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 number of atoms per unit cell are 1 (simple cubic), 4 (FCC), and 2 (BCC), matching option (b) 1, 4, 2.

Step 1: Understanding the Simple Cubic Structure

A simple cubic unit cell is comprised of atoms located only at the corners of the cube. Since there are 8 corners in a cube and each corner atom is shared amongst 8 adjacent unit cells, the number of atoms in a simple cubic unit cell is \(\frac{1}{8} \times 8 = 1\).

Step 2: Analyzing the Face-Centered Cubic Structure

In a face-centered cubic (FCC) structure, atoms are present at each of the 8 corners and at the centers of all 6 faces of the cube. Corner atoms contribute \(\frac{1}{8} \times 8 = 1\) atom per unit cell. Each face-centered atom is shared between 2 unit cells, giving \(\frac{1}{2} \times 6 = 3\) atoms per unit cell. Therefore, the total is \(1 + 3 = 4\) atoms per unit cell in an FCC structure.

Step 3: Evaluating the Body-Centered Cubic Structure

A body-centered cubic (BCC) unit cell has atoms at each of the 8 corners and a single atom entirely within the center of the cube. Corner atoms contribute \(\frac{1}{8} \times 8 = 1\) atom, and the central atom contributes 1 atom itself. So the BCC has a total of \(1 + 1 = 2\) atoms per unit cell.

Step 4: Comparing to Answer Options

From steps 1, 2, and 3, we have found the number of atoms per unit cell in simple cubic, FCC, and BCC to be 1, 4, and 2 respectively. Comparing these results with the options, the correct answer matches option (b): \(1, 4, 2\).

Key Concepts

Simple Cubic Structure

The simple cubic structure is one of the most straightforward crystal lattice types found in materials science. It consists of a cube where each atom is positioned at one of the eight corners.
Understanding the contribution of corner atoms is vital as each atom isn't entirely part of just one unit cell—it is shared by all surrounding unit cells. In practice, each corner atom contributes only \(\frac{1}{8}\) of an atom to the individual unit cell because it is shared equally among eight adjacent cells. With eight corner atoms, the calculation for the total number of atoms per unit cell is: \(\frac{1}{8} \times 8 = 1 \)
Thus, a simple cubic unit cell effectively contains one complete atom. It is important to note that this structure is quite rare compared to others like BCC and FCC; for instance, the element polonium crystallizes in a simple cubic structure.

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is a more complex and frequently encountered crystalline arrangement. Here, atoms sit at each of the eight corners, similar to the simple cubic structure, but additionally, an atom is found at the center of each of the six faces of the cube.
To compute the number of atoms in an FCC unit cell, sum the contributions from the corners and faces:

• Corner atoms: \(\frac{1}{8} \times 8 = 1\) atom, as they are shared among eight unit cells.
• Face-centered atoms: Each face-centered atom is shared between two adjacent unit cells. Thus, each contributes \(\frac{1}{2}\) atom to the unit cell, and for six faces, this becomes \(\frac{1}{2} \times 6 = 3\) atoms.

Therefore, the total number of atoms present in an FCC unit cell is the sum of corner and face contributions, giving us \(1 + 3 = 4\) atoms.
This structure is observed in metals like aluminum, copper, and silver, offering a high packing density that contributes to the material's ductility and conductivity.

Body-Centered Cubic Structure

The body-centered cubic (BCC) structure is another prevalent crystal arrangement, distinguished by its unique atom placement. It features one atom at each of the eight corners of a cube, much like the other structures, but also includes a single atom at the center of the cube.
To determine the total number of atoms in a BCC unit cell, consider:

• Corner atoms: \(\frac{1}{8} \times 8 = 1\) atom, as these atoms are shared among eight unit cells.
• Center atom: This atom is completely contained within the unit cell and contributes an entire atom.

Adding these contributions together yields a total of \(1 + 1 = 2\) atoms per unit cell.
BCC structures are found in metals like iron at certain temperatures, tungsten, and chromium. The arrangement allows for some degree of flexibility with packing efficiency, lending certain mechanical properties like hardness and strength.