Element x,y and z crystallize in primitive , face centered and body centered unit cell respectively. What would be the correct order of metallic radii of each unit cell is ?

Options

A. `r_x gt r_z gt r_y`
B. `r_y gt r_z gt r_x`
C. `r_x gt r_y gt r_z`
D. `r_y gt r_x gt r_z`

Answer

To determine the correct order of metallic radii for elements X, Y, and Z, which crystallize in primitive, face-centered, and body-centered unit cells respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Unit Cells**:
- Element X crystallizes in a **primitive unit cell**.
- Element Y crystallizes in a **face-centered cubic (FCC)** unit cell.
- Element Z crystallizes in a **body-centered cubic (BCC)** unit cell.

2. **Determine the Relationship between Edge Length (a) and Atomic Radius (r)**:
- For **primitive unit cell**:
- The atoms are located at the corners of the cube.
- The relationship is given by \( a = 2r \).
- Therefore, \( r = \frac{a}{2} \).

- For **face-centered cubic (FCC)**:
- The atoms are located at the corners and the centers of the faces.
- The face diagonal can be expressed as \( a\sqrt{2} \).
- The relationship is \( a\sqrt{2} = 4r \) (since there are 4 radii along the face diagonal).
- Therefore, \( r = \frac{a\sqrt{2}}{4} = \frac{0.35a}{1} \).

- For **body-centered cubic (BCC)**:
- The atoms are located at the corners and one atom in the center of the cube.
- The body diagonal can be expressed as \( a\sqrt{3} \).
- The relationship is \( a\sqrt{3} = 4r \).
- Therefore, \( r = \frac{a\sqrt{3}}{4} \approx 0.43a \).

3. **Calculate the Values of r**:
- For X (primitive): \( r_x = \frac{a}{2} = 0.5a \)
- For Y (FCC): \( r_y = \frac{0.35a}{1} \)
- For Z (BCC): \( r_z = \frac{0.43a}{1} \)

4. **Compare the Radii**:
- From the values calculated:
- \( r_x = 0.5a \)
- \( r_z \approx 0.43a \)
- \( r_y \approx 0.35a \)

5. **Order the Radii**:
- The order of the metallic radii from largest to smallest is:
- \( r_x > r_z > r_y \)
- Therefore, the correct order of metallic radii is:
- **X > Z > Y**

### Final Answer:
The correct order of metallic radii is: **X > Z > Y**.