Sodium  crystalises in body centred cubic lattice. The edge length of its unit cell is 0.424 nm. The density of the metal is

Options

A. `1.002xx10^(6)"gm/cc"`
B. `1.002kg//m^(3)`
C. `1.002xx10^(3)kg//m^(3)`
D. `1.002xx10^(3)" gm/cc"`

Answer

To find the density of sodium, which crystallizes in a body-centered cubic (BCC) lattice, we will follow these steps:

### Step 1: Identify the given data
- Edge length of the unit cell (A) = 0.424 nm = \(0.424 \times 10^{-9}\) m
- Type of lattice = Body-Centered Cubic (BCC)
- Number of atoms per unit cell (Z) for BCC = 2
- Molar mass of sodium (m) = 23 g/mol
- Avogadro's number (Na) = \(6.022 \times 10^{23}\) atoms/mol

### Step 2: Convert the edge length to meters
Since the edge length is given in nanometers, we convert it to meters:
\[ A = 0.424 \, \text{nm} = 0.424 \times 10^{-9} \, \text{m} \]

### Step 3: Calculate the volume of the unit cell
The volume (V) of the cubic unit cell can be calculated using the formula:
\[ V = A^3 \]
Substituting the value of A:
\[ V = (0.424 \times 10^{-9})^3 \, \text{m}^3 \]

### Step 4: Calculate the density using the formula
The density (D) of the metal can be calculated using the formula:
\[ D = \frac{Z \cdot m}{N_a \cdot A^3} \]
Substituting the values:
\[ D = \frac{2 \cdot 23 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol} \cdot (0.424 \times 10^{-9} \, \text{m})^3} \]

### Step 5: Calculate the numerical value
1. Calculate \(A^3\):
\[ A^3 = (0.424 \times 10^{-9})^3 = 7.594 \times 10^{-29} \, \text{m}^3 \]
2. Substitute into the density formula:
\[ D = \frac{46 \, \text{g/mol}}{6.022 \times 10^{23} \cdot 7.594 \times 10^{-29}} \]
3. Calculate the denominator:
\[ 6.022 \times 10^{23} \cdot 7.594 \times 10^{-29} \approx 4.576 \times 10^{-5} \]
4. Now calculate density:
\[ D = \frac{46}{4.576 \times 10^{-5}} \approx 1002119.495 \, \text{g/m}^3 \]

### Step 6: Convert density to kg/m³
To convert from g/m³ to kg/m³, divide by 1000:
\[ D \approx 1002119.495 \, \text{g/m}^3 = 1002.119495 \, \text{kg/m}^3 \approx 1.002 \times 10^3 \, \text{kg/m}^3 \]

### Final Answer
The density of sodium is approximately \(1.002 \times 10^3 \, \text{kg/m}^3\).

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