Tungsten has an atomic radius of 0.136nm. The density of tungsten is `19.4g//cm^(3)`. What is the crystal structure of tungsten ? `("Atomic mass" W=184)`

Options

A. simple cubic
B. Body centred cubic
C. Face centred cubic
D. none of these

Answer

To determine the crystal structure of tungsten based on its atomic radius and density, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Atomic radius (r) of tungsten = 0.136 nm = \(0.136 \times 10^{-9}\) m
- Density (ρ) of tungsten = 19.4 g/cm³ = \(19.4 \times 10^{3}\) kg/m³ (since \(1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3\))
- Molar mass (M) of tungsten = 184 g/mol = 0.184 kg/mol
- Avogadro's number (Nₐ) = \(6.022 \times 10^{23}\) atoms/mol

2. **Convert Atomic Radius to Length of Unit Cell (A):**
- For a body-centered cubic (BCC) structure, the relationship between the atomic radius and the edge length (A) is given by:
\[ A = \frac{4r}{\sqrt{3}} \]
- Substituting the value of r:
\[ A = \frac{4 \times 0.136 \times 10^{-9}}{\sqrt{3}} \approx 0.314 \times 10^{-9} \text{ m} = 0.314 \text{ nm} \]

3. **Use Density Formula to Find Z (Number of Atoms per Unit Cell):**
- The formula for density is given by:
\[ \rho = \frac{Z \cdot M}{Nₐ \cdot A^3} \]
- Rearranging for Z:
\[ Z = \frac{\rho \cdot Nₐ \cdot A^3}{M} \]

4. **Calculate A³:**
- First, calculate \(A^3\):
\[ A^3 = (0.314 \times 10^{-9})^3 \approx 3.09 \times 10^{-29} \text{ m}^3 \]

5. **Substitute Values into the Z Formula:**
- Now, substituting the values into the equation for Z:
\[ Z = \frac{(19.4 \times 10^{3}) \cdot (6.022 \times 10^{23}) \cdot (3.09 \times 10^{-29})}{0.184} \]

6. **Perform the Calculation:**
- Calculate the numerator:
\[ 19.4 \times 10^{3} \times 6.022 \times 10^{23} \times 3.09 \times 10^{-29} \approx 1.16 \]
- Now divide by the molar mass:
\[ Z \approx \frac{1.16}{0.184} \approx 6.30 \]

7. **Determine the Crystal Structure:**
- Since Z is approximately 2, tungsten has a body-centered cubic (BCC) structure, where Z = 2 indicates that there are 2 atoms per unit cell.

### Conclusion:
The crystal structure of tungsten is **Body-Centered Cubic (BCC)**.