Pair fraction

The configurational entropy of the solution can now be expressed in terms of the pair fractions, Xjj, defined by... 

Finally, it should be noted that the model can be extended to multi-component systems [6], Furthermore, the Gibbs energy of the pair exchange reaction may depend on the relative proportions of the different pairs. In this case tab is a polynomial function of the pair fractions XAA and XBB [4],... 

Table 2.8 Coefficients A for silicate - pair fractionations (1,000 Ina X —Y = A/T IO (after Chiba et al. 1989)...

A commercial customer had four compounds to recover from a synthesis mixture they separated as two pairs of compounds. They injected and collected each pair together, then diluted each pair with water, reinjected onto the reverse-phase column, and ran shave/recycle. Using this technique they purified 50gm of each compound in the two injections of the pair fractions on the 3-in column. 

Recently a cobaloxime complex with a racemic 1-cyanoethyl group and methyl" (S)-alaninate as axial ligands was prepared [41]. In order to resolve the diastereo-meric pair, fractional crystallization was performed several times. However, the, optical rotation of the complex did not increase after several times of crystalliza-S tion. One of the crystals suitable for x-ray work was picked up and the structure was analyzed. To our surprise, there are two crystallographically independent molecules, which are a pair of diastereomers, in a chiral unit cell. This explains why the fractional resolution of the diastereomeric pair was impossible in the usual way. f... 

During metamorphism and recrystallization, oxygen isotopes are redistributed among mineral phases, according to the mass-dependent equilibrium fractionations corresponding to the peak metamorphic temperature. The measured mineral-pair fractionations (usually for major minerals olivine, pyroxene, and feldspar) can then be used for metamorphic thermometry, yielding temperatures of 600 °C for an L4 chondrite, and 850 50 °C for several type-5 and type-6 chondrites (Clayton et al., 1991). Isotopic equilibration, even in type-6 chondrites, involves oxygen atom transport only over distances of a few millimeters (Olsen et al., 1981). 

Even though there were some differences in experimental methodologies, the agreement between the various studies was in general excellent for some of the major mineral-water systems (e.g. albite-water, quartz- and calcite-water systems). Matthews et al. (1983a) and Matthews (1994) provide compilations of mineral-pair fractionation factors derived from the high-temperature hydrothermal experiments. 

Coefficients for mineral-pair fractionation factors of the form 1000 In a = A x 10 /T (K), where the coefficient A is given in the table. 

Mineral-pair fractionation factors obtained with the carbonate-exchange technique have gained wide, but not universal, acceptance. In particular. Sharp and Kirschner... 

The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives... 

Methane, CH4, for example, has a central carbon atom bonded to four hydrogen atoms and the shape is a regular tetrahedron with a H—C—H bond angle of 109°28, exactly that calculated. Electrons in a lone pair , a pair of electrons not used in bonding, occupy a larger fraction of space adjacent to their parent atom since they are under the influence of one nucleus, unlike bonding pairs of electrons which are under the influence of two nuclei. Thus, whenever a lone pair is present some distortion of the essential shape occurs. 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

An example of a commercial semibatch polymerization process is the early Union Carbide process for Dynel, one of the first flame-retardant modacryhc fibers (23,24). Dynel, a staple fiber that was wet spun from acetone, was introduced in 1951. The polymer is made up of 40% acrylonitrile and 60% vinyl chloride. The reactivity ratios for this monomer pair are 3.7 and 0.074 for acrylonitrile and vinyl chloride in solution at 60°C. Thus acrylonitrile is much more reactive than vinyl chloride in this copolymerization. In addition, vinyl chloride is a strong chain-transfer agent. To make the Dynel composition of 60% vinyl chloride, the monomer composition must be maintained at 82% vinyl chloride. Since acrylonitrile is consumed much more rapidly than vinyl chloride, if no control is exercised over the monomer composition, the acrylonitrile content of the monomer decreases to approximately 1% after only 25% conversion. The low acrylonitrile content of the monomer required for this process introduces yet another problem. That is, with an acrylonitrile weight fraction of only 0.18 in the unreacted monomer mixture, the low concentration of acrylonitrile becomes a rate-limiting reaction step. Therefore, the overall rate of chain growth is low and under normal conditions, with chain transfer and radical recombination, the molecular weight of the polymer is very low. 

Binary interaction parameters are determined for each pq pair p q) from experimental data. Note that = k and k = k = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. 

Ternary-phase equilibrium data can be tabulated as in Table 15-1 and then worked into an electronic spreadsheet as in Table 15-2 to be presented as a right-triangular diagram as shown in Fig. 15-7. The weight-fraction solute is on the horizontal axis and the weight-fraciion extraciion-solvent is on the veriical axis. The tie-lines connect the points that are in equilibrium. For low-solute concentrations the horizontal scale can be expanded. The water-acetic acid-methylisobutylketone ternary is a Type I system where only one of the binary pairs, water-MIBK, is immiscible. In a Type II system two of the binary pairs are immiscible, i.e. the solute is not totally miscible in one of the liquids. 

Solution Henstebeck diagrams were produced based upon using different key component pairs. These showed where pinch points were occurring and the resulting understanding allowed a proper computer simulation to be selected. (See Section 3 Fractionators Graphical... 

The technique of INS is probably the least used of those described here, because of experimental difficulties, but it is also one of the physically most interesting. Ions of He" of a chosen low energy in the range 5-10 eV approach a metal surface and within an interaction distance of a fraction of a nanometer form ion-atom pairs with the nearest surface atoms. The excited quasi molecule so formed can de-excite by Auger neutralization. If unfilled levels in the ion fall outside the range of filled levels of the solid, as for He", an Auger process can occur in which an electron from the va-... 

In practice, inverse transformation is most easily achieved by using partial fractions to break down solutions into standard components, and then use tables of Laplace transform pairs, as given in Table 3.1. 

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