Iron oxide (FeO) has the rock salt crystal structure and a density of \(5.70 \mathrm{g} / \mathrm{cm}^{3}\) (a) Determine the unit cell edge length. (b) How does this result compare with the edge length as determined from the radii in Table \(12.3,\) assuming that the \(\mathrm{Fe}^{2+}\) and \(\mathrm{O}^{2-}\) ions just touch each other along the edges?

The rock salt crystal structure is named for its resemblance to the crystal form of common table salt, sodium chloride (NaCl). This structure is characterized by an arrangement where each sodium ion is surrounded by six chloride ions and vice versa, forming a cubic lattice. In the case of iron oxide (FeO), which also adopts this structure, the Fe²⁺ ions and O²⁻ ions take the places of the sodium and chloride ions, respectively.

The ions are arranged in a face-centered cubic (FCC) lattice, where aside from the ions at the corners of the cube, there's an ion at the center of each face. Since each corner ion is shared by eight unit cells and each face-centered ion by two, only four Fe²⁺ and four O²⁻ ions belong exclusively to one unit cell, leading to a formula unit count of four for FeO. This arrangement allows for high packing efficiency and contributes to the physical properties, like density, of the material.

Density, defined as mass per unit volume, is a critical property in material science, influencing both mechanical and thermal attributes of materials. To calculate the density of a crystal, like iron oxide with a rock salt crystal structure, we must understand the relationship between the mass contained within a unit cell and the volume that unit cell occupies.

The mass of a unit cell is calculated by summing the masses of each constituent ion, multiplied by their respective quantity within the cell. Since we know the mass and density, we can rearrange the density equation \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) to solve for volume. This provides insight into the spatial arrangement of ions within the crystal lattice, which directly impacts the material's overall density. Through these calculations, we can determine how closely theoretical models align with experimental measurements, such as in the case of the edge length of a unit cell.

Determining the unit cell edge length is fundamental in crystallography as it gives insight into the atomic or ionic spacing within a crystal. For a cubic system, which includes the face-centered cubic (FCC) lattice of iron oxide's rock salt structure, the volume of the unit cell (V) can be expressed as \( V = a^3 \) where 'a' is the edge length of the cube.

Once the volume is known from the density calculation, we can extract the edge length by taking the cube root of the volume. This edge length reveals crucial information, such as the possible packing of ions and the potential porosity of the material. Furthermore, comparing the calculated edge length with observed values, like those inferred from ionic radii, validates our understanding of ionic sizes and their arrangement within the lattice. This comparison is an essential part of crystallographic studies, as it can confirm or challenge established theories regarding the structure-property relationships in crystalline materials.