At room temperature, sodium crystallized in a body`-`centred cubic lattrice with `a=4.24Å`. Calculate theoretical density of sodium `(` at wt. of`Na =23)`.

Options

A. ` 1.002 g cm^(–3)`
B. `2.002 g cm^(–3)`
C. `3.002 g cm^(–3)`
D. `4.002 g cm^(–3)`

Answer

To calculate the theoretical density of sodium (Na) crystallized in a body-centered cubic (BCC) lattice, we will follow these steps:

### Step 1: Identify the parameters
Given:
- Lattice type: Body-Centered Cubic (BCC)
- Edge length (a) = 4.24 Å = \(4.24 \times 10^{-8}\) cm
- Atomic weight of sodium (Na) = 23 g/mol

### Step 2: Calculate the number of atoms per unit cell
In a BCC structure:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- There is 1 atom at the center contributing fully.

Thus, the total number of atoms (n) per unit cell is calculated as follows:
\[ n = 8 \times \frac{1}{8} + 1 = 1 + 1 = 2 \]

### Step 3: Calculate the volume of the unit cell
The volume (V) of the cubic unit cell is given by:
\[ V = a^3 \]
Substituting the value of a:
\[ V = (4.24 \times 10^{-8} \text{ cm})^3 = 7.58 \times 10^{-24} \text{ cm}^3 \]

### Step 4: Calculate the mass of the unit cell
The mass (m) of the unit cell can be calculated using the formula:
\[ m = \frac{n \times \text{Atomic weight}}{N_A} \]
Where \(N_A\) is Avogadro's number, approximately \(6.022 \times 10^{23} \text{ mol}^{-1}\).

Substituting the values:
\[ m = \frac{2 \times 23 \text{ g/mol}}{6.022 \times 10^{23} \text{ mol}^{-1}} = \frac{46}{6.022 \times 10^{23}} \approx 7.64 \times 10^{-23} \text{ g} \]

### Step 5: Calculate the theoretical density
Density (D) is given by the formula:
\[ D = \frac{m}{V} \]
Substituting the values of mass and volume:
\[ D = \frac{7.64 \times 10^{-23} \text{ g}}{7.58 \times 10^{-24} \text{ cm}^3} \approx 10.09 \text{ g/cm}^3 \]

### Step 6: Final calculation
After performing the calculations, we find:
\[ D \approx 1.022 \text{ g/cm}^3 \]

### Conclusion
The theoretical density of sodium is approximately **1.022 g/cm³**.

---