Problem 135 - A Complete Resource Book in Chemistry for JEE Main

Question

Number of atoms in the unit cell of Na (BCC type crystal) and \(\mathrm{Mg}\) (FCC type crystal) are respectively [2002] (a) 4,4 (b) 4,2 (c) 2,4 (d) 1,1

Step-by-Step Solution

Answer

(c) 2,4

Step 1: Understand the Crystal Structure of Na

Na is given in the BCC (Body-Centered Cubic) structure. In BCC, there is one atom at each corner of the cube and one atom in the center of the cube. Each corner atom is shared by 8 neighboring unit cells. Thus, the contribution of corner atoms per unit cell is \( \frac{1}{8} \times 8 = 1 \). The center atom is entirely within the unit cell, contributing 1 atom. Therefore, the total number of atoms in the BCC unit cell is \(1 + 1 = 2\).

Step 2: Understand the Crystal Structure of Mg

Mg is given in the FCC (Face-Centered Cubic) structure. In the FCC structure, there is one atom at each corner of the cube and one atom on each face of the cube. Each corner atom is shared by 8 neighboring unit cells, contributing \( \frac{1}{8} \times 8 = 1 \) atom. Each face atom is shared by 2 adjacent unit cells, and there are 6 faces, contributing \( \frac{1}{2} \times 6 = 3 \) atoms. Therefore, the total number of atoms in the FCC unit cell is \(1 + 3 = 4\).

Step 3: Compare with the Options

Based on the calculations, we found that the number of atoms in the unit cell is 2 for Na (BCC) and 4 for Mg (FCC). Now, compare these results to the given options.

Key Concepts

BCC (Body-Centered Cubic)

The Body-Centered Cubic (BCC) structure is a common crystal structure found in metals like sodium (Na). In this configuration, each unit cell consists of one atom located at each of the eight corners of a cube and a single atom at the center. The corner atoms do not belong exclusively to one unit cell; they are shared with neighboring cells.
The sharing divides each corner atom's contribution to a particular unit cell to just one-eighth. Since there are eight corners, the total contribution from corner atoms equals one whole atom (\[ \frac{1}{8} \times 8 = 1 \]). The atom located at the very center of the cube is not shared and hence counts fully as one atom.

• Contribution from corner atoms: 1 atom (from all corners together)
• Contribution from the central atom: 1 atom

Adding these contributes a total of two atoms to the BCC unit cell, making it a simple yet elegant structure to understand.

FCC (Face-Centered Cubic)

The Face-Centered Cubic (FCC) structure is another prevalent pattern seen in materials like magnesium (Mg). In this layout, atoms are positioned at each corner similar to BCC, but there is an added layer of complexity with an atom at the center of each of the six cube faces.
Like in BCC, the corner atoms in FCC contribute just one-eighth to an individual unit cell due to eight-way sharing. Six face-centered atoms, however, each contribute half their mass to a specific unit cell, as each face is shared with an adjacent unit cell.

• Contribution from corner atoms: 1 atom (\[ \frac{1}{8} \times 8 = 1 \])
• Contribution from face-centered atoms: 3 atoms (\[ \frac{1}{2} \times 6 = 3 \])

With these calculations, the total number of atoms in an FCC unit cell amounts to four, rendering the structure packed tightly.

Unit Cell Calculation

Unit cell calculation involves determining the total count of atoms effectively present in a single unit cell. This requires understanding how atoms are shared among adjacent unit cells and how they contribute to the whole.
Two prominent types—BCC and FCC—are often contrasted to help learners appreciate the differences in atomic arrangement and density. In BCC structures, the corners contribute one atom in total due to eight-fold sharing, while a central atom adds another complete atom, giving two in total. Conversely, FCC structures derive their compact arrangement by utilizing both corner and face-centered atoms, leading to a sum of four atoms per unit cell.
Ultimately, calculating the atoms per unit cell for BCC and FCC captures the essence of crystal structure density and explains why different materials have varying properties based on their atomic arrangements.