Diffusional time constant determination

In the chromatographic method a pulse or step change in sor-bate concentration is introduced into the carrier stream at the inlet of a packed adsorption column and the diffusional time constant is determined from the dispersion of the response signal at the column outlet. Since heat transfer in a packed bed is much faster than in a closed system the chromatographic method may, in principle, be used to follow somewhat faster sorption processes. 

As an alternative to conventional sorption rate measurements it is also possible to derive diffusional time constants from the dynamic response of a packed column to a change in sorbate concentration. In a chromatographic system the broadening of the response peak results from the combined effects of axial dispersion and mass transfer resistance. By making measurements over a range of gas velocities it is possible to separate the dispersion and mass transfer effects and so to determine the effective overall mass transfer coefficient or the diffusional time constant. Further details are given in Section 8.5. 

In the long time region a plot of ln(l - mjm versus t should be linear with slope -v D fr] and intercept ln(6/ir ), as illustrated in Figure 6.2. Such a plot provides, in principle, a simple meUiod of both checking the conformity of an experimental uptake curve with the diffusion equation and determining the diffusional time constant. 

The ease with which the individual mass transfer parameters and the axial dispersion coefficient can be determined depends on the relative magnitude of the various resistances. If mass transfer is rapid the dispersion of the chro matogram will be caused mainly by axial dispersion, jand under these conditions it is not possible to derive any information coiicerning the diffusional time constants. In the low Reynolds number regime Sh = D , w2.0 so... 

This method of estimating Rc is useful when it can be applied, since the determination is not based on any presumed model of the corrosion damage process or any of the assumptions that come with assignment of an equivalent circuit model. This method is particularly helpful when there is more than one time constant in the spectrum, or the impedance spectrum is particularly complicated. Caution is warranted however. This method of estimation can be in serious error for samples with large capacitance-dominated low-frequency impedances. As a general rule, for this estimation method to be reasonably accurate, the impedance function must exhibit a clear DC limit, or a diffusional response that can be modeled by a constant phase element in equivalent circuit analysis (75). 

Despite fission being orders of magnitude slower than the solid state in this diffusion-limited system, fission was equally efficient. The triplet exciton yield determined from the relative extinction coefficients of the triplet and singlet excitons was 197%. Such a high yield indicates that once the molecules collide in solution, fission proceeds rapidly in this exothermic system. Indeed the observed rate of triplet formation in the concentrated solution suggests that fission occurs as fast as the diffusional timescales allow the rise of the triplet exciton absorption had a 500 ps time constant, reaching 30% of the expected diffusion limit for the system. 

For most zeoUtes, when NH3 is used as the probe molecule at a given temperature, the time needed to estabhsh thermal equihbrium after each dose at first increases with increasing adsorbed amount, passes through a maximum, then decreases rapidly and finally reaches a value close to the time constant of the calorimeter. For example. Fig. 3 shows the time constant (in seconds) versus the amount of NH3 adsorbed for samples of H-ZSM-5 (Si/Al = 10.3) pretreated at 673 K or 1073 K and possessing a very small particle size (0.02 to 0.05 xm). The maximum time constant is higher for the sample pretreated at 1073 K than for the sample pretreated at 673 K, because increasing the pretreatment temperature causes dealumination, i.e. creates extra-framework aluminum species which restrict the access to the channels and creates diffusional limitations. The time constant of the calorimeter was close to 300 s. The heat transfer is determined by the mass transfer which becomes slower. 

It is obvious that from any experimental dependence of the total layer thickness upon time it is only possible to determine the sum of the chemical constants as well as the sum of physical (diffusional) constants. The former sum is to be found from an initial portion of this dependence plotted in the coordinates x - t, while the latter from its long-time portion plotted in the coordinates x2- t or x - /l/2. For their separate determination, it is necessary to measure the increases in thickness of the ApBq layer at its both interfaces with initial substances A and B. 

To determine the physical (diffusional) constants k]m and k]A2, it is necessary to establish the conditions under which the total thickness of the ApBq layer increases with time parabolically. Then, like the previous case, the increases, Axm and AxA2, in thickness of the layer at its interfaces 1 and 2 are to be measured. The values of the physical (diffusional) constants k m and k A2 are computed from the equations... 

The ApBq and A Bn layers are seen to grow parabolically, whereas the thickness of the ArBs layer will gradually decrease with passing time. Eventually, this layer will disappear. It is easy to notice that in this case the values of the diffusional constants k[A2 and kim can readily be determined from the experimental dependences x2- t and z2- t, respectively, using an artificially prepared specimen A-ApBq-ArBs-A iB,-B or A-A,B-B. It is essential to mention that both the ApBq and AtBn layers must be the first to occur at the A-B interface. The diffusional constant k[A2 thus obtained is the reaction-diffusion coefficient of the A atoms in the ApBq lattice, while the diffusional constant klB3 is the reaction-diffusion coefficient of the B atoms in the Afin lattice, to be compared with respective self-diffusion coefficients determined using radioactive tracers. 

Measurements of Tj have been made (269,270) to probe the structure of the second coordination sphere around Cr(acac)3. Acetone, chloroform, and methylene chloride were chosen as second sphere ligands. The shortening of their Tj values is found to be independent of solute concentration, and the value measured is determined by the diffusional correlation time of the solute molecule. No detectable second coordination sphere therefore exists in these solvents. However, methanol forms a discrete second sphere, and above a certain concentration the measured Ty values vary linearly with solute concentration. The observed Tj values are compatible with a model having a coordination number of 8, a Cr-CH3 separation of 700 pm, and an equilibrium constant for displacing solvent (CHCI3) of ca. 10. This equilibrium constant value is consistent with values previously... 

If the polymer network formation is performed at a constant temperature (Tc ), vitrification takes place at a time (conversion) when the glass transition temperature (Tg) of the partially reacted system becomes equal to Teu - This requires that Tcu < Tgoj (ultimate glass transition temperature of the polymer network). When the system vitrifies there is a drastic drop in both the polymerization rate and the eventual diffusional processes associated with phase separation. Therefore it is important to determine the conditions where vitrification may take place. 

An example of determining the corrosion rate by this method is shown in Figure 20.12 (Behrens 1987). However, for very small corrosion currents that are independent of potential in some ranges (e.g., in passive metals), this method cannot be used to determine the corrosion rate. Nor is the method applicable when nontransfer-related part reactions occur, such as when the corrosion current is dependent on diffusional processes at the specimen surface. Also, the constant B can be time-dependent when dense protective films form, which further restricts the licability of the metal (Evans 1965 Hertz 1968). 

Determination of Diffusion Coefficients. Even though diffusional characteristics of polymers can be complex, there are many situations where simple concentration-independent Flckian diffusion occurs for small molecular species (18.19). For a polymer film of constant thickness L, the concentration C(x,t) of species at position x (measured perpendicular to the surface) and time t is governed by the one-dimensional Pick s law (20) ... 

---