Diffusion dependence on concentration

It is difficult to obtain how the diffusivity depends on concentration using the bulk mass loss or gain method, although it is possible to verify specific concentration dependence by conducting experiments from small degrees of mass loss to almost complete mass loss (Wang et al., 1996). On the other hand, the shape of diffusion profiles reveals the dependence of diffusivity on concentration. 

Fig. 11-2 Sketch illustrating diffusion dependence on concentration profile.

Both facilitated and simple diffusion depend on concentration gradients net solute transport always occurs from high to low concentration. Unlike diffusion, facilitated transport systems are specific, since they depend on binding of the solute to a site on the transport protein. For example, the D-glucose... 

Molecular Transport of a Property with a Variable Diffusivity. A property is being transported through a fluid at steady state through a constant cross-sectional area. At point 1 the concentration F is 2.78 x 10 amount of property/m and 1.50 x 10 at point 2 at a distance of 2.0 m away. The diffusivity depends on concentration T as follows. 

Back-diffusion is the transport of co-ions, and an equivalent number of counterions, under the influence of the concentration gradients developed between enriched and depleted compartments during ED. Such back-diffusion counteracts the electrical transport of ions and hence causes a decrease in process efficiency. Back-diffusion depends on the concentration difference across the membrane and the selectivity of the membrane the greater the concentration difference and the lower the selectivity, the greater the back-diffusion. Designers of ED apparatus, therefore, try to minimize concentration differences across membranes and utilize highly selective membranes. Back-diffusion between sodium chloride solutions of zero and one normal is generally [Pg.173]

The diffusion coefficient in these phases D,j is usually considerably smaller than that in fluid-filled pores however, the adsorbate concentration is often much larger. Thus, the diffusion rate can be smaller or larger than can be expected for pore diffusion, depending on the magnitude of the flmd/solid partition coefficient. 

Laminar flow reactors have concentration and temperature gradients in both the radial and axial directions. The radial gradient normally has a much greater effect on reactor performance. The diffusive flux is a vector that depends on concentration gradients. The flux in the axial direction is... 

Solutions for a number of typical cases are reported below. To simplify our task we use the assumption that reactant migration is not observed (a large excess of foreign electrolyte), that the diffusion coefficients Dj do not depend on concentration, and that for the reactant v = 1. (The subscript j is dropped in what follows.)... 

Koizumi and Higuchi [18] evaluated the mass transport of a solute from a water-in-oil emulsion to an aqueous phase through a membrane. Under conditions where the diffusion coefficient is expected to depend on concentration, the cumulative amount transported, Q, is predicted to follow the relationship... 

Table IV includes theoretical transition times (C2, R14, SI7c) in laminar flow between parallel plates, following a concentration step at the wall (Leveque mass transfer). Clearly, in laminar flow (Re 100 or lower), transition times are comparable to those in laminar free convection. Here, however, the dependence on concentration (through the diffusivity) is weak. The dimensionless time variable in unsteady-state mass transfer of the Leveque type is...

The second major difference found in vapor-liquid extraction of polymeric solutions is related to the low values of the diffusion coefficients and the strong dependence of these coefficients on the concentration of solvent or monomer in a polymeric solution or melt. Figure 2, which illustrates how the diffusion coefficient can vary with concentration for a polymeric solution, shows a variation of more than three orders of magnitude in the diffusion coefficient when the concentration varies from about 10% to less than 1%. From a mathematical viewpoint the dependence of the diffusion coefficient on concentration can introduce complications in solving the diffusion equations to obtain concentration profiles, particularly when this dependence is nonlinear. On a physiced basis, the low diffiisivities result in low mass-transfer rates, which means larger extraction equipment. 

The diffusion theory states that interpenetration and entanglement of polymer chains are additionally responsible for bioadhesion. The intimate contact of the two substrates is essential for diffusion to occur, that is, the driving force for the interdiffusion is the concentration gradient across the interface. The penetration of polymer chains into the mucus network, and vice versa, is dependent on concentration gradients and diffusion coefficients. It is believed that for an effective adhesion bond the interpenetration of the polymer chain should be in the range of 0.2-0.5 pm. It is possible to estimate the penetration depth (/) by Eq. (5),... 

