Copper crystallises in a structure of face centerd cubic unit cell. The atomic radius of copper is `1.28 Å`. What is axial length on an edge of copper ?

Options

A. `2.16Å`
B. `3.62Å`
C. `3.94Å`
D. `4.15Å`

Answer

To find the axial length (edge length) of the face-centered cubic (FCC) unit cell of copper, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) unit cell, atoms are located at each of the corners and at the centers of each face of the cube. Each unit cell has:
- 8 corner atoms (each contributes 1/8 to the unit cell)
- 6 face-centered atoms (each contributes 1/2 to the unit cell)

### Step 2: Relate Atomic Radius to Edge Length
For an FCC unit cell, the relationship between the atomic radius (r) and the edge length (a) can be derived from the geometry of the unit cell. The diagonal of the face of the cube can be expressed in terms of the edge length and the atomic radius.

The diagonal of the face of the cube can be calculated using the Pythagorean theorem:
\[ \text{Diagonal} = \sqrt{a^2 + a^2} = a\sqrt{2} \]

### Step 3: Express the Diagonal in Terms of Atomic Radius
In the FCC structure, the diagonal of the face is also equal to the distance across four atomic radii (2 radii for each atom at the corners):
\[ \text{Diagonal} = 4r \]

### Step 4: Set the Two Expressions Equal
Equating the two expressions for the diagonal:
\[ a\sqrt{2} = 4r \]

### Step 5: Solve for Edge Length (a)
Rearranging the equation to solve for the edge length (a):
\[ a = \frac{4r}{\sqrt{2}} \]

### Step 6: Substitute the Given Radius
Given that the atomic radius of copper (r) is 1.28 Å:
\[ a = \frac{4 \times 1.28 \, \text{Å}}{\sqrt{2}} \]

### Step 7: Calculate the Edge Length
Calculating the value:
\[ a = \frac{5.12 \, \text{Å}}{1.414} \]
\[ a \approx 3.62 \, \text{Å} \]

### Final Answer
The axial length (edge length) of the copper FCC unit cell is approximately **3.62 Å**.

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