Problem 66 - A Complete Resource Book in Chemistry for JEE Main

Question

A substance \(\mathrm{A}_{x} \mathrm{~B}_{y}\) crystallizes in a face-centred cubic (FCC) lattice in which atoms 'A' occupy each corner of the cube and atoms 'B' occupy the centres of each face of the cube. Identify the correct composition of the substance \(\mathrm{A}_{\mathrm{x}} \mathrm{B}_{y}\). (a) \(\mathrm{AB}_{3}\) (b) \(\mathrm{A}_{4} \mathrm{~B}_{3}\) (c) \(\mathrm{A}_{3} \mathrm{~B}\) (d) composition cannot be specified

Step-by-Step Solution

Answer

The composition of the substance is \(\mathrm{AB}_{3}\).

Step 1: Understand FCC Lattice Structure

In a face-centered cubic (FCC) lattice, atoms are positioned at each corner as well as the centers of all the faces of a cube. We need to determine how many atoms contribute to the lattice from both corners and faces for this structure.

Step 2: Calculate Atoms 'A' Contribution

Atoms 'A' are located at the corners of the cube. There are 8 corners in a cube, and each corner atom is shared by 8 adjacent unit cells. Thus, the contribution of 'A' to one unit cell is \(\frac{1}{8} \times 8 = 1\).

Step 3: Calculate Atoms 'B' Contribution

Atoms 'B' are located at the face centers of the cube. Each cube has 6 faces with 'B' atoms, and each face-centered atom is shared by 2 cells. Therefore, the contribution of 'B' to one unit cell is \(\frac{1}{2} \times 6 = 3\).

Step 4: Determine the Composition

Based on the contributions calculated, 1 atom of 'A' and 3 atoms of 'B' make up a single unit cell of the substance. Hence, the composition is \(\mathrm{AB}_{3}\).

Key Concepts

Atoms Contribution

In a face-centered cubic (FCC) lattice, different atoms contribute to the overall unit cell based on their positions within the lattice. Understanding this contribution is key to determining the formula of the compound. Atoms can occupy different positions, such as corners and face centers in the unit cell. Each position has unique sharing characteristics that affect how much an atom contributes to a single unit cell. For instance, corner atoms contribute less to a single unit cell because they're shared among several cells, while face-centered atoms have a different sharing pattern.

Corner Atoms

Corner atoms in a crystal lattice are those located at the vertices of the cube. In a face-centered cubic structure, there are exactly 8 corners.When considering their contribution to the unit cell, it's important to note that each corner atom is shared by 8 other unit cells in the overall lattice. This sharing condition means each corner atom only contributes \(\frac{1}{8}\) of itself to a single unit cell. Therefore, for a full cube with 8 corner atoms:

• Total corner atoms = 8
• Contribution = 8 corners \(\times \frac{1}{8} = 1\) atom per unit cell.

This calculation explains why corner atoms, though numerous in appearance, contribute minimally on a per unit cell basis.

Face-Centered Atoms

The face-centered atoms are placed at the center of each face of the cube in a face-centered cubic lattice. With 6 faces in a cube, and an atom located at the center of each, these atoms play a significant role in the composition of the unit cell. Each face-centered atom is shared between two adjacent unit cells, which means their contribution per unit cell is higher than that of corner atoms. Specifically, the contribution from each face-centered atom is:

• Shared atoms per face = 1/2
• Total faces = 6
• Total contribution = 6 faces \(\times \frac{1}{2} = 3\) atoms per unit cell.

Thus, face-centered atoms significantly impact the total atomic composition of the unit cell, contributing 3 atoms per unit cell in total.

Unit Cell Composition

The unit cell composition is derived from adding the contributions of all types of atoms present in the lattice. In a substance crystallizing in an FCC lattice, knowing the placements and contributions of different atoms helps figure out its formula. For a substance \( \mathrm{A}_{x} \mathrm{B}_{y} \):

• Corner 'A' atoms contribute 1 atom per unit cell.
• Face-centered 'B' atoms contribute 3 atoms per unit cell.

Adding these contributions gives a complete picture: 1 atom of 'A' and 3 atoms of 'B' per unit cell. This gives the compound a chemical composition expressed as \( \mathrm{AB}_{3} \).Understanding this composition is crucial for various applications, including material science and chemistry, as it clarifies the arrangement and proportion of elements within the material.