In rigor

Clearly there was no lack of devoted adversaries (perhaps a more proper term than enemies) on both sides of the norbornyl ion controversy. It is to their credit that we today probably know more about the structure of carbocations, such as the norbornyl cation, than about most other chemical species. Their efforts also resulted not only in rigorous studies but also in the development or improvement of many techniques. Although many believe that too much effort was expended... 

During the early years of physieal ehemistry, Ostwald did not believe in the existence of atoms... and yet he was somehow ineluded in the wild army of ionists. He was resolute in his scepticism and in the 1890s he sustained an obscure theory of energetics to take the place of the atomic hypothesis. How ions could be formed in a solution containing no atoms was not altogether clear. Finally, in 1905, when Einstein had shown in rigorous detail how the Brownian motion studied by Perrin could be interpreted in terms of the collision of dust motes with moving molecules (Chapter 3, Section 3.1.1), Ostwald relented and publicly embraced the existence of atoms. 

Slightly removed from this in rigor is the use of a substituent to make a pure exchange into a net chemical reaction. No isotopic label is then needed. For example, the first reliable estimate of the rate constant for the exchange of ferrocenium ions and ferrocene was made on the basis of kinetic data for processes such as... 

X-ray diffraction of live muscle (H.E. Huxley and Brown, 1967) showed the structure of the thick and thin filaments and how they changed when the muscle contracted, or was put into rigor (in rigor muscle, ATP is absent from the muscle. 

Figure 9. A schematic representation of crossbridge orientation assumed from electron micrographs of insect flight muscle in relaxed and rigor states by Reedy et al. (1965). The crossbridge is thought to have an orientation of 90° to the thick filament axis in the relaxed state and an orientation of 45° to the thick filament axis in rigor.

Solution Repeat the calculations in Example 2.3 but now reduce At by a factor of 2 for each successive calculation rather than by the factor of 4 used in the examples. Calculate the corresponding changes in a(tmax) and denote these changes by A. Then A should decrease by a factor of 2 for each calculation of a(t ax)- (The reader interested in rigor will note that the error is halved and will do some algebra to prove that the A are halved as well.) If A was the change that just occurred, then we would expect the next change to be A/2, the one after that to be A/4, and so on. The total... 

Electroosmosis is used in medicine for introducing into the body continuously or periodically liquid medications in rigorously defined quantities. 

The diffuse layer is formed, as mentioned above, through the interaction of the electrostatic field produced by the charge of the electrode, or, for specific adsorption, by the charge of the ions in the compact layer. In rigorous formulation of the problem, the theory of the diffuse layer should consider ... 

In rigorous treatments, e.g., Newman (N8a, Ch. 20), the concentration overpotential is defined as the potential difference (excluding ohmic poten-... 

Until recently, advances in calculating the free energy were not accompanied by comparable progress in rigorous error analysis and reduction. Although a variety of methods to estimate the error in calculated free energies were proposed [32, 106], they were usually somewhat heuristic or involved approximations that were not always sufficiently well supported. Only recently, considerable progress has been made on this front, in particular by Daniel Zuckerman and Thomas Woolf [107]. 

In most cases oxidation of uncharged borabenzene complexes produces cations which can only be observed electrochemically. The iron compounds 62 and 63 may serve as an example. Oxidation is fully reversible in rigorously dried CH2C12 but irreversible in more basic solvents such as THF and acetonitrile (62). Preparative oxidation with Ce(IV) salts cleanly produces monosubstituted ferricenium cations 64 (Scheme 10) (66). In contrast to the above mentioned boranediyl extrusions, the substituent at boron is retained here in the newly formed cyclopentadienyl ring. 

Processes reported in Table 1 have been carried out at temperature ranging, typically, between 20 and 40°C. Though the investigations carried out at lower temperature are very few [26, 33], this issue holds a key role in the design and optimization of the conversion processes. Provided that the heating-up of the waste-water streams is not economically feasible, the remediation process should be carried out at low temperature, particularly pressing in rigorous climate countries [50]. 

