Diffusion coefficient, calculation

Sorption curves obtained at activity and temperature conditions which have been experienced to be not able to alter the polymer morphology during the test, i.e. a = 0.60 and T = 75 °C, for as cast (A) and for samples previously equilibrated in more severe conditions, a = 0.99 and T = 75 °C (B), are shown in Fig. 13. According to the previous discussion, the diffusion coefficient, calculated by using the time at the intersection points between the initial linear behaviour and the equilibrium asymptote (a and b), for the damaged sample is lower than that of the undamaged one, since b > a. The morphological modification which increases the apparent solubility lowers, in fact, the effective diffusion coefficient. 

Table II gives the diffusion coefficients calculated by means of the model.

Table II. Diffusion coefficients calculated from the curves in Figure 7...

The self-diffusion coefficient calculated for the three body potential is Z) = 1.3 X lO cm /sec. This is to be compared with the experimental value of 2.3 x 10" cmVsec and to the value of 2.25 x 10 cm /sec for the two-body liquid. It could be said that the three-body liquid shows more rigidity in some sense than the two-body liquid. 

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

Aqueous diffusion coefficient calculated using Eqs. (41) and 42). Note PC = partition coefficient. 

The previous paper (63) also studied the disintegration of solid solutions and for that purpose samples were heated for 300 hours at 250°C, but no signs of disintegration were detected in an X-ray diffractogram. This might be due to the fact that solid state diffusion is still too slow at that temperature. This is supported by the low diffusion coefficient calculated if one extrapolates from the experimental values determined at high temperature (60). 

We have also reported on the ordinate axis of Fig.3 the values of the translational diffusion coefficient calculated from radius values measured by transient electric birefringence, using ... 

To date, D coefficients of carbohydrates established with the PFGSE approactf - " have been undertaken to (1) validate the theoretical self-diffusion coefficients calculated from MD trajectories, (2) demonstrate the complexation of lanthanide cations by sugars,(3) probe the geometry of a molecular capsule formed by electrostatic interactions between oppositely charged P-cyclodextrins, (4) study the influence of concentration and temperature dependence on the hydrodynamic properties of disaccharides, and (5) discriminate between extended and folded conformations of nucleotide-sugars. ... 

Murillo et al. (2004) studied the adsorption of phenanthrene (polycyclic aromatic hydrocarbon -PAH) from helium as carrier gas on a coke fixed-bed adsorber, at 150 °C. The isotherm of the phenanthrene-coke system at 150 °C was found to be of Freundlich type with Fr = 0.161 and KF = 1.9 (mol/kg)(m3/mol)0161. The isotherm has been derived for phenanthrene concentrations between 1.71 X 10 4 and 1.35 X 10-2 mol/m3. Finally, the average solid-phase diffusion coefficient, calculated from several experimental runs, was found to be 6.77 X 10-8 cm2/s. 

The diffusion coefficients calculated from a simulation employing a flexible framework were all between 5 and 10 times larger than those calculated from fixed lattice simulations. A comparison between flexible framework results and NMR measurements (57) illustrated the influence of the cations in the experimental sample calculated diffusion coefficients from the cation-free (flexible) framework were approximately 5 times higher than the experimental results. The increase in diffusion coefficient as a function of loading found in experimental studies was reproduced by the simulations. 

Fixed framework-flexible adsorbate calculations were also reported by Dumont and Bougeard (68, 69). The diffusion coefficient calculated from a 42-ps calculation of 4 molecules per unit cell at 300 K was 1.60 x 10 8 m2/s. Once again the anisotropy of the diffusive process was calculated to be strong transport through the straight channel was found to be twice as fast as through the sinusoidal channel and an order of magnitude faster than motion parallel to the z-axis. 

The slope of the MSD curve as a function of temperature was not constant, indicating that the methane molecules were trapped in a region of the lattice (for a time estimated by Catlow et al. as 5 ps). The diffusion coefficient, calculated as the slope of the MSD graph, is in satisfactory... 

