The fraction of total volume occupied by the atoms present in a face centred cubic unit cell is:-

Options

A. `(pi)/(6)`
B. `(pi)/(3sqrt(2))`
C. `(pi)/(4sqrt(2))`
D. `(pi)/(4)`

Explanation

To find the fraction of total volume occupied by the atoms present in a face-centered cubic (FCC) unit cell, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic unit cell, there are atoms located at each of the corners and at the center of each face. This means that there are a total of 4 atoms per unit cell (1/8 atom from each of the 8 corners and 1/2 atom from each of the 6 faces).

### Step 2: Calculate the Volume Occupied by Atoms
The volume occupied by one atom (assuming it is a sphere) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Since there are 4 atoms in the FCC unit cell, the total volume occupied by the atoms is:
\[ V_{\text{atoms}} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

### Step 3: Calculate the Total Volume of the Unit Cell
The edge length \( a \) of the FCC unit cell can be related to the radius \( r \) of the atoms. The relationship is:
\[ a = 2\sqrt{2}r \]
The volume of the cubic unit cell is:
\[ V_{\text{cell}} = a^3 = (2\sqrt{2}r)^3 = 8 \cdot 2\sqrt{2} \cdot r^3 = 16\sqrt{2}r^3 \]

### Step 4: Calculate the Packing Efficiency
The packing efficiency (fraction of the total volume occupied by the atoms) is given by:
\[ \text{Packing Efficiency} = \frac{V_{\text{atoms}}}{V_{\text{cell}}} = \frac{\frac{16}{3} \pi r^3}{16\sqrt{2}r^3} \]
Now, we can simplify this expression:
\[ \text{Packing Efficiency} = \frac{16\pi}{3 \cdot 16\sqrt{2}} = \frac{\pi}{3\sqrt{2}} \]

### Step 5: Final Answer
Thus, the fraction of total volume occupied by the atoms in a face-centered cubic unit cell is:
\[ \frac{\pi}{3\sqrt{2}} \]