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 \(\mathrm{A}\) 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 formula is \( \mathrm{AB}_{3} \).

Step 1: Determine Effective Atoms in the Cube

In a cubic structure where atoms are at the corners, each corner atom contributes  ext{th} of an atom to the unit cell. Since a cube has 8 corners, the effective number of A atoms in the unit cell is \(8 \times \frac{1}{8} = 1\).

Step 2: Evaluate Atoms at Face Centers

Atoms at the face centers contribute  ext{th} of an atom per face to the unit cell since each face atom is shared by two adjacent unit cells. A cube has 6 faces, so the number of B atoms is \(6 \times \frac{1}{2} = 3\).

Step 3: Write the Formula of the Compound

The total number of A atoms in the unit cell is 1, and the total number of B atoms is 3. Thus, the formula of the compound based on these counts is AB_{3}.

Key Concepts

Cubic Crystal Structure

When elements crystallize in a solid, they often do so in a specific pattern called a cubic crystal structure. This is a very common way that atoms combine and organize themselves. Imagine the atoms as tiny spheres that align in a cube-like pattern. Cool, right? There are different types of cubic crystal structures, but they all share this basic cube form.
Some atoms are positioned at the corners of this cube. Give each corner atom inside a typical cube-like structure its own tiny slice of domain. But remember, these corner atoms might actually belong to several cubes at once.

• Corner atoms are shared among eight cubes, which means that each corner atom only contributes one-eighth of itself to a single unit cell.
• Some structures also have atoms at the center of each face of the cube, contributing differently than corners due to their positioning.

Recognizing where these atoms are in the structure helps us understand how they interact and bond with each other.

Unit Cell Calculation

Understanding the unit cell is crucial when figuring out how many atoms are in this smallest structural portion of the crystal. Picture it as the basic building block that repeats to form the entire crystal. Knowing how to calculate effective atoms within a unit cell sharpens our understanding of the compound's properties.
In the case of our cubic crystal structure:

• We see that each corner atom contributes one-eighth to each unit cell. With eight corners, the total contribution is one whole atom from the corners.
• Atoms located at face centers are shared as well. Each face center atom is divided between two unit cells, contributing one-half of an atom to each. With six faces in a cube, the faces together contribute three whole atoms.

So, by using these logical steps and calculations, we determine the number of each type of atom present.

Stoichiometry of Compounds

Stoichiometry may sound fancy, but it simply refers to the calculation of reactants and products in chemical reactions. Here, we use it to understand the composition of a compound in its solid form.
In a given unit cell, we already know from our calculations:

• Each unit cell contains 1 'A' atom due to its presence at cube corners.
• It also contains 3 'B' atoms because of their presence at face centers.

With this information, we write the empirical formula of the compound as \( \mathrm{AB}_3 \). This simple ratio reveals how "A" and "B" atoms are organized within the compound in its crystalline structure. Understanding stoichiometry is essential for creating formulas that predict how these compounds will behave in different conditions.