A General Approach to Radius Ratios of Simple Ionic Crvstals Hasan Giil Middle East Technical University, Ankara, Turkey Many textbooks of chemistry1 introduce the idea of radius ratio to predict the structure of ionic crystals. The calculations for different structures, however, are carried out separately for each structure in question, and the radius ratios calculated are taken as the limiting values because of the simplicity of the geometrical approach. A general approach to calculate these limiting values can easily be done by applying the Law of Cosines. Such an approach, t o my knowledge, has not been described in common textbooks. T o illustrate this approach, let us take the cross section as shown a t the right. Using the Law of Cosines, the equation for the radius ratio for any structure (considering that the equation holds for 0 < 90") can be related to the cosine of the angle between the lines connecting the centers of two anions and a cation. The angle, in turn, can be calculated for each structure using simple geometry. Note that only the cosine of the angle needs to he known (not the angle itself), which is much easier to compute. The box below shows the angle and the computations for the cosines for different limiting structures. This general approach has the advantage of allowing all structures t o be handled using one equation. The author acknowledges fruitful discussions with $ Siizer.
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Journal of Chemical Education
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See for example: Mahan, B. H., "University Chemistry," 2nd ed., Addison-Wesley. 1972; Huheey, J. E., "inorganic Chemistry," 2nd ed., Harper and Row. New York, 1978; Cotton, F. A,, and Wilkinson, G., "Advanced Inorganic Chemistry," 4th ed., Wiley. 1980.