Table 2.10 Reactions which should be considered in rigorous modeling of PET synthesis, PET processing and PET recycling...

Rates of reaction of urease with deuterated and especially tritiated urea were markedly reduced compared with the rate with the unlabeled ( H) substrate, but usually isotope effects are insignificant biochemically except in rigorous kinetic studies. 

The theoretical interest in the LiH has increased since the electron affinity of LiH and its deuterated counterpart, LiD, were measured with the use of the photoelectron spectroscopy by Bowen and co-workers [126]. The adiabatic electron affinities of LiH and LiD determined in that experiment were 0.342 0.012 eV for the former and 0.337 0.012 eV for the latter system. The appearance of these data posed a challenge for theory to reproduce those values in rigorous calculations based on the first principles. Since the two systems are small, it has been particularly interesting to see if the experimental EAs can be reproduced in calculations where the BO approximation is not assumed [123]. 

Semiempirical methods, on the other hand, utilize minimum basis sets to speed up computations, and the loss in rigor is compensated by the use of experimental data to reproduce important chemical properties, such as the heats of formation, molecular geometries, dipole moments, and ionization potentials (Dewar, 1976 Stewart, 1989a). As a result of their computational simplicity and their chemically useful accuracy, semiempirical methods are widely used, especially when large molecules are involved (see, for example, Stewart, 1989b Dewar et al., 1985 Dewar, 1975). 

Lithium aluminium hydride reacts violently with water, liberating hydrogen, and must therefore be used in rigorously anhydrous conditions, usually in ether solution. In fact, any solvent containing OH or NH groups would destroy the reagent by acting as a proton donor for hydride. In the case of LAH reductions, the addition of water as the proton source has to be carried out with considerable caution, since any unreacted LAH will react violently with this... 

Reduction potentials for polycyclic aromatic hydrocarbons in rigorously dried... 

In rigorous quantum mechanics, something like an electronic base function parametrically dependent on nuclear configuration space cannot be. Such dependence would imply that the electronic quantum number of the base function depends upon the particular selected region of nuclear configuration space. 

The infinite string of 8 functions causes certain difficulties in rigor when its Fourier transform is considered. Nevertheless, the concept is a very useful one. It is well worth defining a special symbol... 

An interesting method of fitting was presented with the introduction, some years ago, of the model 310 curve resolver by E. I. du Pont de Nemours and Company. With this equipment, the operator chose between superpositions of Gaussian and Cauchy functions electronically generated and visually superimposed on the data record. The operator had freedom to adjust the component parameters and seek a visual best match to the data. The curve resolver provided an excellent graphic demonstration of the ambiguities that can result when any method is employed to resolve curves, whether the fit is visually based or firmly rooted in rigorous least squares. The operator of the model 310 soon discovered that, when data comprise two closely spaced peaks, acceptable fits can be obtained with more than one choice of parameters. The closer the blended peaks, the wider was the choice of parameters. The part played by noise also became rapidly apparent. The noisy data trace allowed the operator additional freedom of choice, when he considered the error bar that is implicit at each data point. 

The 3,4-dihydrooxocin (147 X = CH2) is obtained in 20% yield by photolytic ring opening of the bicyclic pyran (146) in rigorously oxygen-free solution (78CB3608). The reaction,... 

This approach is therefore based in rigorous and general geometric tensor theory. The PL vector dual to turns out to be the light-like invariant ... 

This paper discusses three aspects of our extensive but as yet incomplete studies of the reaction oxygen uptake in rigorously purified materials, the effects of added iron complexes, and the influence of a well-known radical capture agent. An interesting feature of the last, anticipated from the work of Rosenwald (19) on inhibitor sweetening, was the large accelerating effect of N,N -di- ec-butyl-p-phenylenediamine on the co-oxidation reaction. 

The BDEs are well-defined thermodynamic quantities that can be employed in rigorous first-law calculations, but their values strongly differ in each of the above steps. Only the average of the four BDE[CH —H] values could be taken as an estimate of bond enthalpy DH° [C—H],... 

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