Diffusion coefficient types

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

Micellization is a second-order or continuous type phase transition. Therefore, one observes continuous changes over the course of micelle fonnation. Many experimental teclmiques are particularly well suited for examining properties of micelles and micellar solutions. Important micellar properties include micelle size and aggregation number, self-diffusion coefficient, molecular packing of surfactant in the micelle, extent of surfactant ionization and counterion binding affinity, micelle collision rates, and many others. 

If tlie diffusion coefficients of tlie chemical species are sufficiently different, new types of chemical instability arise which can lead to tlie fonnation of chemical patterns and ultimately to spatio-temporal chaotic behaviour. 

These must supplement the minimal set of experiments needed to determine the available parameters in the model-It should be emphasized here, and will be re-emphasized later, Chat it is important Co direct experiments of type (i) to determining Che available parameters of some specific model of Che porous medium. Much confusion has arisen in the past frcjci results reported simply as "effective diffusion coefficients", which cannot be extrapolated with any certainty to predict... 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods ate used however, the simplest method uses the Permatran-C instmment (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is appHed to the other side of the film. AH the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeabiUty. Figure 18 shows how the diffusion coefficient can be deterrnined in this type of experiment. The time lag is substituted into equation 21. The solubiUty coefficient can be calculated with equation 2. 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

The process requires the interchange of atoms on the atomic lattice from a state where all sites of one type, e.g. the face centres, are occupied by one species, and the cube corner sites by the other species in a face-centred lattice. Since atomic re-aiTangement cannot occur by dhect place-exchange, vacant sites must play a role in the re-distribution, and die rate of the process is controlled by the self-diffusion coefficients. Experimental measurements of the... 

The diffusion coefficients of the constituent ions in ionic liquids have most commonly been measured either by electrochemical or by NMR methods. These two methods in fact measure slightly different diffusional properties. The electrochemical methods measure the diffusion coefficient of an ion in the presence of a concentration gradient (Pick diffusion) [59], while the NMR methods measure the diffusion coefficient of an ion in the absence of any concentration gradients (self-diffusion) [60]. Fortunately, under most circumstances these two types of diffusion coefficients are roughly equivalent. 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

The diffusion coefficient varies with temperature according to the following Arrhenius-type equation ... 

Strictly the diffusion coefficient D measured for any type of binary system A/B is in fact the resultant effect of two partial diffusivities and D, representing respectively the diffusivity of A into B and of B intO/4. For most practical purposes, however, a single diffusion coefficient is sufficient to define a given diffusion system. 

There is experimental evidence to suggest that anion and cation diffusion can have different mechanisms [70]. The temperature variation of the diffusion coefficients of 1,1 P and 7 Li in aPEO-LiPF6 shows quite different trends. The 31P diffusion coefficients follow a VTF-type de-... 

Increasing temperature permits greater thermal motion of diffusant and elastomer chains, thereby easing the passage of diffusant, and increasing rates Arrhenius-type expressions apply to the diffusion coefficient applying at each temperature," so that plots of the logarithm of D versus reciprocal temperature (K) are linear. A similar linear relationship also exists for solubUity coefficient s at different temperatures because Q = Ds, the same approach applies to permeation coefficient Q as well. 

For gases, both permeation and diffusion data are best measured by permeation tests, many different types been described elsewhere. The same sheet membrane permeation test can quantify permeation coefficient Q, diffusion coefficient D, solubility coefficient s, and concentration c. The membrane, of known area and thickness, must be completely sealed to separate the high-pressure (initial) region from that containing the permeated gas it may need an open-grid support to withstand the pressure. The permeant must be suitably detected and quantified (e.g., by pressure or volume buildup, infrared (IR) spectroscopy, ultraviolet (UV), gas chromatography, etc.). 

The values of hj for different ions are between 0 and 15 (see Table 7.2). As a rule it is found that the solvation number will be larger the smaller the true (crystal) radius of the ion. Hence, the overall (effective) sizes of different hydrated ions tend to become similar. This is why different ions in solution have similar values of mobilities or diffusion coefficients. The solvation numbers of cations (which are relatively small) are usually higher than those of anions. Yet for large cations, of the type of N(C4H9)4, the hydration number is zero. 

To summarize, there is a sizable and self-consistent body of data indicating that rotational and translational mobility of molecules inside swollen gel-type CFPs are interrelated and controlled mainly by viscosity. Accordingly, T, self-diffusion and diffusion coefficients bear the same information (at least for comparative purposes) concerning diffusion rates within swollen gel phases. However, the measurement of r is by far the most simple (it requires only the collection of a single spectrum). For this reason, only r values have been used so far in the interpretation of diffusion phenomena in swollen heterogeneous metal catalysts supported on CFPs [81,82]. 

OPTIMUM OPERATING CONDITIONS FOR DIFFERENT COLUMN TYPES coluan pressure drop 200 bar, viscosity lx 10 V.b/w and diffusion coefficient - 1.0 x 10 w /s... 

Detailed quantitative analyses of the data allowed the production of a mathematical model, which was able to reproduce all of the characteristics seen in the experiments carried out. Comparing model profiles with the data enabled the diffusion coefficients of the various components and reaction rates to be estimated. It was concluded that oxygen inhibition and latex turbidity present real obstacles to the formation of uniformly cross-linked waterborne coatings in this type of system. This study showed that GARField profiles are sufficiently quantitative to allow comparison with simple models of physical processes. This type of comparison between model and experiment occurs frequently in the analysis of GARField data. 

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