$ ext{A metal crystallizes in 2 cubic phases fcc and bcc whose unit lengths are 3.5 } A^\circ ext{ and 3.0 } A^\circ ext{ respectively. The ratio of their densities is :}$

Q: $\text{A metal crystallizes in 2 cubic phases fcc and bcc whose unit lengths are 3.5 } A^\circ \text{ and 3.0 } A^\circ \text{ respectively. The ratio of their densities is :}$

$\text{A metal crystallizes in 2 cubic phases fcc and bcc whose unit lengths are 3.5 } A^\circ \text{ and 3.0 } A^\circ \text{ respectively. The ratio of their densities is :}$

Question

$	ext{A metal crystallizes in 2 cubic phases fcc and bcc whose unit lengths are 3.5 } A^\circ 	ext{ and 3.0 } A^\circ 	ext{ respectively. The ratio of their densities is :}$

Accepted Answer

To determine the ratio of the densities of the metal in its face-centered cubic (fcc) and body-centered cubic (bcc) phases, we need to use the formula for density in terms of the unit cell dimensions and the number of atoms per unit cell.

### Explanation:

1. **Density Formula:**
   $$
   \text{Density} = \frac{\text{Mass of atoms in unit cell}}{\text{Volume of unit cell}}
   $$

2. **Face-Centered Cubic (fcc) Structure:**
   - In an fcc structure, there are 4 atoms per unit cell.
   - The volume of the unit cell is $a^3$, where $a$ is the edge length.
   - Given $a_{\text{fcc}} = 3.5 \, A^\circ$, the volume is $(3.5 \, A^\circ)^3$.

3. **Body-Centered Cubic (bcc) Structure:**
   - In a bcc structure, there are 2 atoms per unit cell.
   - Given $a_{\text{bcc}} = 3.0 \, A^\circ$, the volume is $(3.0 \, A^\circ)^3$.

4. **Density Ratio Calculation:**
   - Let $m$ be the mass of one atom.
   - Density of fcc: $\rho_{\text{fcc}} = \frac{4m}{(3.5)^3}$
   - Density of bcc: $\rho_{\text{bcc}} = \frac{2m}{(3.0)^3}$
   - Ratio of densities: 
     $$
     \frac{\rho_{\text{fcc}}}{\rho_{\text{bcc}}} = \frac{\frac{4m}{(3.5)^3}}{\frac{2m}{(3.0)^3}} = \frac{4 \times (3.0)^3}{2 \times (3.5)^3}
     $$

5. **Simplifying the Ratio:**
   $$
   \frac{\rho_{\text{fcc}}}{\rho_{\text{bcc}}} = \frac{4 \times 27}{2 \times 42.875} = \frac{108}{85.75} \approx 1.25
   $$

### Conclusion:

- **Option 1 (1.25)** is correct because the calculated ratio of densities for the fcc and bcc structures is approximately 1.25.

### Explanation for Incorrect Options:

- **Option 2 (2.04):** This value is too high. The calculation shows that the ratio is closer to 1.25, not 2.04.
- **Option 3 (0.72):** This value is too low. The calculated ratio is greater than 1, indicating that the fcc structure is denser than the bcc structure.
- **Option 4 (3.12):** This value is significantly higher than the calculated ratio, which is approximately 1.25.

By following the given instructions and calculations, Option 1 is justified as the correct answer.