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HINT- We have to keep in mind that, atoms on a corner are shared by eight unit cells and hence contribute only $\dfrac{1}{8}$ atom per unit cell, while atoms on a face are shared by two unit cells, each counts as $\dfrac{1}{2}$ atom per unit cell. Thus, the total number of atoms in each unit cell is $8 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2} = 1 + 3 = 4$
Complete step by step solution:
Now moving onto the question, In a face centered cubic arrangement of A and B atoms in which ‘A’ atoms are at the corners of the unit cell.
There are a total 8 corners and each contributes one eight to the unit cell.
One of the ‘A’ atoms is missing from one corner in the unit cell. So, there are 7 ‘A’ atoms with total contribution $7 \times \dfrac{1}{8} = \dfrac{7}{8}$
‘B’ atoms are at the face centers. There are six face centers. Each contributes one half to the unit cell.
Total contribution= $6 \times \dfrac{1}{2} = 3$
A:B= $\dfrac{7}{8}:3 = 7:24$
The simplest formula of compound is ${A_7}{B_{24}}$
Hence, the correct option is $D.$
NOTE- FCC stands for Face Centered Cubic. It is a type of atomic arrangement and is relatively “tightly” packed. FCC is formally defined as a cubic lattice with the face positions fully equivalent to each of the eight corners. Face Centered Cubic structures are, as the name implies, based around “faces” of a cube with an atom at each of the corners and one at the center of each face. The face centered cubic has a coordination number of 12 and contains 4 atoms per unit cell. Since there are eight corners, there are eight tetrahedral voids in a fcc unit cell.
Complete step by step solution:
Now moving onto the question, In a face centered cubic arrangement of A and B atoms in which ‘A’ atoms are at the corners of the unit cell.
There are a total 8 corners and each contributes one eight to the unit cell.
One of the ‘A’ atoms is missing from one corner in the unit cell. So, there are 7 ‘A’ atoms with total contribution $7 \times \dfrac{1}{8} = \dfrac{7}{8}$
‘B’ atoms are at the face centers. There are six face centers. Each contributes one half to the unit cell.
Total contribution= $6 \times \dfrac{1}{2} = 3$
A:B= $\dfrac{7}{8}:3 = 7:24$
The simplest formula of compound is ${A_7}{B_{24}}$
Hence, the correct option is $D.$
NOTE- FCC stands for Face Centered Cubic. It is a type of atomic arrangement and is relatively “tightly” packed. FCC is formally defined as a cubic lattice with the face positions fully equivalent to each of the eight corners. Face Centered Cubic structures are, as the name implies, based around “faces” of a cube with an atom at each of the corners and one at the center of each face. The face centered cubic has a coordination number of 12 and contains 4 atoms per unit cell. Since there are eight corners, there are eight tetrahedral voids in a fcc unit cell.
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