Proof that

Janak J F 1978 Proof that dEldn- = r . in density-functional theory Phys. Rev. B 18 7165-8... 

The evidence outlined strongly suggests that nitration via nitrosation accompanies the general mechanism of nitration in these media in the reactions of very reactive compounds.i Proof that phenol, even in solutions prepared from pure nitric acid, underwent nitration by a special mechanism came from examining rates of reaction of phenol and mesi-tylene under zeroth-order conditions. The variation in the initial rates with the concentration of aromatic (fig. 5.2) shows that mesitylene (o-2-0 4 mol 1 ) reacts at the zeroth-order rate, whereas phenol is nitrated considerably faster by a process which is first order in the concentration of aromatic. It is noteworthy that in these solutions the concentration of nitrous acid was below the level of detection (< c. 5 X mol... 

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. 

While this is an easy calculation to make, Eq. (3.7) does little to clarify exactly what AS means. Phenomenological proofs that AS as defined by Eq. (3.7) is a state variable often leave us with little more than a lament for the inefficiency of spontaneous processes. 

The integration of Eq. (6.106) is central to the kinetic proof that living polymers follow Poisson statistics. The solution of this differential equatior is illustrated in the following example. 

Tazettine (73) has gained notoriety siace, subsequent to its isolation from Sprekeliaformosissima or Narcisus ta tta and proof that it was generated in vivo from haemanthamine [466-75-1] (76), ia accord with biosynthetic dogma (52), more careful work (53) ia which the strongly basic... 

Manufacture of oBenzoylbenzoic Acid from l-Methyl-3-phenylindane. In 1909 it was reported that treatment of styrene with concentrated sulfuric acid resulted in its dimerization (51). However, the wrong stmcture was assigned to this dimer (52). Years later it was suggested, without proof, that the dimer consisted primarily of l-methyl-3-phenylindane (13) [6416-39-3] and some 1,3-diphenyl-l-butene (53). In 1950, oxidative studies on the dimer proved that this supposition was correct (54) ... 

In studio photography, instant color sHdes exposed simultaneously with conventional color films are used to provide proofs that can be projected immediately for viewing by the customer. Professional photographers also use instant films as proof material to check composition and lighting. Large format Polacolor films are often used directly for exhibition prints. 

Insurance requirements are stipulated in contracts between tollers and client companies. Often, the toller is required to show proof that it obtains and maintains specified insurance coverage. Minimum amounts may be set for ... 

Study has proved that these coliform bacteria indicate the presence of human or animal wastes in water. Coliform bacteria naturally exist in the intestines of humans and certain animals. Thus, the presence of these bacteria in water is accepted proof that the water has been contaminated by human or animal wastes. Although such water may contain no pathogens, an infected person, animal or a carrier of disease, could add pathogens at any moment Thus, immediate corrective action must be taken. The presence of coliform bacteria shows water is contaminated by human... 

Thus, the BLEVE theory predicts that, when the temperature of a superheated liquid is below T, liquid flashing cannot give rise to a blast wave. This theory is based on the solid foundations of kinetic gas theory and experimental observations of homogeneous nucleation boiling. It is also supported by the experiments of BASF and British Gas. However, because no systematic study has been conducted, there is no proof that the process described actually governs the type of flashing that causes strong blast waves. Furthermore, rapid vaporization of a superheated liquid below its superheat limit temperature can also produce a blast wave, albeit a weak... 

Finally we should note that the demonstration of a Br nsted relationship does not constitute proof that general acid or general base catalysis is occurring. Because of the problem of kinetic equivalence of rate terms, we may not be able unequivocally to distinguish between these possibilities ... 

Proof that the second mole of phenyl isocyanate did not react at the nitrogen of 160 was provided by the reaction of the enamine from 2-methylcyclo-hexanone, which gave only the monocarboxanilide on reaction with excess phenyl isocyanate. 

Where unidentate ligands are present, the ability to effect the resolution of an octahedral complex (i.e. to separate 2 optical isomers) is proof that the 2 ligands are cis to each other. Resolution of [PtCl2(en)2] therefore shows it to be cis while of the 2 known geometrical isomers of [CrCl2en(NH3)2] the one which can be resolved must have the cis-cis structure since the trans form would give a superimposable, and therefore identical, mirror image ... 

Demonstration of ligand transfer is crucial to the proof that this purticitlar reaction proceed.s via an inner-sphere mechanism, and ligand transfer i.s indeed a usual feature of inner-sphere redox reaction.s, but it is not an essential feature of oil such reactions. 

Dipole moment data have provided valuable information for the study of the tautomerism of compounds such as isonicotinic acid, pyrid-4-one, and ethyl acetoacetate, However, this method must be used with discretion since it can lead to inconclusive results. Thus, the fact that 4-aminopyridine has a higher dipole moment than the algebraic sum of the dipole moments of pyridine and aniline was originally interpreted as proof that structure 54 exists with a strong contribution from 36, and it was stated that 55 w ould have a very low moment. Later, Angyal and AngyaF pointed out that the... 

It should be stated, however, that there is no sound proof that... 

C and addition of excess of potassium iodide to the product, gives only a small yield of methiodide, but 2-acetamidoquinoxaline 4-methiodide (67) is readily prepared by this method. Proof that quaternization occurs at position 4 is obtained by degradation of... 

The substantial difference between these two chromatograms was a clear proof that CSP 13 interacted differently with the mixtures of l and d enantiomers. This also indicated the presence of at least one pair of enantiomers that interacted selectively with the CSP. Unfortunately, a tedious synthesis of 16 sublibraries (eight l and eight d) containing decreasing numbers of blocks had to be prepared to deconvolute the best selector. A comparison of the chromatograms obtained from these sublibraries in each deconvolution step was used again, and those selectors for which no difference was observed were eliminated. This procedure enabled the identification... 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

---