Apparent diffusion coefficient, equation

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

In the absence of field effects, equations (4.22), (4.23), (4.25), and (4.26) amount to Fick s second and first laws of diffusion with an apparent diffusion coefficient given by equation (4.24). 

It is not unreasonable to use the left-hand side of this equation as the definition of the effective diffusion constant K, the more so as it will be shown that any distribution tends to normality. With this definition K is the sum of the molecular diffusion coefficient, D, and the apparent diffusion coefficient k = oP-U2I 48D, which was discovered by Taylor in his first paper (Taylor 1953, equation (25)). Equation (26), however, is true without any restriction on the value of p, or on the distribution of solute. The constant 1/48 is a function of the profile of flow, and for so-called piston flow with x — 0 this constant is zero and K = D as it should. 

The recorded time is actually f, and the apparent diffusion coefficient U is calculated at t from Equation 3, while D in Equation 5 is the true diffusion coefficient. 

GPPS = general purpose polystyrene HIPS = high impact polystyrene a) 1 1 HIPS GPPS diffusion coefficient equation In D = 15.61 - 14 500- b) apparent diffusion coefficient calculated from experimental migration data. (1 /T(K))... 

Results are shown in Figs. 12 and 13. All blend specimens were set iso-thermally above LCST and kept there for a maximum of 5 min. As will be seen, this corresponds only in some cases to an early stage of spinodal decomposition depending on temperature. The diffusion coefficients governing the dynamics of phase dissolution below LCST are in the order of 10"14 cm2 s"1. Figure 12 reflects the influence of the mobility coefficient on the phase dissolution. As can be seen, the apparent diffusion coefficient increases with increasing temperature of phase dissolution which expresses primarily the temperature dependence of the mobility coefficient. Furthermore, it becomes evident that the mobility obeys an Arrhenius-type equation. Similar results have been reported for phase dis-... 

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. 

The effect of using the sodium adsorption isotherm (equation 6) to determine the apparent diffusion coefficient can be seen by... 

The fluctuations of a self-exdted osdllator have been studied via a model based on the Van der Pol equation. Such an equation has been used to account for the ampUtude and phase fluctuations. A growth of the time coherence of phase above threshold is attributed to a decrease in an apparent diffusion coefficient for phase fluctuations. 

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 ... 

The time, f,-, for the induction period (region I) to end is an important factor in determining the surface tension as a function of time, since only when that period ends does the surface tension start to fall rapidly. The value of f,- has been shown (Gao, 1995 Rosen, 1996) to be related to the surface coverage of the air-aqueous solution interface and to the apparent diffusion coefficient, Dap, of the surfactant, calculated by use of the short-time approximation of the Ward-Tordai equation (Ward, 1946) for diffusion-controlled adsorption (equation 5.6) ... 

As mentioned above, the value of t, has been shown to be related to the coverage of the air-aqueous solution interface by the surfactant and to its apparent diffusion coefficient, Dap (equation 5.7). To calculate the values of Dap at short times, equation 5.8 (Bendure, 1971), based upon the short-time approximation equation of Ward and Tordai (equation 5.6), and using dynamic short-time surface tension data, may be used ... 

Apparent diffusion coefficients may also be calculated from longer-time dynamic surface tension data by use of equation 5.9 (Joos, 1992) ... 

In this equation, A is the area of application, D is the apparent diffusion coefficient and h is the diffusional path length (often taken as the thickness of the membrane). The permeability coefficient (A ) is the steady-state flux per unit area divided by the concentration of drug applied in solution and may be calculated from ... 

The apparent diffusion coefficient, D, was determined for the particular leaching conditions of each of the thirteen experiments. This was accomplished using the measured chloride breakthrough (effluent concentration) curve and the analytical solution to Equation 7 with Kd==0. Examples of the observed and calculated chloride concentrations (determined by adjustment of D until a best fit was obtained) are presented for three different experiments (Experiments 7, 8, 11) in Figures 2-4. Values of D and the pore water velocity (v) determined for each experiment are presented in Table III. The value of D increased for cases with large v, and was different between soils for any particular v. This is consistent with the basic relationship be-... 

The rate of transport of water is thus expressed by equation 8 in terms of an apparent diffusion coefficient, D, and the gradient of the overall concentration of water in the rubber, dC /dx. The theory predicts that the diffusion of water in rubber containing hydrophilic Impurities can be expressed in terms of a concentration dependent diffusion coefficient (equation 9). 

Under these circumstances, the apparent rate at which Q appears to move through the film from electrode to the outer boundary of the film depends upon the rate of the electron-transfer reaction between P and Q. Considerations of analogous reactions in homogeneous solution showed that such a process is equivalent to diffusion (76, 77). The apparent diffusion coefficient observed for a species. Dp, is composed of contributions from the physical movement of the species (governed by its translational diffusion coefficient, D) and the electron-transfer process. When bimolecular kinetics apply and the species can be considered as points, then Dp can be estimated from the Dahms-Ruff equation. 

Figure 3. The ratio jx/Mo can be obtained from the ratio of the apparent diffusion coefficients measured on cylindrical and spherical gels. The theoretical curve is obtained from equation 12. The two experimental points correspond to a polyacrylamide gel in water (circle) and a polydimethylsiloxane gel in toluene (box). The temperature was 22 °C.

This is sometimes called the apparent diffusion coefficients app( ) since it depends on (. If the particle sizes are small compared to A., or if we observe at very small angles, then P(q) -> 1 and Equation 18.72 reduces to... 

In case a, future penetration of chlorides has to be evaluated in order to assess if the chloride threshold will be reached at the surface of the reinforcement before time tf. If this is expected, the concrete with a chloride content higher than the critical value has to be removed (and replaced with a chloride-free material that prevents further penetration of chloride). Equation (1), Chapter 6, may be used to evaluate the future penetration of chlorides. By fitting the present chloride profile, the surface content and the apparent diffusion coefficient Dj j, may be calculated at time Since these parameters are evaluated on the actual structure and usually after a long time of service, it is often reasonable to assume that they will not change significantly in the future (unless the conditions of exposure of the structure will change). Equation (1), Chapter 6, can then be used to plot the expected chloride profile at time tf. 

The apparent diffusion coefficient, Da in Eq. (11) is a mole fraction-weighted average of the probe diffusion coefficient in the continuous phase and the microemulsion (or micelle) diffusion coefficient. It replaces D in the current-concentration relationships where total probe concentration is used. Both the zero-kinetics and fast-kinetics expressions require knowledge of the partition coefficient and the continuous-phase diffusion coefficient for the probe. Texter et al. [57] showed that finite exchange kinetics for electroactive probes results in zero-kinetics estimates of partitioning equilibrium constants that are lower bounds to the actual equilibrium constants. The fast-kinetics limit and Eq. (11) have generally been considered as a consequence of a local equilibrium assumption. This use is more or less axiomatic, since existing analytical derivations of effective diffusion coefficients from reaction-diffusion equations are approximate. 

Equation (12) does not predict any dependence of the apparent diffusion coefficient Da on the probe concentration C.v. Assuming that the probe is almost entirely bound to the particle (droplet or micelle), the following approximate model for Da was obtained by... 

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