal gas, where the diffusivity can be shown to be independent of concentration, integration of Fick s law leads to a linear concentration profile through the film and to the following expression where (P/RT)y is substituted for... 

FIGURE 1.10. Rotating disk electrode voltammetry. A + e B, with a concentration of A equal to C° and no B in the solution a Linearized concentration profiles —, at the plateau (vertical arrow in b), , at a less negative potential (horizontal arrow in b). b Current potential curve, c Concentrations of A and B at the electrode surface, d Logarithmic analysis of the current potential curve. 

Por steady-state diffusion occurring across flat and thin diffusion films, only one dimension can be considered and Eq. (5.1) is greatly simplified. Moreover, by replacing differentials with finite increments and assuming a linear concentration profile within the film of thickness 5, Eq. (5.1) becomes... 

Fig. 5.8 Linear concentration profiles for the partition of A between an aqueous and an organic phase.

Fig. 5.9 Linear concentration profiles for the extraction of by BH(org) in presence of an interfacial chemical reaction.

As a special case, we consider a linear concentration profile along the x-axis C(x) = a0 + atx. Since the second derivative of C(x) of such a profile is zero, diffusion leaves the concentrations along the x-axis unchanged. In other words, a linear profile is a steady-state solution of Eq. 18-14 (dC/dt = 0). Yet, the fact that C is constant does not mean that the flux is zero as well. In fact, inserting the linear profile into Fick s first law (Eq. 18-6) yields ... 

Since according to Eq. 9 of Box 22.1 the absolute value of the other eigenvalue is even smaller, we can replace both exponential functions by their linear approximations. Thus, Eq. 22-9 turns into Eq. 22-20 indicating a linear concentration profile. We call this the case of slow advection (Fig. 22.3, case C). 

At steady state there will be a linear concentration profile of solute across the center fluid region. The concentration gradient dC/dz will be AC/a. In addition there will be a flux of solute species across the fluid from the high-concentration boundary (z = a) to the low-concentration boundary. The flux of species will be proportional to the areas of the bounding walls, proportional to the concentration difference AC, and inversely proportional to the gap distance a. The molar flux per unit area is thus... 

At the RDE, various approximate analytical treatments have been presented by dropping the highest order convective term [237], neglecting convection completely [238], and by assuming a linear concentration profile within a time-dependent mass transfer boundary layer [239]. The last of these gives... 

Equations (2.19) show the concentration profiles for species O and R. The linear concentration profiles of these species correspond to the lines tangent to C (x, f) at the electrode surface (i.e., at x = 0) and are given by... 

From Fig. 2.1a, it can be observed that the Nemst diffusion layer, defined by the abscissa at which the concentration reaches the value r0 in the linear concentration profile, is independent of the potential in all the cases in spite of their having been obtained under transient conditions. This is in agreement with Eqs. (2.20) and... 

In Fig. 2.14 we have plotted the transient accurate concentration profiles for species O, cQ(r, f) (Eq. 2.144) and the linear concentration profiles... 

The concentration profiles are very sensitive to the kinetics of the electrode reaction. In this context, the determination of the diffusion layer thickness is of great importance in the study of non-reversible charge transfer processes. This magnitude can be defined as the thickness of the region adjacent to the electrode surface where the concentration of electro-active species differs from its bulk value, and it can be accurately calculated from the concentration profiles. In the previous chapter, the extensively used concept of Nemst diffusion layer (8), defined as the distance at which the linear concentration profile (obtained from the straight line tangent to the concentration profile curve at the electrode surface) takes its bulk value, has been explained. In this chapter, we will refer to it as linear diffusion layer since the term Nemst can be misunderstood when non-reversible processes... 

Fig. 3.1 Real concentration profiles (solid lines) and linear concentration profiles (dashed lines) of the oxidized species at a planar electrode for the application of a potential step, calculated from...

The zero-order kinetics is characterized by a linear concentration profile, which is however unrealistic at very large reaction times, since it produces a negative reactant concentration this result confirms that a zero-order reaction derives from a complex reaction mechanism that cannot be active at very low reactant concentrations. On increasing the reaction order, the reaction is faster at the highest concentration values... 

FIGURE 5.12 Linear concentration profile in the case of steady-state diffusion. 

Fig. 16. Concentration profile and growth kinetics for the case of the diffusion of uncharged defects, (a) Linear concentration profile (b) parabolic growth kinetics.

In this case, drag release is controlled by dissolution of the matrix. Since the size of the matrix decreases as the dissolution process continues, the amount of drag released also decreases with time. The decrease in drag release can be compensated in part by constructing a non-linear concentration profile in the polymer matrix. This strategy is used in the oral dosage form, Adalat, where the core of the dissolution matrix... 

Nernst layer -> Diffusion of electroactive species from the bulk solution to the -> electrode surface, or vice versa, takes place in a thin layer of stagnant solution close to the electrode/solution interface when the concentration of electroactive species at this interface, q (x = 0), deviates from the bulk concentration c (see Figure). The concentration gradient at the electrode/solution interface drives the diffusion flux of electroactive species. Generally, there is a linear concentration profile close to the electrode surface and at longer distances from the electrode surface the concentration asymptotically approaches the bulk concentration. The Nernst layer is ob-... 

Figure 3.9 Steady state concentration profile due to passive diffusion in a homogeneous membrane of width d. Linear concentration profile is given by Equation (3.59).

Approximate Solution. Assuming a linear concentration profile for reactant B in the liquid-side film (5) ... 

The vividness of our world does not rely on processes that are characterized by linear force-flux relations, rather they rely on the nonlinearity of chemical processes. Let us recapitulate some results for proximity to equilibrium (see also Section VI.2.H.) In equilibrium the entropy production (n) is zero. Out of equilibrium, II = T<5 S/I8f > 0 according to the second law of thermodynamics. In a perturbed system the entropy production decreases while we reestablish equilibrium (II < 0), (Fig. 72). For the cases of interest, the entropy production can be written as a product of fluxes and corresponding forces (see Eq. 108). If some of the external forces are kept constant, equilibrium cannot be achieved, only a steady state occurs. In the linear regime this steady state corresponds to a minimum of entropy production (but nonzero). Again this steady state is stable, since any perturbation corresponds to a higher II-value (<5TI > 0) and n < 0.183 The linear concentration profile in a steady state of a diffusion experiment (described in previous sections) may serve as an example. With... 

Henderson approximation [27], which assumes a linear concentration profile of the ions in the membrane (Equation 6) ... 

The solution of Equation (6.2.8) results in a linear concentration profile through the boundary layer ... 

To empirically adapt Equations 15.65 through 15.67 for the description of drug permeation through thick membranes, it is convenient to introduce a lag time taking in account the time required to get a linear concentration profile in the two stagnant layers and in the membrane. Accordingly, it follows ... 

Introduction of conditions (181) and (182) into these linear concentration profiles gives the relationships between ro, Po, and xfr in Eq. (183). Incorporation of these latter figures into Eq. (180) and reorganization of the latter as a function of the dimensionless current yields the equation of the dimensionless voltammogram in Eq. (184). 

---