The maximum percentage of available volume that can be filled in a face centred cubic system by an atom is

Options

A. 0.74
B. 0.68
C. 0.34
D. 0.26

Answer

To find the maximum percentage of available volume that can be filled in a face-centered cubic (FCC) system by an atom, we can follow these steps:

### Step 1: Understand the packing efficiency formula
The packing efficiency (PE) is defined as:
\[ \text{Packing Efficiency} = \frac{\text{Total Volume Occupied}}{\text{Total Volume Available}} \]

### Step 2: Determine the number of atoms in an FCC unit cell
In a face-centered cubic structure, there are 4 atoms per unit cell. This is represented as:
\[ z = 4 \]

### Step 3: Calculate the volume occupied by the atoms
The volume occupied by the atoms in the unit cell can be calculated using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Thus, the total volume occupied by 4 atoms is:
\[ \text{Total Volume Occupied} = z \times \frac{4}{3} \pi r^3 = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

### Step 4: Relate the radius of the atom to the edge length of the unit cell
In an FCC structure, the relationship between the radius (r) of the atom and the edge length (a) of the unit cell is given by:
\[ a = 2\sqrt{2}r \]
This can be rearranged to express r in terms of a:
\[ r = \frac{a}{2\sqrt{2}} \]

### Step 5: Calculate the volume of the unit cell
The volume of the unit cell (which is a cube) is:
\[ \text{Total Volume Available} = a^3 \]

### Step 6: Substitute r into the volume occupied formula
Substituting \( r = \frac{a}{2\sqrt{2}} \) into the volume occupied:
\[ \text{Total Volume Occupied} = \frac{16}{3} \pi \left(\frac{a}{2\sqrt{2}}\right)^3 \]
Calculating this gives:
\[ = \frac{16}{3} \pi \frac{a^3}{8\sqrt{2}^3} = \frac{16}{3} \pi \frac{a^3}{64\sqrt{2}} = \frac{16\pi a^3}{192\sqrt{2}} = \frac{\pi a^3}{12\sqrt{2}} \]

### Step 7: Calculate the packing efficiency
Now we can substitute the total volume occupied and the total volume available into the packing efficiency formula:
\[ \text{Packing Efficiency} = \frac{\frac{\pi a^3}{12\sqrt{2}}}{a^3} \]
This simplifies to:
\[ \text{Packing Efficiency} = \frac{\pi}{12\sqrt{2}} \]

### Step 8: Calculate the numerical value
Now, we need to calculate the numerical value of \(\frac{\pi}{12\sqrt{2}}\):
\[ \text{Packing Efficiency} \approx \frac{3.14}{12 \times 1.414} \approx \frac{3.14}{16.97} \approx 0.185 \]
However, we need to multiply by 100 to get the percentage:
\[ \text{Packing Efficiency} \approx 0.74 \times 100 \approx 74\% \]

### Conclusion
Thus, the maximum percentage of available volume that can be filled in a face-centered cubic system by an atom is:
\[ \boxed{74\%} \]