Differential diffusion coefficients

The present survey has outlined some properties of porous crystals as diffusion media. It has shown some aspects which have been studied insufficiently, such as concentration dependence of D, which may be very important for catalysis at intracrystalline sites. Many more accurate determinations of differential diffusion coefficients are required to give added understanding of the role of concentration, chain length (of paraffins), polarity, exchange ions, channel geometry, and chemical dam-... 

Kinetic studies of ion exchange on partially ion-exchanged type A zeolites of Mg Ca and Mn " revealed that mini-mums and maximums characterize the differential coefficients of internal diffusion for every exchange of 2 Na " ions for one divalent cation per unit cell of the zeolite. On the basis of these observations, assuming definite interactions between the cations and the zeolite lattice, predictions can be made concerning the distribution and arrangement of cations in the unit cells of a type A zeolite. Research on liquid phase adsorption of n-alkanes on partially ion-exchanged type A zeolites indicated that the differential diffusion coefficients for alkane adsorption are influenced likewise by cation distribution in the unit cells of the zeolite. 

D Differential diffusion coefficient Np Number of divalent cations per cavity y Valence fraction of the divalent cation in the zeolite... 

Figure 1 shows the differential diffusion coefficients in ion exchange as a function of zeolite uptake for the 3 investigated ion exchange pro-... 

R. M. Barrer (Imperial College, London) Can you indicate the way in which you were able to obtain the differential diffusion coefficients... 

F. Wolf As I indicated, the observed D (differential diffusion coefficient ) is not only valid for zeolite A but also for zeolite X. The exchange experiments were carried out stepwise in dependence on the degree of cation exchange (about 20-25 single measurements, involving the whole degree of ion exchange). The slow process was observed. In this case, D decreased by a factor of about 100 the same results were indicated in tliis symposium by the contributions of Barrer and other authors. 

The differential diffusion coefficients are characteristic of any mechanically normal and chemically stable equilibrium mixture they represent properties of state. This reminds us of the necessity in (non-dilute) chemical kinetics of assuming a small change, so that the medium effects will not turn the rate constants into variables of time. This fact regarding the elementary description of a chemical reaction rate is not always explicitly stated in the texts. The reasons may be that a chemical change has often been conveniently measured only in a rather limited concentration range of the reactants and that most experiments have been confined to dilute solutions. If this simplification were not introduced, the kinematics in question would, for instance, contain partial volumes. 

Equation (Eq. 26) to obtain the so-called corrected diffusion, i.e the selfdiffusion coefficient. In these measurements, D is assumed to be constant. In the experiments described here, only very small perturbations from the equihbrium state are involved and, therefore, D is the differential diffusion coefficient for the equilibrium concentration of sorbates and can be safely considered to be constant. 

Differential diffusion coefficients for dilute aqueous solutions at 25°C... 

The resolution of Equation (2) for a unidimensional process is mnch easier if we consider Z) as a constant. This approxiniation is applicable only when there are small differences of concentration, which is the case of open-ended condnctometric technique and of the Taylor technique [20, 21], In these conditions, it is legitimate to consider that measurements of differential diffusion coefficients obtained by the techniques are parameters with a well defined thermodynamic meaning [20, 21]. 

Mutual differential diffusion coefficients of binary (e.g. [6-8]) and pseudo binary systems (such as, e.g., cobalt chloride in aqueous solutions of sucrose [9]), have been measured using a conductometric cell and an automatic apparatus to follow diffusion. This cell uses an open-ended capillary method and a conductometric technique is used to follow the diffusion process by measuring the resistance of the solution inside the capillaries, at recorded times. Figure 1 shows a schematic representation of the open-ended capillary cell. 

Mutual diEFerential diffusion coefficients of several electrolytes 1 1, 2 2, and 2 1, in different media (considering these systems as binary or pseudo binary systems, depending on the circumstances) have been measured using a conductometric cell [1]. The already published mutual differential diffusion coefficients data are average results of, at least, three independent measurements. The imprecision of such average results is, with few exceptions, lower than 1%. 

Thompson SD, Newman J (1989) Differential diffusion coefficients of sodium polysulfide melts. J Electrochem Soc 136(ll) 3362-3369... 

Hamed HS, Nuttall RL (1949) The differential diffusion coefficient of potassium chloride in aqueous solutions. J Am Chem Soc 71(4) 1460-1463... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Frequently, the transport coefficients, such as diffusion coefficient orthermal conductivity, depend on the dependent variable, concentration, or temperature, respectively. Then the differential equation might look Bke... 

It is appropriate to differentiate between polymerizations occuring at temperatures above and below the glass transition point(Tg) of the polymer being produced. For polymerizations below Tg the diffusion coefficients of even small monomer molecules can fall appreciably and as a consequence even relatively slow reactions involving monomer molecules can become diffusion controlled complicating the mechanism of polymerization even further. For polymerizations above Tg one can reasonably assume that reactions involving small molecules are not diffusion controlled, except perhaps for extremely fast reactions such as those involving termination of small radicals. 

Diffusion of the fluid into the bulk. Rates of diffusion are governed by Pick s laws, which involve concentration gradient and are quantified by the diffusion coefficient D these are differential equations that can be integrated to meet many kinds of boundary conditions applying to different diffusive processes. ... 

In the general case, the initial concentration of the oxidized component equals Cqx and that of the reduced component cRed. If the appropriate differential equations are used for transport of the two electroactive forms (see Eqs 2.5.3 and 2.7.16) with the corresponding diffusion coefficients, then the relationship between the concentrations of the oxidized and reduced forms at the surface of the electrode (for linear diffusion and simplified convective diffusion to a growing sphere) is given in the form... 

See Partial Differential Equations. ) If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. 

The electrochemical behavior of niclosamide was described on the basis of d.c. polarography, cyclic voltammetry, a.c. polarography, and differential pulse polar-ography, in the supported electrolytes of pH ranging from 2.0 to 12.0 [32], A tentative mechanism for the reduction of niclosamide is proposed that involves the transfer of 4 e . Parameters such as diffusion coefficients and heterogeneous forward rate constant values were evaluated. 

Fig. 1. (a) left) Profiles at the bump, of the total diffusion coefficient (top) and of the degree of differential rotation (bottom) for model B (solid lines) and model C dotted, lines). Hatched regions correspond to the CE. (b) right) Comparison of our models with observations ([4]). Triangles are lower limits. Dots are actual values. 

For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as a. The first boundary condition follows from the continuity of the material flux ... 

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