Calculation of Cation Radius in an FCC Ionic Solid

Given data:
- Edge length of the FCC structure (a) = 508 pm
- Radius of anion (r-) = 144 pm

Step 1: Find the Effective Edge Length of the FCC Unit Cell
The effective edge length (a') of the FCC unit cell can be calculated using the formula:

a' = a / √2

Substituting the given value of a, we get:
a' = 508 pm / √2

Step 2: Calculate the Face Diagonal of the FCC Unit Cell
The face diagonal (d) of the FCC unit cell can be calculated using the formula:

d = a' * √2

Substituting the value of a' calculated in the previous step, we get:
d = (508 pm / √2) * √2

Simplifying, we get:
d = 508 pm

Step 3: Find the Cation Radius (r+)
In an FCC unit cell, the cations are located at the center of each face. The cation radius (r+) can be calculated using the formula:

r+ = (d - 2 * r-) / 2

Substituting the given value of r-, we get:
r+ = (508 pm - 2 * 144 pm) / 2

Simplifying, we get:
r+ = (508 pm - 288 pm) / 2
r+ = 220 pm / 2
r+ = 110 pm

Therefore, the radius of the cation in the FCC ionic solid is 110 pm.

Explanation:
- The FCC (Face-Centered Cubic) structure is a common arrangement for ionic solids.
- In an FCC unit cell, the anions are located at the corners and the face centers of the cube.
- The effective edge length of the FCC unit cell is obtained by dividing the actual edge length by the square root of 2.
- The face diagonal of the FCC unit cell is equal to the effective edge length.
- The cations are located at the center of each face, and the cation radius can be calculated using the face diagonal and the anion radius.
- By substituting the given values and following the calculations, we find that the cation radius is 110 pm.