Fractionated crystallization definition

Figure 5. A schematic representation of superposed steady-state reservoirs of constant volumes Vi (fractional crystallization is omitted in this schema). At steady-state, Vi/xi=V2/x2=..., where x is the residence time. This is analogous to the law of radioactive equilibrium between nuclides 1 and 2 Ni/Ti=N2/T2=...A further interest of this simple model is to show that residence times by definition depend on the volume of the reservoirs.

Another expression that needs a definition is fractional crystallization. Fractional crystallization is a repetition of a crystallization process performed in order to receive a further purified... 

A variety of 3-D superlattices formed from metal nanopartides has been reported [1-8]. For instance, gold nanopartides, when stabilized by alkylthiolates RS (R = n-Ci2H2s) and rather narrowly distributed in size, could be obtained by the repeated fractional crystallization of a mixture of 1.5-3.5 nm particles from unpolar solvents. The success of fractionation was controlled by mass spectroscopy [6j. Based on these data, definite numbers of atoms could be assigned to the particles of distinct fractions (e.g., 140, 225, 314, and 459), all of which consisted of fee-structured octahedra. Solutions of these remarkably well-defined particles can be used to... 

Isomers of i-B,OH-Bio were separated by fractional crystallization and column chromatography. H, Si, and MALDI-TOF MS and single crystal XRD definitely confirmed the stracture of each isomer. 

But even if we leave the enigmatical radon family alone the situation still remains unclear. In 1900 W. Crookes observed a strange phenomenon. After fractional crystallization of a uranium compound he obtained a filtrate and a precipitate. Uranium remained in the solution but it exhibited no activity. On the contrary, the precipitate did not contain uranium but exhibited a high-intensity radioactivity. On the strength of his observations Crookes made a paradoxical conclusion that uranium was not radioactive by itself, and its radioactivity was due to some admixture which he managed to separate from uranium. As if he had ill premonitions, Crookes refrained from giving the admixture any definite name and referred to it as uranium-x (UX). Later it was found that uranium restores its activity after... 

The dominant crystal size is given by = 3Gr. This quantity is also the ratio mJwhich is often given the symbol 2-(J) Prom the definition of the coefficient of variation given by equation 41, cv = 50% for an MSMPR crystallizer. Such a cp may be too large for certain commercial products, which means either the crystallizer must be altered or the product must be screened to separate the desired fraction. 

Except to note that the occurrence of the coefficients 15 and 10 in 155 and lOD obviate the need for fractions here or elsewhere in crystal-field theory thus, they are there for reasons of convenience and definition only. 

De Man (1983) has reviewed this property of fats. Consistency is defined as (1) an ill-defined and subjectively assessable characteristic of a material that depends on the complex stress-flow relation or as (2) the property by which a material resists change of shape. Spreadabil-ity, a term used in relation to consistency, is the force required to spread the fat with a knife. The definition is similar to that for hardness the resistance of the surface of a body to deformation. The most widely used simple compression test in North America is the cone penetrometer method (AOCS Method Cc 16-60, 1960). More sophisticated rheological procedures are also available. Efforts have been made to calibrate instrumental tests with sensory response. With the cone penetrometer method, penetration depth is used as a measure of firmness. Hayakawa and De Man (1982) studied the hardness of fractions obtained by crystallization of milk fat. Hardness values obtained with a constant speed penetrometer reflected trends in their TG composition and solid fat content. 

The results of the discussion on the phenomenological thermodynamics of crystals can be summarized as follows. One can define chemical potentials, /jk, for components k (Eqn. (2.4)), for building units (Eqn. (2.11)), and for structure elements (Eqn, (2.31)). The lattice construction requires the introduction of structural units , which are the vacancies V,. Electroneutrality in a crystal composed of charged SE s requires the introduction of the electrical unit, e. The composition of an n component crystal is fixed by n- 1) independent mole fractions, Nk, of chemical components. (n-1) is also the number of conditions for the definition of the component potentials juk, as seen from Eqn. (2.4). For building units, we have (n — 1) independent composition variables and n-(K- 1) equilibria between sublattices x, so that the number of conditions is n-K-1, as required by the definition of the building element potential uk(Xy For structure elements, the actual number of constraints is larger than the number of constraints required by Eqn. (2.18), which defines nk(x.y This circumstance is responsible for the introduction of the concept of virtual chemical potentials of SE s. 

Let us now discuss some details of practical relevance. From the Gibbs phase rule, it is evident that crystals consisting of only one component (A) become nonvariant by the predetermination of two thermodynamic variables, which for practical reasons are chosen to be Pand T. In these one-component systems, it is easy to recognize the (isobanc) concentration dependence of the point defects on temperature. From the definition of the vacancy chemical potential for sufficiently small vacancy mole fractions Nv, namely //v = /A (P, T) + RT- In Vv, together with the condition of equilibrium with the crystal s inerL surroundings (gas, vacuum), one directly finds... 

Some very interesting ideas concerning the relationship between free-volume formation and the energy of one mole of hole formation were developed in detail by Kanig42. Kanig introduced some improvements to the definition of free-volume, On the basis of Frenkel s ideas43 he divided the free-volume into two parts, one of which is determined only by the thermal vibrations of atoms in the lattice of a real crystal while the other is connected with inherent free-volume, i.e. voids and holes. It is the latter that makes possible the exchange of particles, i.e. the very existence of the liquid state. He introduced some new definitions of fractions of free-volume ... 

An alternative and perhaps more accurate definition of thermodynamic fragility that can be determined experimentally for almost all liquids would be to determine the fraction of entropy of fusion lost by T/Tm = 0.8 [37,88]. In both the definitions, the property of the crystal phase enters this disadvantage is missing in the kinetic measures of fragility [37,88]. 

When the crystal surface contains sites of different adsorption energies the adsorbate will condense area by area. According to Predali and Cases89) the adsorption isotherm will be steplike in relation to condensation of molecules on areas i, and each step i is compared with sites having the same energy 0 j. Developing the isotherm with condensation on inhomogenous surfaces they start with the definition of the fraction of sites a and /S on which condensation occurs between undersaturation Afi and dAft. When all the... 

Cocoa butter (CB) has a challenging chemistry and has attracted many efforts to develop lower cost, acceptable alternatives. The following definitions provide a quick introduction to this field (1) cocoa butter equivalents (CBEs) are compounded mostly from tropical oils other than palm. Because their melting and crystallization properties closely resemble CB, they are compatible as diluents at all levels of substitution (2) cocoa butter replacers (CBRs) are made from nondairy oils (typically soybean, cottonseed, or palm) partially hydrogenated for maximum trans-CIS isomer formation to acquire a steep melting profile. They are best used for enrobing bakery products, but their melting profiles can be improved by chill fractionation and (3) cocoa butter substitutes (CBSs) are made primarily from... 

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