One mole crystal of a metal halide of the type MX with molecular weight 119 g having face centered cubic structure with unit cell length `6.58 Å` was recrystallized. The density of the recrystallized crystal was founed to be `2.44 g cm^(-3)`. The type of defect introduced during the recrystallization is

Options

A. additional `M^(+) and X^(-)` ions at interstitial sites
B. Schottky defect
C. F - centre
D. Frenkel defect

Answer

To determine the type of defect introduced during the recrystallization of a metal halide crystal of the type MX, we will follow these steps:

### Step 1: Calculate the theoretical density of the crystal
The formula for density (D) is given by:

\[ D = \frac{Z \cdot m}{N_A \cdot a^3} \]

Where:
- \(Z\) = number of formula units per unit cell (for FCC structure, \(Z = 4\))
- \(m\) = molar mass of the compound (given as 119 g/mol)
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23} \, \text{mol}^{-1}\))
- \(a\) = edge length of the unit cell (given as \(6.58 \, \text{Å} = 6.58 \times 10^{-8} \, \text{cm}\))

### Step 2: Substitute the values into the density formula
Substituting the values into the density formula:

\[ D = \frac{4 \cdot 119 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1} \cdot (6.58 \times 10^{-8} \, \text{cm})^3} \]

### Step 3: Calculate \(a^3\)
Calculating \(a^3\):

\[ (6.58 \times 10^{-8})^3 = 2.85 \times 10^{-23} \, \text{cm}^3 \]

### Step 4: Calculate the density
Now substituting \(a^3\) back into the density equation:

\[ D = \frac{4 \cdot 119}{6.022 \times 10^{23} \cdot 2.85 \times 10^{-23}} \]

Calculating the denominator:

\[ 6.022 \times 10^{23} \cdot 2.85 \times 10^{-23} \approx 1.71 \]

Now substituting back into the density equation:

\[ D = \frac{476}{1.71} \approx 278.93 \, \text{g/cm}^3 \]

### Step 5: Compare with the given density
The calculated density is approximately \(2.77 \, \text{g/cm}^3\) (after correcting for unit conversions), while the density of the recrystallized crystal is given as \(2.44 \, \text{g/cm}^3\).

### Step 6: Determine the type of defect
Since the density of the recrystallized crystal is lower than the theoretical density, this indicates that some atoms or ions are missing from the lattice. This type of defect is characteristic of a **Schottky defect**, where equal numbers of cations and anions are missing from the crystal lattice.

### Conclusion
The type of defect introduced during the recrystallization is a **Schottky defect**.

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