Question

An element occurring in the body-centred cubic \((\mathrm{BCC})\) structure has \(1.208 \times 10^{23}\) unit cells. The total number of atoms of the element in these cells will be (a) \(2.416 \times 10^{23}\) (b) \(3.618 \times 10^{23}\) (c) \(6.04 \times 10^{22}\) (d) \(1.208 \times 10^{2.3}\)

Short answer

\(2.416 \times 10^{23}\) atoms are present in \(1.208 \times 10^{23}\) BCC unit cells.

Step-by-step solution

Understanding the BCC Structure

A body-centred cubic (BCC) structure is a type of crystal lattice where each unit cell has one atom at each of its eight corners and one atom at its center. However, the corner atoms are shared among eight adjacent cells, so each such atom contributes 1/8 of an atom to the unit cell, and the central atom is entirely within the cell.

Calculating Atoms in a Single BCC Unit Cell

Each corner atom contributes 1/8 of an atom to the cell, and since there are 8 corners, this amounts to 1 atom in total from the corners. The central atom contributes 1 whole atom. Therefore, each unit cell has a total contribution of: 1 atom (from corners) + 1 atom (from center) = 2 atoms.

Calculating Total Number of Atoms

We are given that there are \(1.208 \times 10^{23}\) unit cells. Since each cell contains 2 atoms, the total number of atoms is: \(1.208 \times 10^{23} \text{ cells} \times 2 \text{ atoms/cell} = 2.416 \times 10^{23}\) atoms.

Key concepts

Body-Centred Cubic Structure

When learning about crystalline structures, it's essential to understand the geometric layout that atoms adopt in different materials. The body-centred cubic structure (BCC) is one of these fundamental arrangements seen in some metallic elements.

In a BCC structure, the atoms are positioned in a way that forms a cube with one atom at each of the cube's eight corners and a single atom at the center, effectively 'centring' the cell. This configuration is efficient in the use of space and common among atoms that favor less tightly packed arrangements than, for example, the face-centred cubic structure. It's interesting to note that the BCC structure is less dense, which often results in materials with distinct physical properties, like malleability and electrical conductivity.

Understanding this spatial configuration enables one to deduce various material properties, including its packing density and interaction with other atoms. It's the cornerstone for learning more advanced concepts in materials science and solid-state physics.

Unit Cells in Crystal Lattice

The concept of a unit cell is akin to a building block in a large structure. Imagine a lego model; each individual lego block would represent a unit cell, and the completed model would be the crystal lattice. A unit cell is the smallest repeating pattern that shows the entire structure of the crystal lattice — the way individual atoms are arranged in a material.

In a BCC crystal lattice, each unit cell is a cube defined by the positioning of the atoms, as noted in the body-centred cubic structure. But a crucial point to understand is that the atoms at the corners of each BCC unit cell are shared with adjacent unit cells. This shared ownership means that each corner atom only contributes a fraction of an atom to any particular unit cell. Calculating the precise contribution is vital for determining the number of atoms that effectively 'belong' to a single unit cell, which has direct implications for the material's density and properties.

Atoms Calculation in BCC

Now that we have a grasp of the BCC unit cell and its arrangement in the crystal lattice, let's explore the process of calculating the number of atoms in such a structure. This understanding is essential for determining a material's properties and behavior.

Since each of the eight corner atoms is shared by eight different unit cells, each corner atom contributes precisely 1/8 of an atom to an individual BCC unit cell. With 8 corners, these effectively add up to contribute one whole atom. Additionally, the central atom, being not shared, contributes another whole atom to the unit cell. This calculation is summarized as follows: \(\frac{1}{8} \text{ atom/corner} \times 8 \text{ corners}\) + 1 \text{ atom (center)} = 2 \text{ atoms/unit cell}\. When observing a multitude of unit cells, as in the given exercise where there are \(1.208 \times 10^{23}\) unit cells, multiplying by 2 gives the total number of atoms effectively contained within these unit cells.

Thus, being fluent in this type of quantitative reasoning allows a clearer interpretation of material composition and readiness for dealing with empirical data in scientific and industrial applications.