Problem 27 - Chemistry The Central Science

Question

What is the minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice? (a) \(1,(\mathbf{b}) 2,(\mathbf{c}) 3,(\mathbf{d}) 4,(\mathbf{e}) 5\)

Step-by-Step Solution

Answer

In a body-centered cubic (BCC) lattice, there are 8 corner atoms with each contributing 1/8th of its volume to the unit cell, and 1 additional atom at the center contributing fully to the unit cell. Thus, the minimum number of atoms contained in the unit cell of an element with a BCC lattice is \(1 + 1 = 2\). So, the answer is (b) \(2\).

Step 1: Understanding a body-centered cubic lattice

A body-centered cubic lattice is a three-dimensional arrangement of atoms where there is an atom at each corner of the cubic unit cell, and an additional atom at the center of the cube.

Step 2: Counting the number of atoms in the BCC unit cell

In a BCC unit cell, there are a total of 9 atoms present. However, only some portion of each corner atom is present inside the unit cell, while the central atom is entirely inside the unit cell. To determine the number of corner atoms that contribute to the unit cell, we take into account only the fraction of each corner atom present in the unit cell. There are 8 corner atoms, and each corner atom contributes only 1/8th of its volume to the unit cell because it is shared between 8 neighboring unit cells. Therefore, the contribution of the 8 corner atoms to the unit cell is: \(8 \times \dfrac{1}{8} = 1\) And there is 1 additional atom at the center which contributes fully to the unit cell. So, the total contribution of atoms to the unit cell is: \(1 + 1 = 2\)

Step 3: Choosing the correct answer

The minimum number of atoms contained in the unit cell of an element with a body-centered cubic lattice is 2. Therefore, the correct answer is (b) 2.

Key Concepts

Unit Cell

A unit cell is the smallest repeating unit of a crystal lattice that retains the overall symmetry and structure of the solid. Think of it like a tiny building block from which the entire crystal is constructed.
It contains a specific arrangement of atoms and can be stacked in three dimensions to form the whole crystal.
In the context of crystal structures, the unit cell is crucial because it provides a systematic way to describe the entire lattice with a minimal amount of information.

• For a body-centered cubic (BCC) lattice, the unit cell includes atoms located at the corners and one atom at the very center of the cube.
• Each corner atom is shared with adjacent unit cells, contributing only a fraction of its volume to any single unit cell.

This concept helps us calculate properties such as density and packing efficiency by using the simplest representation instead of dealing directly with the entire crystal.

Atoms in Crystals

Atoms in crystals are arranged in a very orderly and repeating manner, which defines the crystal lattice structure. Understanding how these atoms are packed is fundamental to grasping material properties like hardness, melting point, and electrical conductivity.
In a body-centered cubic lattice (BCC), the arrangement goes beyond simple stacking.

• The BCC structure has one atom at each corner of the cube and another atom precisely in the middle of the structure.
• This arrangement means there are parts of different atoms contributing to a single cell.

Although it appears complex, each pattern is a key to understanding how materials behave under different conditions. With this, students can visualize how atoms fit together like a three-dimensional jigsaw puzzle.

Crystal Lattice Structures

Crystal lattice structures refer to the geometrical arrangement of atoms, ions, or molecules in a crystalline solid. These structures are defined by their symmetry and consistency across the entire material.
The body-centered cubic (BCC) is one of several possible lattice structures.

• In a BCC lattice, each unit cell is aligned perfectly with its neighbors creating a very stable arrangement.
• This stability is due to the central atom being equidistant from all of the corner atoms in its unit cell.

Lattice structures determine many material properties, including strength and ductility. Understanding these arrangements sheds light on why some metals, like iron, exhibit specific properties when cooled or heated. Each structural type—whether BCC, face-centered cubic (FCC), or hexagonal close-packed (HCP)—offers a unique insight into the behavior of metallic elements and compounds.