Silver forms ccp lattice and X-ray studies of its crystals show that the edge length of its unit cell is 408.6 pm. Calculate the density of silver (Atomic mass = 107.9u).

Explanation

To calculate the density of silver, we will follow these steps:

### Step 1: Convert the edge length from picometers to centimeters
The edge length of the unit cell is given as 408.6 pm. We need to convert this to centimeters.

\[ 1 \text{ pm} = 10^{-10} \text{ cm} \]

So,

\[ \text{Edge length (a)} = 408.6 \text{ pm} = 408.6 \times 10^{-10} \text{ cm} \]

### Step 2: Calculate the volume of the unit cell
The volume (V) of the cubic unit cell can be calculated using the formula:

\[ V = a^3 \]

Substituting the value of \(a\):

\[ V = (408.6 \times 10^{-10} \text{ cm})^3 \]

Calculating this gives:

\[ V = 68.77 \times 10^{-24} \text{ cm}^3 \]

### Step 3: Determine the mass of one silver atom
The atomic mass of silver is given as 107.9 u. To find the mass of one silver atom in grams, we use Avogadro's number (\(N_A = 6.022 \times 10^{23} \text{ atoms/mol}\)):

\[ \text{Mass of one silver atom} = \frac{\text{Atomic mass}}{N_A} = \frac{107.9 \text{ g/mol}}{6.022 \times 10^{23} \text{ atoms/mol}} \]

Calculating this gives:

\[ \text{Mass of one silver atom} \approx 17.93 \times 10^{-23} \text{ g} \]

### Step 4: Calculate the mass of the unit cell
In a CCP (cubic close-packed) structure, there are 4 atoms per unit cell. Therefore, the mass of the unit cell (M) can be calculated as:

\[ M = 4 \times \text{Mass of one silver atom} = 4 \times 17.93 \times 10^{-23} \text{ g} \]

Calculating this gives:

\[ M \approx 71.72 \times 10^{-23} \text{ g} \]

### Step 5: Calculate the density of silver
Density (\(d\)) is defined as mass per unit volume. Thus, we can calculate the density of silver using the mass of the unit cell and the volume of the unit cell:

\[ d = \frac{\text{Mass of unit cell}}{\text{Volume of unit cell}} = \frac{71.72 \times 10^{-23} \text{ g}}{68.77 \times 10^{-24} \text{ cm}^3} \]

Calculating this gives:

\[ d \approx 10.51 \text{ g/cm}^3 \]

### Final Answer
The density of silver is approximately **10.51 g/cm³**.

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