Problem 93 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of atoms per unit cell in a simple cubic, face-centred cubic and body-centred cubic are....... respectively (a) \(1,2,4\) (b) \(1,4,2\) (c) \(4,2,1\) (d) \(1,4,3\)

Step-by-Step Solution

Answer

The answer is (b) 1, 4, 2.

Step 1: Simple Cubic

In a simple cubic structure, there is one atom per unit cell. Each corner atom is shared among eight adjacent unit cells, and since there are eight corners, the contribution is \(\frac{1}{8} \times 8 = 1\).

Step 2: Face-Centred Cubic (FCC)

In an FCC structure, there are 8 corner atoms and 6 face atoms in each unit cell. Each corner atom contributes \(\frac{1}{8}\) of an atom, and each face atom contributes \(\frac{1}{2}\). The calculation is \( (8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4\).

Step 3: Body-Centred Cubic (BCC)

In a BCC structure, there are 8 corner atoms and a single atom in the center. The corner atoms contribute \(\frac{1}{8}\) each, so \(8 \times \frac{1}{8} = 1\). The center atom contributes 1, making a total of \(1 + 1 = 2\) atoms per unit cell.

Key Concepts

Simple Cubic

The simple cubic structure is one of the most basic configurations of atoms in crystals. Imagine the structure like a perfect cube with atoms positioned at each corner. The simple cubic structure is not as tightly packed as some other crystal arrangements. Each corner atom is part of eight neighboring cubes. This means that each atom at a corner is shared between eight different unit cells.

One important point to consider is the calculation of the exact number of atoms per unit cell. Since each corner atom is shared among eight unit cells, each makes a contribution of \( \frac{1}{8} \) to any single unit cell. With eight corners in total, the simple cubic unit cell holds exactly one full atom (\( \frac{1}{8} \times 8 = 1 \)).

While simple cubic structures are easy to visualize, they are not very common in nature due to inefficient use of space, where only about 52% of the space is filled by atoms.

Face-Centred Cubic (FCC)

The face-centred cubic (FCC) structure is a more efficient way for atoms to be stacked together. In this configuration, each unit cell contains atoms at each corner and one atom at the center of each of the six faces of the cube. This setup contributes to a densely packed structure.

Calculating the number of atoms in an FCC cell considers both corner and face-centered atoms. Each corner atom contributes \( \frac{1}{8} \) of itself (with 8 corners total) and each face-centered atom contributes \( \frac{1}{2} \) of itself to the unit cell. As there are six faces, this totals:

• For corners: \( 8 \times \frac{1}{8} = 1 \)
• For faces: \( 6 \times \frac{1}{2} = 3 \)

Combining these, an FCC unit cell contains 4 full atoms. The highly efficient packing means FCC structures are much more common than simple cubic arrangements, filling approximately 74% of the space.

Body-Centred Cubic (BCC)

In the body-centered cubic (BCC) structure, atoms are arranged in a slightly different format. Picture the structure as having atoms at each corner of the cube, similar to the simple cubic configuration, but adding an extra atom in the center of the cube makes the BCC structure distinct.

This central atom is completely contained within its unit cell and does not share space with other unit cells. In addition to the center atom, the corner atoms contribute \( \frac{1}{8} \) of themselves per cube. With eight corners, this totals an entire atom. Therefore, the BCC unit cell comprises:

• 8 corner atoms contributing: \( 8 \times \frac{1}{8} = 1 \)
• 1 total center atom

This results in two atoms per unit cell. The BCC arrangement is also fairly efficient, filling about 68% of the available space, leading to its frequent occurrence in certain metallic elements like iron.