The edge length of face centred cubic unit cell is 508 pm. The nearest distance between two atoms is :

Options

A. 360 pm
B. 288 pm
C. 618 pm
D. 398 pm

Explanation

To find the nearest distance between two atoms in a face-centered cubic (FCC) unit cell with an edge length of 508 pm, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic unit cell, atoms are located at each of the eight corners of the cube and at the center of each of the six faces. This configuration means that atoms touch each other along the face diagonals.

### Step 2: Identify the Relationship Between Edge Length and Atomic Radius
In an FCC unit cell, the relationship between the edge length (A) and the atomic radius (R) can be derived from the geometry of the cube. The face diagonal (which connects two corner atoms through the face-centered atom) can be expressed using the Pythagorean theorem:
\[ \text{Face diagonal} = \sqrt{A^2 + A^2} = \sqrt{2A^2} = A\sqrt{2} \]

### Step 3: Relate the Face Diagonal to Atomic Radii
The face diagonal consists of three atomic radii (two corner atoms and one face-centered atom):
\[ \text{Face diagonal} = 4R \]
Thus, we can set the two expressions for the face diagonal equal to each other:
\[ A\sqrt{2} = 4R \]

### Step 4: Solve for the Atomic Radius (R)
Rearranging the equation gives us:
\[ R = \frac{A\sqrt{2}}{4} \]

### Step 5: Substitute the Given Edge Length
Now we can substitute the given edge length of 508 pm into the equation:
\[ R = \frac{508 \times \sqrt{2}}{4} \]

### Step 6: Calculate the Atomic Radius
Calculating \( R \):
1. Calculate \( \sqrt{2} \approx 1.414 \).
2. Substitute and calculate:
\[ R = \frac{508 \times 1.414}{4} \approx \frac{718.792}{4} \approx 179.698 \text{ pm} \]

### Step 7: Find the Nearest Distance Between Two Atoms
The nearest distance between two atoms in the FCC structure is given by:
\[ \text{Nearest distance} = 2R \]
Substituting the value of \( R \):
\[ \text{Nearest distance} = 2 \times 179.698 \approx 359.396 \text{ pm} \]

### Step 8: Round the Result
Rounding this value gives us approximately:
\[ \text{Nearest distance} \approx 360 \text{ pm} \]

### Final Answer
The nearest distance between two atoms in the face-centered cubic unit cell is approximately **360 pm**.

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