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 Structure of A and B
- A atoms form a cubic close packing (CCP), which is equivalent to a face-centered cubic (FCC) arrangement.
- B atoms occupy all the tetrahedral voids in the crystal.

### Step 2: Calculate the Number of A Atoms in FCC
- In a face-centered cubic (FCC) unit cell:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom to the unit cell.

Therefore, the total number of A atoms in one FCC unit cell is:
\[ \text{Total A atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 3: Determine the Number of Tetrahedral Voids
- In an FCC structure, there are 8 tetrahedral voids.
- Each tetrahedral void can be filled by one B atom.

### Step 4: Relate B Atoms to Tetrahedral Voids
- Let’s denote the number of B atoms as \( N \).
- Since each tetrahedral void is filled by one B atom, the number of B atoms \( N \) is equal to the number of tetrahedral voids, which is 8.

### Step 5: Establish the Relationship Between A and B Atoms
- From the previous steps, we know:
- Number of A atoms in the unit cell = 4
- Number of tetrahedral voids = 8 (which corresponds to \( N \))

- To find the number of A atoms in terms of \( N \):
\[ \text{Number of A atoms} = \frac{4}{8} \times N = \frac{N}{2} \]

### Conclusion
Thus, if a sample of the crystal contains \( N \) number of B atoms, the number of A atoms in that sample is:
\[ \text{Number of A atoms} = \frac{N}{2} \]

### Final Answer
The number of A atoms in the sample is \( \frac{N}{2} \).

---