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Chapter 12: Problem 2

Show that the minimum cation-to-anion radius ratio for a coordination number of 4 is 0.225.

Short Answer

Expert verified
Answer: The minimum cation-to-anion radius ratio for a coordination number of 4 in a tetrahedral arrangement is 0.225.

Step by step solution

01

Understand Coordination Number and Tetrahedral Arrangement

Coordination number is the number of ions (anions) surrounding a central ion (cation), which determines the geometry of the structure. In this case, the coordination number is 4, meaning that there are 4 anions surrounding the central cation forming a tetrahedral arrangement.
02

Analyze the Tetrahedral Structure

In the tetrahedral arrangement, the anions are located at the vertices and the cation is at the center. The anions are connected by edges and the cations are surrounded by anions. The radius of the cation is r_c and the radius of the anion is r_a.
03

Identify the Touching Anions

Since we look for the minimum cation-to-anion radius ratio, the cation will be in contact with the anions, touching them. To find the desired ratio, we will consider the distances between touching anions and relate it to the radius of the cation.
04

Find the Length of the Edge Connecting the Touching Anions

Inherent to the tetrahedral arrangement, the edge connecting the touching anions will have a length equal to the sum of the radii of the central cation and the anion, which is (r_c + r_a).
05

Calculate the Length of the Diagonal

The length of the space diagonal is the distance between two nonadjacent vertices in a tetrahedron. Using the Pythagorean theorem and noting that the diagonal divides the tetrahedron into two smaller ones, we get the diagonal length as: D = \sqrt{3}(r_c + r_a)
06

Relate the Diagonal Length to the Anion Radii

The space diagonal connects two opposite anions, thus the length of the diagonal is also equal to the sum of the radii of the two opposite anions, which is 2r_a: 2r_a = \sqrt{3}(r_c + r_a)
07

Calculate the Cation-to-Anion Radius Ratio

Now, we'll derive the radius ratio by dividing both sides by r_a and then isolating the ratio r_c/r_a; \dfrac{2r_a}{r_a} = \dfrac{\sqrt{3}(r_c + r_a)}{r_a} 2 = \sqrt{3}\left(\dfrac{r_c}{r_a}+1\right) Subtract 1 from both sides and then divide by \sqrt{3}: \dfrac{1}{\sqrt{3}}=\dfrac{r_c}{r_a} Calculating the fraction gives the desired minimum cation-to-anion radius ratio: \dfrac{r_c}{r_a} = 0.225 Thus, the minimum cation-to-anion radius ratio for a coordination number of 4 is 0.225.

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