Final answer:
The density of metallic gold can be calculated using the inter-planar spacing obtained from the X-ray diffraction and Bragg's Law, together with the mass and number of atoms per unit cell of the face-centered cubic lattice.
Step-by-step explanation:
The calculation of the density of metallic gold involves using the information provided by X-ray diffraction from a face-centered cubic (fcc) unit cell. Given that gold crystallizes in an fcc lattice and the second-order reflection (n = 2) from the planes of the unit cell is at an angle of θ = 22.20° with the X-ray wavelength λ = 1.54 Å, one can use Bragg's Law to calculate the inter-planar spacing d. The equation for Bragg's Law is nλ = 2d·sin(θ). Using this formula, we can find the value of d. Then, the density (ρ) of gold can be calculated using the formula ρ = (Z · M) / (N_A · a^3) where Z is the number of atoms per unit cell for fcc (which is 4), M is the molar mass, N_A is Avogadro's number, and a^3 is the volume of the unit cell, which is calculated from the cube of the edge length (a) of the cell, which itself is derived from inter-planar spacing d.