Earth Materials. John O'Brien. Читать онлайн. Newlib. NEWLIB.NET

Автор: John O'Brien
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: География
Год издания: 0
isbn: 9781119512219
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polyhedron is determined by the cation : anion radius ratio.

       Rule 2: An ionic structure is stable when the sum of the strengths of all the bonds that join the cation to the anions in the polyhedron equals (balances) the charge on the cations and on the anions. This rule is called the electrostatic valency rule.

       Rule 3: The sharing of edges and particularly faces by adjacent anion polyhedral elements decreases the stability of an ionic structure. Similar charges tend to repel. If they share components, adjacent polyhedra tend to share corners, rather than edges, to maximize cation spacing.

       Rule 4: Cations with high valence charges and small coordination number tend not to share polyhedral elements. Their large positive charges tend to repel.

       Rule 5: The number of different cations and anions in a crystal structure tends to be small. This is called the rule of parsimony.

      Pauling's rules provide important tools for understanding crystal structures. Especially important is the rule concerning radius ratio and coordination polyhedra. Coordination polyhedra provide a powerful means for visualizing crystal structures and their relationship to crystal chemistry. In fact, they provide a fundamental link between the two. When atoms and ions combine to form crystals, they bond together into geometric patterns in which each atom or ion is bonded to a number of nearest neighbors. The number of nearest neighbor ions or atoms is called the coordination number (CN). Clusters of atoms or ions bonded to other coordinating atoms produce coordination polyhedron structures. Polyhedrons include triangles, cubes, octahedra, tetrahedra, and other geometric forms.

      When ions of opposite charge combine to form minerals, each cation attracts as many nearest neighbor anions as can fit around it as approximate “spheres in contact”. In this way, the basic units of crystal structure are formed which grow into crystals as multiples of such units are added to the existing structure. One can visualize crystal structures in terms of different coordinating cations and coordinated anions that together define a simple three‐dimensional polyhedron structure. As detailed in Chapters 3 and 4, complex polyhedral structures develop by linking of multiple coordination polyhedra.

Radius ratio (Rc/Ra) Coordination number Coordination type Coordination polyhedron
<0.155 2 Linear Line
0.155–0.225 3 Triangular Triangle
0.225–0.414 4 Tetrahedral Tetrahedron
0.414–0.732 6 Octahedral Octahedron
0.732–1.00 8 Cubic Cube
>1.00 12 Cubic or hexagonal closest packed Cubeoctahedron complex

      When predicting coordination number using radius ratios, several caveats must be kept in mind.

      1 The ionic radius and coordination number are not independent. As illustrated by Table 2.6, effective ionic radius increases as coordination number increases.

      2 Since bonds are never truly ionic, models based on spheres in contact are only approximations. As bonds become more covalent and more highly polarized, radius ratios become increasing less effective in predicting coordination numbers.

      3 Radius ratios do not successfully predict coordination numbers for metallically bonded substances.

      The great value of the concept of coordination polyhedra is that it yields insights into the fundamental patterns in which atoms bond during the formation of crystalline materials. These patterns most commonly involve threefold (triangular), fourfold (tetrahedral), sixfold (octahedral), eightfold (cubic) and, to a lesser extent, 12‐fold coordination polyhedra or small variations of such basic patterns. Other coordination numbers and polyhedron types exist, but are rare in inorganic Earth materials.

      Another advantage of using spherical ions to model coordination polyhedra is that it allows one to calculate the size or volume of the resulting polyhedron. In a coordination polyhedron of anions, the cation–anion distance is determined by the radius sum (R). The radius sum is simply the sum of the radii of the two ions (Rc + Ra); that is, the distance between their respective centers. Once this is known, the size of any polyhedron can be calculated using the principles of geometry. Such calculations are beyond the scope of this book but are discussed in Wenk and Bulakh (2016) and Klein and Dutrow (2007).

      2.4.2 Electrostatic valency

      An important concept related to the formation of coordination polyhedra is electrostatic valency (EV). In a stable coordination structure, the total strength of all the bonds that reach a cation from all neighboring anions is equal to the charge on the cation. This is another way of saying that the positive charge on the cation is neutralized by the electrostatic component of the bonds between it and its nearest neighbor anions. Similarly, every anion in the structure is surrounded by some number of nearest neighbor cations to which it is bonded, and the negative charges on each anion are neutralized by the electrostatic component of the bonds between it and its nearest neighbor cations. For a cation of charge Z bonded to a number of nearest neighbor anions (CN), the electrostatic valency of each bond is given by the charge of the cation divided by the number of nearest neighbors to which it is coordinated:

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