Manson and Chin 151) reported that the addition of filler to an epoxy binder reduces the epoxy s permeability coefficient (P), as well as the solubility of water in the resin (S) and that the reduction is stronger than expected from theory 1 2). Diffusion coefficients calculated from P and S for the unfilled resin were found to be somewhat higher than those for filled resin. The difference seems to be due to the formation of ordered layers, up to 4 pm thick, around every filler particle. The layers form because of residual stresses caused by the difference between the binder and filler coefficients of thermal expansion. The effective activation energy for water to penetrate into these materials, calculated in the 0-100 °C temperature range, is 54.3 kJ/mol151). 

The seriousness of this oversight is apparent in Sefcik and Schaefer s analysis of Toi s transport data (24) in terms of their NMR results (28) The value of the so-called "apparent" diffusion coefficient calculated from Toi s time lag data increases by 25% for an upstream pressure range between 100 mm Hg and 500 mm Hg On the other hand, the value of Deff(c) calculated from Toi s data changes by 86% over the concentration range from 100 to 500 mm Hg The difference in the two above coefficients arises from the fact that Da is an average of values corresponding to a range of concentrations from the upstream value to the essentially zero concentration downstream value in a time lag measurement Deff > on t le other hand, has a well-defined point value at each specified concentration and is typically evaluated (independent of any specific model other than Fick s law) by differentation of solubility and permeability data (22) ... 

TABLE 7.2 Comparison of the Thicknesses of the Leached Layer and the Diffusion Coefficients Calculated from the Literature at Room Temperature, Based on Diffusion Models"... 

Table 5-1 Diffusion coefficients calculated by molecular dynamic simulation and from experiment.

Table 5-2 Diffusion coefficients calculated with the Iransition-state approach and from experiment.

Examples of diffusion coefficients calculated with Eq. (6-18) are summarized in Table 6-3 where they are compared with corresponding experimental values (Reid et al., 1987). Diffusion coefficients of water and hydrocarbons in air are especially important parameters in mass transfer processes between plastic materials and the atmospheric environment. 

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))... 

One gets a mild shock therefore when one looks into the experimental tests of these equations, for only under extremely simple conditions (in fact, very dilute solutions) do they work out to be correct. Although the results never hint that the equations are wrong, there is sufficient discrepancy (e.g., between tire diffusion coefficient calculated by using an experimental mobility substituted in the Einstein equation and that determined by direct experiment) for one to take notice and form some idea of a puzzle. How can simple mathematical reasoning based on the existence of movements undeniably present give rise to error ... 

When the diffusion process is not ideal, the diffusion coefficient calculated from Equation 6 for a stationary state process is the mean diffusion coefficient corresponding to a particular concentration difference. Methods of determining the true diffusion coefficient pertaining to a particular concentration were reported by Crank (9). The permeability coefficient calculated from Equation 4 is also a mean value corresponding to a particular vapor pressure difference if the diffusion process is not ideal. 

The concentration dependent diffusion coefficient defined by Eq. (9) can be evaluated by differentiation of steady state permeation data without reference to tile partial immobilization model The concentration dependent diffusion coefficient calculated from the partial immobilization model agrees very well with values calculated in this way, and one can consider them to be essentially identical mathematically The partial inunobilization theory, therefore, serves to explain the source of the concentration dependency of Dgfr in Eq. (9). 

Values too obtained by plotting Equation 19 and the internal diffusion coefficients calculated by Equation 18 are represented in Table II. Column 7 of the table displays diffusion coefficients of substances in nitrogen (6, 7, 10). 

Theophylline Diffusion Coefficient Calculated According to the Linear (Equation 15.80) (Dlin) and Numerical (Dnumeric) Models Data Eitting... 

ZSM-5 zeolite. Transport D values were Consequently, the diffusion coefficient calculated from FR results assuming that particles are spheres of lO-pm diameter (0) Data determined by the MEMBRANE (A), TAP( ), and QENS (H) methods were taken from refs. 13 and 14 and are given for comparison. 

Figure 17. The group velocity (dashed curve) and frequency-dependent diffusion coefficient, calculated with Eqs. (40)-(42) using parameters obtained for myoglobin, are plotted for frequencies up to 150 cm-1.

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