Question

In a fcc arrangement of A and B atoms in which 'A' atoms are at the corners of the unit cell and 'B' atoms are at the face centers. One of the 'A' atoms is missing from one corner in unit cell. What is the simplest formula of a compound ?

• A. \(\text{A}_7\text{B}_8\)
• B. \(\text{A}_7\text{B}_3\)
• C. \(\text{AB}_3\)
• D. \(\text{A}_7\text{B}_{24}\)

Hint:

Corners give 1/8 each, faces give 1/2 each; subtract the missing corner and write A7B24.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In a unit cell, an atom at a corner is shared by 8 cells (contributes 1/8) and an atom at a face centre is shared by 2 cells (contributes 1/2).

Step 2: Count B atoms:
A cube has 6 faces. B atoms at the face centres contribute \(6 \times \tfrac{1}{2} = 3\).

Step 3: Count A atoms:
A cube has 8 corners, but one A is missing, so 7 corners are occupied. A atoms contribute \(7 \times \tfrac{1}{8} = \tfrac{7}{8}\).

Step 4: Write the ratio:
A : B = \(\tfrac{7}{8} : 3 = 7 : 24\). So the simplest formula is \(\text{A}_7\text{B}_{24}\).

Step 5: Why the other options are wrong.
AB\(_3\) is the formula with all 8 corners filled. \(\text{A}_7\text{B}_8\) and \(\text{A}_7\text{B}_3\) do not match the ratio of 7/8 to 3.

Final Answer:
The simplest formula is \(\text{A}_7\text{B}_{24}\). \[ \boxed{\text{(D) }\text{A}_7\text{B}_{24}} \]