Iron crystallises in a body centred cubic structure. Calculate the radius of iron atom if edge length of unit cell is 286 pm.

Explanation

To solve the problem of calculating the radius of an iron atom in a body-centered cubic (BCC) structure given the edge length of the unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given data:**
- Edge length of the unit cell (a) = 286 pm (picometers).

2. **Understand the BCC structure:**
- In a body-centered cubic (BCC) structure, there are two atoms per unit cell: one at each corner and one at the center of the cube.

3. **Relate the radius of the atom to the edge length:**
- In a BCC unit cell, the relationship between the radius (R) of the atom and the edge length (a) is given by the formula:
\[ a = 4R / \sqrt{3} \]
- Rearranging this formula to solve for R gives:
\[ R = \frac{a \cdot \sqrt{3}}{4} \]

4. **Substitute the values:**
- Substitute the given edge length (a = 286 pm) into the formula:
\[ R = \frac{286 \cdot \sqrt{3}}{4} \]

5. **Calculate the value of \(\sqrt{3}\):**
- \(\sqrt{3} \approx 1.732\)

6. **Perform the calculation:**
- Now calculate R:
\[ R = \frac{286 \cdot 1.732}{4} \]
\[ R = \frac{495.592}{4} \approx 123.898 \text{ pm} \]

7. **Final result:**
- The radius of the iron atom is approximately \( R \approx 123.9 \text{ pm} \).

### Final Answer:
The radius of the iron atom is approximately **123.9 picometers**.

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