Diffusion coefficient numerical values

The experimental decays from the infinite rods at high salt concentrations in Fig. 4 fit well to the 1-D model. The same model has been used in several investigations of the sphere-to-rod transition and to determine the diffusion coefficients of the probe and quencher in rodlike micelles. However, equations based on other assumptions could fit the experimental data about equally well there is no distinctive feature in the decay that points to the 1-D case. Often one can reject alternative models due to unreasonable values for estimated parameters. A 2-D model, for example, would probably also fit well but would result in too low a value of the diffusion coefficient. Numerical studies demonstrated that 2-D and (with consideration of transient effects) 3-D models generally fit well to 1-D data but with unreasonable values suggested for the parameters, whereas a decay curve from a 2-D structure does not fit to a 1-D model [15]. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

It will be noted that because of the low self-diffusion coefficients the numerical values for representations of self-diffusion in silicon and germanium by Anhenius expressions are subject to considerable uncertainty. It does appear, however, that if this representation is used to average most of the experimental data the equations are for silicon... 

The same equation applies to other solvents. It is often easier to incorporate an expression for the diffusion coefficient than a numerical value, which may not be available. According to the Stokes-Einstein equation,6 diffusion coefficients can be estimated from the solvent viscosity by... 

The coefficient < turb introduced in Equation (41) (dimension L /T) is called the turbulent, or eddy diffusivity. In the general case the eddy diffusivity is given separate values for the three spatial dimensions. It must be remembered that the eddy diffusivities are not constants in any real sense (like the molecular diffusivities) and that their numerical values are very uncertain. The assumption underlying Equation (41) is therefore open to question. 

Q2 behavior takes place at decreasing Q. The position of the crossover point Q (c) is a direct measure of the dynamic correlation length (c) = 1/Q (c). The plateau value at low Q determines the collective diffusion coefficient Dc. A simultaneous fit of all low Q spectra where a simple exponential decay was found led to the concentration dependence and the numerical values of Dc... 

Substitution of the numerical values for the bulk and Knudsen diffusion coefficients into equation 12.2.8 gives... 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

I wish to ask about the concept of a unit of length in a biological cell. My concern is with the numerical value to be used for the diffusion coefficient. Most of biological space is heavily organized even in a single cell, and therefore a diffusion constant is not a simple property. To put my question simply, it is easiest, however, to consider enzyme sites that are otherwise identical in two situations (1) with water between them, (2) with a biological membrane between them. Is it not the case that the unit of length is quite different, for the diffusion in (1) is virtually free diffusion whereas in (2) the diffusion is constrained most probably as a series of activated hops ... 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

Tyrrell and Watkiss [58] showed that the numerical coefficient for 2-methylpentan-2,4-diol solvent is not 6 or 4 but nearer 1—3 depending on the solute molecular size and shape and temperature, being smaller for smaller and planar molecules. In a study of normal alkanes, Amu [59] found that this numerical factor decreased as the molecular chain length increased. Indeed, if the solute molecules are significantly different from a spherical shape, the diffusion coefficient should be regarded as (a tensor) having different values in different directions. Usage of the microfriction factor mentioned in Sect. 5.2 is an improvement, but still rather unsatisfactory (Alwatter et al. [43]). 

In order to calculate the numerical values of transfer coefficients, values of the molecular properties are required. In the next section, we present estimation methods for viscosity, diffusivity, thermal conductivity and surface tension, for the high-pressure gas. 

For very dilute solid solutions of B in A, the basic physics of diffusional mixing is the same as for (A, A ). An encounter between VA and BA is necessary to render the B atoms mobile. But B will alter the jump frequencies of V in its surroundings and therefore numerical values of the correlation factor and cross coefficient are different from those of tracer A diffusion. Since the jump frequency changes also involve solvent A atoms, in addition to fB, the numerical value of fA must be reconsidered (see next section). 

The question about the difference between the macroscopic and microscopic values of the quantities characterizing the translational mobility (viscosity tj, diffusion coefficient D, etc.) has often been discussed in the literature. Numerous data on the kinetics of spin exchange testify to the fact that, with the comparable sizes of various molecules of which the liquid is composed, the microscopic translational mobility of these molecules is satisfactorily described by the simple Einstein-Stokes diffusion model with the diffusion coefficient determined by the formula... 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

After the substitution of expression (3.66) into Eq. (3.64), the integral was taken numerically at any diffusion coefficient except for its very small values, where the quenching is quasistatic. However, for this very region there is an approximate expression derived in Ref. 99 ... 

Thus, the value of cAi in the denominator of equation (1.70) is numerically equal to the content of A in AB, while cA2 is practically zero. Consequently, this equation yields k]A2 = DA. Therefore, the physical (diffusional) constant k A2 is identified with the reaction-diffusion coefficient, Da, of component A in the lattice of any chemical compound. 

The lower bound for the coagulation coefficient, ft, is calculated from Eq. [81] using the values of the constant A determined from the numerical solution of Eqs. [74], [77], and [78]. The upper bound for the coagulation coefficient, ft, is obtained from Eq. [97]. In the above calculations, the Philips slip correction factor (Eq. [6]), is used for the calculation of the diffusion coefficient of particles. The results are expressed in terms of the dimensionless coagulation coefficient y defined as the ratio between the coagulation coefficient and the Smoluchowski coagulation coefficient... 

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