Question

Lithium forms body centered cubic structure. The length of the side of its unit cell is \(351 \mathrm{pm}\). Atomic radius of the lithium will be (a) \(75 \mathrm{pm}\) (b) \(240 \mathrm{pm}\) (c) \(300 \mathrm{pm}\) (d) \(152 \mathrm{pm}\)

Short answer

The atomic radius of lithium is approximately \(152 \mathrm{pm}\).

Step-by-step solution

Understanding the Problem

Lithium forms a body-centered cubic (bcc) structure, which means that the unit cell contains atoms at the corners and one atom in the center of the cube. We need to calculate the atomic radius given the length of the side of the cube, which is known as the lattice parameter, denoted as \(a = 351 \mathrm{pm}\).

Identify BCC Characteristics

In a bcc structure, the atoms at the center and at the corners touch along the body diagonal. The relationship between the body diagonal \(d\), the atomic radius \(r\), and the lattice parameter \(a\) in a bcc cell is given by \(d = 4r\) because the body diagonal spans across two radii from corner atoms and two radii from the center atom.

Calculate the Body Diagonal

The body diagonal \(d\) of a cube can be calculated using the lattice parameter \(a\) as follows: \[ d = \sqrt{3} \cdot a \]Substitute \(a = 351 \mathrm{pm}\) into this equation to find the body diagonal length.

Substitute and Solve for Radius

With \(d = \sqrt{3} \cdot 351 \mathrm{pm}\), set this equal to \(d = 4r\) from the bcc structure characteristic:\[ \sqrt{3} \times 351 = 4r \]Now solve for \(r\) by dividing both sides by 4.

Key concepts

Body-Centered Cubic (BCC) Structure

The body-centered cubic (bcc) structure is a type of crystal lattice arrangement often seen in metals like lithium. It is one of the simplest and most common crystal structures in solid materials. In a bcc structure, each unit cell has atoms located at all eight corners and a single atom in the center, making it highly symmetrical.
Here's a quick breakdown of the bcc structure:

• Corner Atoms: Eight atoms are positioned at the corners of the cube.
• Center Atom: One atom is located in the middle of the cube.
• Connectivity: The center atom touches only the corner atoms.

These characteristics make bcc different from other structures, such as face-centered cubic (fcc), and lead to specific properties and calculations, like the behavior of the atoms along the diagonals of the cube.

Lattice Parameter

The lattice parameter is a crucial measurement in crystallography. It represents the physical dimension of unit cells in a crystal lattice. In simple terms, it is the length of the sides of the cube in a cubic crystal system.
For a bcc structure:

• Symbolized as: Usually denoted by the letter 'a', for instance, in the case of lithium, \( a = 351 \mathrm{pm} \).
• Determining Factor: The value of the lattice parameter is influenced by the type of atoms and the structure they form.
• Critical for Calculations: It facilitates calculations related to distances between atoms and other geometric features of the crystal.

Understanding the lattice parameter is essential to grasp the layout and the repeating patterns of atoms within the crystal.

Atomic Radius Calculation

Calculating the atomic radius in a bcc structure involves understanding the geometry within the cube. The relationship between the lattice parameter and the atomic radius is key to determining how atoms pack within the crystal.
For a body-centered cubic structure:

• Body Diagonal: The atoms in a bcc structure touch each other along the cube's body diagonal.
• Mathematical Relationship: \( d = 4r \) connects the body diagonal and the atomic radius, where \( d \) is the body diagonal.
• Formula for Body Diagonal: The formula \( d = \sqrt{3} \cdot a \) helps in computing the body diagonal's length, substituting \( a \) for the lattice parameter.
• Solve for Radius: Rearrange \( \sqrt{3} \times a = 4r \) to find the radius, giving a clearer picture of the atom size within the crystal.

This step-by-step calculation is pivotal to fully comprehend the atomic arrangement and the dimensions of atoms within the crystal matrix.

Crystal Structure

Crystal structure is the organized arrangement of atoms in a crystalline solid. It dictates many material properties, such as density, melting point, and hardness.
Here are some aspects related to crystal structures:

• Types of Crystal Structures: Commonly includes body-centered cubic (bcc), face-centered cubic (fcc), and hexagonal close-packed (hcp).
• Implications of Structure: The specific arrangement influences how the material behaves. For instance, bcc structures tend to have higher strength but lower ductility compared to fcc.
• Lattice Arrangement: The repeating pattern of atoms in a crystal lattice emphasizes uniformity and order.
• Impact on Properties: Differences in atomic arrangement lead to varied physical and chemical characteristics.

Recognizing how crystal structures are formed helps in appreciating the material's overall properties and how they contribute to diverse applications in science and industry.