Geometrical considerations

in one lifetime (t = 75 pis) the distance which would be traveled is tvp — 2.5 cm. It is thus obvious that serious errors in the measurement of collision-free decay constants could be made if fluorescence cells with dimensions on the order of tdp are used. The limiting dimensions also apply to the observation region since, once the excited molecule moves out of the detection area, the effect is the same as if it had been quenched. Equations for the time dependence of fluorescence which is influenced by migration of long-lived excited molecules to the boundaries of a cylindrical observation region have been developed by Sackett and Yardley. Also included in these equations is the effect of the variation in detection efficiency over the volume of the fluorescence cell. 

Consider a cylindrical fluorescence cell where the laser excites molecules along an infinitely thin line in the center of the cylinder, i.e. along the z axis of a standard cylindrical coordinate system. The excited molecules then move away from this narrow line, in a collision-free environment, with a Maxwell—Boltzmann distribution of radial velocities 

at time t = 0, N(0) molecules are excited along the laser axis, we can write, as before (p. 14), an expression for the number of excited molecules remaining at any time 

Substituting p = vt gives the number of excited molecules in the given velocity which reach radius p 

The number of molecules in this velocity range emitting per unit radius is 

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Table 2. Minimum wall thicknesses to be measured by different transducer frequencies. Guidance of the tube (geometrical considerations)...

A slightly more general case is that in which the liquid meets the circularly cylindrical capillary wall at some angle 6, as illustrated in Fig. II-7. If the meniscus is still taken to be spherical in shape, it follows from simple geometric consideration that / 2 = r/cos 6 and, since R = / 2, Eq. II-9 then becomes... 

Fomially, we can say that the operations in act on points in space. For example, we show in figure A1.4.2 how a point P in the pq plane is transfomied into another point P by the operation we can say tliat P = The reader can now show by geometrical considerations that if we first reflect a point P in the plane to obtain P = and we then reflect P in the 02 plane to obtain P " = = S2S P, then P" can be... 

The percolation argument is based on the idea that with an increasing Cr content an insoluble interlinked cliromium oxide network can fonn which is also protective by embedding the otherwise soluble iron oxide species. As the tlireshold composition for a high stability of the oxide film is strongly influenced by solution chemistry and is different for different dissolution reactions [73], a comprehensive model, however, cannot be based solely on geometrical considerations but has in addition to consider the dissolution chemistry in a concrete way. 

The fact that the proportionality constant in Eq. (10.102) is not written as, say, k but also includes the factor (47t)" is a recognition of the fact that 47t arises frequently in equations from geometrical considerations and can be conveniently eliminated by this device. 

Theoretical Strength of Agglomerates. Based on statistical-geometrical considerations, Rumpf developed the following equation for the mean tensile strength of an agglomerate in which bonds ate localized at the points of particle contact (9) ... 

For piping arrangements and vessels, the following geometric considerations are recommended. 

Oxygen chelates such as those of edta and polyphosphates are of importance in analytical chemistry and in removing Ca ions from hard water. There is no unique. sequence of stabilities since these depend sensitively on a variety of factors where geometrical considerations are not important the smaller ions tend to form the stronger complexes but in polydentate macrocycles steric factors can be crucial. Thus dicyclohexyl-18-crown-6 (p. 96) forms much stronger complexes with Sr and Ba than with Ca (or the alkali metals) as shown in Fig. 5.6. Structural data are also available and an example of a solvated 8-coordinate Ca complex [(benzo-l5-crown-5)-Ca(NCS)2-MeOH] is shown in Fig. 5.7. The coordination polyhedron is not regular Ca lies above the mean plane of the 5 ether oxygens... 

From the standpoint of geometrical considerations, the major difference is in the far greater steric requirements of the nitro group. This could result in either primary or secondary steric effects. Nevertheless, primary steric effects do not seem to be necessarily distinguishable by direct kinetic comparison. A classic example is the puzzling similarity of the activation parameters of 2-chloropyrimidine and 2,6-dinitrochlorobenzene (reaction with piperidine in ethanol), which has been described by Chapman and Rees as fortuitous. However, that nitro groups do cause (retarding) primary steric effects has been neatly shown at peri positions in the reaction with alkoxides (see Section IV,C, l,c). 

Lipatov et al. [116,124-127] who simulated the polymeric composite behavior with a view to estimate the effect of the interphase characteristics on composite properties preferred to break the problem up into two parts. First they considered a polymer-polymer composition. The viscoelastic properties of different polymers are different. One of the polymers was represented by a cube with side a, the second polymer (the binder) coated the cube as a homogeneous film of thickness d. The concentration of d-thick layers is proportional to the specific surface area of cubes with side a, that is, the thickness d remains constant while the length of the side may vary. The calculation is based on the Takayanagi model [128]. From geometric considerations the parameters of the Takayanagi model are related with the cube side and film thickness by the formulas ... 

Geometric considerations in cyclopolymerization are optimal for 1,6-dienes (see 4.4.1.1). Instances of cyclopolymerization involving formation of larger rings have also been reported (see 4.4.1.4), as have examples where sequential intramolecular additions lead to bicyclic structures within the chain (see 4.4.1.2). Various 1,4- and 1,5-dienes are proposed to undergo cyclopolymerization by a mechanism involving two sequential intramolecular additions (see 4.4.1.3). 

Geometric considerations would seem to dictate that 1,4- and 1,5-dicncs should not undergo cyclopolymerization readily. However, in the case of 1,4-dienes, a 5-hexenyl system is formed after one propagation step. Cyclization via 1,5-backbiling generates a second 5-hexenyl system. Homopolymerization of divinyl ether (22) is thought to involve such a bicyclization. The polymer contains a mixture of structures including that formed by the pathway shown in Scheme 4.18. 

Rate equations expressing the a—time variations resulting from the inward advance of a reaction interface from the existing surfaces of other reactant shapes follow directly by the application of simple geometric considerations. The approach can also include quantitative allowance for any... 

When reaction is absent from certain crystallographic surfaces, the formulation of rate equations based on geometric considerations proceeds exactly as outlined above, but includes only the advance of interfaces into the bulk of the reactant particle from those crystallographic surfaces upon which the coherent reactant/product contact is initially established. When reaction occurs only at the edges of a disc or plate-like particle... 

The exclusion effect of hard-spheres is illustrated in Figure lA., which shows a spherical solute of radius r inside an infinitely deep cylindrical cavity of radius a. Here the exclusion process can be described by straightforward geometrical considerations, namely, solute exclusion from the walls of the cavity. Furthermore, it can be shown thatiQJ... 

Kjekshus, A., Rakke, T. Geometrical Considerations on the Marcasite Type Structure. Vol. 19, pp. 85-104. 

Thus we have illustrated that the number of independent rows, the number of independent columns and the rank of the matrix are all identical. Hence, from geometrical considerations, we conclude that the ranks of the patterns in row- and column-space must also be equal. The above illustration is also rendered geometrically in Fig. 29.7. 

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

The geometric considerations leading to the following values are outlined in Fig. 7.2 ... 

In the structure types for compounds MX so far considered, both anions and cations have the same coordination numbers. In compounds MX2 the coordination number of the cations must be twice that of the anions. The geometric considerations concerning... 

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