The percent of void space in a body - centred cubic lattice is :

Options

A. `32%`
B. `48%`
C. `53%`
D. `68%`

Answer

To find the percent of void space in a body-centered cubic (BCC) lattice, we can follow these steps:

### Step 1: Understand the structure of BCC
In a body-centered cubic lattice, there are atoms located at each of the eight corners of the cube and one atom at the center of the cube.

### Step 2: Calculate the number of atoms per unit cell
Each corner atom contributes 1/8 of its volume to the unit cell. Since there are 8 corners:
- Contribution from corner atoms = \(8 \times \frac{1}{8} = 1\)
- Plus the 1 atom at the center = 1

Thus, the total number of atoms per BCC unit cell = \(1 + 1 = 2\).

### Step 3: Determine the relationship between the radius of the atom and the side length of the cube
For a BCC structure, the relationship between the radius \(r\) of the atom and the side length \(a\) of the cube is given by:
\[ r = \frac{\sqrt{3}}{4} a \]

### Step 4: Calculate the volume of the atoms in the unit cell
The volume of a single atom (assuming it is spherical) is given by:
\[ V_{atom} = \frac{4}{3} \pi r^3 \]

For 2 atoms, the total volume of the atoms in the unit cell is:
\[ V_{total\ atoms} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 5: Substitute the value of \(r\) in terms of \(a\)
Substituting \(r = \frac{\sqrt{3}}{4} a\) into the volume formula:
\[ V_{total\ atoms} = \frac{8}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3 \]
\[ = \frac{8}{3} \pi \left(\frac{3\sqrt{3}}{64} a^3\right) \]
\[ = \frac{8 \cdot 3\sqrt{3}}{192} \pi a^3 \]
\[ = \frac{\sqrt{3}}{8} \pi a^3 \]

### Step 6: Calculate the volume of the cubic unit cell
The volume of the cubic unit cell is:
\[ V_{unit\ cell} = a^3 \]

### Step 7: Calculate the packing fraction
The packing fraction (the fraction of the volume occupied by the atoms) is given by:
\[ Packing\ fraction = \frac{V_{total\ atoms}}{V_{unit\ cell}} = \frac{\frac{\sqrt{3}}{8} \pi a^3}{a^3} = \frac{\sqrt{3}}{8} \pi \]

### Step 8: Convert packing fraction to percentage
To find the percentage of space occupied by the atoms:
\[ Percentage\ occupied = Packing\ fraction \times 100 \]

Calculating this gives:
\[ \text{Percentage occupied} = \frac{\sqrt{3}}{8} \pi \times 100 \approx 68\% \]

### Step 9: Calculate the percent of void space
The percent of void space is:
\[ \text{Percent void space} = 100\% - \text{Percentage occupied} = 100\% - 68\% = 32\% \]

### Final Answer
Thus, the percent of void space in a body-centered cubic lattice is **32%**.

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