A metal crystallizes into two cubic phases BCC and FCC. The ratio of densities of FCC and BCC is equal to 1.5. Calculate the difference between the unit cell lengths of the FCC and BCC crystals if the edge length of the FCC crystal is equal to 4.0 Å.

Options

A. 0.5 Å
B. 0.37 Å
C. 0.28 Å
D. 0.73 Å

Answer

To solve the problem, we need to calculate the difference between the unit cell lengths of FCC (Face-Centered Cubic) and BCC (Body-Centered Cubic) crystals given that the edge length of the FCC crystal is 4.0 Å and the ratio of densities of FCC to BCC is 1.5.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Edge length of FCC, \( a_{FCC} = 4.0 \, \text{Å} \)
- Ratio of densities, \( \frac{\rho_{FCC}}{\rho_{BCC}} = 1.5 \)

2. **Understand the Density Formula:**
The density (\( \rho \)) of a crystal can be calculated using the formula:
\[ \rho = \frac{Z \cdot M}{a^3 \cdot N_A} \]
where:
- \( Z \) = number of atoms per unit cell
- \( M \) = molar mass of the metal (constant for both structures)
- \( a \) = edge length of the unit cell
- \( N_A \) = Avogadro's number (constant for both structures)

3. **Determine the Values of \( Z \) for FCC and BCC:**
- For FCC, \( Z_{FCC} = 4 \)
- For BCC, \( Z_{BCC} = 2 \)

4. **Set Up the Ratio of Densities:**
Using the density formula, we can write:
\[ \frac{\rho_{FCC}}{\rho_{BCC}} = \frac{Z_{FCC} \cdot M / a_{FCC}^3}{Z_{BCC} \cdot M / a_{BCC}^3} \]
Since \( M \) is constant and cancels out, we have:
\[ \frac{\rho_{FCC}}{\rho_{BCC}} = \frac{Z_{FCC} \cdot a_{BCC}^3}{Z_{BCC} \cdot a_{FCC}^3} \]

5. **Substitute Known Values:**
Substituting the known values into the equation:
\[ 1.5 = \frac{4 \cdot a_{BCC}^3}{2 \cdot (4.0)^3} \]
Simplifying gives:
\[ 1.5 = \frac{4 \cdot a_{BCC}^3}{2 \cdot 64} = \frac{4 \cdot a_{BCC}^3}{128} = \frac{a_{BCC}^3}{32} \]

6. **Solve for \( a_{BCC}^3 \):**
Rearranging the equation:
\[ a_{BCC}^3 = 1.5 \cdot 32 = 48 \]

7. **Calculate \( a_{BCC} \):**
Taking the cube root:
\[ a_{BCC} = \sqrt[3]{48} \approx 3.634 \, \text{Å} \]

8. **Calculate the Difference in Unit Cell Lengths:**
Now, we find the difference between the edge lengths of FCC and BCC:
\[ \Delta a = a_{FCC} - a_{BCC} = 4.0 \, \text{Å} - 3.634 \, \text{Å} \approx 0.366 \, \text{Å} \]

### Final Answer:
The difference between the unit cell lengths of FCC and BCC is approximately \( 0.366 \, \text{Å} \).