Final answer:
The radius of a silicon atom in an FCC unit cell is approximately 0.077 nm, and the theoretical density of silicon is approximately 2.32 g/cm³.
Step-by-step explanation:
To calculate the radius of a silicon atom in a face-centered cubic (FCC) unit cell, we need to consider the distances between the atom positions. In an FCC structure, the atoms touch each other along the body diagonal of the unit cell. The distance between the atom at (0,0,0) and (1/4, 1/4, 1/4) is equal to the length of the face diagonal, which is equal to four times the radius of a silicon atom.
Given that the lattice parameter is 0.543 nm, we can calculate the length of the face diagonal as 0.543 * sqrt(2). Dividing this by 4 gives us the radius of a silicon atom, which is approximately 0.077 nm.
To calculate the theoretical density of silicon, we need to know the molar mass and the volume of the unit cell. The volume of an FCC unit cell can be calculated as (4 * r^3) / 3, where r is the radius of the atom. Substituting the value of the radius we calculated earlier, we can find the volume of the unit cell. Dividing the molar mass by the volume of the unit cell gives us the theoretical density of silicon, which is approximately 2.32 g/cm³.