Problem 10 - A Complete Resource Book in Chemistry for JEE Main

Question

Out of seven crystal systems how many can have body-centred unit cell? (a) 4 (b) 2 (c) 3 (d) 5

Step-by-Step Solution

Answer

The answer is 3 (cubic, tetragonal, orthorhombic).

Step 1: Understanding the Crystal Systems

Crystals are organized into seven different systems based on the symmetry and shape of the unit cell. These systems include: cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, and rhombohedral.

Step 2: Defining a Body-Centred Unit Cell

A body-centred unit cell is one where there is an additional lattice point in the center of the cell. This is one type of lattices that can be associated with certain crystal systems.

Step 3: Examining Each Crystal System

Different crystal systems can have multiple forms of lattice structures. Let's explore which can have a body-centred arrangement: 1. Cubic - Can have a body-centred lattice 2. Tetragonal - Can have a body-centred lattice 3. Orthorhombic - Can have a body-centred lattice 4. Monoclinic, Triclinic, Hexagonal, and Rhombohedral - Do not have body-centred lattices due to symmetry restrictions specific to their geometries.

Step 4: Counting the Possibilities

From the analysis, there are three crystal systems among the seven: cubic, tetragonal, and orthorhombic, that can have body-centred unit cells.

Key Concepts

Body-Centred Unit Cell

A body-centred unit cell is a unique arrangement within a crystal lattice. It isn't just a basic cube. Instead, think of it as a cube that has an additional atom right at its center. This central atom plays a significant role in stabilizing the structure and can affect the properties of the crystal, such as its density. In this type of unit cell, each corner atom is shared with adjacent unit cells, but the central atom belongs solely to the unit cell in question. As a result, this setup contributes to the overall structural integrity of the material. Not all crystal systems can accommodate a body-centred unit cell. Only specific geometries permit this type of arrangement. In the vast world of crystalline structures, the body-centred arrangement is one of many, each lending unique properties to the substance they form.

Lattice Structures

Lattice structures are the backbone of crystal formation. Think of them as a repeating pattern that extends in all directions. These structures are based on the geometric symmetry and dimensions of the crystal. Lattice structures can be considered the skeleton of crystalline materials. There are several types of lattice arrangements, dictated by the arrangement of atoms. Some of these include:

• Simple Cubic: Atoms positioned only at the corners.
• Body-Centred Cubic: Atoms at the corners and one in the center.
• Face-Centred Cubic: Atoms at corners and centers of all cube faces.

Such variations in lattice structures lead to different properties. For instance, materials with a body-centred cubic lattice may have different mechanical properties, such as strength and stability, compared to those with a simple cubic structure.

Crystal Symmetry

Crystal symmetry refers to how a crystal's structure can be rotated, reflected, or translated, yet still appear the same. This concept is crucial because it determines the crystal system to which a particular crystal belongs. Crystals are classified into seven symmetry systems: cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, and rhombohedral. Each system has its defined symmetry rules which are

• Cubic: Highly symmetrical with equal axes and angles at 90°.
• Tetragonal: Similar to cubic but with one axis longer or shorter.
• Orthorhombic: All axes are different lengths but intersect at 90°.
• Others (such as monoclinic or rhombohedral) have less symmetry and complex angles.

The symmetry dictates the crystal's external shape and internal properties. Moreover, it also influences the type of lattice structure that can be formed. This is why only certain crystal systems can accommodate the body-centred arrangement, as not all symmetries align with having an atom in the central position of the unit cell.