The percentage of vacant space in bcc unit cell and simple cubic unit cell are 26% and 32% respectively.

Explanation

To determine the percentage of vacant space in a body-centered cubic (BCC) unit cell and a simple cubic unit cell, we can follow these steps:

### Step 1: Understand the Unit Cells
- **Simple Cubic (SC)**: Contains 1 atom per unit cell (z = 1).
- **Body-Centered Cubic (BCC)**: Contains 2 atoms per unit cell (z = 2).

### Step 2: Calculate the Volume of Atoms
- The volume of a single atom is given by the formula:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]

### Step 3: Calculate the Volume of the Unit Cell
- For the **Simple Cubic** unit cell:
- The edge length \( a \) is related to the radius \( r \) as \( a = 2r \).
- Therefore, the volume of the unit cell \( V_{\text{unit cell}} \) is:
\[ V_{\text{unit cell}} = a^3 = (2r)^3 = 8r^3 \]

- For the **BCC** unit cell:
- The relationship between the radius \( r \) and the edge length \( a \) is given by \( 4r = \sqrt{3}a \) or \( a = \frac{4r}{\sqrt{3}} \).
- Therefore, the volume of the unit cell is:
\[ V_{\text{unit cell}} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 4: Calculate the Packing Fraction
- **Packing Fraction (PF)** is given by:
\[ PF = \frac{z \cdot V_{\text{atom}}}{V_{\text{unit cell}}} \]

- For **Simple Cubic**:
\[ PF_{\text{SC}} = \frac{1 \cdot \frac{4}{3} \pi r^3}{8r^3} = \frac{\frac{4}{3} \pi}{8} = \frac{\pi}{6} \approx 0.5238 \quad \text{(or 52.38%)} \]

- For **BCC**:
\[ PF_{\text{BCC}} = \frac{2 \cdot \frac{4}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} = \frac{8 \pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \approx 0.680 \quad \text{(or 68%)} \]

### Step 5: Calculate the Percentage of Vacant Space
- The percentage of vacant space is given by:
\[ \text{Vacant Space} = 100\% - \text{Packing Fraction} \]

- For **Simple Cubic**:
\[ \text{Vacant Space}_{\text{SC}} = 100\% - 52.38\% = 47.62\% \]

- For **BCC**:
\[ \text{Vacant Space}_{\text{BCC}} = 100\% - 68\% = 32\% \]

### Conclusion
- The percentage of vacant space in a BCC unit cell is 32%, and in a simple cubic unit cell, it is approximately 47.62%. Therefore, the statement that the percentage of vacant space in BCC is 26% and in simple cubic is 32% is **false**.