In fcc arrangement of A and B atoms, where A atoms are at corners of the unit cell, B atoms at the face - centers, one of the atoms are missing from the corner in each unit cell then find the percentage of void space in the unit cell.

Answer

To solve the problem of finding the percentage of void space in a face-centered cubic (FCC) arrangement of A and B atoms, where A atoms are at the corners and B atoms are at the face centers, and one corner atom is missing, we can follow these steps:

### Step 1: Determine the number of atoms in the unit cell
In an FCC unit cell:
- There are 8 corner atoms, but since one is missing, we have 7 corner atoms remaining.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell, so the contribution from the corner atoms is:
\[ \text{Contribution from corner atoms} = 7 \times \frac{1}{8} = \frac{7}{8} \]
- There are 6 face-centered atoms, each contributing \( 1 \) atom to the unit cell:
\[ \text{Contribution from face-centered atoms} = 6 \times 1 = 6 \]
- Therefore, the total number of atoms in the unit cell is:
\[ \text{Total number of atoms} = \frac{7}{8} + 6 = \frac{7 + 48}{8} = \frac{55}{8} = 6.875 \text{ atoms} \]

### Step 2: Calculate the volume occupied by the atoms
The volume of a single atom (assuming they are spherical) is given by:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
Thus, the total volume occupied by the atoms in the unit cell is:
\[ V_{\text{occupied}} = \text{Total number of atoms} \times V_{\text{atom}} = 6.875 \times \frac{4}{3} \pi r^3 \]

### Step 3: Calculate the volume of the unit cell
In an FCC structure, the atoms touch along the face diagonal. The relationship between the edge length \( a \) of the unit cell and the radius \( r \) of the atoms is:
\[ a \sqrt{2} = 4r \implies a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r \]
The volume of the unit cell is:
\[ V_{\text{unit cell}} = a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \]

### Step 4: Calculate the packing fraction
The packing fraction (PF) is the ratio of the volume occupied by the atoms to the volume of the unit cell:
\[ \text{Packing Fraction} = \frac{V_{\text{occupied}}}{V_{\text{unit cell}}} = \frac{6.875 \times \frac{4}{3} \pi r^3}{16\sqrt{2}r^3} \]
Simplifying this gives:
\[ \text{Packing Fraction} = \frac{6.875 \times \frac{4}{3} \pi}{16\sqrt{2}} \]

### Step 5: Calculate the percentage of void space
The percentage of void space is given by:
\[ \text{Void Space Percentage} = (1 - \text{Packing Fraction}) \times 100 \]

### Final Calculation
1. Calculate the packing fraction using the values obtained.
2. Substitute the packing fraction into the void space percentage formula.