Question

A solid contains \(X, Y,\) and \(Z\) atoms in a cubic lattice with \(X\) atoms in the corners, \(Y\) atoms in the bodycentered positions, and \(Z\) atoms on the faces of the cell. What is the empirical formula of the compound?

Short answer

The empirical formula of the compound is \(X_1Y_1Z_3\).

Step-by-step solution

Understand the atoms' positions

In the problem, it's given that there are \(X\) atoms at the corners of the lattice, \(Y\) atoms in the body-centered positions, and \(Z\) atoms on the faces of the cell in each unit cell. Each atom contributes differently to the cell. Atoms at the corners are shared by 8 adjacent cells, those at the body-centered positions are not shared and those on the faces are shared by 2 adjacent cells.

Calculate X's contribution to the cell

In this cubic structure, there are 8 corners, and at each corner there is 1 atom of type X. However, only \(1/8\) of each atom is actually within the cell because it is shared among 8 cells. So the contribution of X to one unit cell is \(8*\frac{1}{8}=1\)

Calculate Y's contribution to the cell

There is 1 atom of type Y in the body-centered position of each cell. This atom is not shared so contributes entirely to 1 cell. So the contribution of Y to one unit cell is 1.

Calculate Z's contribution to the cell

In this unit cell, there are 6 faces and at each face there is 1 atom of type Z. However, only \(1/2\) of each atom is actually within the cell because it is shared among two cells. So, the contribution of Z to one unit cell is \(6*\frac{1}{2}=3\)

Derive the empirical formula

The empirical formula for a compound is derived based on the ratio of the atoms present in the compound. Our task was to find the contribution of atoms X, Y and Z to the unit cell, which has been calculated as 1:1:3. Hence, the empirical formula of the compound is \(X_1Y_1Z_3\)

Key concepts

Cubic Lattice

In solid state chemistry, the term "cubic lattice" refers to one of the simplest and most common types of crystal structures. This lattice is characterized by its three-dimensional, symmetrical arrangement in the shape of a cube. Each point in a cubic lattice is called a lattice point, and they are evenly spaced across the crystal.

When we talk about different atoms like X, Y, and Z occupying various positions in a cubic lattice, it's crucial to understand that these positions influence the stability and properties of the material. In a cubic structure:

• Corner atoms are shared by eight adjacent unit cells.
• Face atoms are shared by two adjacent unit cells.
• Body-centered atoms belong entirely to one unit cell and are not shared.

By knowing where atoms reside in the cubic lattice, we can determine how much each atom contributes to a single unit cell, which is vital for calculating the empirical formula.

Empirical Formula

The empirical formula is a simple notation that represents a compound's composition in the smallest whole-number ratio of its atoms. Determining the empirical formula is a key task in chemistry, as it provides essential information about the compound's basic structure.

In the context of a cubic lattice, finding the empirical formula requires calculating each type of atom's contribution to the unit cell. From the example, we see

• X atoms are located at the corners and because there are 8 corners in a cube and each atom is shared among 8 cells, the cell contains 1 X atom.
• Y atoms are at the body center and fully contained within the cell, so the cell contains 1 Y atom.
• Z atoms sit on the faces, with 6 faces in total. Each atom is shared between 2 cells, resulting in 3 Z atoms in one unit cell.

With these calculations established, the empirical formula reflects the 1:1:3 ratio of X:Y:Z and is expressed as \(X_1Y_1Z_3\). This formula is significant in revealing the stoichiometry fundamental to the compound.

Unit Cell

The concept of a "unit cell" is central in solid state chemistry. A unit cell is the smallest repeating unit that makes up the entire crystal. When these cells stack together in three-dimensional space, they create the full lattice structure.

Understanding the unit cell involves recognizing several key factors:

• It defines the crystal symmetry and its dimensions.
• It includes all atomic arrangements within one cell, impacting the properties of the material.
• In cubic lattices, the unit cell can be quite simple, like a cube, but the arrangement of atoms within can vary.

For instance, with X atoms at corners, Y atoms in the body center, and Z atoms on the faces, the specific arrangement within this unit cell leads to the formation of calculated relationships among the atoms, essential for determining the material's empirical formula. Each type of atom's position within the unit cell affects the overall lattice structure and the stoichiometric formula resulting from these arrangements.