Spinel is a mineral that contains 37.9% Al, 17.1% Mg, and 45.0% O, by mass, and has a density of 3.57 g/cm³. The unit cell is cubic with an edge length of 8.09 Å. How many atoms of each type are in the unit cell?

Verified step by step guidance

1. Step 1: Calculate the molar mass of the compound using the given percentages and the atomic masses of Al, Mg, and O. Assume 100 g of the compound to simplify calculations.
2. Step 2: Convert the mass percentages to moles by dividing the mass of each element by its respective atomic mass (Al: 26.98 g/mol, Mg: 24.31 g/mol, O: 16.00 g/mol).
3. Step 3: Determine the mole ratio of Al, Mg, and O by dividing the number of moles of each element by the smallest number of moles calculated in Step 2.
4. Step 4: Use the density and the volume of the unit cell to calculate the mass of the unit cell. The volume of the unit cell is calculated using the edge length: \( V = (8.09 \times 10^{-8} \text{ cm})^3 \).
5. Step 5: Calculate the number of formula units per unit cell by dividing the mass of the unit cell by the molar mass of the compound. Use this to determine the number of atoms of each type in the unit cell based on the mole ratio.