Iridium crystallizes in a face-centered cubic unit cell that has an edge length of 3.833 Å. (a) Calculate the atomic radius of an iridium atom. (b) Calculate the density of iridium metal.

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 of the corners and the centers of all the faces of the cube. Each face-centered atom is shared between two adjacent unit cells.
2. Step 2: Relate the edge length of the FCC unit cell to the atomic radius. In an FCC structure, the face diagonal is equal to four times the atomic radius (4r). The face diagonal can also be expressed in terms of the edge length (a) as \( \sqrt{2}a \). Therefore, set up the equation \( 4r = \sqrt{2}a \) to solve for the atomic radius (r).
3. Step 3: Calculate the atomic radius of iridium using the edge length provided. Substitute the given edge length (3.833 Å) into the equation from Step 2 and solve for r.
4. Step 4: Determine the number of atoms per unit cell in an FCC structure. An FCC unit cell contains 4 atoms per unit cell (1/8 of an atom at each of the 8 corners and 1/2 of an atom at each of the 6 face centers).
5. Step 5: Calculate the density of iridium. Use the formula for density \( \text{Density} = \frac{\text{mass of atoms in unit cell}}{\text{volume of unit cell}} \). Find the mass of the atoms in the unit cell using the molar mass of iridium and Avogadro's number, and calculate the volume of the unit cell using the edge length.