Calculate the efficiency of the packaing in case of face - centered cubic crystal .

Options

A. `52%`
B. `68%`
C. `74%`
D. `92%`

Answer

To calculate the efficiency of packing in a face-centered cubic (FCC) crystal, we will follow these steps:

### Step 1: Determine the number of atoms in the FCC unit cell.
In an FCC unit cell:
- There are 8 corner atoms, and each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- There are 6 face-centered atoms, and each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.

Calculating the total number of atoms:
\[ \text{Total atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 2: Calculate the volume occupied by the atoms.
The volume of one atom (assuming it is spherical) is given by the formula:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Thus, the total volume occupied by 4 atoms is:
\[ \text{Total volume occupied} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

### Step 3: Relate the radius of the atom to the edge length of the cube.
In an FCC structure, the relationship between the edge length \( a \) of the cube and the radius \( r \) of the atom is given by:
\[ a = 2\sqrt{2}r \]
From this, we can express \( r \) in terms of \( a \):
\[ r = \frac{a}{2\sqrt{2}} \]

### Step 4: Calculate the volume of the cube.
The volume of the cube is given by:
\[ \text{Volume of the cube} = a^3 \]

### Step 5: Substitute \( r \) into the volume of the atoms and calculate the packing fraction.
Substituting \( r = \frac{a}{2\sqrt{2}} \) into the volume of the atoms:
\[ \text{Total volume occupied} = \frac{16}{3} \pi \left(\frac{a}{2\sqrt{2}}\right)^3 = \frac{16}{3} \pi \frac{a^3}{16\sqrt{2}} = \frac{4\pi a^3}{3\sqrt{2}} \]

Now, we can calculate the packing fraction:
\[ \text{Packing fraction} = \frac{\text{Total volume occupied}}{\text{Volume of the cube}} = \frac{\frac{4\pi a^3}{3\sqrt{2}}}{a^3} = \frac{4\pi}{3\sqrt{2}} \]

### Step 6: Calculate the packing efficiency percentage.
To find the packing efficiency (percentage of space occupied):
\[ \text{Packing efficiency} = \text{Packing fraction} \times 100 = \left(\frac{4\pi}{3\sqrt{2}}\right) \times 100 \]

Using \( \pi \approx 3.14 \):
\[ \text{Packing efficiency} = \left(\frac{4 \times 3.14}{3 \times 1.414}\right) \times 100 \approx 0.74 \times 100 = 74\% \]

### Final Answer:
The efficiency of packing in a face-centered cubic crystal is **74%**.

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