In any ionic crystal A has formed cubical close packing and B atoms are present at every tetrahedral voids. If any sample of crystal contain 'N' number of B atoms then number of A atoms in that sample is

Options

A. N
B. `(N)/(2)`
C. 2 N
D. `sqrt(2)N`

Answer

To solve the problem, we need to determine the relationship between the number of A atoms and the number of B atoms in the given ionic crystal structure. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Crystal Structure
- A is present in a cubic close packing (CCP) arrangement, which corresponds to a face-centered cubic (FCC) lattice.
- B atoms occupy the tetrahedral voids in the CCP structure.

### Step 2: Determine the Number of Atoms in the Unit Cell
- In an FCC unit cell, the number of atoms (Z) is 4. This means there are 4 A atoms per unit cell.
- The number of tetrahedral voids in an FCC unit cell is 8. Therefore, there are 8 tetrahedral voids available for B atoms.

### Step 3: Establish the Relationship Between A and B
- If the sample contains 'N' number of B atoms, we need to find out how many A atoms correspond to this number.
- Since there are 8 tetrahedral voids in the unit cell, if all these voids are filled with B atoms, then the maximum number of B atoms that can be present in one unit cell is 8.

### Step 4: Calculate the Number of A Atoms
- The ratio of B atoms to A atoms in the unit cell can be established as follows:
- For every 8 B atoms, there are 4 A atoms.
- This gives us a ratio of A to B as 4:8 or simplified to 1:2.

### Step 5: Find the Number of A Atoms for N B Atoms
- If we have 'N' B atoms, we can set up a proportion based on the established ratio:
- If 8 B atoms correspond to 4 A atoms, then N B atoms will correspond to \( \frac{N}{2} \) A atoms.

### Final Answer
- Therefore, the number of A atoms in the sample containing 'N' number of B atoms is \( \frac{N}{2} \).

### Summary of Steps:
1. Identify the structure and number of atoms in the unit cell.
2. Establish the relationship between A and B based on the unit cell.
3. Use the ratio to find the number of A atoms corresponding to N B atoms.