Estimation of the diffusion coefficient

Knowing an experimental value of k, it is possible to evaluate the diffusion coefficient of the atoms of a dissolving solid substance across the diffusion boundary layer at the solid-liquid interface into the bulk of the liquid phase using equations (5.6) and (5.7). Its calculation includes two steps. First, an approximate value of D is calculated from equation (5.6). Then, the Schmidt number, Sc, and the correction factor, /, is found (see Table 5.1). The final, precise value is evaluated from equation (5.7). In most cases, the results of these calculations do not differ by more than 10 %. Values of the diffusion coefficient of some transition metals in liquid aluminium are presented in Table 5.9.303 

Since even at the zero initial concentration of component A in the solution its average concentration in the diffusion boundary layer at the solid-liquid interface varies during dissolution from cJ2 to cs, the values in 

As illustrated in Fig. 5.10, the temperature dependence of the diffusion coefficient of transition metals into liquid aluminium is well described by the Arrhenius equation, D = D0 exp (-E/R.T), giving a linear plot of In I) against T l. Values of the pre-exponential factor, D0, and the activation energy, E, for some of them are given in Table 5.10. 

Note that both Stokes-Einstein s and Swalin s relations usually produce a less satisfactory fit to the experimental data on the solute diffusion. The former predicts D to be roughly proportional to T, while the latter to T (for more detail, see for example Refs 314-322). Hence, according to these relations, in the 700-900°C (973-1173 K) temperature range the ratio of diffusion coefficients would be expected to increase either by 1173/973 = 1.21 or by 11732/9732 = 1.45. However, as seen in Table 5.9, in most cases it increases by a factor of 2 or more, in accordance with the Arrhenius 

It should be emphasised that the diffusion data for liquid metals, obtained by different authors, especially with the use of the capillary-reservoir technique, differ very considerably. If desired, therefore, examples may easily be found in the literature, providing evidence for the validity of either of those three relations and many others. Generally, however, a much wider applicability of the Arrhenius relation can hardly be doubted. It appears to hold in the greater number of cases than all other known dependences taken together. 

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The correct and critical estimate of the diffusion coefficients may be used to get informations on the morphological modifications occurring in presence of the sorbed moisture. 

A simple estimate of the diffusion coefficients can be approximated from examining the effects of molecular size on transport through a continuum for which there is an energy cost of displacing solvent. Since the molecular weight dependence of the diffusion coefficients for polymers obeys a power law equation [206], a similar form was chosen for the corneal barriers. That is, the molecular weight (M) dependence of the diffusion coefficients was written as ... 

Chemical Reaction Engineering Aspects of Homogeneous Hydrogenations Table 45.4 Estimation of the diffusion coefficient in liquid phase. 

Values of ionic mobility for various ions in water are shown in Table 1.8. For the estimation of the diffusion coefficient of a single ion the second term in the right-hand part of the eq. (1.24) is replaced by k.Jnv... 

MD calculations may be used not only to gain important insight into the microscopic behavior of the system but also to provide quantitative information at the macroscopic level. Different statistical ensembles may be generated by fixing different combinations of state variables, and, from these, a variety of structural, energetic, and dynamic properties may be calculated. For simulations of diffusion in zeolites by MD methods, it is usual to obtain estimates of the diffusion coefficients, D, from the mean square displacement (MSD) of the sorbate, (rfy)), using the Einstein relationship (/) ... 

As discussed in Chap. 3 Sect. 2.5, while observation of time-dependent rate coefficients does enable reliable estimates of the diffusion coefficient appropriate to reaction between donors and acceptors, the very ease of observation of these time-dependent effects masks much detail of diffusive motion in liquids. Estimates of i eff reflect more on the parameters appropriate to long-range transfer processes than on collisional events in... 

Among the uses of these relations is their value in providing estimates of the diffusion coefficient of an ion (but watch the units ). 

Estimation of the diffusion coefficient from the data on the electrical conductivity of nitrate glasses has shown that, in the KN03 + Ca(N03)2 matrix, the diffusional decay of N03 radicals at T < 297 K and t < 102s is negligible. A rise of temperature in the low-temperature region results in a small increase in the slope of kinetic curves. Thus, just as in the case of the decay of N03 in the water-alkaline matrix, the decay of N03 radicals in the KN03 + Ca(N03)2 matrix at low temperatures is an activated electron tunneling process. 

In this respect one solution for the estimation of a Dp-value is to correlate the diffusion coefficient with the relative molecular mass, Mr, of the migrant and with matrix specific parameters at a given temperature T in Kelvin. This approach has already been successfully used (Piringer 1993,1994 Limm and Hollifield 1996). The estimation of the diffusion coefficient can be achieved for example using the following heuristic correlation (Piringer 1994 Baner et al. 1996) ... 

The presence of a polymer in the water affects not only the relaxation, but also the diffusion, of the solvent. For an estimation of the diffusion coefficient, we can use the following expression... 

However, we cannot a priori use this model without the previous establishment of conditions which accept the transformation of the three-dimensional and unsteady state model into a one-dimensional model. These conditions can be studied using the simulations as a tool of comparison. At the same time, it is interesting to show the advantages of the dynamic (unsteady) methods for the estimation of the diffusion coefficient of the species through the porous membrane by comparison with the steady state methods. 

