A compound formed by elements \(A\) and \(B\) crystallizes in cubic structure where \(A\) atoms are at the corners of a cube and \(B\) atoms are at the face centre. What is the formula of the compound?

Crystallography is the scientific study of crystals and their formation. At its core, crystallography investigates the arrangement of atoms in solid materials and characterizes their structure using mathematical models and physical principles. The arrangement of atoms in a crystal structure is highly ordered and follows repetitive patterns, known as a crystal lattice. Each point in the lattice represents the position of an atom or a group of atoms.

In the context of our exercise, the cubic crystal structure is one of the seven primary crystal systems in crystallography. It is distinguished by a lattice where the unit cells are shaped like a cube, with equal sides and 90-degree angles between them. Inside the cubic lattice, atoms can be positioned at various sites: corners, face centers, edge centers, and body centers. Understanding where atoms are located within a unit cell is crucial to identifying the stoichiometry and properties of the material.

Stoichiometry in chemistry is the calculation of relative quantities of reactants and products in chemical reactions. For substances forming crystalline solids, stoichiometry can also refer to the ratio of the different atoms within a crystal lattice. This ratio is an intrinsic property of the compound and determines its chemical formula.

The calculated exercise taps into stoichiometry by seeking to identify the formula of a compound, which is based on the number of atoms of each type (A and B) within the unit cell of its crystal structure. The stoichiometric relationship in a compound with a cubic crystal structure is crucial for understanding the compound's composition. In our example, by identifying the locations of atoms A and B and utilizing the principle of stoichiometry, we deduced the compound's formula as AB3. This reflects the 1:3 ratio of A atoms to B atoms within the unit cell.

Unit cell calculations are essential in determining the crystal structure's composition. A unit cell is the smallest repeating unit of a crystal lattice that displays the full symmetry of the crystal's structure. By calculating how many atoms of each element are present in a unit cell, we can derive the empirical formula of the compound.

In the provided exercise, two types of positions are considered: cube corners and face centers. The strategy is to count the atoms occupying each position while considering their sharing with adjacent unit cells. For example, the corner atoms are shared among eight different unit cells, hence, each unit cell effectively contributes 1/8 of every corner atom. Face-centered atoms are shared between two unit cells, counting for a 1/2 contribution per cell. By summing up these contributions, we arrive at the total number of atoms of each kind within a single unit cell and thus the empirical formula. It's a logical process that hinges not only on counting but also on understanding the geometric and shared nature of the crystal lattice.