Percentage of void space in AB solid having rock salt structure if `(r_(+))/(r_(-)) = (1)/(2)` having cation anion contact. Given `pi = 3.15`

Explanation

To find the percentage of void space in an AB solid with a rock salt structure given the ratio of cation to anion radii, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Ratio**:
We are given that the ratio of the cation radius \( r_+ \) to the anion radius \( r_- \) is \( \frac{r_+}{r_-} = \frac{1}{2} \). This implies that \( r_+ = \frac{1}{2} r_- \).

2. **Setting Up the Relationship**:
In a rock salt structure, the cation and anion touch each other along the edge of the unit cell. Therefore, the sum of the radii of the cation and anion is equal to half the edge length of the unit cell \( a \):
\[ r_+ + r_- = \frac{a}{2} \]

3. **Substituting the Radius**:
Substitute \( r_+ = \frac{1}{2} r_- \) into the equation:
\[ \frac{1}{2} r_- + r_- = \frac{a}{2} \]
This simplifies to:
\[ \frac{3}{2} r_- = \frac{a}{2} \]

4. **Finding the Anion Radius**:
Rearranging gives:
\[ r_- = \frac{a}{3} \]

5. **Finding the Cation Radius**:
Now substituting \( r_- \) back to find \( r_+ \):
\[ r_+ = \frac{1}{2} r_- = \frac{1}{2} \cdot \frac{a}{3} = \frac{a}{6} \]

6. **Calculating the Volume of the Cations and Anions**:
The volume occupied by the cations and anions in the unit cell can be calculated as follows:
- The volume of one cation (4 cations in the unit cell):
\[ V_{cation} = 4 \times \frac{4}{3} \pi (r_+)^3 = 4 \times \frac{4}{3} \pi \left(\frac{a}{6}\right)^3 \]
- The volume of one anion (4 anions in the unit cell):
\[ V_{anion} = 4 \times \frac{4}{3} \pi (r_-)^3 = 4 \times \frac{4}{3} \pi \left(\frac{a}{3}\right)^3 \]

7. **Calculating the Total Volume**:
The total volume occupied by the cations and anions in the unit cell is:
\[ V_{total} = V_{cation} + V_{anion} \]

8. **Volume of the Unit Cell**:
The volume of the unit cell is:
\[ V_{unit\ cell} = a^3 \]

9. **Calculating the Packing Efficiency**:
The packing efficiency (PE) is given by:
\[ PE = \frac{V_{total}}{V_{unit\ cell}} = \frac{V_{cation} + V_{anion}}{a^3} \]

10. **Finding the Percentage of Void Space**:
The percentage of void space is:
\[ \text{Void Space Percentage} = 100\% - PE \times 100\% \]

11. **Final Calculation**:
After performing the calculations, we find that the packing efficiency is 70%, which means the void space is:
\[ \text{Void Space Percentage} = 100\% - 70\% = 30\% \]

### Final Answer:
The percentage of void space in the AB solid with a rock salt structure is **30%**.