Sodium crystallises in bcc arrangement with the interfacial separation between the atoms at the edge length of 53 pm. The density of the solid is

Options

A. 1.23 g/cc
B. 485 g/cc
C. 4.85 g/cc
D. 123 g/cc

Answer

To calculate the density of sodium crystallizing in a body-centered cubic (BCC) arrangement with an edge length of 53 pm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the structure and parameters:**
- Sodium (Na) crystallizes in a BCC arrangement.
- In BCC, the effective number of atoms per unit cell (Z) is 2.
- The edge length (a) is given as 53 pm (picometers).

2. **Relate the edge length to atomic radius:**
- In a BCC lattice, the relationship between the edge length (a) and the atomic radius (R) is given by:
\[ 2R = \frac{\sqrt{3}}{2} a \]
- Rearranging gives:
\[ R = \frac{\sqrt{3}}{4} a \]

3. **Calculate the atomic radius:**
- Substitute the edge length (a = 53 pm) into the equation:
\[ R = \frac{\sqrt{3}}{4} \times 53 \text{ pm} \approx 22.95 \text{ pm} \]

4. **Calculate the edge length in centimeters:**
- Convert the edge length from picometers to centimeters:
\[ a = 53 \text{ pm} = 53 \times 10^{-12} \text{ m} = 53 \times 10^{-10} \text{ cm} \]

5. **Calculate the volume of the unit cell:**
- The volume (V) of the unit cell is given by:
\[ V = a^3 = (53 \times 10^{-10} \text{ cm})^3 \]

6. **Calculate the molar mass of sodium:**
- The molar mass (M) of sodium is approximately 23 g/mol.

7. **Use Avogadro's number:**
- Avogadro's number (N_A) is approximately \(6.022 \times 10^{23} \text{ mol}^{-1}\).

8. **Calculate the density (D):**
- The formula for density is:
\[ D = \frac{Z \times M}{N_A \times V} \]
- Substitute the values:
\[ D = \frac{2 \times 23 \text{ g/mol}}{6.022 \times 10^{23} \text{ mol}^{-1} \times (53 \times 10^{-10} \text{ cm})^3} \]

9. **Perform the calculations:**
- Calculate the volume:
\[ V = (53 \times 10^{-10})^3 \approx 1.48 \times 10^{-28} \text{ cm}^3 \]
- Now substitute back into the density formula:
\[ D = \frac{46}{6.022 \times 10^{23} \times 1.48 \times 10^{-28}} \approx 1.23 \text{ g/cm}^3 \]

10. **Conclusion:**
- The density of solid sodium is approximately **1.23 g/cm³**.