Diffusion, concentration dependent constant

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

However, there is another operative timescale in solution. This is that timescale for reaction with other photolytically generated species or with added reactants. This reaction cannot take place faster than the diffusion-limited reaction rate which is concentration dependent (59). Typical diffusion-controlled reaction rate constants are 109-1010 dm3 mol"1 second-1. By comparison, a typical gas-kinetic rate con-... 

It can be seen from the table that, in dilute solutions, the diffuse layer may extend some hundreds of angstroms out from the electrode. In contrast, in more concentrated solutions, i.e. 0.1 M, the diffuse layer thickness decreases to < 10 A not much more than the thickness of the Helmholtz layer. As CH has no concentration dependence it remains constant on changing the concentration however, from equations (2.22) and (2,23), CGC decreases as the concentration of the electrolyte increases. Thus, at low concentration ... 

Pressure effects The diffusion through liquids is governed by the number of defects or atomic-sized holes in the liquid. A high external pressme can reduce the concentration of holes and slow diffusion. Therefore, in a liquid, a diffusion-controlled rate constant also depends on the pressure. 

As for the rate of diffusion, the equilibrium constant for a reaction in a biphasic system is not determined by the overall concentration of each reagent, but by their concentrations in the reaction phase. In some cases this can drive the forward reaction to completion, and in other cases it can be inhibitory, depending on the relative concentrations of the reactants and products. In model 1, where the reaction takes place at the phase boundary, the effective concentration of the reactants and products will be that in phase 1, and assuming each has an equivalent solubility, the equilibrium position will approach that of a homogeneous system. Where the reaction takes place in the bulk solvent, as in model 2, the equilibrium position is very much dependent on the solubility of the reagents in phase 2. For example, if the product is less soluble in phase 2 than the reactant, as the product is formed it will diffuse back into phase 1, reducing its concentration in phase 2 where the reaction is occurring and therefore the reaction will... 

The ionic conductivity of a solution depends on the viscosity, diffusivity, and dielectric constant of the solvent, and the dissociation constant of the molecule. EFL mixtures can carry charge. The conductivity of perfluoroacetate salts in EFL mixtures of carbon dioxide and methanol is large (10 to 10 " S/cm for salt concentrations of 0.05-5 mM) and increases with salt concentration. The ionic conductivity of tetra-methylammonium bicarbonate (TMAHCO3) in methanol/C02 mixtures has specific conductivities in the range of 9-14 mS/cm for pure methanol at pressures varying from 5.8 to 14.1 MPa, which decreases with added CO2 to a value of 1-2 mS/cm for 0.50 mole fraction CO2 for all pressures studied. When as much as 0.70 mole fraction... 

As the concentration of MeOH increases, the divergent diffusion behavior between the two membrane types is a reflection of fhe difference in MeOH solubility and its concentration dependence within each membrane. This was verified by solvenf upfake measurements. Upon increasing MeOH concentration, Nafion 117 showed a steady increase in mass, while a sharp drop in total solution uptake was observed for BPSH 40. The lower viscosity of MeOH also affecfs fhe fluidity of the solution within the pores. The constant solvent uptake and the increased fluidity of the more concentrated MeOH solutions accounted for fhe slight increase in diffusion coefficienf of Nafion 117. For BPSH 40, increasing the MeOH concentration resulted in a decrease in MeOH diffusion. The solvent uptake measurements showed very similar behavior, indicating that the membrane excludes the solvent upon exposure to higher MeOH concentrations. 

In an isotropic medium, D is a scalar, which may be constant or dependent on time, space coordinates, and/or concentration. In anisotropic media (such as crystals other than cubic symmetry, i.e., most minerals), however, diffusivity also depends on the diffusion direction. The diffusivity in an anisotropic medium is a second-rank symmetric tensor D that can be represented by a 3 x 3 matrix (Equation 3-25a). The tensor is called the diffusivity tensor. Diffusivity along any given direction can be calculated from the diffusivity tensor (Equation 3-25b). Each element in the tensor may be constant, or dependent on time, space coordinates and/or concentration. 

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. 

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. 

Microresearch Diffusion control in constant potential mode. The ion concentration gradient in proximity to an electrode surface depends on how the electrode potential state is manipulated by an external electronic device, either... 

Preliminaries. In the previous chapter we dealt with locally electro-neutral time-dependent electro-diffusion under the condition of no electric current in a medium with a spatially constant fixed charge density (ion-exchangers). It was observed that under these circumstances electrodiffusion is equivalent to nonlinear diffusion with concentration-dependent diffusivities. 

Eq. (18). Their experiment provided an opportunity only to fit three of the four unknown constants, namely Dg, c2, and Tg. Tg in the solution was found to be depressed below that known for the rubber. An examination of the concentration dependence of oil diffusion in the same rubber host confirmed the applicability ofEq. (17) below vdil < 0.9 and permitted a measurement ofthe fractional free volume of one or both components of the oil-rubber solution. 

The theoretical lines in Figure 2 are calculated assuming constant values of D0 with the derivative d In p/d In c calculated from the best fitting theoretical equilibrium isotherm (Equation 8). The theoretical lines give an adequate representation of the experimental data suggesting that the concentration dependence of the diffusivity is caused by the nonlinearity of the relationship between sorbate activity and concentration as defined by the equilibrium isotherm. The diffusivity data for other hydrocarbons showed similar trends, and in no case was there evidence of a concentration-dependent mobility. Similar observations have been reported by Barrer and Davies for diffusion in H-chabazite (7). 

The sorption of a weak electrolyte by a charged polymer membrane is another case where Nernst + Langmuir-like dual mode sorption, involving the undissociated and dissociated species respectively, may be expected. The concentration of each species in solution follows, of course, from the dissociation constant of the electrolyte. The sorption isotherms of acetic acid and its fluoroderivatives have been analysed in this manner, and the concentration dependence of the diffusion coefficient of acetic acid interpreted resonably successfully, using Nylon 6 as the polymer substrate 87). In this case the major contribution to the overall diffusion coefficient is that of the Nernst species consequently DT2 could not be determined with any precision. By contrast, in the case of HC1, which was also investigated 87 no Nernst sorption or diffusion component could be discerned down to pH = 2 and the overall diffusion coefficient obeyed the relation D = DT2/( 1 — >1D), which is the limiting form of Eq. (25) when p — 00. 

When Fick s law applies, the concentration profile generally contains information about the concentration dependence of the diffusivity. For constant D, step-function initial conditions have the error function (Eq. 4.31) as a solution to dc/dt = Dd2c/dx2. When the diffusivity is a function of concentration,... 

Matano developed a graphical method which, for certain classes of boundary value problems, relates the form of the diffusion profile with the concentration dependence of the interdiffusivity, D(c), introduced in Section 3.1.3 [5]. This method can determine D(c) from the diffusion profile in chemical concentration-gradient diffusion experiments where atomic volumes are sufficiently constant so that changes in overall specimen volume are insignificant and diffusion can be formulated in a F-frame. The method uses scaling, as discussed in Section 4.2.2. 

The results obtained by the present mechanical measurements are also consistent with the previous experimental results of the dynamic light scattering studies of the collective diffusion coefficient of gels and the rheological studies of the shear modulus of gels. The studies published by different researchers indicate that the concentration dependence of the collective diffusion constant of the polymer networks of gel and that of the elastic modulus are well represented by the following power law relationships [2, 3, 5]... 

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