Element X crystallizes in 12 co - ordination fcc lattice. On applyng high temperature it changes to bcc lattice. Find the ratio of the density of the crystal lattice before and and after applying high temperature

Options

A. `1:1`
B. `3:2`
C. `sqrt2:sqrt3`
D. `2(sqrt2)^(3):(sqrt3)^(3)`

Answer

To find the ratio of the density of the crystal lattice before and after applying high temperature, we need to analyze the face-centered cubic (FCC) and body-centered cubic (BCC) structures of element X.

### Step-by-Step Solution:

1. **Understanding the Structures**:
- In FCC, the coordination number is 12, and the effective number of atoms (Z) in the unit cell is 4.
- In BCC, the effective number of atoms (Z) in the unit cell is 2.

2. **Relations for Edge Length**:
- For BCC, the relation between the edge length (a1) and atomic radius (r) is given by:
\[ a_1 \sqrt{3} = 4r \quad \Rightarrow \quad a_1 = \frac{4r}{\sqrt{3}} \]
- For FCC, the relation between the edge length (a2) and atomic radius (r) is given by:
\[ a_2 \sqrt{2} = 4r \quad \Rightarrow \quad a_2 = \frac{4r}{\sqrt{2}} \]

3. **Density Formula**:
- The density (ρ) of a crystal lattice can be calculated using the formula:
\[ \rho = \frac{Z \cdot M}{a^3 \cdot N_A} \]
where:
- Z = number of effective atoms in the unit cell
- M = molar mass (atomic weight)
- a = edge length
- \(N_A\) = Avogadro's number

4. **Calculating Densities**:
- **Density of FCC (ρ_FCC)**:
\[ \rho_{FCC} = \frac{4M}{\left(\frac{4r}{\sqrt{2}}\right)^3 \cdot N_A} = \frac{4M}{\frac{64r^3}{8} \cdot N_A} = \frac{32M}{64r^3 \cdot N_A} = \frac{M}{2r^3 \cdot N_A} \]
- **Density of BCC (ρ_BCC)**:
\[ \rho_{BCC} = \frac{2M}{\left(\frac{4r}{\sqrt{3}}\right)^3 \cdot N_A} = \frac{2M}{\frac{64r^3}{27} \cdot N_A} = \frac{54M}{64r^3 \cdot N_A} = \frac{27M}{32r^3 \cdot N_A} \]

5. **Finding the Ratio of Densities**:
- Now, we can find the ratio of the densities:
\[ \frac{\rho_{FCC}}{\rho_{BCC}} = \frac{\frac{M}{2r^3 \cdot N_A}}{\frac{27M}{32r^3 \cdot N_A}} = \frac{32}{54} = \frac{16}{27} \]

### Final Answer:
The ratio of the density of the crystal lattice before and after applying high temperature is:
\[ \frac{16}{27} \]