Numerical fractionation

Throughout this discussion we have used the numerical fraction of molecules in a class as the weighting factor for that portion of the population. This restriction is not necessary some other weighting factor could be used equally well. As a matter of fact, one important type of average encountered in polymer chemistry is the case where the mass fraction of the ith component is used as the weighting factor. Defining the mass of material in the ith class as mj, we write... 

But by applying the process of fractional distillation to a sample, and determining the characters of numerous fractions, the actual number of available determinations may be, and frequently is, so largely increased that there is practically no chance of the adulterator being able to standardise his oils to meet all these figures. This proposition, of course, fails where it is possible to use a substance obtained from a cheaper source which is identical with one contained naturally in the oil, and which is added in such amount as not to materially exceed, in the total, the amount naturally occurring in the essential oil in question. 

Coal Tar.—This coal tar is the crude source of many compounds of the benzene series. On redistillation of the coal tar numerous fractional distillates are obtained from which different compound are isolated, (p. 497). 

Most problems with this procedure have involved tracer impurities and the separation of bound and free labeled fractions. Several separation techniques have been used, including equilibrium dialysis, membrane ultrafiltration, and steady-state gel filtration. Their deficiencies include a requirement for a large sample volume, the need for complicated correction of sample volume changes that occur during the separation, and difficulties of collecting and measuring radioactivity in numerous fractions of each sample. Equilibrium dialysis has been used most often in the past, but serious errors often arise from the sample dilution required by this method. Symmetrical dialysis of undiluted samples is reported to be less susceptible to tracer contamination and dilution effects. Ultrafiltration also appears to overcome these problems and to obviate errors caused by dilution. 

Finally, Bertaut imagined the homogeneous domains of coherent diffraction as constituted by columns of elementary cells juxtaposed orthogonally to the diffracting planes, and he defined a size distribution P(n) as the numerical fraction of columns of length n cells. From that size distribution function, the size-only Fourier coefficients can be defined " ... 

In the following sections some of the numerous fraction techniques that have been used for biochemical separation will be examined with particular reference to modification that could occur to the distribution of trace element-containing species. 

Numerous fractionator or absorber foaming case histories have been reported in the literature (24, 33, 50, 85, 135, 203, 209, 239, 318, 373, 408). These suggest several s3miptoms which are typical of a foaming problem. These s3rmptoms are not universal a foaming problem may exist even if some of these symptoms are not observed. The following symptoms have been observed ... 

In order to circumvent the problem of unwieldy numerical fractions having several digits (e.g., the clay-silt bonndary at 0.001953mm) and to simplify (at that time) the laborious hand calculations of various statistical parameters for the particle size distributions, Krumbein (1934) subseqnently proposed that the size of particles should be measured in dimensionless units of 0 (phi). Phi is defined as ... 

Assuming a statistical nature of the deviation of AC from the maximum, the distribution can be subdivided into separate Gaussian functions (Figure 3.32). The area occupied by each Gaussian curve corresponds to the numerical fraction of polymer synthesised on a respective centre or the fraction of monomer consumed by this type of active centre. Therefore, the area inside the Gaussian curves corresponds to the kinetic activity of this type of active centre. 

A new series of samples was analyzed by the GPC device (including two SHOWA DENKO columns) fitted with the dynamic li t-scattering detector and RI detector. This procedure allowed to obtain by dynamic LS and by GPC on numerous fractions (slices) characterized by... 

Preparative GC offers the greatest resolving power of the fractionation methods. Traditionally, a flavor concentrate was chromatographed on a 0.125-inch or 0.25-inch o.d. packed column and the effluent collected via cold trap in numerous fractions. The solvent-free fractions could be subjected to sensory evaluation at this point to focus on a specific sensory property (e.g., an off-flavor or desirable note) or subjected to additional chromatographic separations. Today it is more common to use large... 

Teymour E, Campbell JD. Analysis of the dynamics of gelation in polymerization reactors using the numerical fractionation technique. Macromolecules 1994 27 2460-2469. 

Selective precipitation is a more advanced method of precipitating proteins out of solution than simply adding add and heat or proteolytic enzymes. There are numerous fraction-spedfic methods currently in nse to precipitate and pnrify the proteins withont applying as much heat or acid, thus limiting the denatnring of the protein fractions. 

From extensive TLC investigations of the external lipids of the Austin strain and analysis of the numerous fractions by IR and NMR spectroscopy, it was evident that several other ethers were also present however attempts to isolate these compounds in pure form have been so far unsuccessful. These unresolved compounds are likely to be botryal-botryal ethers, botryal-alkylphenol ethers and isomers of (74) and (75) differing in the relative location of hydroxyl groups and ester in the two hydrocarbon chains. Moreover, the presence of an epoxide ring at C((o9)-C((d10), as shown by the NMR spectra, in some of the botryal... 

We can calculate A/ac within the framework of random models of mixtures according to the numerical fraction A/a of grains A and their co-ordination number... 

Unlike polymers, characterized by continuous functions of molecular weight distribution, oligomers are characterized by discrete functions. This means that while for polymers the numeric molecular weight distribution (M)AM represents the numeric fraction of molecules with molecular weight from M to M+AM, for oligomers, it represents the numeric fraction of molecules with molecular weight Mj, this fraction corresponds to a certain degree of... 

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