A metallic solid with atoms in a face-centered cubic unit cell with an edge length of 392 \(\mathrm{pm}\) has a density of 21.45 $\mathrm{g} / \mathrm{cm}^{3} .$ Calculate the atomic mass and the atomic radius of the metal. Identify the metal.

In an fcc structure, there are 8 corner atoms shared by 8 neighboring unit cells, and 6 face atoms shared by 2 neighboring unit cells. Therefore, the number of atoms per unit cell in an fcc structure can be calculated as: Number of atoms per unit cell = (8 corner atoms ÷ 8) + (6 face atoms ÷ 2) = 1 + 3 = 4

We are given that the edge length of the unit cell is 392 pm. To find the volume, take the cube of the edge length and convert from pm³ to cm³: Volume of unit cell = (392 pm)³ * (1 cm / 10^10 pm)³ = 6.027 * 10^(-23) cm³

We are given the density (ρ) of the metal as 21.45 g/cm³. We can now calculate the molar mass by using the formula: ρ = (mass of unit cell) × (Avogadro's number) / (volume of unit cell × molar mass) Rearranging to solve for molar mass, we have: Molar mass = (mass of unit cell) × (Avogadro's number) / (ρ × volume of unit cell) The mass of the unit cell can be calculated as follows: mass of unit cell = (number of atoms per unit cell) × (atomic mass) Now we can substitute this formula back into the molar mass formula: Molar mass = ((number of atoms per unit cell) × (atomic mass)) × (Avogadro's number) / (ρ × volume of unit cell) Plugging in the values we know: Molar mass = (4 × atomic mass) × (6.022 * 10^23 atoms/mol) / (21.45 g/cm³ * 6.027 * 10^(-23) cm³) From this equation, we can calculate the atomic mass: Atomic mass = 195.8 g/mol

We know the atomic radius (r) of a face-centered cubic unit cell can be determined using the edge length (a) and the following formula: 4r = √(2) × a Rearrange the formula to solve for r and plug in the known values: Atomic radius (r) = (edge length / √(2)) / 4 r = (392 pm / √2) / 4 = 137.9 pm

Based on the calculated atomic mass of 195.8 g/mol and atomic radius of 137.9 pm, we can identify the metal as Platinum (Pt), which has an atomic mass of approximately 195 g/mol. In conclusion, the metallic solid has an atomic mass of 195.8 g/mol and an atomic radius of 137.9 pm, indicating that it is likely Platinum.