Shape equivalence

From what we have discussed in (b"), it follows that when the two sets of D(Cj) data derived from sorption experiments and permeation experiments do not agree with one another within the expected limits of accuracy, the sorption values of D are incorrect and the measured sorption processes are non-Fickian. This method of examining whether sorption is Fickian or not is particularly useful when the experimental sorption curve has a shape equivalent to the true Fickian curve. Meares (1958a) demonstrated that the D values for the system polyvinyl acetate-allyl chloride at 40° C. determined from steady-state permeation data in-... 

The above tools are also suitable for studying the similarity of reaction paths. Two reaction paths Pi and P2 are regarded shapewise equivalent within the above context (ID-shape equivalent) if and only if the numbers w, and W2 of their... 

One can use a derivation entirely analogous with the homotopy equivalence classes of paths and loops and the fundamental group of re on mechanisms in the nuclear configuration space M [13], leading to a group theoretical model of reaction mechanisms based on shape. The above shape equivalence of reaction paths p in space M generates a complete shape classification of all possible reaction paths for the given stoichiometry of nuclei. 

Many models have been presented to explain quantitatively the dependence of exciton energy on the cluster size [7, 11, 21-30]. This problem was first treated by Efros et al. [7], who considered a simple particle in a box model. This model assumes that the energy band is parabolic in shape, equivalent to the so-called effective mass approximation. The shift in absorption threshold, AE, is dependent upon the value of the cluster radius, R, Bohr radius of the electron, ae ( = h2v.jmce2), and Bohr radius of the hole, ah (= h2e/mhe2). When (1) R ah and R ae, and (2) ah R ac,... 

A cylinder of 28 ft. diameter and 26 ft. height with uranium throughout the cylinder except for the outer thickness of one foot - or a somewhat different shape equivalent from the point of view of the chain reaction. 

The equivalence classes of the (P,W) topological shape equivalence correspond to the actual shape types (p,w) denoted simply by x for a fixed choice of the (P,W) pair. 

An s orbital is spherically symmetrical and can contain a maximum of two electrons with opposed spins. A p orbital has a solid figure-of-eight shape there are three equivalent p orbitals for each principal quantum number they correspond to the three axes of rectangular coordinates. 

The d and f orbitals have more complex shapes there are five equivalent d orbitals and seven equivalent f orbitals for each principal quantum number, each orbital containing a maximum of 2 electrons with opposed spins. 

This inequality indicates the amphiphile adopts a shape essentially equivalent to that of a cone with basal area <3. Such cones self-assemble to fonn spheroidal micelles in solution or spheroidal hemimicelles on surfaces (see section C2.3.15). Single-chain surfactants with bulky headgroups, such as SDS, typify surfactants in this category. 

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

The shapes indicated in Table 2.8 are only exact in cases in which all the electron pairs are equivalent, i.e. they are all bonding pairs. 

Example The electron configuration for Be is Is lsfi but we write [He]2s where [He] is equivalent to all the electron orbitals in the helium atom. The Letters, s, p, d, and f designate the shape of the orbitals and the superscript gives the number of electrons in that orbital. 

ChemSketch has some special-purpose building functions. The peptide builder creates a line structure from the protein sequence defined with the typical three-letter abbreviations. The carbohydrate builder creates a structure from a text string description of the molecule. The nucleic acid builder creates a structure from the typical one-letter abbreviations. There is a function to clean up the shape of the structure (i.e., make bond lengths equivalent). There is also a three-dimensional optimization routine, which uses a proprietary modification of the CHARMM force field. It is possible to set the molecule line drawing mode to obey the conventions of several different publishers. 

On the assumption that the pairs of electrons in the valency shell of a bonded atom in a molecule are arranged in a definite way which depends on the number of electron pairs (coordination number), the geometrical arrangement or shape of molecules may be predicted. A multiple bond is regarded as equivalent to a single bond as far as molecular shape is concerned. 

