Question

A substance \(A_{x} B_{y}\) crystallizes in FCC lattice in which atom \(A\) occupies each corner of cube and atom \(B\) occupies the centres of each face of cube. Identify the correct composition.

Short answer

The composition is \(AB_{3}\).

Step-by-step solution

Identify the positions of atoms in the FCC lattice

In a Face-Centered Cubic (FCC) lattice, atoms are located at each corner of the cube and at the centers of all the cube faces. Atom A occupies each corner, and atom B occupies the centers of each face.

Count the number of atoms at the corners

A cube has 8 corners. Since each corner atom is shared by 8 adjacent cubes, each corner contributes \frac{1}{8} of an atom to the unit cell. So, the total number of atoms A in the FCC lattice is:\[ 8 \times \frac{1}{8} = 1 \]

Count the number of atoms at the face centers

A cube has 6 faces, and each face-centered atom is shared by 2 adjacent cubes. Therefore, each face-centered atom contributes 1/2 of an atom to the unit cell. So, the total number of atoms B in the FCC lattice is:\[ 6 \times \frac{1}{2} = 3 \]

Determine the chemical formula from the atom count

From the previous steps, we have 1 atom of A and 3 atoms of B in the unit cell. Therefore, the chemical composition of the substance is \(AB_{3}\).

Key concepts

Crystal Structure

In solid materials, the arrangement of atoms, ions, or molecules is called the crystal structure. This structure determines many physical properties of the material, such as density, conductivity, and strength.
Crystals can have various kinds of structures, such as simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). It's not just about the type of atoms present but also their spatial arrangement.
For example, diamond and graphite are both made of carbon, but they have different crystal structures leading to vastly different properties.

Face-Centered Cubic lattice

The Face-Centered Cubic (FCC) lattice is one of the most common types of crystal structures found in metals and alloys.
In an FCC lattice, atoms are positioned at each of the 8 corners and the centers of the 6 faces of the cube. This arrangement allows atoms to pack together very efficiently.
Here's how FCC lattice mathematically works:

• There are 8 corner atoms, each shared by 8 adjacent cubes, contributing 1/8 atom per cube corner. Total contribution from corners is: \[ 8 \times \frac{1}{8} = 1 \]
  
• There are 6 face-centered atoms, each shared by 2 adjacent cubes, contributing 1/2 atom per cube face. Total contribution from faces is: \[ 6 \times \frac{1}{2} = 3 \]
  

Thus, a single FCC unit cell has 4 atoms in total. This dense packing results in high metallic bonding and common occurrence in metals like aluminum, copper, and gold.

Chemical Composition

Chemical composition refers to the types and amounts of atoms present in a substance. It's commonly expressed in the form of a chemical formula, which shows the ratio of different atoms.
For the given exercise, we have the compound \(A_{x} B_{y}\) with atoms A and B arranged in an FCC lattice.

• Atom A is located at each of the 8 cube corners, contributing to: \[ 8 \times \frac{1}{8} = 1 \] atom.
  
• Atom B is located at each of the 6 face centers, contributing to: \[ 6 \times \frac{1}{2} = 3 \] atoms.
  

So, for every unit cell, we have one atom of A and three atoms of B. Hence, the chemical formula for this substance is \(AB_{3}\).
This formula indicates a compound where the number of B atoms is three times that of A atoms. Recognizing such patterns helps in understanding material properties and predicting behaviors under different conditions.