Point-area method

While the point - area method is very convenient in terms of computational efforts, it has a serious drawback. The matrix of the linear system (5.6B) is inherently ill - conditioned (ref. 25), and the result is very sensitive to the errors in the observations. 

Since the convolution integral is symmetrical in u(t) and hit), this problem is similar to the one of system identification considered in the previous section. Nevertheless, it is usually easier to find the weighting function h(t) since its form is more - or - less known (e.g., as a sum of polyexponentials), and hence parametric methods apply, whereas the input function u(t) is a priori arbitrary. Therefore, the non - parametric point -area method is a popular way of performing numerical deconvolution. It is really simple evaluating the integral means h, f, . .., fy, of the weighting function over the subinterval [t. ., t ] we can easily solve the set (6.68)... 

The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

One is obliged to conclude that this method, like those which derive the cumulative surface area from pore size calculations, can be regarded as no more than ancillary to the BET or Point B methods. The few cases where reasonable agreement with the BET area is obtained are probably to be explained by compensation of opposing effects. 

The specific surface areas of the samples were measured by the single point BET method (p/pQ=0.3). 

The Cr203 content of each catalyst was determined by atomic absorption spectroscopy (Varian/Spectr AA-20 plus) on acid-digested samples. Total surface areas were determined by a single point BET method (nitrogen adsorption-desorption at 77.5 K) using a mixture of 29.7% N2 in helium. Samples were wet-loaded into the flow tube and dried at 423 K in a hydrogen flow for 15 minutes and then for another 30 minutes at 513 K before cooling in helium. 

Calculating the P/S ratio from the corresponding peak area ratios or by using one point calibration method leads to erroneous interpretations [54] because the P/S ratio depends on sample dilution. An accurate quantitation of FA is thus needed to evaluate this important parameter correctly. 

The simplest measure of surface area is that obtained by the so-called point B method. Many isotherms of Type II or IV show a straight section at intermediate relative pressures. The more pronounced the section, the more complete is the adsorbed monolayer before multilayer adsorption begins. As may be seen from Figure 17.9, the lower limit of the straight section is the point B which was identified by Emmett 24 and Yang 3 as corresponding to a complete monolayer. This interpretation is accepted as a convenient empiricism. 

Figure 17.9. The point B method of estimating surface area...

Data taken from the adsorption leg of the isotherm of Figure 17.11 are listed in the first two columns of the following table. Test the applicability of the following equilibrium theories (a) Langmuir (b) infinite BET and (c) Harkins and Jura. From (a) and (b) obtain estimates of the surface area of the adsorbent and compare the values with that obtained by the point B method. One molecule of nitrogen adsorbed on alumina occupies 0.162 nm2. 

For FTS data, artifact removal is a consideration that is as important as resolution improvement for most researchers in this field. Interferogram continuation methods are not as yet widely known in this area. Methods currently in widespread use that are effective in artifact removal involve the multiplication of the interferogram by various window functions, an operation called apodization. A carefully chosen window function can be very effective in suppressing the artifacts. However, the peaks are almost always broadened in the process. This can be understood from the uncertainty principle. A window that reduces the function most strongly closest to the end points will yield a transform for the modified function that must be broader than it was originally. Alternatively we may employ the convolution... 

Zeolite samples that were exhaustively treated were gray in color (materials A). Portions of each treated catalyst (0.3 gram) were calcined at 500°C in oxygen for 6 hours, giving a white product (material B). X-ray analysis showed no deterioration in crystal structure on treatment with TMS and subsequent calcination. Surface area measurements using N2 adsorption at — 196°C and the Point B method are also given in Table I. The value for HY zeolite, activated as described, prior to TMS treatment was 840 m2/gram. 

As before, surface areas were measured by the single point BET method using a Micromeritics Flowsorb II 2300 analyser and a mixture of 30% N2 in He as the adsorbate gas. The observed values are 16 m2g 1 and 5 m2g 1... 

Catalyst surface areas were measured using the multi-point BET method on a Carlo-Erba Ins. Sorpty 1750. Before the measurements, the samples were heated under dynamic vacuum at 573 K for 1 h in order to remove adsorbed water and impurities. Measurements were made at liquid nitrogen temperature with nitrogen as the adsorbate gas. Powder X-ray diffraction measurements were performed on a Siemens Model D-500 diffractometer with Co Kc monochromatic radiation (X = 1.78901 A) and the high resolution electron microscopy was carried out on a Topcon EM-002B microscope. To prevent artefacts no solvents were used in the preparation and mounting of samples for HRTEM. 

Analytical methods including surface area and vanadium levels were also performed. Nitrogen BET surface areas were evaluated with an Autosorb-6 by the multi point BET method and the matrix contribution determined. An ARE 3410 ICP was used to determine vanadium, sodium, and rare earth content after decon )osing the san le in HF. 

Alternatively, areas can be obtained from isotherms using the "point B method". This point can be read from the isotherm if C is high enough (see fig. 1.13, type II) and interpreted as corresponding to 0 = 1. In reality, if the isotherm is twice differentiated to find the inflection point, the latter is found to occur up to 15% above 0 = 1. 

Vaughan DP, Dennis M. Mathematical basis of point-area deconvolution method for determining in vivo input functions. J Pharm Sci 1978 67 663-5. 

The specific surface area (ssa) of the solids (m g ) obtained after hydrolysis and drying at various temperatures was measured routinely by multi-point BET method at 77K by N2 adsorption in a Fisons Sorptomatic 1900 system. From these isotherms the BET ssa s were determined and the corresponding pore size distribution was also found. Typical results including the adsorption-desorption isotherms as well as the corresponding pore size distributions are shown in Fig.2... 

Supports and Catalysts. The catalyst supports used in this work are described in Table I. The surface areas, except for the sillcalite, were measured by the multi-point BET method. The surface area for the sillcalite was obtained from the manufacturer. Silicalite is an essentially aluminum-free pentasil zeolite (14) manufactured by Union Carbide. The chlorine contents of the supports were determined by neutron activation analysis, and sulfur contents were obtained with a Leco sulfur analyser. Sulfur and chlorine contents were measured since these elements may influence subsequent hydrogen adsorption on the supported platinum catalysts (15). 

Macheras P, Symillides M, Reppas C. The cut-off time point of the partial area method for assessment of rate of absorption in bioequivalence studies. Hiarm Res 1994 11 831-834. 

The point-R method of estimating surface areas was frequently used prior to the development of the Brunauer-Emmett-Teller approach. It entailed choosing from an absorption diagram such as Fig. 8-3 the point at which the central linear section begins. This procedure worked well for some systems, but it was extremely difficult, if not impossible, to select a reliable point B on an isotherm such as that shown for n-butane in Fig. 8-3. In contrast, the Brunauer-Emmett-Teller method was found to be reasonably satisfactory for this type of isotherm. 

Specific surface areas were determined by N2 (Air Liquide, 30% N2 in He) adsorption at -196°C (one point BET method) with a Micromeritics Flow Sorb II. 

The cross section of the films were observed and their porosity were calculated by the point counting method. Cracking of the films were also observed by optical microscope. The phase structure of the film was analyzed by XRD analysis using Cu-Ka radiation. Chemical dispersion of a cross section was evaluated by EPMA. An area of a cross section of the thick film was also calculated with a outline including porosity. Adhesive strength was qualitatively analyzed whether the film can be peeled with a knife or not. 

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