Aluminum metal crystallizes in a face-centered cubic unit cell. (a) How many aluminum atoms are in a unit cell? (b) Estimate the length of the unit cell edge, a, from the atomic radius of aluminum (1.43 Å). (c) Calculate the density of aluminum metal.

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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. Calculate the total number of atoms per unit cell by considering the contribution of each atom at the corners and faces.
2. Step 2: Calculate the number of atoms in the FCC unit cell. Each corner atom is shared by eight adjacent unit cells, and each face-centered atom is shared by two unit cells. Therefore, the total number of atoms in one FCC unit cell is calculated as follows: 8 corner atoms * (1/8) + 6 face atoms * (1/2).
3. Step 3: Use the atomic radius to estimate the length of the unit cell edge, a. In an FCC structure, the face diagonal is equal to four times the atomic radius. Use the relationship between the face diagonal and the edge length: face diagonal = a√2 = 4 * atomic radius.
4. Step 4: Rearrange the equation from Step 3 to solve for the edge length, a. Substitute the given atomic radius of aluminum (1.43 Å) into the equation to find the edge length.
5. Step 5: Calculate the density of aluminum metal. Use the formula for density: density = mass/volume. First, calculate the mass of the aluminum atoms in the unit cell using the molar mass of aluminum and Avogadro's number. Then, calculate the volume of the unit cell using the edge length from Step 4. Finally, divide the mass by the volume to find the density.