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? \((\mathbf{a}) 1,(\mathbf{b}) 2,(\mathbf{c}) 3,(\mathbf{d}) 4,(\mathbf{e}) 5 .\)

Short answer

The minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice is \(2\). This includes the contributions of \(1\) atom from the 8 corner atoms and \(1\) atom from the center. Therefore, the correct answer is \((\mathbf{b}) 2\).

Step-by-step solution

Understand the body-centered cubic lattice structure

A body-centered cubic lattice has atoms at the eight corners of the unit cell and one atom at the center. It's important to note that each of the corner atoms is shared by 8 adjacent unit cells. Therefore, only one-eighth of each corner atom belongs to the unit cell.

Calculate the total number of atoms in the unit cell

To find the total number of atoms in a unit cell of a BCC lattice, first, we need to find the contribution of atoms from the corners and the center: 1. Atoms at the corners: There are 8 corners, and each corner atom is shared by 8 unit cells. So, the contribution of corner atoms to the unit cell is \(8 \times \frac{1}{8} = 1\). 2. Atom at the center: The center atom is entirely inside the unit cell. Therefore, its contribution to the unit cell is equal to 1.

Add the contributions of corner and center atoms

The total number of atoms in the unit cell of a body-centered cubic lattice can be found by adding the contributions of corner atoms and the center atom: Total atoms in unit cell = corner atoms contribution + center atom contribution = 1 + 1 = 2 The minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice is 2. So, the correct answer is: \((\mathbf{b}) 2\).

Key concepts

Unit Cell

The concept of a unit cell is fundamental in understanding crystal structures. A unit cell is the smallest repeating unit that can be used to construct the entire crystal lattice. Imagine it as a tiny building block that makes up the whole structure. In the context of a body-centered cubic (BCC) lattice, the unit cell is organized with atoms situated strategically at specific positions.

For a BCC unit cell, one atom resides at each of the eight corners, and a single atom is located at the center. This configuration means that the unit cell encapsulates the basic structural and spatial properties of the entire material. It's through this tiny volume that the material's atomic arrangement and symmetry are defined. Recognizing the arrangement within a unit cell is key to deciphering various material properties, including density and stability.

Atomic Structure

Atomic structure refers to how atoms are arranged within a substance. This arrangement is pivotal in defining the properties of the material. Atoms can be thought of as the fundamental building blocks of matter, comprised of a nucleus and electrons moving around it.
In a BCC arrangement, the position of each atom is determined by its relationship with adjacent atoms. This structure leads to each corner atom being shared across several unit cells, which is why only a portion of each corner atom belongs to any one cell. This strategic placement affects how densely atoms pack, contributing directly to physical characteristics such as the material's strength and ductility.
The atomic structure in a BCC lattice is defined by the contribution of these atoms—one full atom from the center and a net of one atom from all eight corners. Understanding their placement helps in visualizing the interconnected nature of atomic assemblies.

Crystal Lattice

A crystal lattice is a systematic, three-dimensional arrangement of atoms inside a crystal. It is like a grand structure built entirely from repetitive unit cells, much like a complex building made of uniform bricks. Understanding this arrangement helps in making sense of a material’s physical and chemical properties.
The BCC crystal lattice features a unique geometric configuration. This lattice has a high degree of symmetry because of its organized framework of atoms. Each atom in the BCC crystal lattice touches other atoms at specific points, giving rise to distinct structural properties.

• Stability: Provides a balanced structure where each atom is equidistant.
• Density: Typically less dense than face-centered cubic (FCC) lattices due to its unique configuration.

The type of lattice a material has can be directly linked to its hardness, electrical conductivity, and melting point.

Chemistry Education

Chemistry education is not just about memorizing formulas and structures; it's about understanding the building blocks of materials and how they interact. This knowledge is crucial for grasping more complex concepts in chemistry and materials science.
In learning about body-centered cubic lattices, students not only practice mathematical calculations, like determining the number of atoms in a unit cell, but they also learn about spatial geometry. Understanding these patterns helps leverage concepts such as molecular interactions and the reasons behind the physical properties observed in everyday materials.

• Visualization: Enhances the ability to mentally visualize atomic structures.
• Problem-solving: Develops skills to tackle challenges related to material properties and behavior.
• Application: Links theoretical knowledge with practical applications, from metallurgy to pharmaceutical development.

A strong grasp of these foundational concepts equips students with the tools they need for future scientific endeavors.