The `gamma`-form of iron has `fcc` structure (edge length `386 pm`) and `beta`-form has `bcc` structure (edge length `290 pm`). The ratio of density in `gamma`-form and `beta`-form is

Options

A. `0.9788`
B. `1.02`
C. `1.57`
D. `0.6344`

Explanation

To find the ratio of densities of the gamma-form and beta-form of iron, we will follow these steps:

### Step 1: Understand the Structures
- The gamma-form of iron has a face-centered cubic (FCC) structure.
- The beta-form of iron has a body-centered cubic (BCC) structure.

### Step 2: Identify the Number of Atoms per Unit Cell (Z)
- For FCC, the number of atoms per unit cell (Z) is 4.
- For BCC, the number of atoms per unit cell (Z) is 2.

### Step 3: Use the Density Formula
The density (ρ) of a crystal can be calculated using the formula:
\[ \rho = \frac{Z \cdot M}{N_A \cdot a^3} \]
where:
- \( \rho \) = density
- \( Z \) = number of atoms per unit cell
- \( M \) = molar mass of the substance (for iron, \( M = 55.85 \, \text{g/mol} \))
- \( N_A \) = Avogadro's number (\( 6.022 \times 10^{23} \, \text{mol}^{-1} \))
- \( a \) = edge length of the unit cell

### Step 4: Calculate the Density for Each Form
- **Gamma-form (FCC) Density**:
\[ \rho_{\gamma} = \frac{4 \cdot 55.85}{6.022 \times 10^{23} \cdot (386 \times 10^{-10})^3} \]

- **Beta-form (BCC) Density**:
\[ \rho_{\beta} = \frac{2 \cdot 55.85}{6.022 \times 10^{23} \cdot (290 \times 10^{-10})^3} \]

### Step 5: Find the Ratio of Densities
To find the ratio of densities \( \frac{\rho_{\gamma}}{\rho_{\beta}} \):
\[ \frac{\rho_{\gamma}}{\rho_{\beta}} = \frac{Z_{\gamma} \cdot M}{N_A \cdot a_{\gamma}^3} \cdot \frac{N_A \cdot a_{\beta}^3}{Z_{\beta} \cdot M} \]
This simplifies to:
\[ \frac{\rho_{\gamma}}{\rho_{\beta}} = \frac{Z_{\gamma}}{Z_{\beta}} \cdot \frac{a_{\beta}^3}{a_{\gamma}^3} \]
Substituting the values:
\[ \frac{\rho_{\gamma}}{\rho_{\beta}} = \frac{4}{2} \cdot \frac{(290)^3}{(386)^3} \]
\[ = 2 \cdot \frac{290^3}{386^3} \]

### Step 6: Calculate the Final Ratio
Calculating \( \frac{290^3}{386^3} \):
\[ \frac{290^3}{386^3} = \left(\frac{290}{386}\right)^3 \]
Calculating \( \frac{290}{386} \):
\[ \frac{290}{386} \approx 0.7516 \]
Then,
\[ \left(0.7516\right)^3 \approx 0.422 \]
Thus,
\[ \frac{\rho_{\gamma}}{\rho_{\beta}} = 2 \cdot 0.422 \approx 0.844 \]

### Conclusion
The ratio of the density in the gamma-form to the beta-form of iron is approximately **0.844**.