Problem 86 - A Complete Resource Book in Chemistry for JEE Main

Question

A compound formed by elements A and B crystallizes in cubic structure where A atoms are at the corners or a cube and \(\mathrm{B}\) atoms are at the face centre. The formula of the compound is (a) \(\mathrm{AB}\) (b) \(\mathrm{AB}_{2}\) (c) \(\mathrm{AB}_{3}\) (d) \(\mathrm{AB}_{4}\)

Step-by-Step Solution

Answer

The compound is \( \mathrm{AB}_3 \).

Step 1: Count Atoms at the Corners

Each "A" atom is located at the corners of a cube. A cube has 8 corners, so there are 8 "A" atoms per unit cell. However, each corner atom is shared among 8 adjacent unit cells. Therefore, each unit cell contains \( \frac{1}{8} \times 8 = 1 \) "A" atom.

Step 2: Count Atoms at the Face Centers

Each "B" atom is located at the center of a face of the cube. Since a cube has 6 faces and each face-centered atom is shared between 2 unit cells, each unit cell contains \( \frac{1}{2} \times 6 = 3 \) "B" atoms.

Step 3: Determine the Compound Formula

The unit cell contains 1 "A" atom and 3 "B" atoms. Therefore, the formula of the compound is \( \mathrm{AB}_3 \).

Key Concepts

Cubic Crystal Structure

In solid state chemistry, understanding the arrangement of atoms in a crystal is key to predicting the properties of a compound. One of the fundamental structures is the cubic crystal structure. It is highly symmetrical, which means that atoms are organized in a three-dimensional checkerboard pattern. There are several types of cubic structures, namely simple cubic, body-centered cubic, and face-centered cubic (FCC).

• In a simple cubic structure, atoms are located only at the corners of the cube.
• For body-centered cubic, there is an additional atom located in the center of the cube in addition to the corner atoms.
• The face-centered cubic structure, relevant to our compound, places atoms not only at the corners but also in the center of each face of the cube.

In our exercise, the A atoms are positioned at the cube corners, while the B atoms occupy the face centers. This systematic arrangement plays a significant role in determining both the compactness and density of the material.

Unit Cell Calculation

The unit cell is the smallest repeating unit that shows the entire structure of a crystal. Calculations involving unit cells are crucial for deriving the formula of a compound based on its crystal structure. For a cubic unit cell:

• Each corner atom is shared among 8 neighboring cubes. Hence, each unit cell contains \( \frac{1}{8} \) of each corner atom.
• Atoms located at the face center are shared among 2 unit cells; thus, each face contributes \( \frac{1}{2} \) of an atom to a single unit cell.

In the given exercise, we have A atoms at all 8 corners and B atoms at each of the 6 faces. By counting the atoms from these parts:

• Corners: \( \frac{1}{8} \times 8 = 1 \) A atom per unit cell.
• Face centers: \( \frac{1}{2} \times 6 = 3 \) B atoms per unit cell.

Hence, the molecular formula of the compound is derived as \( AB_3 \). This calculation is a vital step in crystallography when determining the composition of complex structures.

Coordination Number

In crystallography, the coordination number is a measure of how closely packed the atoms are in a crystal structure. It refers to the number of nearest neighboring atoms or ions surrounding a central atom. This number gives important insights into the stability and bonding characteristics of the compound.
For cubic structures, the coordination number can vary:

• In a simple cubic structure, the coordination number is 6 because each atom touches 6 others in the direct surroundings.
• In a body-centered cubic, this increases to 8 due to an additional central atom.
• For a face-centered cubic structure, the coordination number elevates to 12 since each atom at the face center is closely packed with the surrounding atoms.

Understanding the coordination number allows scientists to infer other properties such as density and bonding strength. In the face-centered structure discussed in the exercise, each B atom situated at a face center touches surrounding A atoms, contributing to its characteristic properties.