The percentage of void space of a metallic element crystallising in a ABCABC .....type lattice pattern is 

Options

A. `24%`
B. `26%`
C. `34%`
D. `74%`

Answer

To find the percentage of void space in a metallic element crystallizing in an ABCABC type lattice pattern (which corresponds to face-centered cubic or FCC), we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) lattice, there are atoms located at each of the eight corners of the cube and at the centers of each of the six faces.

### Step 2: Calculate the Total Number of Atoms in the Unit Cell
The total number of atoms in the FCC unit cell can be calculated as follows:
- Atoms at corners: 8 corners × (1/8 atom per corner) = 1 atom
- Atoms at faces: 6 faces × (1/2 atom per face) = 3 atoms
- Total atoms = 1 + 3 = 4 atoms

### Step 3: Relate Atomic Radius to Edge Length
In the FCC structure, the atoms touch along the face diagonal. The relationship between the atomic radius (r) and the edge length (a) of the unit cell is given by:
\[ 4r = \sqrt{2}a \]
From this, we can express the atomic radius:
\[ r = \frac{a}{2\sqrt{2}} \]

### Step 4: Calculate the Volume of the Unit Cell
The volume of the cubic unit cell is given by:
\[ \text{Volume of unit cell} = a^3 \]

### Step 5: Calculate the Volume Occupied by Atoms
The volume occupied by the atoms in the unit cell can be calculated using the formula for the volume of a sphere:
\[ \text{Volume occupied} = \text{Number of atoms} \times \text{Volume of one atom} \]
\[ = 4 \times \left(\frac{4}{3}\pi r^3\right) \]
Substituting for r:
\[ = 4 \times \left(\frac{4}{3}\pi \left(\frac{a}{2\sqrt{2}}\right)^3\right) \]
\[ = 4 \times \left(\frac{4}{3}\pi \frac{a^3}{8\sqrt{2^3}}\right) \]
\[ = \frac{16\pi a^3}{24\sqrt{2}} = \frac{2\pi a^3}{3\sqrt{2}} \]

### Step 6: Calculate the Volume Percentage Occupied
The volume percentage occupied by the atoms is given by:
\[ \text{Volume percentage} = \left(\frac{\text{Volume occupied}}{\text{Volume of unit cell}}\right) \times 100 \]
\[ = \left(\frac{\frac{2\pi a^3}{3\sqrt{2}}}{a^3}\right) \times 100 \]
\[ = \left(\frac{2\pi}{3\sqrt{2}}\right) \times 100 \]

### Step 7: Calculate the Percentage of Void Space
The percentage of void space is given by:
\[ \text{Percentage void} = 100 - \text{Volume percentage occupied} \]
Using the calculated volume percentage occupied, we find:
\[ \text{Percentage void} = 100 - 74 = 26\% \]

### Final Answer
Thus, the percentage of void space in the FCC lattice is **26%**.

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