In the characterization of porous membranes by liquid or gaseous permeation methods, the interpretation of data by the hyperbolic model can be of interest even if the parabolic model is accepted to yield excellent results for the estimation of the diffusion coefficients in most experiments. This type of model is currently applied for the time-lag method, which is mostly used to estimate the diffusion coefficients of dense polymer membranes in this case, the porosity definition can be compared to an equivalent free volume of the polymer [4.88, 4.89]. 

The equilibrium constant iTj can be determined using electrochemical techniques (Valenta, 1960) for the computation of the hypothetical diffusion current an estimate of the diffusion coefficient is necessary and a two-electron process is assumed. 

It is possible to use alternative formulations considering mole fractions rather than mass fractions. For most cases, mass fraction formulations will be adequate. An estimation of the diffusion coefficient (of component k) in a multicomponent mixture Dkm) however, is not straightforward. For mixtures of ideal gases, the diffusion coefficient in a mixture can be estimated as (Hines and Maddox, 1985)... 

Here the factor ip accounts for effects of composition (cf. Sect. 8.23.6), tortuousity and psd and the square root is the mean thermal velocity of molecules of mass M,-. The value D serves as a reasonable estimate of the diffusion coefficient. 

Equation 5.9 has been derived from a correlation of data obtained with compounds with molecular weights between 100 and 500, and it does not give good estimates of the diffusion coefficient of polymers, especially proteins. Nevertheless, as the specific volume of proteins is nearly constant, around 0.73 [17], their molecular volume is proportional to their molecular weight, and Eq. 5.9 predicts values of the diffusion coefficient between 0.5 and 1 x 10 ... 

The diffusion coefficient ot the chain is controlled by the reptation time [Eq. (9.12)]. The linear polybutadiene chain with M= 130000gmol has A =1240 Kuhn monomers, with Kuhn length /)=10A and coil size R — hy/N = 350 A. Since linear polymers move a distance of order their own size in their reptation time, the reptation time of ri-ep = 0.2s at 25 °C enables estimation of the diffusion coefficient 6x... 

The diffusion coefficient in a gas is proportional to -jP, the constant of proportionality being a rather slowly increasing function of temperature. The estimation of the diffusion coefficient in liquids is discussed briefly by Sherwood and Satterfield it is proportional to r// , where /z. is the viscosity. At atmospheric pressure and ordinary temperatures, the order of magnitude of D for a gas is 0.1-1 cm-/sec and for liquids it is smaller by a factor of about 10. ... 

In the recleaning of the heat exchanger, several of the parallel, return-bend U-tubes (1 inch in diameter) were purposely pitched to assure a long section (approximately 10 feet) of trapped sodium. As would be expected from the previous estimate of the diffusion coefficient for water through concentrated caustic, the humid gas was not effective in removing these sodium pockets. The humid gas did successfully reclean the balance of the heat-exchanger circuits. In this phase of the operation, the maximum temperature rise did not exceed 130 °F. 

Equations that allow a more accurate calculation can be found in the literature [Baerns 1999]. Eor the description and estimation of the diffusion coefficients of liquids, the reader is referred to the specialist literature [Eei 1998]. The follovdng orders of magnitude can be given for the self-diffusion coefficients ... 

In the last section we have reported what we have found to be typical coefficients for all three equations. Different values for the coefficients have been reported in the literature as well. To some degree, the coefficients depend on the estimation of the diffusion coefficient, which is needed to calculate the reduced velocity. To some degree, the differences in the reported values may also reflect the properties of different packing materials. We should take them clearly as estimates. 

For the estimation of the diffusion coefficient in isotropic macroporous media, the relation... 

ATR-FTIR spectroscopy was used to monitor the uptake of urea into a silicone polymer. Analysis of the time-dependent changes in the IR absorbances of urea and silicone leads to an estimate of the diffusion coefficient for urea that is in close agreement with a value obtained using a bulk transport method (involving radiolabelled permeant). The silicone polymer was medical grade silicone pressure-sensitive adhesive (X7 201). ATR-FTIR is proposed as a rapid and accurate method of rapidly and accurately determining solute diffusion within a polymer matrix. 12 refs. 

To convert to different reference conditions and to therefore compute a rough estimate of the diffusion coefficient at different temperatures and pressures, a diagram is given by Slattery and Bird [8.15]). 

From equation (4.49) we obtain k = 6nr DJ for the bimolecular chain termination rate constant. By estimation of the diffusion coefficient of a macroradical in a range of three orders of magnitude (from the diffusion coefficient in the liquid phase to the diffusion coefficient in the solid phase) Di=10 -10" m s" we find A ,=5x(10 -10 ) m moP. s". In the liquid monomeric phase the value tv, in accordance with the literature, is estimated to be in the range / .=10 -10 m moP s . Comparison of the obtained estimations does not confirm this, but at the same time does not exclude the possibility of the diffusion control on the rate of bimolecular chain termination in the liquid monomeric phase. Simultaneously, the values k and ka from Table 4.3 are 6-7 orders of magnitude smaller than kx and convincingly confirm that they do not present the bimolecular chain termination in the interface layer. That is why the question about the diffusion controls on /t,i and ka should be discussed in another way. 

Dynamic Structure Factor and Mean Square Displacement Here we learn how gi(T) obtained in DLS gives an estimate of the diffusion coefficient. We are concerned with dilute solutions here. Hence, gi(T) = Si(k, t). 

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