Question

A face-centered cubic cell contains \(8 \mathrm{X}\) atoms at the corners of the cell and \(6 \mathrm{Y}\) atoms at the faces. What is the empirical formula of the solid?

Short answer

The empirical formula of the solid is \(XY_{3}\).

Step-by-step solution

Understand the Structure of FCC

In a face-centered cubic (fcc) structure, there are 8 corner atoms and 6 face atoms. In such a structure, each corner atom is shared by eight adjacent cubes and each face atom is shared by two adjacent cubes.

Calculate the Contribution of X Atoms

Given that there are 8 corner atoms of type X, and each corner atom is shared by eight adjacent cubes, the contribution of X atoms to one unit cell is \(8*(1/8)\) = 1 atom.

Calculate the Contribution of Y Atoms

Given that there are 6 face atoms of type Y, and each face atom is shared by two adjacent cubes, the contribution of Y atoms to one unit cell is \(6*(1/2)\) = 3 atoms.

Determine the Empirical Formula

The empirical formula is decided by the ratio of the number of X atoms to the number of Y atoms in the unit cell. Since the unit cell contains 1 X atom and 3 Y atoms, the empirical formula of the solid is \(X_{1}Y_{3}\), or simply \(XY_{3}\).

Key concepts

Face-Centered Cubic

The face-centered cubic (fcc) structure is a common crystal arrangement found in metals and some ionic compounds. In this structure, atoms are located at each of the corners and the centers of all the faces of the cube. This arrangement has several interesting attributes:

• Each cube has eight corners, and there is one atom at each corner. However, because these corner atoms are shared among eight neighboring unit cells, each corner contributes only one-eighth of an atom to a single cell.
• There is an atom at each of the six faces, shared between two adjacent cells, giving a contribution of one-half of an atom per face to each cell.

The efficiency of packing atoms in an fcc structure results in a high packing density. This means there's minimal empty space, making the structure stable and common in metals like aluminum and copper.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of atoms of each element in a compound. It provides important clues about the composition without revealing the actual number of atoms.
The derivation of an empirical formula from a face-centered cubic structure involves calculating contributions from different types of atoms:

• Count the effective number of atoms from the corners and faces within the unit cell.
• Determine the ratio of these atoms to find the simplest whole number ratio.

In our exercise, we found that the unit cell contains 1 atom of X and 3 atoms of Y. Therefore, the empirical formula for our compound is \(XY_3\), representing one atom of X for every three atoms of Y.

Unit Cell

A unit cell is the smallest repeating portion of a crystal structure that, when repeated in all directions, results in the bulk crystal. Unit cells define the symmetry and structure of the crystal.
There are different types of unit cells, such as simple cubic, body-centered cubic, and face-centered cubic, each having a unique arrangement and number of atoms.

• The fcc unit cell has atoms at eight corners and centers of the faces, balancing structural efficiency and atom usage.
• Understanding the unit cell is crucial for calculating the empirical formula because it determines the number of each type of atom per unit.

In solid-state chemistry, the concept of a unit cell helps in visualizing how atoms, ions, or molecules are organized in crystalline solids.