What is the density of `Na_(2)O` having antifluorite-type crystgal stryctutre, if the edge length of cube is `100 gm` and what is the effect on density by `0.05%` Frenkel defect?

Options

A. `823.5 g cm^(-3)`, density decreases
B. `4.14.16 g cm^(-3)`, density decreases
C. `823.5 g cm^(-3)`, density remains same
D. `414.16 g cm^(-3)`, density remains same

Explanation

To find the density of Na₂O with an antifluorite-type crystal structure, we will follow these steps:

### Step 1: Understand the Formula for Density
The density (ρ) of a substance can be calculated using the formula:
\[ \rho = \frac{Z \cdot M}{N_A \cdot a^3} \]
where:
- \(Z\) = number of formula units per unit cell
- \(M\) = molar mass of the substance (in g/mol)
- \(N_A\) = Avogadro's number (\(6.022 \times 10^{23} \, \text{mol}^{-1}\))
- \(a\) = edge length of the unit cell (in cm)

### Step 2: Determine the Number of Formula Units (Z)
For Na₂O with an antifluorite structure, \(Z = 4\). This means there are 4 formula units of Na₂O per unit cell.

### Step 3: Calculate the Molar Mass (M)
The molar mass of Na₂O can be calculated as follows:
- Molar mass of Na = 22.99 g/mol (approximately 23 g/mol)
- Molar mass of O = 16 g/mol

Thus, the molar mass of Na₂O:
\[ M = 2 \times 23 + 16 = 46 + 16 = 62 \, \text{g/mol} \]

### Step 4: Convert Edge Length to cm
The edge length of the cube is given as \(100 \, \text{gm}\). However, this seems to be a misprint. The edge length should be in picometers or centimeters. Assuming it is \(100 \, \text{pm}\):
\[ a = 100 \, \text{pm} = 100 \times 10^{-10} \, \text{cm} = 1 \times 10^{-8} \, \text{cm} \]

### Step 5: Calculate the Density (ρ)
Substituting the values into the density formula:
\[ \rho = \frac{Z \cdot M}{N_A \cdot a^3} = \frac{4 \cdot 62}{6.022 \times 10^{23} \cdot (1 \times 10^{-8})^3} \]
Calculating \(a^3\):
\[ a^3 = (1 \times 10^{-8})^3 = 1 \times 10^{-24} \, \text{cm}^3 \]
Now substituting:
\[ \rho = \frac{248}{6.022 \times 10^{23} \cdot 1 \times 10^{-24}} = \frac{248}{6.022 \times 10^{-1}} \approx 411.6 \, \text{g/cm}^3 \]

### Step 6: Effect of 0.05% Frenkel Defect
Frenkel defects do not change the density significantly because they involve the displacement of ions within the crystal lattice without changing the overall mass or volume of the crystal. Therefore, the density remains approximately the same.

### Final Answer
The density of Na₂O with an antifluorite structure is approximately **411.6 g/cm³**, and the density is unaffected by a 0.05% Frenkel defect.