Demonstrate that the minimum cation-to-anion radius ratio for a coordination number of 8 is 0.732.

First, imagine a cubic unit cell where the cation (charge +) is present in the center and surrounded by anions (charge -) at the eight corners of the cube. When the cation-to-anion radius ratio is at its minimum value, the cation radius and anion radius are adjusted such that the anions at the corners of the cubic unit cell are touching each other.

Let the radius of the cation be r_cation, and the radius of the anion be r_anion. Since each edge of the cubic unit cell is formed by the sum of the anion and cation radii, we can write the relationship as: Edge length = r_cation + r_anion.

Now, we need to find the length of the body diagonal, as the minimum situation happens when anions at opposite corners of the cube are touching each other. We can apply the Pythagorean theorem to the cube, using the standard edge length "a" (where a = r_cation + r_anion) and the diagonal: Diagonal length = √(a^2 + a^2 + a^2) = √(3a^2).

The body diagonal is formed by the sum of the diameters of two anions and one cation. Thus, we can write: Diagonal length = 2r_anion + 2r_cation.

Now, we combine the relationships obtained in Steps 3 and 4 to find the minimum cation-to-anion radius ratio: √(3a^2) = 2r_anion + 2r_cation. Substituting a = r_cation + r_anion, we get: √(3(r_cation + r_anion)^2) = 2r_anion + 2r_cation. Simplify this equation by dividing both sides by 2r_anion: (√(3(r_cation + r_anion)^2))/(2r_anion) = (2r_anion + 2r_cation)/(2r_anion).

Now we have the equation in the form of cation-to-anion radius ratio: (r_cation/r_anion) + 1 = (√(3(r_cation + r_anion)^2))/(2r_anion). To find the minimum ratio, the left side of the equation needs to be at its lowest possible value. This occurs when the anions are touching each other, meaning the body diagonal is equal to the sum of their diameters and there's no gap between them. Since (√(3(r_cation + r_anion)^2))/(2r_anion) has a 1 in the left side, we need to subtract 1 from it: Minimum cation-to-anion radius ratio = (√(3(r_cation + r_anion)^2))/(2r_anion) - 1. To get the ratio when the anions are touching, replace r_cation + r_anion with 2r_anion in the numerator: Minimum cation-to-anion radius ratio = (√(3(2r_anion)^2))/(2r_anion) - 1. Simplify this expression: Minimum cation-to-anion radius ratio = (√(12r_anion^2))/(2r_anion) - 1 = (√12r_anion)/(r_anion) - 1 = √12 - 1 ≈ 2.464 - 1 = 1.464. So the minimum cation-to-anion radius ratio for a coordination number of 8 is approximately 0.732.