Question

A compound contains two types of atoms: \(\mathrm{X}\) and \(\mathrm{Y}\). It crystallizes in a cubic lattice with atoms \(\mathrm{X}\) at the corners of the unit cell and atoms \(\mathrm{Y}\) at the body centres. The simplest possible formula of this compound is: (a) XY (b) \(\mathrm{X}_{2} \mathrm{Y}_{2}\) (c) \(\mathrm{XY}_{6}\) (d) \(\mathrm{X}_{\mathrm{g}} \mathrm{Y}\)

Short answer

The simplest formula is XY.

Step-by-step solution

Understanding the Position of Atoms in the Unit Cell

Atoms \((\mathrm{X})\) are at each of the 8 corners of the cubic unit cell. Since each corner atom is shared by 8 adjacent unit cells, each corner contributes \((\frac{1}{8})\) of an atom per unit cell. Atoms \((\mathrm{Y})\) are located at the body center of the unit cell and fully belong to one unit cell.

Calculating Contribution of Atom X

There are 8 corners in a cubic lattice. Therefore, the contribution of atom \(\mathrm{X}\) in the unit cell is \[ 8 \times \frac{1}{8} = 1. \] Hence, there is a total of 1 atom of \(\mathrm{X}\) per unit cell.

Calculating Contribution of Atom Y

There is 1 body-centered atom \(\mathrm{Y}\) in the unit cell. Since it is fully inside the unit cell, its contribution is \[ 1 \times 1 = 1. \] Thus, there is a total of 1 atom of \(\mathrm{Y}\) per unit cell.

Determining the Chemical Formula

According to the contributions calculated: there is 1 atom of \(\mathrm{X}\) and 1 atom of \(\mathrm{Y}\) per unit cell. This gives us the simplest formula \(\mathrm{XY}.\)

Key concepts

Cubic Lattice

A cubic lattice is a three-dimensional arrangement of atoms, molecules, or ions, forming a repeating pattern in space. It is structured like a cube, meaning it has equal dimensions along its length, width, and height. This symmetry allows for a straightforward and predictable pattern of arrangements.
In a cubic lattice, the primary cells that repeat throughout the structure are known as unit cells. Each of these unit cells acts like a building block for the entire lattice, repeating to extend the pattern infinitely without altering the geometry. The uniformity and simplicity of cubic lattices make them crucial for understanding many crystal structures when studying chemistry and materials science.

Unit Cell

The unit cell is the smallest portion of a crystal lattice that shows the entire pattern of the structure. Understanding how atoms are arranged within the unit cell is vital to predicting the properties of the material. In a cubic lattice, the unit cell has a cube shape.
In the context of this exercise, the unit cell is particularly relevant for determining how atoms

• contribute to the overall lattice
• affect the chemical composition

Each unit cell in the cubic lattice repeats in three dimensions to build up the complete structure of the material. By examining the unit cell closely, you can gather information about how atoms like X and Y interact and form compounds. This is essential for calculating quantities like the simplest formula of a compound.

Corner Atom

Corner atoms are atoms located at the corners of a unit cell. In a cubic structure, each unit cell shares its corner atoms with adjacent unit cells, meaning that each corner atom is only partially considered as part of a single unit cell. Specifically, a corner atom is shared among 8 unit cells. Thus, each contributes only \( \frac{1}{8} \) of an atom to each unit cell.
This sharing of atoms across the unit cells affects the way you calculate the overall number of atoms. Thus, in any calculations, the fraction of each atom's contribution is crucial to determine the actual number of atoms present in a single complete unit cell. This insight ensures accuracy when figuring out the chemical formula for any compound involving corner atoms.

Body-Centered Atom

A body-centered atom is located at the center of the unit cell. In a body-centered cubic structure, this atom is not shared with any other unit cells. This full ownership allows it to contribute entirely (a full 1 atom) to the unit cell it resides in.
This makes calculations simpler compared to corner atoms because there's no need to account for sharing among multiple cells. Understanding the role of body-centered atoms in a lattice helps to determine the composition of the chemical formula accurately by knowing exactly how many such atoms exist in a unit volume of the crystal. This was seen in the exercise with atom \( Y \), where its full contribution was easier to account for because it was centrally located in the unit cell.