Silver crystallises in a face centred cubic unit cell. Each side of the unit cell has a length of 500 pm. Calculate radius of the silver atoms (Assume that the atoms just touch each other on the diagonal across the face of the unit cell i.e. each face atom is touching four atoms)

Explanation

To solve the problem of calculating the radius of silver atoms in a face-centered cubic (FCC) unit cell, we will follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) unit cell, atoms are located at each of the eight corners and at the center of each of the six faces. The atoms at the corners contribute 1/8 of an atom to the unit cell, while the face-centered atoms contribute 1/2 of an atom each.

### Step 2: Identify the Relationship Between Edge Length and Atomic Radius
In an FCC structure, the atoms touch each other along the face diagonal. The face diagonal can be expressed in terms of the edge length (a) of the cube. The relationship is given by:
\[ \text{Face diagonal} = a\sqrt{2} \]

### Step 3: Determine the Arrangement of Atoms Along the Face Diagonal
Along the face diagonal, there are three atomic radii involved:
- One radius from the face-centered atom,
- Two radii from the corner atoms on either end.

Thus, the total length of the face diagonal can also be expressed as:
\[ \text{Face diagonal} = 4R \]
where \( R \) is the radius of the silver atom.

### Step 4: Set Up the Equation
Equating the two expressions for the face diagonal, we have:
\[ a\sqrt{2} = 4R \]

### Step 5: Solve for the Radius \( R \)
Rearranging the equation to solve for \( R \):
\[ R = \frac{a\sqrt{2}}{4} \]

### Step 6: Substitute the Given Edge Length
Given that the edge length \( a = 500 \) pm, we substitute this value into the equation:
\[ R = \frac{500 \sqrt{2}}{4} \]

### Step 7: Simplify the Expression
Calculating \( \sqrt{2} \approx 1.414 \):
\[ R = \frac{500 \times 1.414}{4} \]
\[ R = \frac{707}{4} \]
\[ R = 176.75 \text{ pm} \]

### Step 8: Final Result
Thus, the radius of the silver atom is approximately:
\[ R \approx 176.75 \text{ pm} \]

### Summary
The radius of the silver atom in a face-centered cubic unit cell with an edge length of 500 pm is approximately 176.75 pm.

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