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

Question

A substance \(A_{x} B_{y}\) 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 \(A_{x} B_{y^{\prime}}\) (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 correct composition for the substance is (a) \(AB_3\).

Step 1: Identify face-centered cubic (FCC) lattice arrangement

In a face-centered cubic (FCC) lattice, atoms are located at each corner and at the center of each face of the cube.

Step 2: Count atoms at corners for 'A'

Each atom 'A' is located at a corner of the cube. There are 8 corners in a cube. However, each corner belongs to 8 adjacent cubes, so each corner atom contributes 1/8th to one cube. Thus, the total contribution of 'A' in the cube is: \(8 \times \frac{1}{8} = 1\).

Step 3: Count atoms at face centers for 'B'

Atoms 'B' are located at the centers of each face of the cube. There are 6 faces for a cube, and each face-center atom is shared between 2 adjacent cubes. Therefore, each face contributes \(\frac{1}{2}\). Thus, the total contribution of 'B' in the cube is: \(6 \times \frac{1}{2} = 3\).

Step 4: Derive the chemical formula

From the calculations above, there is 1 atom of 'A' and 3 atoms of 'B'. Thus, the chemical formula derived from the lattice structure is \(AB_3\).

Step 5: Identify correct composition option

Compare the derived composition \(AB_3\) with given options: (a) \(\mathrm{AB}_{3}\), (b) \(\mathrm{A}_4 \mathrm{~B}_3\), (c) \(\mathrm{A}_3 \mathrm{~B}\), (d) composition cannot be specified. The correct option is (a) \(\mathrm{AB}_{3}\).

Key Concepts

Crystallography

Crystallography is the scientific study of crystal structures and the arrangement of atoms within a solid. It involves analyzing how atoms are packed in ordered structures called lattices. A lattice is a repeating pattern of points in space and is crucial in determining the physical properties of materials.
In the face-centered cubic (FCC) lattice, features are characterized by atoms positioned at the corners and center of each face of the cube. This particular lattice arrangement is one of the most efficient ways to fill a space, maximizing the density of the structure.

• Each cube has 8 corners and 6 faces.
• The corner atoms contribute partially as each corner atom is shared by 8 adjacent cubes.
• The face-centered atoms are shared between 2 adjacent cubes.

Understanding these arrangements helps us derive important details about a compound’s properties, like how they interact with energy and how they might behave in different temperatures or pressures.

Chemical Formula Derivation

Chemical formula derivation involves using the structural data of a crystal lattice to understand the ratio of elements in a compound. It provides a concise way to convey complex information about molecules.
A face-centered cubic (FCC) lattice helps us understand this by offering a visual guide to the location and contribution of atoms:

• For atoms 'A' situated at the corners, each contributes \( \frac{1}{8} \) to the unit cell, since they are shared with 8 surrounding cubes.
• Thus, the total contribution of 'A' amounts to \( 8 \times \frac{1}{8} = 1 \).
• For atoms 'B' located at the centers of the cube faces, each atom contributes \( \frac{1}{2} \) as they are shared between 2 cubes.
• This results in the total contribution of 'B' being \( 6 \times \frac{1}{2} = 3 \).

Therefore, from these contributions, the chemical formula for this particular crystal arrangement is derived as \( AB_3 \). This formula illustrates the 1:3 ratio in which elements A and B are combined.

Ionic Compounds

Ionic compounds are formed when electrons are transferred from one atom to another, resulting in charged particles called ions. These ions come together to form a compound that is electrically neutral overall.
In the context of crystal structures, particularly the FCC lattice mentioned, ionic compounds often involve atoms organized in a way that maximizes attractions between oppositely charged ions.

• Ions arrange in repetitive patterns to form crystals that minimize the repulsive forces and maximize attractive forces.
• The FCC lattice is ideal for forming ionic compounds because of its dense packing arrangement, where ions of opposite charges can be closer together, leading to more stable bonds.

Such an arrangement not only influences the overall stability and strength of materials but also affects properties like melting point, solubility, and conductivity. Understanding how ions are arranged in structures can thus offer significant insights into the nature of ionic compounds and their practical applications.