Diffusion constant, dependence on the

The equation obtained can be used when the electrode potential can be varied independent of solution composition (i.e., when the electrode is ideally polarizable). For practical calculations we must change from the Galvani potentials, which cannot be determined experimentally, to the values of electrode potential that can be measured E = ( q + const (where the constant depends on the reference electrode chosen and on the diffusion potential between the working solution and the solution of the reference electrode). When a constant reference electrode is used and the working solutions are sufficiently dilute so that the diffusion potential will remain practically constant when their concentration is varied, dE (i(po and... 

The first density correction to the rate constant depends on the square root of the volume fraction and arises from the fact that the diffusion Green s function acts like a screened Coulomb potential coupling the diffusion fields around the catalytic spheres. 

Thus, as described by Equation (2.1), the equilibrium dissociation constant depends on the rate of encounter between the enzyme and substrate and on the rate of dissociation of the binary ES complex. Table 2.1 illustrates how the combination of these two rate constants can influence the overall value of Kd (in general) for any equilibrium binding process. One may think that association between the enzyme and substrate (or other ligands) is exclusively rate-limited by diffusion. However, as described further in Chapter 6, this is not always the case. Sometimes conformational adjustments of the enzyme s active site must occur prior to productive ligand binding, and these conformational adjustments may occur on a time scale slower that diffusion. Likewise the rate of dissociation of the ES complex back to the free... 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

In equation 11.77, 4 is a numerical constant depending on the geometry and decay constant of the parent radionuclide. If the half-life of the parent is long in comparison with the cooling period, A takes a value of 55, 27, or 8.7 for volume diffusion from a sphere, cylinder, or plane sheet, respectively. If decay rates are faster, A progressively diminishes (see table 1 in Dodson, 1973, for numerical values). 

Specific diffusion control The rate constants depend on the size and topology of the molecule the group is bound to i.e., they depend on the translation diffusion coefficient of the species. 

In an isotropic medium, D is a scalar, which may be constant or dependent on time, space coordinates, and/or concentration. In anisotropic media (such as crystals other than cubic symmetry, i.e., most minerals), however, diffusivity also depends on the diffusion direction. The diffusivity in an anisotropic medium is a second-rank symmetric tensor D that can be represented by a 3 x 3 matrix (Equation 3-25a). The tensor is called the diffusivity tensor. Diffusivity along any given direction can be calculated from the diffusivity tensor (Equation 3-25b). Each element in the tensor may be constant, or dependent on time, space coordinates and/or concentration. 

If polyurethanes are used to entrap cells, the diffusion wiU depend on the polyol used to build the polyurethane since the polyol defines equilibrium moisture. Later in this chapter, we will discuss a number of entrapment systems, including acrylates and polysaccharides. Each has its own equilibrium moisture and therefore unique diffusion constant. Only polyurethanes, however, offer the opportunity to affect changes in the constants. Conventional hydrophilic polyurethanes have equilibrium moisture levels around 70%. It is possible, however, to increase the molecular weight of a polyol (an ethylene glycol of 1000 molecular weight) to 3000 or more. This increases the equilibrium moisture to greater than 90%. 

In various applications the following model has been used, which is of more general interest. Consider a molecule having a number of internal states or levels i. From each i it can jump to any other level j with a fixed transition probability yjti per unit time. Moreover the molecule is embedded in a solvent in which it diffuses with a diffusion constant depending on its state i. The probability at time t for finding it in level i at the position r with margin d3r is P,(i% t) d3r. While the molecule resides in i the probability obeys... 

This generic term may denote either the measured value of the independent variable under the experimental conditions employed or some function of that value as with the characteristic potential, the nature of the response constant depends on the technique used, and in addition it depends on the behavior of the system being studied. For a diffusion-controlled process the preferred polarogrephic response constant is the diffusion current constant iH/Cm2 3t1 6. but the ratio id/ is often given instead when a value of nj2 3 1 could not be obtained from the or glnal, and even values alone are sometimes quoted for want of anything better values... 

Figure 5. Segmental diffusion constant depending on temperature and solvent viscosity data are calculated according to Equation 1 and the graph according to Equation 2...

Under this condition, there is complete depletion of O2 at the electrode next to the porous layer of Fig.6a or inside the cavity of Fig.6b. The constant depends on the diffusion constant of O2 (in its particular carrier gas), Dq, and the geometrical characteristics of the diffusion barrier. In the device of Fig.6b (sensor with integral cavity), the diameter of the aperture C (usually greater than 50 microns) is much larger than the mean free path of the gas molecules at 1 atm (about 1 micron) and bulk diffusion dominates. In this case(ll-12). D0 - K Ta/P and cr - (DgA)/(kTd), where K] is a constant, P is the absolute pressure, a is a constant having a value between 1.5 and 2 and A and d are the cross-sectional area and length of the aperture C. Representative values for D0 are about 1.5 cm2/s at 700 °C and 0.15 cm2/s at 20 °C. Since Pg — cP with c the percentage of O2 molecules in the gas, we have... 

For the moment, it is sufficient to know that the sample response, which is the time dependent diffraction efficiency after switching off the optical grating, contains at least a fast contribution from heat and a slow one from mass diffusion. The corresponding diffusion time constants depend on the grating constant and are typically of the order of 10 /rs and 100 ms, respectively. 

A typical concentration profile of lsO oxygen is shown in Fig. 1. It is characterized by some specific features as compared to a simple case of the diffusion equation for a constant source. Firstly, it is seen that the concentration cs on the surface of the sample is lower than the equilibrium concentration in the test samples. Therefore, a solution of the diffusion equation depending on the boundary conditions must be used [18]. Secondly, the concentration cj of the lsO isotope was almost constant at a large depth x and was much higher than the initial concentration of the lsO isotope in samples. It is not improbable that this... 

It is seen that the effective rate constant depends on the diffusion constant D and the intrinsic rate constant ks in a rather complicated way, and that it is a function of time. The time dependence is a consequence of the transient approach to stationarity of the concentration profile of B (see Fig. 9.2.2). At stationarity, the rate constant is independent of time, which is also seen from the asymptotic expansion... 

We now consider the factors that affect the variation of kc (or ka) and kd. The kinetic rate constants depend on the applied potential and on the value of the standard rate constant, k0. As was seen in Chapter 5, kd is influenced by the thickness of the diffusion layer, which we can control through the type of experiment and experimental conditions, such as varying the forced convection. By altering kc (or ka) and kd we can obtain kinetic information as will be described below. At the moment we note that there are two extremes of comparison between k0 and kd. 

Standard enthalpies and entropies for the ketone to enol equilibrium have been determined from data obtained by the kinetic halogenation method (see Section 3) at 5, 15, 25 and 35°C. Since the keto-enol equilibrium constants depend on the absolute rate constant arbitrarily chosen for the diffusion-controlled halogen addition to the enol, only the differences in from one ketone to another must be considered... 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

The transfer of gases at the base of the euphotic zone is by a combination of advection and diffusion processes and in reality is probably dependent on mechanisms that are intermittent rather than constant in time. Here we write the flux as a simple one-dimensional diffusion process dependent on the concentration gradient at the base of the euphotic zone and a parameter that is assumed to be analogous to molecular diffusion, an eddy diffusion coefficient, K ... 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

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