Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of atoms per unit cell in a BCC, a FCC and a simple cubic cell are respectively (a) \(1,4,2\) (b) \(2,4,1\) (c) \(4,1,2\) (d) \(4,2,2\)

Step-by-Step Solution

Answer

The correct choice is (b) (2,4,1).

Step 1: Understanding BCC Structure

In a Body-Centered Cubic (BCC) lattice, there is one atom at each corner of the cube and one atom at the center of the cube. Each corner atom is shared among 8 different cubes. Therefore, the contribution of corner atoms is \( \frac{1}{8} \times 8 = 1 \) atom. The center atom is entirely within one cube. Thus, the total number of atoms per unit cell in BCC is \(1 (from\ corners) + 1 (center) = 2 \).

Step 2: Understanding FCC Structure

A Face-Centered Cubic (FCC) lattice has one atom at each corner of the cube and one atom at the center of each cube face. Each corner atom is shared among 8 cubes while each face-centered atom is shared between 2 cubes. So, the contribution of corner atoms is \( \frac{1}{8} \times 8 = 1 \) atom. The contribution of face-centered atoms is \( \frac{1}{2} \times 6 = 3 \) atoms. Therefore, the total for FCC is \(1 (from\ corners) + 3 (from\ faces) = 4 \).

Step 3: Understanding Simple Cubic Structure

A Simple Cubic lattice has atoms only at the corners of the cube. Each corner atom is shared by 8 unit cells, so the contribution is \( \frac{1}{8} \times 8 = 1 \). Therefore, the number of atoms in a simple cubic cell is \(1\).

Step 4: Conclusion

The number of atoms per unit cell is \(2\) for BCC, \(4\) for FCC, and \(1\) for Simple Cubic. This matches the option (b) \((2,4,1)\).

Key Concepts

FCC

The Face-Centered Cubic (FCC) structure is an efficient and compact way atoms are arranged in space. Imagine a cube. In the FCC structure, there is one atom situated at each corner of the cube. Additionally, each face of the cube—six faces in total—has one atom right at its center. This makes the FCC one of the most closely packed unit cell structures.

- The corner atoms of FCC are shared among 8 different cubes, contributing \( \frac{1}{8} \,times\8 = 1 \) atom to each unit cell.- Each of the face-centered atoms is shared between two cubes, contributing \( \frac{1}{2} \,times\6 = 3 \) atoms per unit cell.

Adding these contributions together results in \(4\) atoms per face-centered cubic unit cell, making it highly dense and stable for various metal structures like aluminum, copper, and gold.

Simple Cubic

The Simple Cubic structure, while being the least dense of the main crystalline structures, is essential for understanding atomic arrangements. Envision a cube where atoms are placed solely at the corners. This straightforward arrangement allows each atom to be shared among 8 unit cells. Consequently, the total contribution of corner atoms per unit cell for simple cubic arrangements is \( \frac{1}{8} \,times\8 = 1 \) atom.

Such a unit cell configuration is rare in metals but manifests in polonium. Because of the minimal atomic packing, the Simple Cubic structure tends to be less stable compared to other structures such as FCC or BCC, making it less common in nature. Understanding it helps in grasping the basics of crystal lattice structures.

Unit Cell

A Unit Cell is the smallest repeatable pattern that reflects the entire structure of a crystal lattice. These three-dimensional cells repeat continuously in three dimensions to form an entire crystal. Understanding the unit cell concept is fundamental to solid-state chemistry and physics.

Unit cells can be in various shapes, such as cubic or tetragonal, depending on how atoms or molecules pack together. The geometric and symmetry properties of a unit cell collectively determine the physical properties of the material, including its density and stability. Unit cells are characterized by lattice parameters such as edge lengths and angles, specific to each crystal system.

Each of the different cubic structures—simple cubic, body-centered cubic, and face-centered cubic—has a unique configuration of atoms that define the nature of the lattice.

Atoms per Unit Cell

The number of atoms per unit cell indicates how densely a crystal structure is packed. This measure varies depending on the arrangement of the atoms within the unit cell. Let's break down three primary cubic structures:

- **Body-Centered Cubic (BCC):** There are two atoms per unit cell. One comes from the atom at the center and another from the contribution of corner atoms. - **Face-Centered Cubic (FCC):** Here, the unit cell contains four atoms in total, calculated from atoms at corners and each face. - **Simple Cubic:** Contains only one atom per unit cell, with atoms solely at the corners.

Understanding atoms per unit cell helps in calculating the theoretical density of materials, as well as informs about atomic connectivity and crystal robustness. Each distinct count reflects how atoms share their positions within a lattice.