Virtue

The basic phenomenon involved is that particles of ore are carried upward and held in the froth by virtue of their being attached to an air bubble, as illustrated in the inset to Fig. XIII-4. Consider, for example, the gravity-free situation indicated in Fig. XIII-5 for the case of a spherical particle. The particle may be entirely in phase A or entirely in phase B. Alternatively, it may be located in the interface, in which case both 7sa nnd 7sb contribute to the total surface free energy of the system. Also, however, some liquid-liquid interface has been eliminated. It may be shown (see Problem XIII-12) that if there is a finite contact angle, 0sab> the stable position of the particle is at the interface, as shown in Fig. XIII-5Z>. Actual measured detachment forces are in the range of 5 to 20 dyn [60]. 

Although it is hard to draw a sharp distinction, emulsions and foams are somewhat different from systems normally referred to as colloidal. Thus, whereas ordinary cream is an oil-in-water emulsion, the very fine aqueous suspension of oil droplets that results from the condensation of oily steam is essentially colloidal and is called an oil hydrosol. In this case the oil occupies only a small fraction of the volume of the system, and the particles of oil are small enough that their natural sedimentation rate is so slow that even small thermal convection currents suffice to keep them suspended for a cream, on the other hand, as also is the case for foams, the inner phase constitutes a sizable fraction of the total volume, and the system consists of a network of interfaces that are prevented from collapsing or coalescing by virtue of adsorbed films or electrical repulsions. 

A great deal of tax money is spent in support of fundamental research, and this is often defended as having an intrinsic virtue. To take the present topic as an example, however, the study of just how molecules adsorb and react on a surface is fascinating and challenging, yet the tax-paying public should not be asked merely to support the esoteric pleasures of a privileged few. The public should expect the occasional major practical advance whose benefits more than pay for the overall cost of all research. The benefits in the present case come from the discovery and development of catalytic processes of major importance to an industrial society. 

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

MW frequency of 10 Hz. There are various considerations that influence the choice of the radiation frequency. Higher frequencies, which require higher magnetic fields, give inlierently greater sensitivity by virtue of a more favourable Boltzmaim factor (see equation (b 1.15.11)). However, several factors place limits on the frequency employed, so that frequencies in the MW region of the electromagnetic spectrum remain favoured. One limitation is the sample size at frequencies around 40 GHz the dimensions of a typical... 

As we have seen, the electron is the easiest probe to make surface sensitive. For that reason, a number of hybrid teclmiques have been designed that combine the virtues of electrons and of other probes. In particular, electrons and photons (x-rays) have been used together in teclmiques like PD [10] and SEXAFS (or EXAFS, which is the high-energy limit of XAES) [2, Hj. Both of these rely on diffraction by electrons, which have been excited by photons. In the case of PD, the electrons themselves are detected after emission out of the surface, limiting the depth of sampling to that given by the electron mean free path. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

Sorbic acid could theoretically, by virtue of the two olefine linkages, exi.st in four geometrical I y-isomeric forms. The above synthesis gives only one form, which is undoubtedly the trans-trans form, analogous to the acu-diphenylpolyenes (p. 238) which are also normally produced solely in one form. 

Aromatic halides can, in virtue of their aromatic character, be nitrated, etc. 

Two simple methods for the semi-micro estimation of halogens are available, (a) the Carius method, and (b) the Parr bomb method. It should be emphasised that there are other methods available for these estimations on the true micro scale, but they do not lend themselves, by virtue of the balances, apparatus and manipulation required, to semi-micro work, or to the intermittent usage which class-work necessarily entails. 

If students are carefully trained in accurate work, accidents in the laboratory should be of very rare occurrence. Since, however, they can never be entirely eliminated, it is hoped that the First Aid directions given in the Appendix will prove of value, particularly to the junior staff of laboratories, who by virtue of their duties as demonstrators are frequently the first to be called upon to help injured students. 

The Grignard reagent RMgX is nucleophilic by virtue of the potential car banion (alkyl anion) R. It will react with the electrophilic carbonyl group as follows ... 

Aromatic aldehydes usually have relatively high boiling points, but distil with little or no decomposition. The vapours burn with a smoky flame. They are easily oxidised on standing in the air into the corresponding acids the odours are often pleasant and characteristic. Aromatic aldehydes, by virtue of their high molecular weight, yield... 

One of the virtues of the Fischer indole synthesis is that it can frequently be used to prepare indoles having functionalized substituents. This versatility extends beyond the range of very stable substituents such as alkoxy and halogens and includes esters, amides and hydroxy substituents. Table 7.3 gives some examples. These include cases of introduction of 3-acetic acid, 3-acetamide, 3-(2-aminoethyl)- and 3-(2-hydroxyethyl)- side-chains, all of which are of special importance in the preparation of biologically active indole derivatives. Entry 11 is an efficient synthesis of the non-steroidal anti-inflammatory drug indomethacin. A noteworthy feature of the reaction is the... 

TABLE 4.4 Electron Affinities of Atoms, Molecules, and Radicals Electron affinity of an atom (molecule or radical) is defined as the energy difference between the lowest (ground) state of the neutral and the lowest state of the corresponding negative ion in the gas phase. A(g) + e = A-(g) Data are limited to those negative ions which, by virtue of their positive electron affinity, are stable. Uncertainty in the final data figures is given in parentheses. Calculated values are enclosed in brackets. ... 

Several of the lower alkanes, from C2 to C7, have been used from time to time for surface area determination. They possess the virtue of chemical inertness towards the majority of adsorbents, and their saturation pressures... 

Proof. By virtue of the convexity of J, the operator J is monotonous. Thus... 

By virtue of the uniform convergence of Pm there exists a function such that X G for all m. Substituting this function in (2.137) as Xm implies... 

The angular brackets ( , ) denote the integration over flc- In virtue of the linearity, boundedness, and coercivity of the form a(-, ), there exists a unique solution to (2.165). 

It follows from (2.275) that the function U and, hence, the solution W belong to the space Then, by virtue of the properties of U and... 

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