Question

A solid structure in which the$\text{W}$ atoms are located at the corners of a cubic lattice, $\text{O}$ atoms are located at the face edges and $\text{Na}$ atoms are at the centre of the cube. The formula of the compound is:
A.$\text{N}{{\text{a}}_{\text{2}}}\text{W}{{\text{O}}_{\text{3}}}$
B. $\text{N}{{\text{a}}_{\text{2}}}\text{W}{{\text{O}}_{2}}$
C.$\text{NaW}{{\text{O}}_{2}}$
D.$\text{NaW}{{\text{O}}_{\text{3}}}$

Answer 1

To determine the structure of a compound, the number of the unit cells present in the lattice of that compound needs to be known. A cubic lattice can be of three types; simple cubic, body-centred cubic, and face-centred cubic.

Complete answer:
As given here, the Tungsten atoms are located at the corners of the cubic lattice. We know that there are 8 corners of a cube. Hence there are 8 atoms at the eight corners.
Therefore, the unit cells of Tungsten in the lattice =$\left( 8\times \dfrac{1}{8} \right)=1$, this is because each atom at the corner is shared by eight other unit cells.
Each cube has 12 edges, therefore the number of oxygen atoms located in the unit cell =$\left( 12\times \dfrac{1}{4} \right)=3$, because each atom is shared by 4 unit cells.
Each cube has 1 centre, therefore the number of sodium atoms present in the cube = 1.
Therefore the ratio of the atoms of the elements are in the ratio, $\text{Na}$:$\text{W}$:$\text{O}$= 1: 1: 3.

Therefore, the formula of the compound is $\text{NaW}{{\text{O}}_{\text{3}}}$, so the correct answer is option D.

Note:
A Lattice is defined as a set of points that have been arranged in an order that define the structure of a crystal forming lattice. The lattice points in the lattice define the unit cell of a crystal, which are the smallest group of atoms, ions, or molecules which have the overall symmetry of a crystal and from which the overall lattice can be built up by repeating in the three dimensions.