Darken equation

The equation conesponding to die Darken equation quoted above is then... 

The need to predict mutual diffusion coefficients from self-diffusion coefficients often arises, and many efforts have been made to understand and predict mutual diffusion data, through approaches such as, for example, the following extension of the Darken equation [5j ... 

At low sorbate concentrations f-+o and eqn. 6 reduces to the familiar Darken equation(33). in comparing the results of sorption and NMR diffusivity measurements it is logical to compare D0 with Ds, even though exact agreement can be expected only at low sorbate concentrations. At higher concentrations one may expect D0 Ds/(l-f). 

Rathbun and Babb [20] suggested that Darkens equation could be improved by raising the thermodynamic correction factor PA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult s law and found n = 0.3. Furthermore, for polar-nonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas [22] followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polar-nonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi, Krahn, and Lucas (ibid.) examined a few other mixtures and... 

In a kinetically controlled separation system using CMS or zeolite 4A as adsorbents, it is necessary to use more accurate rate model. Therefore, concentration dependent diffusivity model based on Darken equation combined with Langmuir-Freundlich isotherm was applied and each result was compared with the experimental data. 

Adsorption rate model Darken equation + L-F isotherm>... 

Combining Eqs. (5)-(7) leads to the definition of the corrected or self-diffusivity, the Darken equation ... 

On comparing Eqs. (20) and (5) and (8), it can be seen that for Langmuir adsorption, the Maxwell-Stefan diffusivity in a one-component system is identical to corrected diffu-sivity in the Darken equation. 

The diffusivity measured by the FR technique, D, is a transport diffu-sivity which has to be corrected, by using the Darken Equation (Eq. 26), to obtain the so-called corrected diffusion coefficient where the diffusion is measured at an equilibriiun pressure, Pe, which is outside the Henry s law range. This corrected diffusivity is generally taken to be the equivalent of the self-diffusion coefficient Dq ... 

NiO/CoO less at the other Need diffusion data Darken equation... 

In this equation, which is known as the Darken equation, X indicates the mole fraction of Co or of Ni. The equation assumes local equilibrium everywhere and that D is a chemical or interdiffusion coefficient in a chemical potential gradient. The matrix is Co Ni O. D is plotted as a function of concentration for both Ni and Co diffusing in the mixed oxide at 1300°C and 1445°C in Figure 25.15a. 

The corrected diffusion coefficients, Db, obtained using the Darken equation are given in Table 5. The diffiisivities of 2-butyneare shown in this table to be 3-5 times faster in the straight chaimels than those of n-butane and 2-3 times faster in the sinusoidal channels. Both n-butane and 2-butyne diffuse some 6-8 times faster in the straight channels than in the sinusoidal channels. The faster diffusivity of 2-butyne over n-butane is most probably due to a) its smaller dynamic dimensions and b) its inflexibility. 

Principally speaking, the chemical potential is the decisive driving force (and not the concentration). The Darken equation takes this effect into account. With d as the self-diffusion coefficient the Darken diffusion coefficient Dj)ark results in... 

However, if we can assiune that the non-diffusive relaxation of the less mobile polymers is fast compared with the diffusion of the more mobile chains, we can derive a simple equation relating the mutual diffusion coefficient to the tracer diffusion coefficient and a thermodynamic factor. This equation has been applied to liquid systems, in which context it is known as the Hartley-Crank equation (Tyrell and Harris 1984), and to metals, in which context it is known as the Darken equation (Haasen 1984). The polymer version was first stated by Kramer (Kramer et al. 1984) and is often known as the fast theory . It is most compactly written in the form... 

Farooq et al. [3] inb oduced a variable diilusivity model to a kinetically controlled PSA separation process. They pointed out that the Darken equation widi die Langmuir isotherms predicted the experimental data better dun die constant difiiisivity assumption. Based on those results, following concentration dependent diflusivity model was presented as the adsorption rate models. 

A slightly modified form of expression was obtained by Ash and Barrer who used a somewhat different definition of the transport diffusivity. If the cross coefficient can be neglected (I. wO), Eq. (5.9) reduces to Eq. (5.6) with Dq = S), which is the familiar Darken equation/ originally derived for the interdiffusion of two alloys. While Eq. (5.6), being essentially a definition of Dq, is always valid it is evident that the assumption that Dq- is only true in the limiting case where In general both Z)q and are... 

The second possible situation which could occur for this type of reaction is the following 2. The metal atoms A and B in the reaction product A B occupy the same lattice, just as in the case of a substitutional solution, or else the sublattices are energetically so similar that both types of atoms can move in every sublattice. Then, in contrast to the case just discussed, the fluxes and are once again coupled, since the condition of site-balance for an observer in the external system must be satisfied. The condition is the same as that used to derive the Darken equation in section 5.5.3. Accordingly, a general diffusion coefficient for A and B is obtained which is a combination of the component diffusion coefficients as given by the Darken equation, just as in the case of the formation of ionic crystals or in the case of diffusion in simple metallic systems. The calculations which were just performed above may now be repeated with the condition ... 

From the viewpoint of diffusion conductance, the NG and Darken equations describe extreme, marginal cases. Darken s equation corresponds to parallel connection when the interdiffusion rate is usually determined by the more mobile component. The NG equation conforms to series (consecutive) connection at which the interdiffusion rate is mainly determined by the less mobile component - the more mobile one has to wait until slower atoms accomplish their migration. The discrepancy between equations becomes most apparent whenever the components mobility ratio is much larger or much less than one. In the general case, we may expect the correct description of interdiffusion to correspond to a certain combination of parallel and series (consecutive) connection, depending on vacancies sinks/sources effectiveness. We will see that the NG and Darken equations conform to different spatial and time scales. 

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