In a ccp structure of X atoms, Y atoms occupy all the octahedral holes. If 2X atom are removed from corners and replaced by Z, then the formula of the compound will be

Options

A. `X_(15)Y_(16)Z`
B. `X_(7)Y_(8)Z`
C. `X_(7.5)Y_(8)Z`
D. `X_(8)Y_(8)Z_(3)`

Answer

To solve the problem step by step, we will analyze the given information and derive the formula of the compound based on the changes in the structure.

### Step 1: Understand the CCP Structure
In a cubic close-packed (CCP) structure:
- The X atoms are located at the corners of the cube.
- The octahedral voids are present in the structure, and Y atoms occupy these voids.

### Step 2: Determine the Number of Atoms
In a CCP structure:
- There are 8 corner atoms, and each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Thus, the total contribution of X atoms from the corners is:
\[ \text{Total X atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 3: Calculate the Number of Octahedral Voids
In a CCP structure, the number of octahedral voids is equal to the number of atoms present in the unit cell. Since there is 1 X atom, there will be 4 octahedral voids:
- Therefore, Y atoms occupy all 4 octahedral voids:
\[ \text{Total Y atoms} = 4 \]

### Step 4: Analyze the Replacement of X with Z
According to the problem, 2 X atoms are removed from the corners and replaced by Z:
- Since each corner atom contributes \( \frac{1}{8} \), removing 2 corner atoms contributes:
\[ \text{Contribution of removed X atoms} = 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \text{ atom} \]
- This means we are effectively reducing the total number of X atoms by \( \frac{1}{4} \).

### Step 5: Update the Count of X Atoms
Initially, we had 1 atom of X. After removing \( \frac{1}{4} \):
\[ \text{Remaining X atoms} = 1 - \frac{1}{4} = \frac{3}{4} \text{ atom} \]

### Step 6: Count the Z Atoms
Since 2 X atoms are replaced by Z, we have:
- Total Z atoms = 2.

### Step 7: Write the Formula of the Compound
Now we can express the formula of the compound in terms of X, Y, and Z:
- X: \( \frac{3}{4} \)
- Y: \( 4 \)
- Z: \( 2 \)

Thus, the formula can be represented as:
\[ \text{Formula} = X_{\frac{3}{4}} Y_{4} Z_{2} \]

### Step 8: Simplify the Formula
To simplify the formula, we can multiply through by 4 to eliminate the fraction:
\[ \text{Formula} = X_{3} Y_{16} Z_{8} \]

### Final Formula
The final formula of the compound is:
\[ \text{Formula} = X_{3} Y_{16} Z_{8} \]