packing fraction of a unit cell is drfined as the fraction of the total volume of the unit cell occupied by the atom(s).
`P.E=("Volume of the atoms(s) present in a unit cell")/("Volume of unit cell")=(Zxx(4)/(3)pir^(3))/(a^(3))`
and `%` of empty space = `100- P.F.xx100 `
where Z= effective number of stoms in s cube .
r= radius of a an atoms
a = edge lenght of the cube
Packing fraction in face centered cubic unit cell is :

Options

A. 0.7406
B. 0.6802
C. 0.5236
D. None of these

Answer

To calculate the packing fraction of a face-centered cubic (FCC) unit cell, we will follow these steps:

### Step 1: Determine the number of effective atoms (Z) in the FCC unit cell
In a face-centered cubic unit cell, there are:
- 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell (since each corner atom is shared by 8 unit cells).
- 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom to the unit cell (since each face-centered atom is shared by 2 unit cells).

Calculating the effective number of atoms:
\[ Z = \left(8 \times \frac{1}{8}\right) + \left(6 \times \frac{1}{2}\right) = 1 + 3 = 4 \]

### Step 2: Relate the edge length (a) to the atomic radius (r)
In an FCC unit cell, the face diagonal consists of 4 atomic radii (2 radii from the corner atoms and 2 radii from the face-centered atom). The relationship can be derived using the Pythagorean theorem:
\[ \text{Face diagonal} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]
Since the face diagonal equals \( 4r \):
\[ 4r = a\sqrt{2} \implies a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r \]

### Step 3: Calculate the volume of the atoms in the unit cell
The volume of one atom is given by:
\[ \text{Volume of one atom} = \frac{4}{3}\pi r^3 \]
Thus, the total volume of the atoms in the unit cell (for Z = 4) is:
\[ \text{Volume of atoms} = Z \times \text{Volume of one atom} = 4 \times \frac{4}{3}\pi r^3 = \frac{16}{3}\pi r^3 \]

### Step 4: Calculate the volume of the unit cell
The volume of the unit cell is given by:
\[ \text{Volume of unit cell} = a^3 = (2\sqrt{2}r)^3 = 8 \cdot 2\sqrt{2}^3 \cdot r^3 = 16\sqrt{2}r^3 \]

### Step 5: Calculate the packing fraction (P.F.)
The packing fraction is given by:
\[ P.F. = \frac{\text{Volume of atoms}}{\text{Volume of unit cell}} = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2}r^3} \]
Simplifying this expression:
\[ P.F. = \frac{16\pi}{48\sqrt{2}} = \frac{\pi}{3\sqrt{2}} \approx 0.7406 \]

### Step 6: Calculate the percentage of empty space
The percentage of empty space in the unit cell can be calculated as:
\[ \text{Percentage of empty space} = 100 - P.F. \times 100 \]
Substituting the value of the packing fraction:
\[ \text{Percentage of empty space} = 100 - 0.7406 \times 100 \approx 25.94\% \]

### Final Answer
The packing fraction of a face-centered cubic unit cell is approximately **0.7406**.

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