Effective thickness

The quantity 1 /k is thus the distance at which the potential has reached the 1 je fraction of its value at the surface and coincides with the center of action of the space charge. The plane at a = l//c is therefore taken as the effective thickness of the diffuse double layer. As an example, 1/x = 30 A in the case of 0.01 M uni-univalent electrolyte at 25°C. 

The rate of dissolving of a solid is determined by the rate of diffusion through a boundary layer of solution. Derive the equation for the net rate of dissolving. Take Co to be the saturation concentration and rf to be the effective thickness of the diffusion layer denote diffusion coefficient by . 

A typical semiconductor laser, shown in Figure 9.11, is small, only a few millimetres long and with an effective thickness of about 2 pm. 

The extent of the interaction between the evanescent field and the absorbing medium is formally described by the effective thickness, of this... 

Cg = the concentration of the saturated solution in contact with the particles, D = a diffusion coefficient (approximated by the Hquid-phase diffusivity), M = the mass of solute transferred in time t, and S = the effective thickness of the liquid film surrounding the particles. For a batch process where the total volume H of solution is assumed to remain constant, dM = V dc and... 

A method of estimating original gas in place using the results of drilling (structural assessment, effective thickness, porosity, gas saturation, pressure, temperature, gas characteristics, and the boundaries of the accumulation). These data may be supplemented by geological or geophysical data on the shape of the reservoir. 

The protein recovery was found to be 95% of the amount injected, whereas, on the untreated carrier they were almost totally irreversibly adsorbed. Meanwhile, some reduction in the pore volume of the carrier could be deduced from the results of the chromatographic test. The calculated pore volume available for phtalic acid was 0.67 cm2/g (V) whereas for cytochrome C — 0.5 cm2/g. A detailed description of the experiment allows the evaluation of the effective thickness (teff) of the polymeric stationary phase. The tcff calculated as V/Ssp is 2.3 nm. The value... 

The effective thickness x is not generally known and therefore the equation is usually rewritten in the form ... 

The double layer is described by its effective thickness, d, and by its field strength E (Fig. 6.15). The adsorbed moleculeJias a dipole moment P. It is well documented100 that the local field strength E can affect strongly not only the chemisorptive bond strength but also the preferred orientation of the adsorbate (Fig. 6.16). 

The streamlines of this flow are shown by Peters and Smith (12). In this case, the effective thickness of this layer appears to be about equal to the gap with the wall, indicating a pressure flow about equal to the drag flow. It can be calculated that this would increase the maximum shear rate on the fluid passing under the agitator blade by a factor of seven. 

Measurement of the differential capacitance C = d /dE of the electrode/solution interface as a function of the electrode potential E results in a curve representing the influence of E on the value of C. The curves show an absolute minimum at E indicating a maximum in the effective thickness of the double layer as assumed in the simple model of a condenser [39Fru]. C is related to the electrocapillary curve and the surface tension according to C = d y/dE. Certain conditions have to be met in order to allow the measured capacity of the electrochemical double to be identified with the differential capacity (see [69Per]). In dilute electrolyte solutions this is generally the case. 

We may note in passing that the intrinsic viscosity of a fully extended rod molecule, for which is proportional to the square of the length, should depend on the square of the molecular weight, in the free-draining approximation. In a more accurate treatment which avoids this approximation, the simple dependence on is moderated by a factor which depends on the effective thickness of the chain (or bead density along the chain) compared with the chain length. 

Most Mossbauer spectra are split because of the hyperfine interaction of the absorber (or source) nuclei with their electron shell and chemical environment which lifts the degeneracy of the nuclear states. If the hyperfine interaction is static with respect to the nuclear lifetime, the Mossbauer spectrum is a superposition of separate lines (i), according to the number of possible transitions. Each line has its own effective thickness t i), which is a fraction of the total thickness, determined by the relative intensity W of the lines, such that t i) = Wit. 

Although Lorentzian line shapes should be strictly expected only for Mossbauer spectra of thin absorbers with effective thickness t small compared to unity, Margulies and Ehrman have shown [9] that the approximation holds reasonably well for moderately thick absorbers also, albeit the line widths are increased, depending on the value of t (Fig. 2.7). The line broadening is approximately... 

The strength of a Mossbauer signal is determined by the effective thickness of the absorber, t = (2.27). This dimensionless factor includes the number Nyi of... 

Using the value t = 0.2 for the effective thickness, the amount of resonance nuclei ( Fe) for a good thin absorber can be easily estimated according to the relation = tl(fA-(to)- For a quadrupole doublet with two equal absorption peaks of natural width and a recoil-free fraction of the sample/a = 0.7 one obtains... 

It is difficult to give an exact limit because the impact of thickness broadening depends on the intrinsic width of experimental lines [31], which often exceeds the natural width 2r at by 0.05—0.1 mm s for Fe as studied in inorganic chemistry. This inhomogeneous broadening, which is due to heterogeneity and strain in the sample, causes a reduction of the effective thickness. Rancourt et ai. have treated this feature in detail for iron minerals [32]. 

Conclusions. A Mossbauer sample with a low content of the resonance nuclide has ideal thickness when it attenuates the incident radiation by ca. 63-85% (/ie t = 1-2, C /Co e - e ). However, the optimization should be subordinated to the requirement of a thin absorber having an effective thickness / < 1 to avoid excessive line broadening. 

Fig. 9.3 Mossbauer spectra (A q = 2 mm s ) in the energy domain and in the time domain. High effective thickness t ff appears in the energy domain as line broadening and in the time domain as dynamical beats which are superimposed over the quantum beats. (Taken from [7])...

In (9.2), AEy is the bandwidth of the incoming radiation and Cei is the electronic absorption cross section. The exponential decay is modulated by the square of a Bessel function of the first order (/j), giving rise to the aforementioned dynamical beats. The positions of their minima and maxima (i.e., the slope of the envelope of the time-dependent intensity) can be determined with high accuracy and thus give precise information about the effective thickness of the sample. 

Frozen solution of deoxymyoglobin (Mb) has been the subject of an NFS investigation in the temperature range 3.2-230 K (Fig. 9.6) [15]. The synchrotron pulses were transmitted through the entrance window of the sample with a size of 12 mm 2 mm (width height). By scanning the sample area with a narrow beam (about 1 mm 0.3 mm), the homogeneity of the effective thickness was determined as 2.5%, which is more than ten times better than in the aforementioned case. 

The envelope of the spectra is mainly determined by the effective thickness of the sample. For small effective thicknesses, the envelope can be approximated by an exponential decay of the measured intensity with time [13]. Therefore, on the semilogarithmic scale shown in Fig. 9.6, the slope of the envelope provides the effective thickness (see dashed line for T = 3.2 K). The slower decay with increasing temperature corresponds to a decrease in the effective thickness which is equivalent to an increase in the msd x ) of iron with temperature... 

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