If three elements \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) crystallized in cubic solid lattice with \(\mathrm{A}\) atoms at corners, \(\mathrm{B}\) atoms at cube centres and \(\mathrm{C}\) atoms at the edges, the formula of the compound is: (a) \(\mathrm{ABC}\) (b) \(\mathrm{ABC}_{3}\) (c) \(\mathrm{AB}_{3} \mathrm{C}\) (d) \(\mathrm{A}_{3} \mathrm{~B}_{2} \mathrm{C}_{3}\)

Understanding the structure of crystal lattices is critical, and central to this is calculating the number of atoms within a cubic unit cell. This involves considering the geometry of the unit cell and the position of each atom within it.

In a simple cubic lattice, atoms are positioned at the corners, and sometimes at the faces or center of the cube. The calculation starts by recognizing that each corner atom is shared by eight surrounding unit cells, thus only contributing 1/8th of an atom to the individual unit cell's count. If an atom is in the center, it belongs entirely to that unit cell, and if it's on an edge, it's shared by four unit cells and contributes 1/4th. When assessing face-centered atoms, each contributes 1/2, as two unit cells share such positions.

Counting the atoms correctly is crucial as it determines the chemical formula of the crystal lattice compound. An accurate count requires applying these fractional contributions systematically to ensure a clear understanding of the compound's stoichiometry.

The precise positioning of elements within a crystal lattice is key to understanding its properties. The exercise provided is a prime example, with three different types of atoms positioned uniquely within a cubic lattice.

An atom at the corner, like element A, is effectively cut into eight pieces, as it's positioned at a frontier shared by eight different unit cells. Element B, located at the cube center, sits snugly within the boundaries of a single unit cell, claiming full residency. Meanwhile, element C atoms arch over the middle of each edge, touching four different unit cells, dividing their allegiance into quarters.

Each position—corner, edge, and center—affects the atom’s contribution to the formula of the compound. By understanding the spatial distribution of the different atoms, we begin to see the framework of the crystal lattice and are better positioned to predict its physical and chemical properties.

Stoichiometry in a crystal lattice speaks to the fixed ratio of atoms and is a direct reflection of the lattice's repetitive structure. This ratio is critical and requires meticulous attention when determining the compound's formula from its unit cell composition.

The calculate stoichiometry, it's important to account for the fraction of each atom that resides within a single unit cell. For example, corner atoms contribute 1/8, edge atoms 1/4, and center atoms contribute fully, as we have seen with elements A, B, and C in our exercise. Multiplying these fractional values by the number of corresponding positions (corners, edges, centers) and summing them gives us the ratio of atoms in the compound.

The stoichiometry of the crystal lattice is not just a count of atoms, it represents the underpinnings of the material's identity. Through stoichiometry, scientists can predict the behavior of the material under various conditions and can engineer materials with desired properties for practical applications.