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.

Key Concepts

Molar Mass and Composition

To determine the number of atoms in the unit cell, one must first calculate the molar mass of each element present in the mineral. The percentages by mass of aluminum (Al), magnesium (Mg), and oxygen (O) can be used to find the empirical formula, which indicates the ratio of atoms in the compound. This is essential for converting mass percentages into moles, which will ultimately help in finding the number of atoms in the unit cell.

Unit Cell and Atomic Arrangement

The unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. In a cubic unit cell, the arrangement of atoms can vary, affecting how many of each type of atom are present. Understanding the geometry and the edge length of the unit cell is crucial for calculating the total number of atoms based on the volume and density of the mineral.

Density and Volume Calculations

Density is defined as mass per unit volume and is a key factor in determining the number of atoms in a unit cell. By using the given density of spinel and the calculated volume from the edge length of the cubic unit cell, one can find the total mass of the unit cell. This mass can then be related back to the number of moles and subsequently the number of atoms of each element present in the unit cell.