Problem 139 - A Complete Resource Book in Chemistry for JEE Main

Question

Total volume of atoms present in a face-centred cubic unit cell of a metal is ( \(\mathrm{r}\) is atomic radius) [2006] (a) \(\frac{20}{3} \pi \mathrm{r}^{3}\) (b) \(\frac{24}{3} \pi \mathrm{r}^{3}\) (c) \(\frac{12}{3} \pi r^{3}\) (d) \(\frac{16}{3} \pi \mathrm{r}^{3}\)

Step-by-Step Solution

Answer

The total volume is \( \frac{16}{3} \pi r^3 \); option (d) is correct.

Step 1: Understanding the Structure of a Face-Centred Cubic (FCC) Unit Cell

A face-centred cubic (FCC) unit cell contains 4 atoms. These atoms are positioned at each corner of the cube and in the centre of each face. However, each corner atom is shared by 8 unit cells, and each face atom is shared by 2 unit cells.

Step 2: Calculating the Total Number of Atoms in the FCC Unit Cell

Each face contributes half an atom to the unit cell (since it's shared between two cells), contributing a total of 3 atoms from the 6 faces. Each corner contributes 1/8th of an atom (for the 8 shared corners), totaling 1 atom from the corners. Thus, the total is 3 (from faces) + 1 (from corners) = 4 atoms in one FCC unit cell.

Step 3: Calculating the Volume of a Single Atom

The volume of a single atom is calculated using the formula for the volume of a sphere: \( V = \frac{4}{3}\pi r^3 \). Here, \( r \) is the atomic radius.

Step 4: Calculating the Total Volume of Atoms

Since there are 4 atoms in the unit cell, the total volume is given by multiplying the volume of one atom by the number of atoms: \( V_{total} = 4 \times \frac{4}{3}\pi r^3 = \frac{16}{3} \pi r^3 \).

Step 5: Identifying the Correct Answer

Among the given options, the one that matches our calculated total volume of atoms in the unit cell is option \( (d) \frac{16}{3} \pi r^3 \).

Key Concepts

Atomic Radius

The atomic radius is a crucial concept in understanding crystal structures, particularly in face-centred cubic (FCC) unit cells. It represents the distance from the center of an atom to its outermost electron shell. In a crystal, it helps determine how atoms pack together.
In FCC structures, the atoms touch each other along the face diagonals. By knowing the atomic radius, one can infer the size of the entire unit cell and determine how atoms are organized spatially. This value aids in calculating other properties like the volume of each atom and, subsequently, the total volume occupied by all atoms in the unit cell.

Volume of a Sphere

Calculating the volume of an atom is based on treating it as a sphere. The formula for the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. This formula helps measure how much space a single atom occupies.
In a face-centred cubic unit cell, once you determine the atomic radius, you can find the volume of a single atom using this formula. This measure is fundamental when calculating the total volume of atoms in complex structures like FCC.

Number of Atoms in FCC

In a face-centred cubic (FCC) unit cell, understanding the number of atoms involved is vital. An FCC unit cell appears with atoms at all corners and centers of each face. It is essential to recognize how these atoms contribute to the unit cell's total atom count.
Each of the 8 corner atoms is shared among eight adjacent unit cells, contributing 1/8th of an atom per unit cell. Each of the 6 face-centered atoms is shared between two cells, contributing half an atom from each face. Adding these contributions gives a total of 4 atoms per FCC unit cell:

• 3 atoms from the faces
• 1 atom from the corners

This understanding forms the basis for further calculations related to material density and atomic packing.

Geometry of Crystal Structures

The geometry of crystal structures dictates how atoms are arranged in space. In face-centred cubic (FCC) systems, the geometry ensures that atoms are tightly packed. This packing plays a critical role in determining the properties of materials, like density and structural stability.
In FCC arrangements, the atoms are aligned along the face diagonals, creating a compact structure. This arrangement maximizes the number of neighboring atoms, or the coordination number, which for FCC is 12. Understanding this geometry is crucial when predicting how metals behave and interact with other substances. It also aids in comprehending crystal defects and their influence on material properties.