Chemistry: cubic-close-packing unit problem?

Question

You are given a small bar of an unknown metal X, whose density is found to be 10.5 g/cm3. An X-ray diffraction experiment measures the edge of its cubic-close-packing unit cell as 4.09 Å. Identify X.

Answer 1

There are three different cubic unit cells: simple cubic, which contains a single atom in the cell; body-centered cubic, which contains two atoms in the cell; and face-centered cubic, which contains four atoms in the cell. The atomic packing factors for there cubic cells are also different: simple cubic has the lowest factor of 0.52, and face-centered cubic has the highest factor of 0.74. Hence the close-packing cubic unit cell must be the face-centered cubic, which contains four atoms in the cell.
Considering 1 cm^3 such metal, it must be of 10.5g. Also,
1 cm^3 = {1/(4.09e-8)^3} unit cells = 1.462e22 unit cells, that contains 5.846e22 atoms or 0.0971 moles. Hence the molar mass of this metal is: 10.5g/0.0971 moles = 108g/mol.
This metal is silver, with molar mass 107.9 g/mol.