When the diffusion coefficient D is dependent on concentration, the diffusion process is said to be Fickian. In such cases, D is inversely related to solubility, S, and to permeability, P, as follows ... 

We have thus far written unimolecular surface reaction rates as r" = kCAs assuming that rates are simply first order in the reactant concentration. This is the simplest form, and we used it to introduce the complexities of external mass transfer and pore diffusion on surface reactions. In fact there are many situations where surface reactions do not obey simple rate expressions, and they frequently give rate expressions that do not obey simple power-law dependences on concentrations or simple Arrhenius temperatures dependences. 

The above equation is known as the three-dimensional diffusion equation. One can also write the diffusion in terms of the first component. Hence, C2 is replaced by C (concentration of either component 1 or component 2) below. The above equation is general and accounts for the case when D depends on concentration (such as chemical diffusion to be discussed later). 

If D is constant, an experimental diffusion profile can be fit to the analytical solution (such as an error function) to obtain D. If it depends on concentration and the functional dependence is known. Equation 3-9 can be solved numerically, and the numerical solution may be fit to obtain D (e.g., Zhang et al., 1991a Zhang and Behrens, 2000). However, if D depends on concentration but the functional dependence is not known a priori, other methods do not work, and Boltzmann transformation provides a powerful way (and the only way) to obtain D at every concentration along the diffusion profile if the diffusion medium is infinite or semi-infinite. Starting from Equation 3-58a, integrate the above from Po to 00, leading to... 

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

Even in the absence of uphill diffusion, a trace element concentration profile often does not match that for a constant diffusivity by using the effective binary diffusion treatment. Hence, the effective binary diffusivity depends on the chemical composition, which is expected. 

If D depends on concentration, the first indication would come from the asymmetry of the diffusion-couple concentration profile. That is, there is no center symmetry with respect to x = 0, which means that one side approaches the end concentration more rapidly than the other side (Figure 3-28a). In such cases, D as a function of C can be obtained by Boltzmann analysis (Equation 3-58e) ... 

Diffusivity of a species in a phase is an intrinsic property and does not depend on the experimental method. If diffusivity does not depend on the concentration of the species, then diffusivity extracted from different methods has the same meaning and hence should all agree within experimental error. ITowever, if diffusivity depends on the concentration of the species or component, the meaning of diffusivity extracted using different techniques may differ, leading to difference in diffusivity values. 

If temperature or pressure varies during crystal dissolution, the problem becomes more complicated because both the diffusivity and the interface melt concentration vary, causing the dissolution rate to vary. Although the diffusivity dependence on time is not difficult to tackle anal3dically, the variation in the interface condition and the consequent change in dissolution rate cannot be treated simply. Hence, the treatment here is for constant temperature and pressure. Numerical method is necessary to handle crystal dissolution with variable temperature and pressure. 

The rate at which they diffuse depends on the concentration gradient, dN dx the larger the gradient, the faster the rate of diffusion. This is the basis of the well-known Fick s first law of diffusion ... 

The mobility of metals in soil solutions is controlled by several processes (1) desorption or dissolution (rate depends on the solubility of metal-mineral form) (2) diffusion (depends on speciation of metal, soil oxidation/reduction potential, and pH) (3) sorption or precipitation (depends on soil solution concentration and rhi-zosphere effects) and (4) translocation in the plants (depends on plant species, soil solution concentration, and competing ions) (McBride... 

The onset of thermal diffusion depends on the gas concentrations, the sample surface area, the rate at which the sample cools to bath temperature, and the packing efficiency of the powder. In many instances, using a conventional sample cell, surface areas less than 0.1 m can be accurately measured on well-packed samples that exhibit small interparticle void volume. The use of the micro cell (Fig. 15.10b) is predicated on the latter of these observations. Presumably, by decreasing the available volume into which the lighter gas can settle, the effects of thermal diffusion can be minimized. Although small sample quantities are used with a micro cell, thermal conductivity detectors are sufficiently sensitive to give ample signal. 

The Debye-Huckel-Onsager equation for the dependence of the ordinary diffusion conductance on concentration has been extended by Falkenhagen, Leist and Kelbg [3] to apply to more concentrated solutions and their equation for a 1 1-electrolyte may be written... 

Phosphoric acid Acidic Depends on concentration Diffusion... 

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