Chapter 12: Problem 24

Compute the atomic packing factor for cesium chloride using the ionic radii in Table \(12.3\) and assuming that the ions touch along the cube diagonals.

Short Answer

Expert verified

Answer: To find the atomic packing factor for the cesium chloride crystal structure, follow these steps: 1. Determine the edge length of the cesium chloride unit cell by dividing the diagonal length, which is equal to 2*(radius of Cs+) + 2*(radius of Cl-), by the square root of 3. 2. Calculate the volume of the unit cell using the formula: unit cell volume = a^3, where a is the edge length from the previous step. 3. Calculate the volume occupied by the Cs+ and Cl- ions by finding the volume of each ion using the volume formula for a sphere and then adding them together. 4. Compute the atomic packing factor by dividing the total volume occupied by the ions (from step 3) by the unit cell volume (from step 2). Using the given ionic radii from Table 12.3 and applying the steps mentioned, you will find the atomic packing factor for the cesium chloride crystal structure.

Step by step solution

Determine the edge length of the cesium chloride unit cell

With the assumption that the ions touch along the cube diagonals, the diagonal length of the cube can be determined as follows: diagonal length = 2*(radius of Cs+) + 2*(radius of Cl-). We know that the diagonal length of a cube is related to its edge length (a) by the formula: diagonal length = a*sqrt(3), where sqrt(3) stands for the square root of 3. So, we can find the edge length of the unit cell by dividing the diagonal length by sqrt(3).

Calculate the volume of the unit cell

The volume of the cubic unit cell is calculated using the formula: unit cell volume = a^3, where a is the edge length determined in the previous step.

Calculate the volume occupied by the Cs+ and Cl- ions

The total unit cell volume is filled with one Cs+ ion and one Cl- ion. The volume of an ion can be calculated using the formula for the volume of a sphere: ion volume = (4/3) * pi * r^3, where pi is a mathematical constant approximately equal to 3.14159 and r is the ion radius. Calculate the volumes of Cs+ and Cl- ions using their radii from Table 12.3, and then find their total volume.