Triangular coordinates

Dreieck, n. triangle, dreieckig, a. triangular, three-cornered. Dreiecks-koordinaten, f.pl. triangular coordinates. -lehre, /. trigonometry. Dreier-gemisch, n. triple mixture, -gruppe, /. 

In Figure 4.9 Snyder s solvent selectivity triangle is presented. The solvents of Table 4.2 are marked in the plot with triangular coordinates for the eight groups. 

The methods for doing so are described in Chapter 9. The basic principles remain unchanged—the primary difference is the choice of a consistent basis for calculation, such as a solvent-free basis. Graphic techniques based on triangular coordinates provide approximate answers, but modern computational techniques are to be preferred. 

Three-component mixtures can be represented on triangular coordinates and the reasoning used in three-component designs can be extended to four or more components. Some typical three-component designs follow. 

This will allow use of the familiar Gibbs triangular coordinates for displaying data. 

In metal peroxide chemistry, the heterolytic or homolytic nature of catalytic oxidation seems to be strongly dependent on the heterolytic or homolytic dissociation mode of the peroxide intermediate, for which the triangular coordination mode of the peroxide moiety of the metal appears to be a key feature. Heterolytic oxidations require attainable coordination sites on the metal, involve strained metallacyclic reaction intermediates, and are highly selective. In contrast, homolytic oxidations involve bimolecular radical processes with no metal-substrate interactions and are less selective. In the important field of palladium oxidation chemistry, hydroperoxo... 

In the molecules of [InOAr]2 and InSn(OBu )j the indium atoms display a /-triangular coordination [ ln 02]. 

If inlet conditions (F, S, xF, and ys) are known, we have four unknown variables (R, E, x, and y). However, since we have only three equations, we need additional information to be able to solve for the unknown variables, which are the equilibrium data of the ternary system solute, solvent, and diluent, which are usually described graphically in triangular coordinates (Treybal, 1980). 

Only oxides A2O3, AO2, and A2O5 satisfy these requirements. Triangular coordination with A2O3 and tetrahedral coordination with AO2 and A2O5 are possible. 

Construction of a complete diagram which represents all these variables would require a four-dimensional space. However, if the pressure is assumed constant (customarily at 1 atm), the system can be represented by a three dimensional diagram with three independent variables, i.e., temperature and two composition variables. In plotting three dimensional diagrams, it is customary that the compositions are represented by triangular coordinates in a horizontal plane and the temperature in a vertical axis. 

Treybal, in his book Liquid Extraction [1], works equilibrium material balances with triangular coordinates. The most unique and simple way to show three-phase equilibrium is a triangular diagram (Fig. 7.1), which is used for extraction unit operation in cumene synthesis plants [2], In this process benzene liquid is used as the solvent to extract acetic acid (the solute) from the liquid water phase (the feed-raffinate). The curve D,S,P,F,M is the equilibrium curve. Note that every point inside the triangle has some amount of each of the three components. Points A,... 

The synthesis of zeolite A, mixtures of A and X, and zeolite X using batch compositions not previously reported are described. The synthesis regions defined by triangular coordinates demonstrate that any of these materials may be made in the same area. The results are described in terms of the time required to initiate crystallization at a given reaction temperature. Control of the factors which can influence the crystallization time are discussed in terms of "time table selectors" and "species selectors . Once a metastable species has preferentially crystallized, it can transform to a more stable phase. For example, when synthesis conditions are chosen to produce zeolite A, the rate of hydroxysodalite formation is dependent on five variables. These variables and their effect on the conversion of zeolite A to hydroxysodalite are described mathematically. 

Fig. 4.1.8. Triangular coordinate system displaying the three types of lignin (C, GS, and HGS) in realtion to the composition of the lignins (H, G, and S units expressed as mol % and the methoxyl contents as the number/100 phenylpropane units). According to Faix 1991...

In general, the crystal chemistry of borates is similar to that of silicates differences arise from the fact that boron combines with oxygen not only in four-fold (tetrahedral) but also in three-fold (trigonal planar = triangular) coordination. As a result, silicate chemistry is considerably less complicated than borate chemistry. 

The phase behaviour of a mixed biopolymer solution is quantitatively characterized by a phase diagram, which graphically describes the boundary conditions of phase separation, the partitioning of the components between the phases and the effects of different variables (temperature, pH, salt concentration, etc.) on the phase behaviour of biopolymers. Conventionally, phase diagrams of three-component (ternary) systems are presented in triangular coordinates. However, an excess of solvent water... 

If, on the other hand, the radius ratio rules are violated on the downside, that is to say the hard sphere cation and anion no longer touch, the structure is longer stable and another structure should be adopted. For instance, if r+/r < 0.224, then tetrahedral coordination is no longer possible but a threefold planar triangular coordination can be found by locating the cation in the trigonal hole centered in the plane of the close-packed layer. Such coordination should be stable for 0.15 < r+/r < 0.224. In fact, however, trigonal coordination is rare in chemical systems other than those of boron. 

Tossell, J. A. (1978b). Theoretical studies of the electronic structure of copper in tetrahedral and triangular coordination with sulfur. Phys. Chem. Minerals 2, 225-36. 

Vaughan, D. J., and J. A. Tossell (1980). The chemical bond and the properties of sulfide minerals I. Zn, Fe and Cu in tetrahedral and triangular coordinations with sulfur. Canad. Mineral. 18, 157-63. 

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