Three metals X, Y and Z are crystallised in simple cubic, B. C. C and F.C.C lattices respectively. The number of unit cells in one mole each of the metals respectively

Options

A. `N,(N)/(2),(N)/(4)`
B. `N, 2N, 4N`
C. `4N, 2N, N`
D. `(N)/(4),(N)/(2),N`

Answer

To solve the problem of determining the number of unit cells in one mole of each of the metals X, Y, and Z crystallized in simple cubic, BCC, and FCC lattices respectively, we will follow these steps:

### Step 1: Determine the number of atoms in a simple cubic unit cell (Metal X)
- In a simple cubic unit cell, there are 8 corner atoms.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Therefore, the total number of atoms in a simple cubic unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 2: Calculate the number of unit cells in one mole of Metal X
- One mole of atoms contains Avogadro's number of atoms, \( N_A \).
- The number of unit cells in one mole of Metal X is given by:
\[ \text{Number of unit cells} = \frac{N_A}{\text{Number of atoms per unit cell}} = \frac{N_A}{1} = N_A \]

### Step 3: Determine the number of atoms in a BCC unit cell (Metal Y)
- In a BCC unit cell, there are 8 corner atoms and 1 body-centered atom.
- The corner atoms contribute \( \frac{1}{8} \) each, and the body-centered atom contributes 1 atom.
- Therefore, the total number of atoms in a BCC unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} + 1 = 2 \text{ atoms} \]

### Step 4: Calculate the number of unit cells in one mole of Metal Y
- The number of unit cells in one mole of Metal Y is given by:
\[ \text{Number of unit cells} = \frac{N_A}{\text{Number of atoms per unit cell}} = \frac{N_A}{2} \]

### Step 5: Determine the number of atoms in an FCC unit cell (Metal Z)
- In an FCC unit cell, there are 8 corner atoms and 6 face-centered atoms.
- The corner atoms contribute \( \frac{1}{8} \) each, and the face-centered atoms contribute \( \frac{1}{2} \) each.
- Therefore, the total number of atoms in an FCC unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4 \text{ atoms} \]

### Step 6: Calculate the number of unit cells in one mole of Metal Z
- The number of unit cells in one mole of Metal Z is given by:
\[ \text{Number of unit cells} = \frac{N_A}{\text{Number of atoms per unit cell}} = \frac{N_A}{4} \]

### Final Answer
Thus, the number of unit cells in one mole of each metal is:
- Metal X (simple cubic): \( N_A \)
- Metal Y (BCC): \( \frac{N_A}{2} \)
- Metal Z (FCC): \( \frac{N_A}{4} \)

### Summary
The final answer is:
- For Metal X: \( N_A \)
- For Metal Y: \( \frac{N_A}{2} \)
- For Metal Z: \( \frac{N_A}{4} \)