Similarity relationships

The selectivity relationship merely expresses the proportionality between intermolecular and intramolecular selectivities in electrophilic substitution, and it is not surprising that these quantities should be related. There are examples of related reactions in which connections between selectivity and reactivity have been demonstrated. For example, the ratio of the rates of reaction with the azide anion and water of the triphenylmethyl, diphenylmethyl and tert-butyl carbonium ions were 2-8x10 , 2-4x10 and 3-9 respectively the selectivities of the ions decrease as the reactivities increase. The existence, under very restricted and closely related conditions, of a relationship between reactivity and selectivity in the reactions mentioned above, does not permit the assumption that a similar relationship holds over the wide range of different electrophilic aromatic substitutions. In these substitution reactions a difficulty arises in defining the concept of reactivity it is not sufficient to assume that the reactivity of an electrophile is related... 

Each equation is independent of impeller type. As pointed out eadier, the absolute kpi values vary considerably from Hquid to Hquid. However, similar relationships have been found for other fluids, including fermentation broths, and also for hold-up, 8. Therefore, loss of power reduces the abiHty of the Rushton turbines to transfer oxygen from the air to the broth. 

Eig. 5. Target efficiency of spheres, cylinders, and ribbons. The curves apply for conditions where Stokes law holds for the motion of the particle (see also N j ia Table 5). Langmuir and Blodgett have presented similar relationships for cases where Stokes law is not vaUd (149,150). Intercepts for ribbon or... 

Activators enhance the adsorption of collectors, eg, Ca " in the fatty acid flotation of siUcates at high pH or Cu " in the flotation of sphalerite, ZnS, by sulfohydryl collectors. Depressants, on the other hand, have the opposite effect they hinder the flotation of certain minerals, thus improving selectivity. For example, high pH as well as high sulfide ion concentrations can hinder the flotation of sulfide minerals such as galena (PbS) in the presence of xanthates (ROCSS ). Hence, for a given fixed collector concentration there is a fixed critical pH that defines the transition between flotation and no flotation. This is the basis of the Barsky relationship which can be expressed as [X ]j[OH ] = constant, where [A ] is the xanthate ion concentration in the pulp and [Oi/ ] is the hydroxyl ion concentration indicated by the pH. Similar relationships can be written for sulfide ion, cyanide, or thiocyanate, which act as typical depressants in sulfide flotation systems. 

Specific gravity is direcdy related to the bulk density of waste fuels prepared ia a variety of ways. Solid oven-dry (OD) wood, for example, has a typical bulk density of 48.1 kg/m (301b/ft ). In coarse hogged form, eg, <1.9-cm minor dimension, this bulk density declines to about24kg/m (151b/ft ). In pulverized form, at a particle size <0.16 cm, this bulk density declines to 16—19 kg/m (10—12 Ib/ft ). Similar relationships hold for municipal waste, agricultural wastes, and related fuels. 

A similar relationship guides the change in performance when the impeller outside diameter (OD) is cut ... 

If detailed manufacturer-specified performance curves are not available for a different size of the pump or operating condition, a best estimate of the off-design performance of pumps can be obtained through similarity relationship or the affinity laws. These are ... 

Droplet breakup via impingement appears to follow a similar relationship, but much less data is available. This type of breakup can result from impingement on equipment walls or compressor blades. In general, there is less tendency to shatter on wetted surfaces. 

We have described a general relationship between structure and function for the a/p-barrel structures. They all have the active site at the same position with respect to their common structure in spite of having different functions as well as different amino acid sequences. We can now ask if similar relationships also occur for the open a/p-sheet structures in spite of their much greater variation in structure. Can the position of the active sites be predicted from the structures of many open-sheet a/p proteins ... 

Neighbouring diaxial protons of cyclohexane can be clearly identified by their large coupling constants 11-13 Hz, Table 2.10) which contrast with those of protons in diequatorial or axial-equatorial configurations ( Jee 2-4 Hz). Similar relationships hold for pyranosides as oxy-... 

Radiolytic oxidation alters most of the important properties of graphite, including strength, elastic modulus, work of fracture, thermal conductivity, permeability, and diffusivity but does not affect the thermal expansion coefficient or Poisson s ratio. The effects of radiolytic oxidation on the properties of a wide range of graphites have been studied in the U.K. [7,73,74] where it was found that, to a first approximation, they can be described by similar relationships ... 

