The unit cell of tungsten is a face-centred cube having a volume of 31.699 A3. The atom at the centre of each face just touches the atoms at the corners. Calculate the radius and atomic volume of tungsten.

Answer

To solve the problem of calculating the radius and atomic volume of tungsten given that its unit cell is a face-centered cubic (FCC) structure with a volume of 31.699 ų, we can follow these steps:

### Step 1: Calculate the length of the edge of the unit cell (a)
The volume of a cubic unit cell is given by the formula:
\[ V = a^3 \]
Where \( V \) is the volume and \( a \) is the edge length.

Given:
\[ V = 31.699 \, \text{Å}^3 \]

To find \( a \):
\[ a = V^{1/3} \]
\[ a = (31.699)^{1/3} \]

Calculating \( a \):
\[ a \approx 3.165 \, \text{Å} \]

### Step 2: Relate the edge length to the atomic radius (r)
In a face-centered cubic (FCC) structure, the atoms at the corners and the face centers touch each other along the face diagonal. The relationship between the edge length \( a \) and the atomic radius \( r \) in an FCC unit cell is given by:
\[ \text{Face diagonal} = 4r \]
The face diagonal can also be expressed in terms of the edge length:
\[ \text{Face diagonal} = a\sqrt{2} \]

Setting these equal gives:
\[ a\sqrt{2} = 4r \]

Now, we can solve for \( r \):
\[ r = \frac{a\sqrt{2}}{4} \]

Substituting the value of \( a \):
\[ r = \frac{3.165 \times \sqrt{2}}{4} \]

Calculating \( r \):
\[ r \approx \frac{3.165 \times 1.414}{4} \]
\[ r \approx \frac{4.48}{4} \]
\[ r \approx 1.12 \, \text{Å} \]

### Step 3: Calculate the atomic volume
The atomic volume \( V_a \) can be calculated using the formula:
\[ V_a = \frac{V}{N} \]
Where \( N \) is the number of atoms per unit cell. For an FCC structure, there are 4 atoms per unit cell.

Thus:
\[ V_a = \frac{31.699 \, \text{Å}^3}{4} \]

Calculating \( V_a \):
\[ V_a \approx 7.92475 \, \text{Å}^3 \]

### Final Results
- The radius of tungsten (r) is approximately \( 1.12 \, \text{Å} \).
- The atomic volume of tungsten (V_a) is approximately \( 7.92475 \, \text{Å}^3 \).