HW-07%20Solution

# HW-07%20Solution - #1 12.2 In this problem we are asked to...

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12.2 In this problem we are asked to show that the minimum cation-to-anion radius ratio for a coordination number of four is 0.225. If lines are drawn from the centers of the anions, then a tetrahedron is formed. The tetrahedron may be inscribed within a cube as shown below. The spheres at the apexes of the tetrahedron are drawn at the corners of the cube, and designated as positions A , B , C , and D . (These are reduced in size for the sake of clarity.) The cation resides at the center of the cube, which is designated as point E . Let us now express the cation and anion radii in terms of the cube edge length, designated as a . The spheres located at positions A and B touch each other along the bottom face diagonal. Thus, AB = 2 r A But ( AB ) 2 = a 2 + a 2 = 2 a 2 or AB = a 2 r A And a 2 r A 2

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There will also be an anion located at the corner, point F (not drawn), and the cube diagonal AEF will be related to the ionic radii as AEF = 2 ( r A + r C ) (The line AEF has not been drawn to avoid confusion.) From the triangle ABF ( AB ) 2 + ( FB ) 2 = ( AEF ) 2
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## This note was uploaded on 09/15/2010 for the course MATSCIE 250 taught by Professor Yalisove during the Fall '08 term at University of Michigan.

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HW-07%20Solution - #1 12.2 In this problem we are asked to...

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