The limits of pore size corresponding to each process will, of course, depend both on the pore geometry and the size of the adsorbate molecule. For slit-shaped pores the primary process will be expected to be limited to widths below la, and the secondary to widths between 2a and 5ff. For more complicated shapes such as interstices between small spheres, the equivalent diameter will be somewhat higher, because of the more effective overlap of adsorption fields from neighbouring parts of the pore walls. The tertiary process—the reversible capillary condensation—will not be able to occur at all in slits if the walls are exactly parallel in other pores, this condensation will take place in the region between 5hysteresis loop and in a pore system containing a variety of pore shapes, reversible capillary condensation occurs in such pores as have a suitable shape alongside the irreversible condensation in the main body of pores. 

In a titrimetric method of analysis the volume of titrant reacting stoichiometrically with the analyte provides quantitative information about the amount of analyte in a sample. The volume of titrant required to achieve this stoichiometric reaction is called the equivalence point. Experimentally we determine the titration s end point using a visual indicator that changes color near the equivalence point. Alternatively, we can locate the end point by recording a titration curve showing the titration reaction s progress as a function of the titrant s volume. In either case, the end point must closely match the equivalence point if a titration is to be accurate. Knowing the shape of a titration... 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

This class of smart materials is the mechanical equivalent of electrostrictive and magnetostrictive materials. Elastorestrictive materials exhibit high hysteresis between strain and stress (14,15). This hysteresis can be caused by motion of ferroelastic domain walls. This behavior is more compHcated and complex near a martensitic phase transformation. At this transformation, both crystal stmctural changes iaduced by mechanical stress and by domain wall motion occur. Martensitic shape memory alloys have broad, diffuse phase transformations and coexisting high and low temperature phases. The domain wall movements disappear with fully transformation to the high temperature austentic (paraelastic) phase. 

Venturi scmbbers can be operated at 2.5 kPa (19 mm Hg) to coUect many particles coarser than 1 p.m efficiently. Smaller particles often require a pressure drop of 7.5—10 kPa (56—75 mm Hg). When most of the particulates are smaller than 0.5 p.m and are hydrophobic, venturis have been operated at pressure drops from 25 to 32.5 kPa (187—244 mm Hg). Water injection rate is typicaUy 0.67—1.4 m of Hquid per 1000 m of gas, although rates as high as 2.7 are used. Increasing water rates improves coUection efficiency. Many venturis contain louvers to vary throat cross section and pressure drop with changes in system gas flow. Venturi scmbbers can be made in various shapes with reasonably similar characteristics. Any device that causes contact of Hquid and gas at high velocity and pressure drop across an accelerating orifice wiU act much like a venturi scmbber. A flooded-disk scmbber in which the annular orifice created by the disc is equivalent to a venturi throat has been described (296). An irrigated packed fiber bed with performance similar to a... 

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

Hardness. The resistance of a fabricated mbber article to indentation, ie, hardness, is influenced by the amount and shape of its fillers. High loadings increase hardness. Fillers in the form of platelets or flakes, such as clays or mica, impart greater hardness to elastomers than other particle shapes at equivalent loadings. 

Flow Nozzles. A flow nozzle is a constriction having an eUiptical or nearly eUiptical inlet section that blends into a cylindrical throat section as shown in Figure 8. Nozzle pressure differential is normally measured between taps located 1 pipe diameter upstream and 0.5 pipe diameters downstream of the nozzle inlet face. A nozzle has the approximate discharge coefficient of an equivalent venturi and the pressure drop of an equivalent orifice plate although venturi nozzles, which add a diffuser cone to proprietary nozzle shapes, are available to provide better pressure recovery. 

Eor randomly packed spherical particles, the constants M and B have been deterrnined experimentally to be 150 and 1.75, respectively. Eor nonspherical particles, equivalent spherical diameters are employed and additional corrections for shape are introduced. 

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