Compute the percentage void space per unit volume of unit cell in zinc-fluoride structure .

Options

A. `25.07 %`
B. `74.93 %`
C. `52%`
D. `48%`

Explanation

To compute the percentage void space per unit volume of the unit cell in the zinc-fluoride structure, we can follow these steps:

### Step 1: Understand the Structure
Zinc fluoride (ZnF₂) has a structure where the fluoride ions (F⁻) occupy the face-centered cubic (FCC) positions, and the zinc ions (Zn²⁺) occupy the tetrahedral voids.

### Step 2: Determine the Number of Ions in the Unit Cell
In the FCC structure, there are 4 fluoride ions per unit cell (since each corner contributes 1/8 and each face contributes 1/2).

### Step 3: Calculate the Number of Tetrahedral Voids
In a face-centered cubic unit cell, there are 8 tetrahedral voids. However, in the zinc-fluoride structure, only half of these tetrahedral voids are occupied by zinc ions. Therefore, there are 4 occupied tetrahedral voids.

### Step 4: Calculate the Volume of the Ions
To find the volume occupied by the ions, we need to know the radius of the ions. The radius ratio (r/R) is given as 0.255, where r is the radius of the zinc ion and R is the radius of the fluoride ion.

Assuming the radius of the fluoride ion (R) is known, we can calculate the radius of the zinc ion (r):
\[ r = 0.255 \times R \]

### Step 5: Calculate the Volume of the Ions
The volume of a single ion can be calculated using the formula for the volume of a sphere:
\[ V_{ion} = \frac{4}{3} \pi r^3 \]

For fluoride ions:
\[ V_{F} = 4 \times \frac{4}{3} \pi R^3 \]

For zinc ions:
\[ V_{Zn} = 4 \times \frac{4}{3} \pi r^3 = 4 \times \frac{4}{3} \pi (0.255R)^3 \]

### Step 6: Calculate the Total Volume Occupied by Ions
Total volume occupied by ions in the unit cell:
\[ V_{total} = V_{F} + V_{Zn} \]

### Step 7: Calculate the Volume of the Unit Cell
The volume of the unit cell (V_cell) can be calculated based on the edge length (a) of the cubic unit cell:
\[ V_{cell} = a^3 \]

### Step 8: Calculate the Void Space
The void space can be calculated as:
\[ V_{void} = V_{cell} - V_{total} \]

### Step 9: Calculate the Percentage Void Space
Finally, the percentage void space per unit volume can be calculated using:
\[ \text{Percentage Void Space} = \left( \frac{V_{void}}{V_{cell}} \right) \times 100\% \]

### Summary of Steps:
1. Understand the structure of zinc fluoride.
2. Determine the number of ions in the unit cell.
3. Calculate the number of tetrahedral voids.
4. Calculate the volume of the ions.
5. Calculate the total volume occupied by ions.
6. Calculate the volume of the unit cell.
7. Calculate the void space.
8. Calculate the percentage void space.