Question

A compound formed by elements X,Y,Z has a cubic structure in which X atoms are at the corner of the cube and also at alternate face centre Y atoms are present at the body centre and Z atoms are present at the alternate edge centre then what will be the molecular formula of the compound?
A) XYZ
B) $X{Y_2}Z$
C) $XY{Z_3}$
D) ${X_2}YZ$

Answer 1

Unit cell is defined as the smallest repeating unit of a large crystal, with the same geometry. The no. of atoms per unit cell can be identified by the lattice structure and its corresponding unit cell. To answer this question, we meet to know the concept of close packing in a solid crystal. The gaps between the constituent particles are known as voids.

Complete answer:
Consider the structure of the cube. It has eight corners. Six faces and 12 edges.

We are given that element X is present at the corners and also at alternate face centres. In a simple cubic lattice there at eight corners present. The atoms at the corner are shared by eight adjacent cubes, therefore the contribution of each atom will be $\dfrac{1}{8}th$ atom per unit cell. Thus the no. of X atoms at the 8 corners will be $= 8 \times \dfrac{1}{8} = 1atom$
There are two alternate phases. And the atom at the alternate face is shared by two adjacent atoms. Therefore, the no. of X atoms at alternate faces will be $= 2 \times \dfrac{1}{2} = 1$ atom.
Total no. of X atoms will be $= 1 + 1 = 2$ X atoms in the cubic lattice.
We are given that Y atoms are present at the body centre. There is only 1 atom as a whole present in the body centre. Therefore no. of Y atoms in the unit cell $= 1$ atom
Z atoms are present at the alternate edge centre. In a cube there are 4 alternate edges. And one atom at the edge is shared by 4 adjacent atoms. Therefore, no. of Z atoms at alternate edges $= 4 \times \dfrac{1}{4} = 1$ atom.
Now we know that there are 2 atoms of X, 1 atom of Y and 1 atom of Z. The molecular formula will thus become: ${X_2}YZ$ .
The correct answer is option D).

Note:
The no. of atoms at the corners, faces and centre can be identified by the name of the crystal lattice. Simple cubic has only eight atoms at the corners, Face centre has atoms on every face (6 faces) and body centre has one atom at the centre of the lattice. The no. of atoms per unit cell can be given when we know the no. of adjacent atoms that share one atom.