Tetrahedron spacing

Figure 4.20 Tetrahedron space model for the phase diagram of the quaternary system A-B-C-D. The isotherms T], 72, T3 are shown for the primary phase volume of component A [17]. Reprinted with permission of The American Ceramic Society, www.ceramics.org. Copyright [1984]. All rights reserved.

Table B8 The directions of the four nucleic bases in the tetrahedron space as proposed by [Randic, Vracko et al., 2000].

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. 

It is often difficult to represent inorganic compounds with the usual structure models because these structures are based on complex crystals space groups), aggregates, or metal lattices. Therefore, these compounds are represented by individual polyhedral coordination of the ligands such as the octahedron or tetrahedron Figure 2-124d). 

Stereochemistry refers to chemistry in three dimensions Its foundations were laid by Jacobus van t Hoff and Joseph Achille Le Bel m 1874 Van t Hoff and Le Bel mde pendently proposed that the four bonds to carbon were directed toward the corners of a tetrahedron One consequence of a tetrahedral arrangement of bonds to carbon is that two compounds may be different because the arrangement of their atoms m space IS different Isomers that have the same constitution but differ m the spatial arrangement of their atoms are called stereoisomers We have already had considerable experience with certain types of stereoisomers—those involving cis and trans substitution patterns m alkenes and m cycloalkanes... 

A most important early addition to organic structure theory was made by the first Nobel Laureate in Chemistry, van t Hoff, who in 1874 recognized that the optical activity of carbon compounds can be explained by the postulate that the four valence bonds of the carbon atom are directed in space toward the comers of a tetrahedron. 

The valence bonds of carbon have bxed directions and are equidistant in space, pointing from the center to the corners of a tetrahedron forming an angle of 109°. Thus, in propane, which we usually write as CH3-CH2 CH3, the carbon atoms are not connected in a straight line, but are actually as shown in the above three-dimensional diagram. 

The concept of metric tensor becomes central whenever distances and projections are considered, particularly when least-square criterion are used, a point that will be discussed in Chapter 5. Let us ask the frequently raised question of how to find an expression in terms of old coordinates (e.g., oxide proportions) for a projection made in the non-Euclidian space. This could be the case for finding oxide abundances of a basalt composition projected in the Yoder and Tilley tetrahedron, or the oxide abundance of a metamorphic rock composition projected into an ACF diagram assuming that quartz is present. 

But what about the three-dimensional images or formulations of molecules What about "la chimie en l espace" introduced by Joseph Achille Le Bel, van t Hoff, and Wislicenus toward the end of the nineteenth century Were these carbon tetrahedra realistic "models" of real molecules in space Van t Hoff argued in favor of the carbon tetrahedron that if atoms were arranged in a plane, there would be more isomers of the type CR1R2R3R4 predicted in principle than are actually observed. With the tetrahedral structure, only two isomers are possible, related to each other as mirror images. 102... 

It often is noted that van t Hoff made the apparently "positivist" statement that "the representations themselves, atom, molecule, their dimensions, and perhaps their shapes, are after all something doubtful, as is the tetrahedron itself. "103 However, as the theoretical chemist Roald Hoffmann has aptly suggested, this does not necessarily mean that van t Hoff doubted "chemistry in space" but perhaps that he did not feel committed to the exact geometrical form that is the Platonic tetrahedron, a form that is not only idealized but static. 104... 

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