Solid empirical rule

The importance of the solid-liquid interface in a host of applications has led to extensive study over the past 50 years. Certainly, the study of the solid-liquid interface is no easier than that of the solid-gas interface, and all the complexities noted in Section VIM are present. The surface structural and spectroscopic techniques presented in Chapter VIII are not generally applicable to liquids (note, however. Ref. 1). There is, perforce, some retreat to phenomenology, empirical rules, and semiempirical models. The central importance of the Young equation is evident even in its modification to treat surface heterogeneity or roughness. ... 

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

Most of the topochemical reactions, including the first finding of the topochemical polymerization of 1, were found accidentally or beyond any expectation [ 14]. During subsequent and extensive investigation, numerous studies have been carried out by trial-and-error approaches to establish any empirical rule accounting for a correlation between the geometry of functional moieties and reactions in the solid... 

Both L coefficients and / factors can, in principle, be calculated from microscopic models. For the evaluation of L,j, the random-alloy model [J. R. Manning (1968) A. R. Allnatt, A. B. Lidiard (1987)] is sometimes used. For the evaluation of thermodynamic factors, one takes advantage of the empirical rule that in extended solid solutions AO-BO, the cation vacancy concentration and the oxygen potential are related to each other as... 

The theory of chemical bonding is overwhelmed by a host of insurmountable obstacles the real orbitals and hybrids of LCAO have no physical, chemical or mathematically useful attributes - certainly not in the quantum-mechanical sense the distribution of electron density between atoms, in the form of spin pairs, is an overinterpretation of the empirical rules devised to catalogue chemical species the structures, assumed in order to generate free-molecule potential fields, are only known from solid-state diffraction experiments the assumption of directed bonds is a leap of faith, not even supported by crystal-structure analysis. The list is not complete. 

One feature that should be noted in passing is that the bulk modulus varies approximately as d in contrast to covalent solids, in which it varies as d (We shall see that in metals it also varies as d. ) As a consequence, the bulk modulus for the alkali halides can be given the universal value of 6.7 eV per ion pair. It is not clear what the significance of this empirical rule is. 

Another empirical rule is the Heat of Fusion Rule, which states that if the higher melting form has a lower heat of fusion relative to the lower melting form, then the two forms bear an enantiotropic relationship. Less well obeyed is the Density Rule, which states that the most dense form will be the most stable at absolute zero. Strictly speaking, the Density Rule is only properly applied to polymorphs of molecular solids where intramolecular hydrogen bonding is not a significant factor. 

A selection of values of the Debye function are tabulated in table 12.3. As T increases D Tj0) tends to unity, so that at temperatures above the characteristic temperature we have a theoretical justification of the empirical rule of Dulong and Petit, namely that 3R for atomic solids. On the other hand at low temperatures D Tj ) tends to zero, and for T/ <0T we have the simple approximate formula... 

The estimation of the free enthalpy of formation using the data from Tables 2.2.8 to 2.2.10 leads to values for the gaseous state of the molecule (in some instances hypothetical state). However, few compounds and no polymers are in gas phase. For this reason, correction must be made to the gaseous state of a compound in order to estimate the free enthalpy for a specific reaction involving compounds in liquid or solid (crystalline or amorphous) state. A number of empirical rules are available for the change from gas phase to a condensed phase, and a selection of such corrections is given in Table 2.2.11 [35]. 

At extremely high and at extremely low temperatures a complete breakdown of the purely empirical rule is to be expected according to our approximation formula. Two cases of this sort, which are particularly telling, have been calculated out by Wintemitz (98a), namely, the sublimation pressure of solid tungsten and that of solid hydrogen. 

The photophysical properties of a molecule are influenced by its immediate surroundings. Often the effects follow a qualitative empirical rule or, in favorable situations, a well-established quantitative law. Thus, it becomes possible to use photophysical parameters in order to get insight into the microenvironment (e.g. polarity, viscosity) of the emitting molecule. Fluorescence dyes have been used as molecular reporters, not only in solution, but also in membranes, in the solid state, in mixtures of low molecular compounds or... 

The study of the coagulation of aqueous dispersions of sparingly soluble or insoluble solids (lyophobic colloids) has been one of the main streams of colloid research for over a century. This arises both from its practical importance and from the fact that it was the first colloidal phenomenon for which empirical rules governing its onset were established. Consequently it was also the first for which attempts were made to develop a theoretical interpretation. 

Hume-Rothery rules - A set of empirical rules for predicting the occurrence of solid solutions in metallic systems. The rules involve size, crystal structure, and electronegativity. 

Markownikoff s rule states that during the addition of H X to an asymmetrically substituted olefin the negative part becomes attached to the unsaturated carbon carrying the smaller number of hydrogen atoms Originally this was an empirical rule, but in the meantime quantuni chemistry and molecular orbital theory have provided a solid theoretical background to it. In the particular case of propylene, for instance, it has been calculated that the rr-electron density distribution is as follows ... 

Specific heat capacities of solid substances near normal atmospheric temperature can be estimated with a reasonable degree of accuracy by combining two empirical rules. 

Hume-Rothery rules A set of empirical rules put forward in 1926 by the British metallurgist William Hume-Rothery (1899-1968) to describe how one metal dissolves in another metal. He found that one metallic element wUl not dissolve in another if the difference in their atomic radii Is greater than 15 per cent or if the electronegativities of the two elements differ substantially. Also, a metal with a lower valency is more likely to dissolve In one with a higher valency, than vice versa. These rules were subsequently justified by the quantum theory of electrons in solids. 

They have analyzed Rl-dispersion data in more than a hundred different solids and liquids and established that the parameter Ed, which is a measure of the strength of inter-band optical transitions, obeys a simple empirical rule... 

The carbides and nitrides of group 4-6 transition metals belong to the so-called Hagg compovmds (6). Hagg formulated a set of empirical rules for the crystal structures of interstitial solid solutions of transition metals. According to the rules, when the radius ratio r = rx/fmetai is less than 0.59, small atoms (such as carbon and nitrogen) occupy the interstitial sites of the simple crystal structures formed by the metal atoms. For the carbides and nitrides of Ti, V, Nb, Mo, and W, which have been extensively applied as catalysts, the radius ratios fall in between 0.491 r lr i) and 0.576 (rc/ry). They are indeed Hagg compounds (6). Table 1 summarizes the crystal structures and lattice parameters of carbides of Mo, W, V, Nb, and Ti. 

At room temperature, the molar heat capacity of most solids, especially metals, can be approximated using the empirical rule of Dulong and Petit, which is based on many experimental studies and states that the molar heat capacity is three times the constant for ideal gases ... 

In practice, in the technical Hterature many tables Hst the latent enthalpies of fusion and vaporization of usual compounds, but few data are given regarding the latent enthalpy of sublimation. When such a value is missing, a rule of thumb for assessing the latent enthdpy consists in using the Trouton s first empirical rule, i.e., the latent molar enthalpy of sublimation of a solid corresponds to the sum of its latent molar enthalpies of fusion and of vaporization as follows ... 

There is no statistical theory of the solid-liquid transition, even for hard spheres. So, to estimate volume fractions of coexisting phases, Marcelja et al. resorted to an intuitive approach based on Lindemann s empirical rule, which states that a solid will melt when the root mean-square displacement of particles about their equilibrium positions exceeds some characteristic fraction /l of the lattice spacing. For potentials of functional form r " (n >4) it is found that /l 0.10. For particles of charge zq occupying volume 4ira /3 on a f.c.c. lattice with spacing b, the Lindemann ratio is... 

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