Qualitative arguments

The final equation obtained by Becker and Doting may be written down immediately by means of the following qualitative argument. Since the flux I is taken to be the same for any size nucleus, it follows that it is related to the rate of formation of a cluster of two molecules, that is, to Z, the gas kinetic collision frequency (collisions per cubic centimeter-second). 

From Eq. (9.1) we see that the viscous force associated with this motion equals [i7(dv/dr)] (area), where the pertinent area is proportional to the surface of the sphere and varies as. This qualitative argument suggests that the viscous force opposing the relative motion of the liquid and the sphere is propor tional to [t7(v /R)] (R ). The complete solution to this problem reveals that both pressure and shear forces arising from the motion are proportional tc 77Rvj., and the total force of viscous resistance is given by... 

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

In 1925, before the development of the Schrodinger equation, Franck put forward qualitative arguments to explain the various types of intensity distributions found in vibronic transitions. His conclusions were based on an appreciation of the fact that an electronic transition in a molecule takes place much more rapidly than a vibrational transition so that, in a vibronic transition, the nuclei have very nearly the same position and velocity before and after the transition. 

Complete understanding of the shapes of the curves in Fig. 8.4 requires a kinetic expression somewhat more complicated than we wish to deal with here. The nature of the extremities of the curves can be understood, however, on the basis of qualitative arguments. The rate decreases with a decrease in pH in the acidic region because formation of the zwitterionic tetrahedral intermediate is required for expulsion of the... 

The qualitative arguments we shall produce in the next sections are not affected by the lack of d functions in the sulphur basis set. This statement will be supported in detail by comparison of PP results with all-electron plus 3d ones, for the three-membered ring sulphur compounds (see Section IV.C.3). 

For purely qualitative arguments, on the other hand, the former is quite convenient. The description of the theory up to this point has been, in fact, merely a statement of the qualitative con elusions of the valence bond theory. 

We conclude that qualitative arguments regarding s-p-d hybridization can be safely based on the simple bond-strength treatment, but that this treatment cannot be expected to have the same quantitative significance in this case as for s-p bond orbitals. 

It should be noted that when we compare the brightness of a LGS to a NGS, the result depends on the spectral bandwidth, because the LGS is a line source, whereas the NGS is a continuum one. The magnitude scale is a logarithmic measure of flux per spectral interval (see Ch. 15). This means that a (flat) continuum source has a fixed magnitude, no matter how wide the filter is. In contrast, the magnitude of a line source is smaller for narrower bandpasses. It is therefore advisable to use the equivalent magnitude only for qualitative arguments. The photon flux should be used in careful system analyses. 

Whatever the advantages of the SCF calculations with the Cl s, they do not always give good results, as is the case for M2 molecules with quintuple and sextuple M-M bonds [135,156], Nevertheless, some deviations from the simple MO scheme can be explained using simpler qualitative arguments. For example, the increased ability of technetium to form d4-d5 complexes and their greater stability in comparison to that of the d4-d4 complexes are explained on the basis of a model of the electrostatic repulsion of M atoms with like charges in a binuclear cluster [10,90,150] or on the basis of different diffusivities of cr, o, n, 5 and S metallic MO s of the clusters [63,141,157]. 

Qualitative arguments at the Hiickel level can be used to predict positions of minima (funnels) and barriers in the hypersurfaces, but do not distin-... 

Computed structures for model /V-acyloxy-/V-alkoxyamides, X-ray diffraction data for a number of congeners, as well as spectroscopic evidence fully support these qualitative arguments. 

We move from the qualitative argument that r(meit) decreases as p increases, and next look for a quantitative measure of the changes in melt temperature with pressure. We will employ the Clapeyron equation ... 

We shall demonstrate explicitly the equivalence between the results of Bogolubov, Choh and Uhlenbeck, and Cohen (BCUC) and the generalized Boltzmann equation in Prigogine s theory. But it seems useful to us to indicate beforehand some qualitative arguments which allow a physical understanding of the grounds on which this equivalence rests (for more details see ref. 24). 

Another Lagrangian-based description of micromixing is provided by multienvironment models. In these models, the well macromixed reactor is broken up into sub-grid-scale environments with uniform concentrations. A four-environment model is shown in Fig. 5.16. In this model, environment 1 contains unmixed fluid from feed stream 1 environments 2 and 3 contain partially mixed fluid and environment 4 contains unmixed fluid from feed stream 2. The user must specify the relative volume of each environment (possibly as a function of age), and the exchange rates between environments. While some qualitative arguments have been put forward to fit these parameters based on fluid dynamics and/or flow visualization, one has little confidence in the general applicability of these rules when applied to scale up or scale down, or to complex reactor geometries. 

Is it possible to say something about the bonding pattern of a molecule like U2 based on qualitative arguments Before undertaking the study of the... 

R3 R2 and R2 Ri gauche interactions however, for the same set of substituents, an increase in the steric requirements of either Rj or R3 will influence only one set of vicinal steric interactions (Rj R2 or R3 R2). Some support for these conclusions has been cited (eqs. [6] and [7]). These qualitative arguments may also be relevant to the observed populations of hydrogen- and nonhydrogen-bonded populations of the aldol adducts as well (see Table 1, entries K, L). Unfortunately, little detailed information exists on the solution geometries of these metal chelates. Furthermore, in many studies it is impossible to ascertain whether the aldol condensations between metal enolates and aldehydes were carried out under kinetic or thermodynamic conditions. Consequently, the importance of metal structure and enolate geometry in the definition of product stereochemistry remains ill defined. This is particularly true in the numerous studies reported on the Reformatsky reaction (20) and related variants (21). 

In this chapter we will deal with those parts of acoustic wave theory which are relevant to chemists in the understanding of how they may best apply ultrasound to their reaction system. Such discussions tvill of necessity involve the use of mathematical concepts to support the qualitative arguments. Wherever possible the rigour necessary for the derivation of the basic mathematical equations has been kept to a minimum within the text. An expanded treatment of some of the derivations of key equations is provided in the appendices. For those readers who would like to delve more deeply into the physics and mathematics of acoustic cavitation numerous texts are available dealing with bubble dynamics [1-3]. Others have combined an extensive treatment of theory with the chemical and physical effects of cavitation [4-6]. 

We now show that as the number n of carbon atoms in the acid increases, diss converge to a constant value. The qualitative argument is simple. Adding a methylene group far from the carboxylic group is not expected to affect the binding properties of the carboxylic group. 

There are other ways to appreciate the catalytic potency of enzymes in addition to that provided above. A second way to understand the same point is to accurately measure ratios between rates of enzyme-catalyzed reactions and the corresponding uncatalyzed reactions under the same conditions, a refinement of the qualitative argument just made. These ratios are frequently not easy to obtain since the rates of the uncatalyzed reactions may be so slow as to make them exceedingly difficult to measure. Nonetheless, a number of these ratios are known and they typically vary between abont 10 (one thousand) and 10 (one quadrillion), truly enormous values. To help understand just how large 10 is, consider a chemical reaction begun at the time of the creation of our solar system whose progress would be barely detectable at the present day. That same reaction would be nearly complete in 1 minute if catalyzed IQi -fold. 

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