Calculate the atomic radius of elementary silver which crystallises in face centred cubic lattice with unit cell edge length `4.086 xx 10^(-10) m`.

Explanation

To calculate the atomic radius of elementary silver which crystallizes in a face-centered cubic (FCC) lattice with a unit cell edge length of \( a = 4.086 \times 10^{-10} \) m, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic lattice, atoms are located at each corner of the cube and at the center of each face. The arrangement of the atoms allows us to derive a relationship between the atomic radius \( r \) and the edge length \( a \) of the unit cell.

### Step 2: Derive the Relationship
For an FCC lattice, the relationship between the atomic radius \( r \) and the edge length \( a \) can be derived using the geometry of the cube. The diagonal of the face of the cube can be expressed in terms of \( a \) and \( r \):

- The diagonal of the face (let's call it \( AC \)) can be calculated using Pythagoras' theorem:
\[ AC = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

- In terms of atomic radii, this diagonal \( AC \) is equal to four times the atomic radius \( r \) (since there are two radii from the corner atom to the face-centered atom):
\[ AC = 4r \]

### Step 3: Set Up the Equation
From the above relationships, we can equate the two expressions for \( AC \):
\[ 4r = a\sqrt{2} \]

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

### Step 5: Substitute the Value of \( a \)
Now, substitute the given value of \( a \):
\[ a = 4.086 \times 10^{-10} \text{ m} \]
\[ r = \frac{4.086 \times 10^{-10} \times \sqrt{2}}{4} \]

### Step 6: Calculate \( r \)
Calculating \( \sqrt{2} \approx 1.414 \):
\[ r = \frac{4.086 \times 10^{-10} \times 1.414}{4} \]
\[ r = \frac{5.775 \times 10^{-10}}{4} \]
\[ r = 1.44375 \times 10^{-10} \text{ m} \]

### Final Answer
The atomic radius of elementary silver is approximately:
\[ r \approx 1.44 \times 10^{-10} \text{ m} \]