Quinoxalin-2-ones are in tautomeric equilibrium with 2-hydroxy-quinoxalines, but physical measurements indicate that both in solution and in the solid state they exist as cyclic amides rather than as hydroxy compounds. Thus quinoxalin-2-one and its A -methyl derivative show practically identical ultraviolet absorption and are bases of similar strength. In contrast, the ultraviolet spectra of quinoxalin-2-one and its 0-methyl derivative (2-methoxyquinoxaIine) are dissimilar. The methoxy compound is also a significantly stronger base (Table II). Similar relationships also exist between the ultraviolet absorption and ionization properties of 3-methylquinoxalin-2-one and its N- and 0-methyl derivatives. The infrared spectrum of 3- (p-methoxy-benzyl)quinoxalin-2-one (77) in methylene chloride shows bands at 3375 and 1565 cm" which are absent in the spectrum of the deuterated... 

There are several similar relationships for centrifugal pumps that can be used if the effects of viscosity of the pumped fluid can be neglected. These relate the operating performance of any centrifugal pump for one set of operating conditions to those of another set of operating conditions, say conditions, and conditions 2. 

The arbitrary division of behaviour has been made because of the extreme behaviour of some chemicals that initiate small areas of attack on a well-passivated metal surface. The form of attack may manifest itself as stress-corrosion cracking, crevice attack or pitting. At certain temperatures and pressures, minute quantities of certain chemicals can result in this form of attack. Chloride ions, in particular, are responsible for many of the failures observed, and it can be present as an impurity in a large number of raw materials. This has led to the development of metals and alloys that can withstand pitting and crevice corrosion, but on the whole these are comparatively expensive. It has become important, therefore, to be able to predict the conditions where more conventional materials may be used. The effect of an increase in concentration on pitting corrosion follows a similar relationship to the Freundlich equation where... 

It would appear that the tradeoffs between these two requirements are optimized at the phase transition. Langton also cites a very similar relationship found by Crutchfield [crutch90] between a measure of machine complexity and the (per-symbol) entropy for the logistic map. The fact that the complexity/entropy relationship is so similar between two different classes of dynamical systems in turn suggests that what we are observing may be of fundamental importance complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity. We will have many occasions to return to this basic idea. 

Similar relationships hold for ionic solids containing polyatomic ions (Table 2.2) or transition metal cations (Figure 4.2). 

To understand how position in the periodic table relates to the filling of sublevels, consider the metals in the first two groups. Atoms of the Group 1 elements all have one s electron in the outermost principal energy level (Table 6.4). In each Group 2 atom, there are two s electrons in the outermost level. A similar relationship applies to the elements in any group ... 

A similar relationship at constant pressure relating dH and Sq can be derived. By definition see equation (1.18) ... 

Similar relationships can be obtained involving G by starting with the fourth Gibbs equation, The results are... 

A similar relationship relating the two activity coefficients can also be derived. We defined activity coefficients such that a =7i-V and a2 — f2x2 where the activities and activity coefficients are established for the standard state that corresponds to and p2, respectively. For both components, changes in the activity at constant temperature and pressure are given by... 

Thus, 0 and 7 have a similar relationship to /m in the limiting law. We usually rewrite equation (7.57) as... 

From Figure A2.1 we can see that differences between and 8 in the temperature range from -100 to 200 °C vary almost linearly with temperature. Similar relationships occur with /48. One of the authors5 has reported that correction of r48 and 8 measurements to 7% in the temperature range from 140 to 400 K can be made with an accuracy better than 0.002 K by using the equation... 

The electrospray mass spectrnm of metabolite 2 indicates it has a molecular weight of 522 Da, while the MS-MS spectrum of the (M + H)+ ion contains an intense ion at m/z 422, 1 Da greater than the base peak of the MS-MS spectrnm of the protonated molecnlar ion of the parent drug. If we assume a similar relationship between these ions as assnmed for m/z 465 and m/z 466 above, it is not nnreasonable to postnlate the strnctnre of metabolite 2 to be that shown in Fignre 5.44. 

The p P values are well defined for both the Brown and Eaborn sets. In view of the approximate but general relationship, Pj = Pj, obtained for other types of reactivities, a similar relationship was assumed. It was then noted that if value were calculated by the relationship p P - or = [log(/ /... 

A similar relationship between AS and ACp, the heat capacity of activation (or reaction), follows also from eq. (79) ... 

The possibility of correlating these fermentation parameters with the turbulent stress equation shows again that obviously similar relationships exist for both the biological systems and the model particle systems used here. 

It is to assume that similar relationships are valid for sparging of animal cell cultures. 

Although such a fragment-based measure clearly provides a very simple picture of the similarity relationships between pairs of structures, it is both efficient (because it involves just the application of logical operations to pairs of bit-strings) and effective (in that it is able to bring together molecules that are judged by chemists to be structurally similar to each other) in operation. 

This equation describes the second fundamental feature of the attraction field, and it can be also treated as the bridge between sources (masses) and the field. It may be appropriate to notice that the similar relationship holds for any field and is often called Gauss s formula. 

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