Final answer:
The atomic packing factor of platinum, which crystallizes in a face-centered cubic structure, is 0.74. This indicates that 74% of the space in the crystal lattice is filled with atoms.
Step-by-step explanation:
The question asks to calculate the atomic packing factor (APF) of platinum, which is a measure of how densely packed the atoms are in a crystal structure. Platinum crystallizes in a face-centered cubic (fcc) structure, which means each unit cell contains 4 atoms. The APF for an fcc structure is 0.74, indicating that 74% of the space in the crystal structure is occupied by atoms. To find this, we use the formula APF = (N×V_s) / V_c where N is the number of atoms per unit cell, V_s is the volume of a single atom assumed to be a sphere, and V_c is the volume of the unit cell. For an fcc lattice, N = 4. The volume of the unit cell (V_c) is a^3 where 'a' is the edge length, and the volume of a sphere (V_s) is (4/3)πr^3 where 'r' is the atomic radius. In an fcc lattice, the atomic packing factor is calculated using the radius 'r' that relates to the edge length 'a' as r = (\sqrt{2}/4)a. Hence, substituting these values gives us the APF.