Apparent diffusion coefficient, equation describing

For a given sequence, Bloch equations give the relationship between the explanatory variables, x, and the true response, i]. The / -dimensional vector, 0, corresponds to the unknown parameters that have to be estimated x stands for the m-dimensional vector of experimental factors, i.e., the sequence parameters, that have an effect on the response. These factors may be scalar (m — 1), as previously described in the TVmapping protocol, or vector (m > 1) e.g., the direction of diffusion gradients in a diffusion tensor experiment.2 The model >](x 0) is generally non-linear and depends on the considered sequence. Non-linearity is due to the dependence of at least one first derivative 5 (x 0)/50, on the value of at least one parameter, 6t. The model integrates intrinsic parameters of the tissue (e.g., relaxation times, apparent diffusion coefficient), and also experimental nuclear magnetic resonance (NMR) factors which are not sufficiently controlled and so are unknown. 

Recently, kinetic models have been combined with the equilibrium data of the interfacial processes, taking into account that soils and rocks are heterogeneous and consequently have different sites. These models are called nonequilibrium models (Wu and Gschwend 1986 Miller and Pedit 1992 Pedit and Miller 1993 Fuller et al. 1993 Sparks 2003 Table 7.2). These models describe processes when a fast reaction (physical or chemical) is followed by one or more slower reactions. In these cases, Fick s second law is expressed—that the diffusion coefficient is corrected by an equilibrium thermodynamic parameter of the fast reaction (e.g., by a distribution coefficient), that is, the fast reaction is always assumed to be in equilibrium. In this way, the net processes are characterized by apparent diffusion coefficients. However, such reactions can be equally well described using Equation 1.126. 

Crank (16) showed that equation 5 can be solved analytically for the boundary condition (equation 6) where (1) P = 1 and the surface concentration is directly proportional to the aqueous concentration, and (2) R = 0 and the surface concentration is zero. The first solution results in diffusion which is dependent on the aqueous concentration, but produces mass transfer which is nonparabolic with time. The second solution results in diffusion which is independent of aqueous concentrations but is parabolic. This latter case has been used by Luce and others ( ), Busenberg and Clemency (26), and others to describe diffusion and to calculate apparent diffusion coefficients for silicate minerals. 

Dahms (1968) and Botar and Ruff (1985) studied exchange reactions such as those represented by Equation (2.6), stating that such processes can be described in terms of a second-order reaction kinetics, so that the apparent diffusion coefficient, D pp, measured in electrochemical experiments (e.g., CA) under diffusion-controlled conditions, can be expressed by ... 

This has some characteristics of diffusion, but is not well described by a simple diffusion equation. Both the apparent diffusion coefficient and the apparent recombination time are electron density dependent. 

When small electroactive ions or molecules are bound to larger aggregates like micelles or microemulsion droplets, the reactant (probe) is transported to the electrode along with the larger, slower diffusing aggregate. Equation (12) describes the influence of concentration of surfactant or reactant on electrochemically measured diffusion coefficients. At [X] >2 mM, the measured apparent diffusion coefficient D approaches the diffusion coefficient of the micelle D. This implies electrolysis of one reactant X per micelle. This electrolyzed X could reside within MX , or be released by dissociation, as illustrated in Eqs. (17 and 18) for an oxidation ... 

Chronocoulometry (CC) is much less frequently used in CP work. It can also be used to calculate, among other parameters, apparent diffusion coefficients. The relevant equation, again under the "Cottrell conditions" described above, is... 

In Equations 6.4-3 to 6.4-5, the quantity in square brackets is an apparent diffusion coefficient. However, in Equation 6.4-3, it is actually due to convective flow, and in Equation 6.4-5, it represents diffusion to the pore, not diffusion in the pore. Only Equation 6.4-4 actually describes diffusion in the pore. These differences seem obvious in this theoretically based discussion. However, when we have experimental data. 

Inasmuch as we have seen that the basic diffusion equations lead to a Gaussian profile, we may in most cases describe Gaussian spreading as an apparent diffusion process, with an effective diffusion coefficient related to [Pg.93]

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

Caroline and co-workers have recently reported measurements of translational diffusion coefficients in solutions of PS in two mixed-solvent systems at or near theta conditions. In the solvent CCb-methanol (85), they observed the diffusion theta state, defined when the coefficient y of Equation 41 equals 0.5, to occur at 25°C and a volume fraction of CCI4, (fyCCU = 0.8025. In this system there is strong preferential adsorption of the polymer for CCI4, and it is not possible to define a true theta state such that y = a = V2 and A2 = 0 simultaneously. Under diffusion theta conditions, the concentration dependence of Dt apparently is closely described by the Pyun-Fixman hard-sphere model. In the mixed solvent benzene—2 propanol, polystyrene exhibits a true theta condition at T = 25.5°C and (benzene) = 0.04. Frost and Caroline confirmed that y = 0.5 within experimental error in this system (86) and report that values of the parameter fcf are scattered between the extreme values corresponding to the predictions of Yamakawa (and Imai) and the soft-sphere model of Pyun-Fixman (or the Freed theory). 

The Nernst-Einstein equation describes the link between the molar conductivity and the diffusion coefficient in an ideal case. However, the equation can also be stretched to apply to other cases, provided that one remembers that D, is the true diffusion coefficient and not the apparent one (first Pick s law, see section 4.2.1.4) ... 

Diffusion in ZSM-5 supported membranes was studied from ambient to 723K. The Maxwell-Stefan equations were used for the interpretation of the results. From these, micropore diffusion coefficients and apparent diffusion activation energies were obtained. The results were described by ... 

---