For a cell of the CsCl type (figure 9.18), how is the length of one side of the cell,s, related to the sum of the radii of the ions, r subscript cation + r subscript anion?

Figure 9.18 :
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Another question : Consider the NaCl unit shell in figure 9.18. looking at only the front face, five large cl- ions, 4 small na+ ions, how many cubes share each of the na+ ions in this face? how many cubes share each of the Cl- ions in this face?

To determine the relationship between the length of one side of a cell (s) in the CsCl structure and the sum of the radii of the ions (r_cation + r_anion), you can follow these steps:

1. Examine the arrangement of ions in the CsCl unit cell (Figure 9.18).

2. Identify that the CsCl structure consists of chloride (Cl-) ions at the corners of the cube and a cesium (Cs+) ion in the center of the cube.

3. Visualize the relationship between the cube length (s) and the sum of the ion radii (r_cation + r_anion).

4. Observe that the Cs ion is located at the cube center, with Cl ions touching each of its faces.

5. Realize that the sum of the radii of the ions determines the distance from the center of the Cs ion to the face of the Cl ion.

6. Understand that this distance is equivalent to half the length of one side of the cube (s/2).

7. Recognize that the sum of the radii (r_cation + r_anion) is equal to (s/2).

Therefore, the length of one side of the CsCl cell (s) is twice the sum of the radii of the ions (2 * (r_cation + r_anion)).

Now, let's move on to your second question:

In Figure 9.18, focusing on the front face of the NaCl unit cell, we observe:

- 5 Cl- ions: Each Cl- ion is present at the center of its respective cube face.

- 4 Na+ ions: Each Na+ ion is located at the corner of one cube face.

To determine the number of cubes shared by each Na+ and Cl- ion on this face:

1. Analyze the arrangement of the ions in the NaCl unit cell.

2. Identify the position of each ion and determine which cubes they are part of.

3. Observe that each Na+ ion shares its cube with only one other Na+ ion located on the adjacent face.

Therefore, each Na+ ion shares one cube with another Na+ ion on the opposite face of the unit cell.

4. Note that each Cl- ion is positioned at the center of its face, surrounded by four Na+ ions located at the corners of the adjacent cubes.

As a result, each Cl- ion shares four cubes with the surrounding Na+ ions.

To summarize:

- Each Na+ ion on the front face of the NaCl unit cell shares one cube with another Na+ ion on the opposite face.

- Each Cl- ion on the front face of the NaCl unit cell shares four cubes with the four surrounding Na+ ions on adjacent faces.