The empty space in the h.c.p. structure is

Options

A. 0.74
B. `47.6%`
C. 0.32
D. 0.26

Answer

To determine the empty space in the hexagonal close-packed (hcp) structure, we can follow these steps:

### Step 1: Understand the Packing Fraction
The packing fraction is defined as the ratio of the volume occupied by the particles (atoms or molecules) to the total volume of the unit cell. It is given by the formula:

\[ \text{Packing Fraction} = \frac{\text{Volume occupied by particles}}{\text{Total volume of unit cell}} \]

### Step 2: Determine the Number of Atoms in hcp
In the hcp structure, there are 6 atoms per unit cell. This is crucial for calculating the volume occupied by the atoms.

### Step 3: Calculate the Volume Occupied by Atoms
The volume occupied by one atom can be calculated using the formula for the volume of a sphere:

\[ V = \frac{4}{3} \pi r^3 \]

Thus, the total volume occupied by 6 atoms is:

\[ \text{Volume occupied by 6 atoms} = 6 \times \frac{4}{3} \pi r^3 = 8 \pi r^3 \]

### Step 4: Calculate the Total Volume of the Unit Cell
The total volume of the hcp unit cell can be expressed in terms of the radius \( r \). For hcp, the total volume of the unit cell is given as:

\[ \text{Total volume of unit cell} = 24 \sqrt{2} r^3 \]

### Step 5: Calculate the Packing Fraction
Now we can substitute the volumes into the packing fraction formula:

\[ \text{Packing Fraction} = \frac{8 \pi r^3}{24 \sqrt{2} r^3} \]

This simplifies to:

\[ \text{Packing Fraction} = \frac{8 \pi}{24 \sqrt{2}} = \frac{\pi}{3 \sqrt{2}} \approx 0.74 \]

### Step 6: Determine the Empty Space
To find the empty space in the hcp structure, we subtract the packing fraction from 1 (or 100%):

\[ \text{Empty Space} = 1 - \text{Packing Fraction} = 1 - 0.74 = 0.26 \]

### Conclusion
The empty space in the hcp structure is approximately 26%.