N+ values

If the DHT is evaluated numerically from its defining expression, the time taken for long data sequences of length N is proportional to as with the DFT, because for each value of v the are N evaluations of the product /(r)cas(2nvT/ N) and there are N values of v. 

The derivation of Eq. (1.62) explicitly requires that n be large, while the notion of nonentanglement which underlies (2.54) precludes large values of n. Since we have not confronted the question of how large is large in either of these derivations, we proceed with this substitution, hoping for a compatibility of n values. 

The foregoing conclusion does not mean that the rate of the reaction proceeds through Table 5.1 at a constant value. The rate of reaction depends on the concentrations of reactive groups, as well as on the reactivities of the latter. Accordingly, the rate of the reaction decreases as the extent of reaction progresses. When the rate law for the reaction is extracted from proper kinetic experiments, specific reactions are found to be characterized by fixed rate constants over a range of n values. 

As p increases, the proportion of molecules with smaller n values decreases and the proportion with larger n values increases. 

The number average degree of polymerization for these mixtures is easily obtained by recalling the definition of the average from Sec. 1.8. It is given by the sum of all possible n values, with each multiplied by its appropriate weighting factor, provided by Eq. (5.24) ... 

As p increases, the maximum in the curves shifts to larger n values and the tail of the curve extends to higher values of n. 

The effect in item (2) is not merely a matter of shifting curves toward higher n values as p increases, but reflects a distinct broadening of the distribution of n values as p increases. 

We saw in Chap. 1 that the ratio M /M is widely used in polymer chemistry as a measure of the width of a molecular weight distribution. If the effect of chain ends is disregarded, this ratio is the same as the corresponding ratio of n values ... 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

A plot of these values is shown in Fig. 8.1. Note the increase in the entropy of mixing over the ideal value with increasing n value. Also note that the maximum occurs at decreasing mole fractions of polymer with increasing degree of polymerization. 

The ratio of Eq. (8.66) to Eq. (8.67) gives the ratio of the concentrations of n-mers in phases P and Q f p/f g = Re ". Taking this ratio to be unity for n = 200 gives Re (200) = which is readily solved for A using the R values given. Once these A values are obtained, f p/f g can be evaluated for the required n values. For the phase volume ratios under consideration, the corresponding values of A are listed below also tabulated are the ratios f p/f g for the various n s ... 

In both of these pieces of apparatus, isothermal operation and optimum membrane area are obtained. Good temperature control is essential not only to provide a value for T in the equations, but also because the capillary attached to a larger reservoir behaves like a thermometer, with the column height varying with temperature fluctuations. The contact area must be maximized to speed up an otherwise slow equilibration process. Various practical strategies for presetting the osmometer to an approximate n value have been developed, and these also accelerate the equilibration process. 

The ratio n/c2 is called the reduced osmotic pressure-and can be plotted with or without the RT-and the zero-intercept value (subscript 0) is the limiting value of the reduced osmotic pressure. Quite an assortment of different pressure units are used in the literature in reporting n values, and the units of R in Eq. (8.88) must be reconciled with these pressure (as well as concentration) units. 

Bisphenol F Resin. Bisphenol F [2467-02-9] epoxy resin is of the same general stmcture as the epoxy phenol novolaks. Bisphenol F is 2,2Emethylene bisphenol. Whereas the epoxy phenol novolaks vary from viscous Hquids to soHd materials, the bisphenol F resin has a low viscosity (ca 4 Pa-s (40 P)) and 165 epoxy equivalent weight. Its n value (degree of polymerization) is about 0.15 and crystallization, often a problem with low viscosity conventional bisphenol A resins, is reduced with the bisphenol F resin. 

Subsequent epoxidation with epichl orohydrin yields the highly functional epoxy novolak. The product can range from a high viscosity Hquid of n = 0.2 to a soHd of n value greater than 3. 

SoHd epoxy resins are sometimes designated as 1-, 4-, 7-, or 9-type resins these approximate the degree of polymerization. Commercial products are designated similarly, eg, Epon 1001, 1004, 1007, and 1009 (SheU Chemical Co.). The relationship between n value, epoxy equivalent weight, and melting point is shown in Table 5. 

Gel-permeation chromatography studies of epoxy resins prepared by the taffy process shown n values = 0, 1, 2, 3, etc, whereas only even-numbered repeat units are observed for resins prepared by the advancement process. This is a consequence of adding a difunctional phenol to a diglycidyl ether derivative of a difunctional phenol in the polymer-forming step. 

The term moving is applied because the filter software maintains a storage array with the previous n values of the input. When a new value is received, the oldest value in the storage array is replaced with... 

The (x, i )), values in Eq. (13-37) are minimum-reflux values, i.e., the overhead concentration that would be produced by the column operating at the minimum reflux with an infinite number of stages. When the light key and the heavy key are adjacent in relative volatihty and the specified spht between them is sharp or the relative volatilities of the other components are not close to those of the two keys, only the two keys will distribute at minimum reflux and the Xi D),n values are easily determined. This is often the case and is the only one considered here. Other cases in which some or all of the nonkey components distribute between distillate and bottom products are discussed in detail by Henley and Seader (op. cit.). 

If air quality data at a receptor for any one averaging time are lognormally distributed, these data will plot as a straight line on log probability graph paper (Fig. 4-9) which bears a note Sg = 2.35. Sg is the standard geometric deviation about the geometric mean (the geometric mean is the Nth root of the product of the n values of the individual measurements). 

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