Diffusion coefficients empirical relations

Under water-saturated conditions, solute diffusion in natural porous media is hindered by the tortuous nature of the pores and reduced cross-sectional area available for diffusion both factors are influenced by the pore size distribution within the medium. If the pores are larger than the mean free path of the molecules in solution ( 1 nm for organic chemical contaminants in water), the molecular diffusivity applies and Knudsen diffusion is negligible (Grathwohl, 1998). Lerman (1979) also consider these factors along with laboratory measurements for water-filled sediments, natural clays, silty-clay soil and sedimentary rocks using sodium chloride, trichloroethylene and iodine. Generally, tortuosity and pore size distribution are difficult to assess, so overall volumetric porosity si (m m ), which can be readily determined from relatively simple measurements, is used to assess the porous media effects on the diffusive process. Therefore, the effective diffusion coefficient is related to the aqueous diffusivity, Daq (m s ) and an empirical function of s alone ... 

Diffusion coefficients may be estimated using the Wilke-Chang equation (Danckwerts, 1970), the Sutherland-Einstein equation (Gobas et al., 1986), or the Hayduk-Laudie equation (Tucker and Nelken, 1982), which state that Dw values decrease with the molar volume (Vm) to the power 0.3 to 0.6. Alternatively, the semi-empirical Worch relation may be used (Worch, 1993), which predicts diffusion coefficients to decrease with increasing molar mass to the power of 0.53. These four equations yield very similar D estimates (factor of 1.2 difference). Using the estimates from the most commonly used Hayduk-Laudie equation... 

There are many transport conditions where experiments are needed to determine coefficients to be used in the solution. Examples are an air-water transfer coefficient, a sediment-water transfer coefficient, and an eddy diffusion coefficient. These coefficients are usually specific to the type of boundary conditions and are determined from empirical characterization relations. These relations, in turn, are based on experimental data. 

The most well-known empirical formula for diffusion coefficients is that due to Gilliland (G3, S9), which is written in terms of the molecular volumes of the diffusion species and gives SDab proportional to TH. Since this temperature dependence has since been shown to be incorrect the Gilliland formula should not be used. Another empirical formula has been suggested by Arnold (Al). Recently Slattery (S10) has proposed the following relation on the basis of corresponding states arguments ... 

A number of useful empirical and semiempirical relations have been proposed for predicting diffusion coefficients in liquids a rather complete summary of these has been given by Treybal (T6, pp. 102 et seg.). Most of these are valid only for nonionic systems at very low concentrations. 

Several other empirical relations for diffusion coefficients have been suggested Olson and Walton (01) have devised a means for estimating diffusion coefficients of organic liquids in water solution from surface-tension measurements. Hill (H5) has proposed a method based on Andrade s theory of liquids which allows for the concentration dependence of the diffusion coefficient in a binary liquid mixture. The formula of Arnold (A2, T6, p. 102) does not seem generally useful inasmuch as it contains two constants ( abnormality factors ) characteristic of the solute and of the solvent. 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

An attempt to correlate experimental diffusion data with free-volume, for the system of organic vapors with polyvinyl acetate, has been made in (57). The experiments showed that in this system, for T > Tg, the diffusion is Fickian and that the measured average diffusion coefficient steeply increases with the concentration, cs, of penetrant in the polymer. To quantify such a finding, an empirical relation has been proposed earlier (58) ... 

The concept of measuring such rates is not new, particularly in the pharmaceutical field. Van de Waterbeemd [14] measured rates of transfer of various drugs from octanol to water and empirically related these rates to the partition coefficient. Similarly Brodin [15], using a different experimental method, obtained rates of transfer for another series of compounds between cyclohexane and water. The rotating diffusion cell has been introduced for similar purposes [16-18]. It is necessary to look into the broader background of liquid-liquid interfacial kinetics, in order to illustrate aspects of the issues under consideration. The subject has been reviewed in part by Noble [19]. 

Even the binary system diffusivities in liquid mixtures are composition dependent. Therefore, in multicomponent liquid mixtures with n components, predictions of the diffusion coefficients relating flows to concentration gradients are empirical. The diffusion coefficient of dilute species i in a multicomponent liquid mixture, Dim, may be estimated by Perkins and Geankoplis equation... 

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. 

Both Tyson [ 125] and Matsuo [ 127] measured the diffusion coefficients in gelatins that had been treated with different amounts of hardener and therefore had different swellability. Tyson found that the diffusion coefficient in gelatin, Z)gei, could be related to the diffusion coefficient in water, Dwater, and the swelled to dry thickness ratio, Z, of the hardened gelatin. The empirical relationship was found to be Eq. (91). 

As reviewed in the previous section, measurements of Ti and Tip can provide an estimation of the length scale of miscibility of polymer blends. Compared with such kinds of experiments, the results of the spin-diffusion experiments are more quantitative and straightforward. The accuracy of the results of spin-diffusion experiments relies, to a large extent, on the values of spin-diffusion coefficients (7)) employed in calculation of the constituent phase components. Despite efforts that have been made, there still lacks a suitably applicable method of directly measuring the spin-diffusion coefficients, at least for polymers. For rigid polymer below Tg, 0.8 nm /ms has been turned out to be a reliable value of spin-diffusion coefficient. The difficulty left then concerns how to determine the coefficient of the mobile phase, which is very sample dependent. Recently, through studies on diblock copolymers and blend samples with known domain sizes, Mellinger et al established empirical relations between the T2 and D as follows ... 

The diffusion coefficients are obtained from the mobilities of the particles, ft, use being made of the Einstein relation D, = kBTb(. For spherical particles the dependence of the mobility on r and A has been derived empirically from careful measurements by Knudsen and Weber (1911) and by Millikan (1923) in the form... 

We have already described diffusion as a thermally activated process that is commonly represented by the Arrhenius relations (see Eqn. 25). A method of estimating diffusion coefficients takes advantage of the linear interdependence of the activation energy. Eg and the pre-exponential factor. Do (or Aq). This compensation law is an empirical observation that indicates that diffusion rates of different species tend to converge at a particular temperature, generally for materials with similar structure. In addition, the validity of diffusion compensation rests on the assumption that the diffusion mechanism is the same for all the minerals and species being considered (Fortier and... 

For multicomponent liquid mixtures, it is usually very difficult to obtain numerical values of the diffusion coefficients relating fluxes to concentration gradients. One important case of multicomponent diffusion for which simplified empirical correlations have been proposed is when a dilute solute diffuses through a homogeneous solution of mixed solvents. Perkins and Geankoplis (1969) evaluated several methods and suggested... 

Stood to be the electrochemical potential, which is a function of pressure, temperature, chemical composition, and the electrical state of the phase. The characterization of the electrical state of phases of different composition itself introduces a number of subtle questions (Newman 1991). Inevitably the phenomenological relations for concentrated solutions rest largely on empirical determinations of the generalized diffusion coefficients. 

From measurements of the sensitised fluorescence, one can determine diffusion coefficients D and diffusion lengths I of excitons. The ratio of the intensities from the guest molecules and the host, Ig/Ih as a function of the concentration cg (in molecules/molecule) can be empirically described by a dimensionless transfer constant k (cf also Fig. 6.20c). The following relation holds ... 

The diffusion coefficient of oxygen was first obtained theoretically in its relation to the degree of nonstoichiometry by Thorn and Winslow (246, 260,300) for nonstoichiometric UOj+ t, while more recently Contamin et at. (292) have reported for the temperature range from 400 to 900 C with X = 0.006 0.16 the empirical relation... 

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