Copper crystallises in face-centred cubic lattice with a unit cell length of 361 pm. What is the radius of copper atom in pm?

Options

A. 128
B. 157
C. 181
D. 108

Answer

To find the radius of a copper atom in a face-centered cubic (FCC) lattice, we can use the relationship between the edge length of the unit cell and the radius of the atom.

### Step-by-Step Solution:

1. **Identify the given data:**
- The unit cell length (edge length) \( a = 361 \) pm.

2. **Understand the geometry of the FCC lattice:**
- In a face-centered cubic (FCC) lattice, the atoms are located at each corner of the cube and at the center of each face. The relationship between the radius \( r \) of the atom and the edge length \( a \) is given by:
\[ 4r = \sqrt{2}a \]
- This relationship arises because the face diagonal of the cube contains 4 atomic radii (2 radii from each atom at the face center and 1 radius from each corner atom).

3. **Rearranging the formula to find the radius:**
- To find the radius \( r \), we can rearrange the equation:
\[ r = \frac{\sqrt{2}}{4} a \]

4. **Substituting the value of \( a \):**
- Now, substitute \( a = 361 \) pm into the equation:
\[ r = \frac{\sqrt{2}}{4} \times 361 \]

5. **Calculating the radius:**
- First, calculate \( \sqrt{2} \approx 1.414 \):
\[ r = \frac{1.414}{4} \times 361 \]
- Now calculate \( \frac{1.414}{4} \approx 0.3535 \):
\[ r \approx 0.3535 \times 361 \approx 127.7 \text{ pm} \]

6. **Rounding to the nearest whole number:**
- Rounding \( 127.7 \) pm gives approximately \( 128 \) pm.

### Final Answer:
The radius of the copper atom is approximately **128 pm**.

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