Calculate the volume in ų of a face-centered cubic unit cell if it is composed of atoms with an atomic radius of 1.82 Å.

Verified step by step guidance

1. Step 1: Understand the structure of a face-centered cubic (FCC) unit cell. In an FCC unit cell, atoms are located at each corner and the centers of all the faces of the cube.
2. Step 2: Recognize that in an FCC unit cell, the face diagonal is equal to four times the atomic radius (4r), because the face diagonal passes through the centers of two corner atoms and one face-centered atom.
3. Step 3: Use the relationship between the face diagonal and the edge length (a) of the cube. The face diagonal can be expressed as \( \sqrt{2}a \). Therefore, set \( \sqrt{2}a = 4r \) and solve for the edge length \( a \).
4. Step 4: Substitute the given atomic radius (1.82 Å) into the equation from Step 3 to find the edge length \( a \) of the unit cell.
5. Step 5: Calculate the volume of the cubic unit cell using the formula \( V = a^3 \), where \( a \) is the edge length obtained in Step 4.