Breadth

Polymerization reactions. Polymers are characterized by the distribution of molecular w eight about the mean as well as by the mean itself. The breadth of this distribution depends on whether a batch or plug-flow reactor is used on the one hand or a continuous well-mixed reactor on the other. The breadth has an important influence on the mechanical and other properties of the polymer, and this is an important factor in the choice of reactor. 

There are thousands of scientists whose work can be classified as vibrational spectroscopy. The following examples are meant to show the breadth of the field, but cannot be expected to constitute a complete representation of all the fields where vibrational spectroscopy is important. 

The breadth of the peaks in an x-ray diffractogram provide a detennination of the average crystallite domain size, assuming no lattice strain or defects, tlirough the Debye-Scherr fonnula ... 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

The scope of electrophilic aromatic substitution is quite large both the aromatic com pound and the electrophilic reagent are capable of wide variation Indeed it is this breadth of scope that makes electrophilic aromatic substitution so important Elec trophilic aromatic substitution is the method by which substituted derivatives of benzene are prepared We can gam a feeling for these reactions by examining a few typical exam pies m which benzene is the substrate These examples are listed m Table 12 1 and each will be discussed m more detail m Sections 12 3 through 12 7 First however let us look at the general mechanism of electrophilic aromatic substitution... 

The breadth or spread of the curve indicates the precision of the measurements and is determined by and related to the standard deviation, a relationship that is expressed in the equation for the normal curve (which is continuous and infinite in extent) ... 

Parameter k of Equation (4.10) is an expression of the breadth of the Gaussian distribution of the cumulative micropore volume IF over the normalized work of adsorption sfifi, and is therefore determined by the pore structure. Thus B also (cf. Equation (4.13)) is characteristic of the pore structure of the adsorbent, and has accordingly been termed the structural constant of the adsorbent. ... 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

The legally trained member of the interdisciplinary committee should provide insight as to the significance of the technological advance and as to whether any commercial product ultimately derived from the invention could be protected by an issued patent. Another important function of this person is to determine the scope of the invention based on preceding events, pubHcations, or activities which may have otherwise limited the breadth of the invention. To this end, U.S. law requires that an invention satisfy a number of prerequisites or requirements before issuing a patent novelty, nonobviousness, utiHty, and disclosure. 

An inventor may estabHsh utiHty by providing several working examples which disclose preparation, appHcation, and even some or all of the benefits of the invention. UtiHty may also be substantiated by merely disclosing several appHcations for the invention. One method of determining the breadth or scope of an invention is to define the invention by only those elements essential to performing the intended task. This definition should then become the broadest cl aim of the patent appHcation. 

The disclosure requirement provides that the patent be a teaching document, and enhance the breadth of knowledge held by the pubHc. By increasing the breadth of knowledge within the pubHc sector, a given patent faciHtates further technological development and growth, which in turn results in the issuance of additional patents. 

Many copolymerization studies have been made. A detailed discussion and critique of the results has been pubHshed (1) and the breadth of the comonomers studied has been summarized (6). Among the comonomers used are oxiranes, oxetanes, 1,3-dioxolane, substituted tetrahydrofurans. 

It may be shown that M > M. The two are equal only for a monodisperse material, in which all molecules are the same sise. The ratio MI /MI is known as the polydispersity index and is a measure of the breadth of the molecular weight distribution. Values range from about 1.02 for carefully fractionated samples or certain polymers produced by anionic polymerization, to 20 or more for some commercial polyethylenes. 

Particles in the gradient may be separated on the basis of sedimentation rate a sample introduced at the top of the preformed gradient setties according to density and si2e of particles, but the mn is terminated before the heaviest particles reach the bottom of the tube. If the density of all the particles ties within the range of the density limits of the gradient, and the mn is not terminated until all particles have reached an equiUbtium position in the density field, equiUbtium separation takes place. The steepness of the gradient can be varied to match the breadth of particle densities in the sample. 

Surface properties are generally considered to be controlled by the outermost 0.5—1.0 nm at a polymer film (344). A logical solution, therefore, is to use self-assembled monolayers (SAMs) as model polymer surfaces. To understand fully the breadth of surface interactions, a portfoHo of chemical functionahties is needed. SAMs are especially suited for the studies of interfacial phenomena owing to the fine control of surface functional group concentration. 

The antibacterial spectmm of moxalactam (Table 9) is similar in breadth and potency to that of cefotaxime (36). Hence, moxalactam (48) is classified with the third-generation cephalosporins. In general 1-oxacephalosporins are considerably more susceptible to P-lactamases than their sulfur counterparts... 

The breadth of reactions catalyzed by cobalt compounds is large. Some types of reactions are hydrotreating petroleum (qv), hydrogenation, dehydrogenation, hydrodenitrification, hydrodesulfurization, selective oxidations, ammonoxidations, complete oxidations, hydroformylations, polymerizations, selective decompositions, ammonia (qv) synthesis, and fluorocarbon synthesis (see Fluorine compounds, organic). 

Condensation Mechanisms Condensation occurs when a saturated vapor comes in contact with a surface whose temperature is below the saturation temperature. Normally a film of condensate is formed on the surface, and the thickness of this film, per unit of breadth, increases with increase in extent of the surface. This is called film-type condensation. 

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic shape beyond which the wave will not spread. The wave is said to be self-sharpening. (If a wave is initially broader than the constant pattern, it will sharpen to approach the constant pattern.) Thus, for an initially uniformly loaded oed, the constant pattern gives the maximum breadth of the MTZ. As bed length is increased, the constant pattern will occupy an increasingly smaller fraction of the bed. (Square-root spreading for a linear isotherm gives this same qualitative result.)... 

All of these solutions are very similar and show, for large N, a wave with breadth proportional to the square root of the bed depth through which it has passed. 

For the charac terization of wave shape and breakthrough cui ves, three methods are popular. The MTZ method [Michaels, Jnd. E/ig. Chem., 44,1922 (1952)] measures the breadth of a wave between two chosen concentrations (e.g., cf = 0.05 and 0.95 or cf = 0.01 and 0.99). Outside of a laboratory, the measurement of full breakthrough cui ves is rare, so the breadth of the MTZ is often estimated from an independently determined stoichiometric capacity and a measured small... 

Spreadsheet Structure There are three principal sections to the spreadsheet. The first has tables of as-reported and normalized composition measurements. The second section has tables for overall and component flows. These are used to check the overall and component material balance constraints. The third has adjusted stream and component flows. Space is provided for recording the basis of the adjustments. The structure changes as the breadth and depth of the analysis increases. 

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