Diffusion coefficient concentration dependence

S. Zugmann, M. Fleischmann, M. Amereller, R. M. Gschwind, M. Winter, H. J. Gores, J. Chem. Eng. Data 2011, 56, 4786-4789. Salt diffusion coefficients, concentration dependence of cell potentials, and transference numbers of lithium dilluoromono(oxalato)borate-based solutions. 

This model is compared with experimental data of CCI4 in ethylene-propylene copolymers of different composition, obtaining good agreement for the diffusion coefficient concentration dependence [3]. The same experimental data are contrasted by Kreituss and Frisch [4] using another free volume model. 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

The diffusion behaviour of organic vapours is much more complicated than that of simple gases. Normally the interaction is much stronger, so that the diffusion coefficient becomes dependent on the concentration of the penetrant ... 

The effective diffusion coefficients, which up till now have been assumed constant, become a function of the concentration and, therefore, due to the internal concentration gradients, the effective diffusion coefficients will depend on the distance inside the catalyst pellet. 

An alternative method to take surface diffusion into account consists in lumping pore diffusion and concentration-dependent surface diffusion together, thus creating an apparent effective diffusion coefficient, which is concentration dependent. This approach was used by Ma et al [53], by Pigtkowski et al. [28] and by Zhou et al. [10]. This method is also an approximation, but it is still an improvement over the simpler HSDM model. 

When is the diffusion coefficient s dependence on concentration significant When is it dependent on the size and shape of the molecule ... 

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

Using Eq. (2.58), from the mean relaxation rate T the average apparent translational diffusion coefficient D can be calculated. The measured apparent diffusion coefficient D depends on the concentration [C] of the scattering particles. When [C] is not too large (4> < 0.1), one has... 

Unlike conventional rubbers, however, the results are time dependent since reduced sorption curves for films of varying thicknesses do not coincide with each other (Figure 2). This anomaly, which is characteristic of glassy materials, is a sign of the inability of the polymer molecules to respond quickly to the changing concentration. The diffusion coefficient (D) depends on the time for which the polymer and penetrant have been in contact. D has more time in which to approach its equilibrium value in thicker films, and therefore sorption proceeds relatively more quickly the thicker the film. 

It should be noted that the apparent diffusion coefficient, which depends on the various species concentrations by means of the activity coefficient, does not follow the equation involving A, derived from the Nernst-Einstein equation ... 

If the temperature cannot be assumed constant, then the equations have to be solved numerically. The same is true if the diffusion coefficient is dependent on the concentration. 

The formulae and terms also apply when the permeation experiment is performed with the pure liquid chemical. The partition coefficient <7 must be replaced with solubility s. Since organic substances can be feirly extensively enriched in a thermoplastic geomembrane, it must be considered in the analysis of mass transport through geomembranes exposed to pure substances that the diffusion coefficient can depend greatly on the concentration as will be discussed later. 

From the results it can be seen that the diffusion coefficient is dependent on the concentration of CO in the polymer, O increases with increasing upstream pressure. 

Thus from this recall of heat transfer, with the similarity between the two processes of heat transfer and mass transfer controlled by diffusion, the necessity of admitting without ambiguity that the course for the mass transfer should emerge as follows diffusion through the thickness of the sheet associated with the convection into the liquid. Finally, the parameters of main importance for a polymer package in contact with a liquid food are the diffusivity and the coefficient of convection. The diffusivity is concentration-dependent, as for example the case of highly plasticised polyvinylchloride where the plasticiser concentration may reach up to 50% of the polymer, but in the present case of the low concentration of the additives distributed in the polymer of the packaging -which are necessary to provide its qualities - the diffusivity can be considered as constant. 

The ternary diffusion coefficient strongly depends on the solution concentration. In order to calculate accurate mass transfer coefficients, experimental data of diffusion coefficients at the interest concentrations and temperatures are necessary. However, data are not available at concentrations and temperature used at the present study, it was assumed that the ternary diffusion coefficients were equal to the binary diffusion coefficients. The binary diffusion coefficients of the KDP-water pairs and the urea-water pairs were taken from literature (Mullin and Amatavivadhana, 1967 Cussler, 1997). The values were transformed into the Maxwell-Stefan diffiisivities using the thermodynamic correction factor. 

Hydrodynamic scaling model for self-diffusion and viscosity As shown in Chapter 8 on polymer self-diffusion, the concentration dependence of the polymer self-diffusion coefficient Ds is uniformly given by... 

Chapter 9 considers the experimental literature on diffusion of mesoscopic probe particles through polymer solutions. A coherent description was obtained, namely that the probe diffusion coefficient generally depends on polymer concentration as Dp = Dpoe p(—ac ). The parameters a and v both depend strongly on polymer molecular weight, with a M > and y close to 1. Parameter u tends toward 1.0 -with much variation - at small matrix M, but reached v 0.5 at large M. 

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