To find the radius of the platinum atom, we can use the formula:
Density = (number of atoms per unit cell) * (atomic mass) / [(volume of unit cell) * (Avogadro's number)]
The face-centered cubic crystal structure has 4 atoms per unit cell, so the number of atoms per unit cell is 4.
The atomic mass of platinum is approximately 195.08 g/mol.
Avogadro's number is approximately 6.022 x 10^23 atoms/mol.
Since the face-centered cubic unit cell has atoms at the corners and in the center of each face, it can be thought of as having 8 atoms at the corners and 6 atoms in the center of each face. This means that half of each corner atom and one-sixth of each face-centered atom is within the unit cell.
The volume of a face-centered cubic unit cell can be calculated using the formula:
Volume of unit cell = (side length)^3 / 4
Since the face diagonal of the face-centered cubic unit cell is equal to 4 times the radius of the atom, we can find the side length by:
Side length = (sqrt(2) * face diagonal) / 4
For a face-centered cubic unit cell, the face diagonal is equal to 2 * radius * sqrt(2).
Now, we can put all these values into the formula to solve for the radius:
Density = (4 * atomic mass) / [(volume of unit cell) * Avogadro's number]
Rearranging the equation to solve for the radius, we have:
Radius = [4 * atomic mass / (density * Avogadro's number * (sqrt(2) * face diagonal) / 4)^3]^(1/3)
Substituting the values of atomic mass (195.08 g/mol), density (21.5 g/cm^3), and Avogadro's number (6.022 x 10^23 atoms/mol) into the formula, we can calculate the radius of the platinum atom.