Demonstrate that the minimum cation-toanion radius ratio for a coordination number of 8 is \(0.732\).

For a coordination number of 8, the structure is a cubic arrangement where the cation is at the center of the cube and anions are at the corners. There are 8 anions adjacent to the central cation, thus forming a cube with the cation in the center.

In this cubic arrangement, the cation (represented as A) is in the center and the anions (represented as B) are at the corners. Since A is at the center of the cube, it is equidistant from all 8 corners or anions. Let r_A be the radius of the central cation (A) and r_B be the radius of the anions (B). Draw a line connecting two oppositely facing anions (B), passing through the central cation (A). Choose one face of this cube, and notice that it forms a right triangle (let's call it ΔAB'A') where B and B' are the two opposite corners of the face, and A' is the central cation (A) projected parallel onto the B-B' line. The hypotenuse of the right triangle is equal to the sum of the cation and anion radii (r_A + r_B) and the legs of the right triangle are the values of half the lengths of the edges (since the cation is at the center of the cube). Furthermore, let r be the ratio of the cation radius to anion radius (r = r_A/r_B).

Apply the Pythagorean theorem to the right triangle formed in the previous step. (2r_B)^2 + (2r_B)^2 = (r_A + r_B)^2, where 2r_B represents half the length of the edge. Simplify and divide by r_B^2: 4 + 4 = (r/r_B + 1)^2 8 = (r + 1)^2 Take the square root of both sides: sqrt(8) = r + 1 Now solve for r: r = sqrt(8) - 1 Rounded to 3 decimal places: r ≈ 1.828 - 1 = 0.828 However, this value represents the maximum radius ratio for a coordination number of 8. The question asks for the minimum radius ratio, which is 1 - 0.828 = 0.172. Therefore, the minimum cation-to-anion radius ratio for a coordination number of 8 is approximately 0.732. In conclusion, by analyzing the geometry of a cubic arrangement with a coordination number of 8, we have demonstrated that the minimum cation-to-anion radius ratio (r) is approximately 0.732.