Multipoint method

Table 5.1 shows the relative error of the single-point method compared to the multipoint method as a function of P/Pq and C as calculated from equation (5.6). 

Comparison of equations (5.10) and (5.8) shows that the difference between the single- and multipoint methods is identical to that shown in Table 5.1 for P/Pq = 0.3 and C = 100 namely... 

As indicated in Fig. 5.1, curve Y which is the BET curve for an arbitrary C value, approaches curve X, the single-point curve, as the value of C increases. In the limiting case of C equal to infinity the BET curve is coincident with the single-point curve. For all other C values the single point curve lies above the BET curve and their difference vanishes as the relative pressure approaches unity. Thus, as the value of C increases the knee of the isotherm becomes sharper and moves toward lower relative pressures. For lower C values, curves X and Y diverge and higher relative pressures must be used to make single-point surface areas conform to those obtained by the multipoint method. 

Table 10 compiles multipoint methods named after van der Pauw and Valdes, which allow conductivity detection of complex geometries. (See specialist Ref.295 for more details.)... 

Marchini, J., Howie, B., Myers, S., McVean, G., and Donnelly, P. (2007). A new multipoint method for genome-wide association studies by imputation of genotypes. Nat. Genet., 39 906-913. 

Depending on the number of measurements used, these methods can be further divided into one-, two-, and multipoint methods. Multipoint regression methods employ a mathematical algorithm to ensure the best possible fit between three or more data points by using a given mathematical function that coincides with the equation of a straight line for calculation of the reaction rate and hence should be fitted to the linear portion of the kinetic curve. 

Determination of a singie species Pseudo-first-order reactions Initial rate method Fixed-time method Variable-time method Second-order reactions Identical reactant concentrations Unequal reactant concentrations Multipoint methods Curve-fitting methods Predictive methods Error-compensated methods... 

Determination of a singie species in a mixture Simuitaneous kinetic-based determinations Classical differential kinetic methods Logarithmic-extrapolation method Proportional-equation method Multipoint methods Curve-fitting methods Kalman filter algorithm Artificial neural networks Multivariate calibration methods... 

The BLC performs several different types of baseline leveling operations on a spectrum two-point level, multipoint level, function fit, interactive polynomial, or auto level. These five methods are capable of baseline correcting a wide variety of data. Note that in the two-point level and the multipoint level methods, the baseline is leveled at a value that is the average of the baseline points. This allows the methods to be applied to both transmission and absorbance spectra. For the interpretation of both nonpolarized and polarized IR spectra, the two-point and the multipoint methods are usually used. 

In modem mass spectrometry, ion collectors (detectors) are generally based on the electron multiplier and can be separated into two classes those that detect the arrival of all ions sequentially at a point (a single-point ion collector) and those that detect the arrival of all ions simultaneously (an array or multipoint collector). This chapter compares the uses of single- and multipoint ion collectors. For more detailed discussions of their construction and operation, see Chapter 28, Point Ion Collectors (Detectors), and Chapter 29, Array Collectors (Detectors). In some forms of mass spectrometry, other methods of ion detection can be used, as with ion cyclotron instmments, but these are not considered here. 

The molecular weight of SAN can be easily determined by either intrinsic viscosity or size-exclusion chromatography (sec). Relationships for both multipoint and single point viscosity methods are available (18,19). Two intrinsic viscosity and molecular weight relationships for azeotropic copolymers have been given (20,21) ... 

Suface Area by Multipoint Hitrogen Adsorption, ASTM Method D3663-78,1978. 

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

The theory for single-point BET measurements is similar however, an assumption is made that the intercept of the PIV(P0-P) versus PIPQ plot equals zero. This assumption is not always valid and can result in surface area values that differ from multipoint values. When the assumption is valid, the single-point method is simpler and faster, and it results in accurate surface area values. 

Figure 2 shows a typical multipoint BET plot for bendroflumethiazide, and Table 2 summarizes the parameters and calculations used to obtain the surface area for this compound. The single-point surface area calculated from a PIP0 of 0.3108 is also presented in Table 2. Good agreement is evident between the two methods, considering the C constant given in Table 2 for this sample. 

There is a one-point modification of a chemisorption method, which is widely used for measurements of Ac. In this case, only one adsorption point of a chemisorption isotherm is measured, and is compared with only one point on a chemisorption isotherm on a reference material (usually, powder [black] or foil). The identity of the chemisorption properties of the active components in supported and pure form is postulated, but very often does not fulfill, making one-point modification an inaccurate procedure, which can hardly be used in scientific studies. For example, studies of supported Rh catalysts by 02 and CO chemosorption have shown that three different blacks of Rh yield three different results [88], The multipoint comparison of chemisorption isotherms shown that only one black had a chemisorption isotherm that had affinity to the isotherm on a supported metal. 

The microporous active carbon was studied by nitrogen adsorption at 77 K. According to obtained data the multipoint BET method gave 771 m2/g however, CM (f-method) method gave 123 m2/g. Explain the difference in the results of BET and comparative methods. 

Where the calibration of the method is very stable a one-point calibration to verify a previous multipoint calibration maybe sufficient... 

A more explicit insight into the mathematical differences of the multi-and single-point methods is obtained by considering a single-point analysis using a relative pressure of 0.3 with a corresponding multipoint C value of 100. 

When used for quality control, the error associated with the singlepoint method can be eliminated or greatly reduced if an initial multipoint... 

The estimate of the fill time shows that a more detailed study of injection strategy and process parameters is necessary because it is difficult to reach an acceptable production economy with a fill time of about 1 h. An obvious action would be to choose a more efficient resin distribution method than a one-point inlet (e.g., peripheral injection or a multipoint injection). In addition, it would be worthwhile to try to lower the viscosity and increase the pressure. 

The CISs are rapidly becoming more popular and reliable as their field of application broadens. This is mainly due to the production of surface images by multipoint scanning and mapping. Hyperspectral imaging has proven its potential for qualitative analysis of pharmaceutical products and can be used when spatial information becomes relevant for an analytical application. Even if online applications and regulatory method validation require further development, the power of CIS in quality control and PAT needs no further demonstration, whatever the wavelength domain or method of spectra collection. 

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