Question

A solid compound contains X, Y and Z atoms in a cube lattice with X atoms occupying the corners, Y atoms in the body centered position and Z atoms at the center of the faces of the unit cell. What is the empirical formula of the compound?
(A) $X{{Y}_{2}}{{Z}_{3}}$
(B) $XY{{Z}_{3}}$
(C) ${{X}_{2}}{{Y}_{2}}{{Z}_{3}}$
(D) XYZ

Answer 1

Hint: If an atom is at the corner of the cube in a unit cell, then it is shared with other eight unit cells. If an atom is at any of the faces of the cube, then it is shared by two unit cells. Body centered position is not shared with another unit cell.

Complete answer:
We are given a cubic lattice in which X, Y and Z atoms are situated at different positions in the cubic lattice. The number of atoms involved in the lattice unit will form the empirical formula of the compound.
- We are given that X atoms occupy the corners of the cube. We will need to identify the total number of X atoms present in the lattice. We know that atoms at corners are shared between a total of eight lattices. So, one atom will have its $\dfrac{1}{8}$ of its part involved in a particular lattice if it is at the corner of the cube. We know that there are a total eight corners of the cube. So, we are given that eight molecules of X are at corners of the cube. So, the total number of X atoms in a single lattice = $8\times \dfrac{1}{8}=1$ .
- Thus there is one atom of X in a single unit cell.
- We are given that Y atoms are at body centered position. As the name suggests, Y atom is at the center of the unit cell and it is not shared with any other unit cell. So, the total number of Y atoms in a single unit cell is 1.
- We are given that Z atoms are at the faces of the cube. Now, the atoms at the face of the cube will be shared by two unit cells. So, only $\dfrac{1}{2}$ of the atom at the face will be covered in a single unit cell. We know that there are 6 faces of a cube. Thus, the total number of Z atoms involved in a unit cell is equal to $6\times \dfrac{1}{2}=3$. Thus, the number of Z atoms in a single unit cell is 3.
We can say that the number of atoms involved in a unit cell forms its empirical formula. Thus, as there are one X atom, one Y atom and three Z atoms present in a unit cell, the empirical formula will be $XY{{Z}_{3}}$.

Therefore, the correct answer is (B).

Note:
Remember that we need not to multiply or divide the number of atoms that are actually present in the unit cell to represent the empirical formula of the compound. The atom situated at the body centered position is not shared by any other unit cell.