The nearest neighbour silver atoms in silver crystal are `2.87xx10^(-10)` m apart. What is the density of the silver metal ? Silver crystallises in fcc form. (Atomic mass of Ag = 108)

Explanation

To find the density of silver metal given the nearest neighbor distance and the crystallization structure, we can follow these steps:

### Step 1: Understand the Structure
Silver crystallizes in a face-centered cubic (FCC) structure. In an FCC lattice, there are atoms at each corner of the cube and one atom at the center of each face.

### Step 2: Relate Nearest Neighbor Distance to Edge Length
The nearest neighbor distance (d) in an FCC structure is related to the edge length (a) of the cube by the formula:
\[ d = \frac{a\sqrt{2}}{2} \]
This means that the distance between two nearest neighbor atoms is half the length of the face diagonal.

### Step 3: Calculate Edge Length (a)
Given the nearest neighbor distance \( d = 2.87 \times 10^{-10} \) m, we can rearrange the formula to solve for \( a \):
\[ a = \frac{2d}{\sqrt{2}} \]
Substituting the value of \( d \):
\[ a = \frac{2 \times 2.87 \times 10^{-10}}{\sqrt{2}} \]

### Step 4: Perform the Calculation
Calculating \( a \):
\[ a = \frac{5.74 \times 10^{-10}}{1.414} \approx 4.06 \times 10^{-10} \text{ m} \]

### Step 5: Convert Edge Length to Centimeters
To convert \( a \) from meters to centimeters:
\[ a = 4.06 \times 10^{-10} \text{ m} \times 100 = 4.06 \times 10^{-8} \text{ cm} \]

### Step 6: Calculate Density
The density \( D \) can be calculated using the formula:
\[ D = \frac{Z \cdot M}{a^3 \cdot N_A} \]
Where:
- \( Z \) = number of atoms per unit cell (for FCC, \( Z = 4 \))
- \( M \) = molar mass of silver = 108 g/mol
- \( N_A \) = Avogadro's number \( = 6.022 \times 10^{23} \) atoms/mol
- \( a \) = edge length in cm

### Step 7: Substitute Values
Substituting the values into the density formula:
\[ D = \frac{4 \cdot 108}{(4.06 \times 10^{-8})^3 \cdot 6.022 \times 10^{23}} \]

### Step 8: Calculate \( a^3 \)
Calculating \( a^3 \):
\[ a^3 = (4.06 \times 10^{-8})^3 \approx 6.73 \times 10^{-23} \text{ cm}^3 \]

### Step 9: Final Calculation of Density
Now substituting back into the density formula:
\[ D = \frac{432}{6.73 \times 10^{-23} \cdot 6.022 \times 10^{23}} \]
\[ D \approx \frac{432}{4.05} \approx 10.72 \text{ g/cm}^3 \]

### Final Answer
The density of silver metal is approximately \( 10.72 \text{ g/cm}^3 \).

---