Steady-state diffusion concentration-dependent

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The carbon isotope ratios of pedogenic carbonates have been used to infer atmospheric CO2 concentrations from calculations based on a model for steady-state diffusive mixing with soil-respired CO2 in the soil profile (Cerling, 1991 Cerling, 1992 Ekart et al., 1999 Ghosh et al., 2001). Significant sources of uncertainty in this approach include the dependence of soil CO2 diffusion on temperature and moisture, the contribution of C3 versus C4 plants to respired CO2, and the somewhat arbitrary choice of values for the mole fraction of respired CO2 at depth in the soil. 

To describe the kinetics of sand-grain dissolution in the system Si02 —CaO—Na20, Sasek in the study mentioned above applied the Hixson-Crowell (1931) solution which takes into consideration the variable solvent concentration during mass transfer by steady-state diffusion. His results indicate that solution of this type describes well isothermal dissolution kinetics however, the temperature dependence of the rate constant does not have the expected simple behaviour. According to the author s explanation, this is due to the change in the character of the rate--controlling step which is surface reaction up to about 1 ISO and diffusion only above 1400 °C. 

Fig. 12.2. Steady-state solute concentration profiles in simultaneous diffusion and convection across a membrane of uniform properties. Numbers adjacent to profiles indicate values of the Peclet number whose sign depends upon the direction of the volumetric flux relative to the extemed solution...

All the time-dependent terms in Equation 50 assembled between the braces are independent of the oxygen consumption rate, v. This means that the rate at which the concentration C at any depth approaches a steady-state value (dC/dt) is independent of the zero order reaction rate constant. The time-dependent terms contain, however, the eddy diffusion coefficient and advection velocity, and the rate of approach to steady-state is therefore dependent on these two physical characteristics of the environment. 

No further difficulties arise when we come to discuss steady state diffusion in a system in which the activity coefficient is dependent upon concentration. In such a system, Di (1 +... 

In convective systems, the situation is different, because the thickness of the nonsteady-state diffusion layer, is limited by that of the steady-state diffusion layer 5, which depends only on the prevailing hydrodynamic conditions (see Sect. 4.3). As the non-steady-state diffusion layer grows, convection contributes increasingly to the transport of the reacting species. Eventually, when becomes comparable to the steady-state diffusion layer thickness 5 the concentration profile no longer changes with time. 

It is worthwhile to note that Equations (A.22) and (A. 23) have genuine steady state solutions without the need to introduce a boundary layer. This is because the chemical reaction (A. 19) causes the formation of a steady state kinetic layer. Only within this boundary layer is R present in solution, and the concentration of 0 perturbed from its initial concentration. The thickness of the kinetic layer depends on k the larger k the thinner the layer. Certainly for high values of k, the kinetic layer will lie well within the normal steady state diffusion layer defined by natural convection. The time required to reach the steady state (form the kinetic layer) also depends inversely on k. 

For ionomeric systems in which the strong interactions between ionic sites and the penetrant (water) result in concentration dependent diffusion coefficients, the Fickian diffusion model, which assumes the solubility coefficient is independent of the concentration, is not valid. The commonly used Fickian diffusion constants actually contain mobility and solubility gradient contributions [19], In addition, the concentration dependent solubilities lead to nonlinear concentration profiles during steady state diffusion. Therefore, mobility measurements which generate average diffusion coefficients are generally not satisfactory. 

Simple steady-state diffusion through a film is modeled in Figure 4.18. The flux J gives the quantity of permeant passing through a unit cross section of membrane per unit time. Thus equation (4.22) leads to first-order transport kinetics, where the quantity transported at any instant depends on the concentration on the high concentration side to the first power. 

These considerations can be explained from a theoretical point of view as follows considering the response in the transient time and at the steady state. The concentration of the substrate and product at the transducer depends on the enzyme Michaelis constant K, the activity of the enzymatic layer, the thickness of the layer, and the diffusion coefficients of the substrate and of the product. The product concentration at the external surface of the electrode is assumed to be zero because the diffusion of the species occurs in the active layer. The temperature, K , and coefficient of diffusion and Dp (diffusion... 

C. Amatore, S. Szunerits, L. Thouin, J.S. Warkocz. Mapping concentration profiles within the diffusion layer of an electrode. Part III. Steady-state and time-dependent profiles via amperometric measurements with an ultramicroelectrode probe. Electrochem Commun. 2 353 (2000). 

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