A commonly occurring mineral has a cubic unit cell in which the metal cations \(\mathrm{M}\) occupy the comers and face centers. Inside the unit cell, there are anions \(\mathrm{A}\) that occupy all the tetrahedral holes created by the cations. What is the chemical formula of the \(M_{x} A_{y}\) compound?

When visualizing the structures of solids at an atomic level, a cubic unit cell is often employed as the building block of these structures. Imagine the unit cell as a tiny box with atoms at the corners and various positions within. In the context of a crystalline solid, the entire lattice can be constructed by stacking these unit cells in all three dimensions.

A cubic unit cell can be categorized into three types: simple cubic, body-centered cubic, and face-centered cubic. The exercise in question refers to a face-centered cubic (FCC) unit cell where the corners and centers of each face are occupied by metal cations.

Understanding the arrangement of these cations is crucial for determining the formula of the compound. By accounting for the shared contribution of cations at the corners and faces, we can ascertain the number of cations within a single unit cell. This approach simplifies understanding complex structures and enables us to predict their physical properties.

In the context of salt-like structures, metal cations and anions are the oppositely charged ions that constitute the compound. Metal cations usually have a positive charge due to the loss of electrons, while anions have a negative charge as they gain electrons.

These ions arrange themselves in a crystal lattice in such a way as to balance the electrostatic forces between them. In our mineral example, metal cations occupy specific positions in the cubic unit cell while anions fit into the spaces between them, called tetrahedral holes.

The ratio in which cations and anions combine depends on maintaining electrical neutrality in the compound. The exercise requires us to calculate the numbers of each type of ion within the unit cell to determine the correct stoichiometry and ultimately, the chemical formula of the compound.

In crystalline solids, a tetrahedral hole is a space where an anion could reside, and it is formed by four cations in a tetrahedral configuration. These holes are significant as they determine how closely the ions pack together and hint at the efficiency of space usage within the crystal lattice.

For every metal cation in an FCC lattice, there exists a corresponding tetrahedral hole. When anions fill these holes, they interact with the cations, stabilizing the structure's overall integrity. It's essential to understand the implications of ion packing in tetrahedral holes as it impacts the stoichiometry and, therefore, the chemical formula of the solid.

In the given exercise, the 1:1 correspondence of cations to tetrahedral holes filled by anions allows students to infer that the overall formula of the compound is 1:1, simplifying to MA. This understanding is vital for grasping the principles of crystal chemistry and property predictions.