Crystalline Solid with fcc Packing

In a crystalline solid, the atoms are arranged in a highly ordered and repeating pattern. In this case, the atoms of element X are arranged in a face-centered cubic (fcc) packing, which is a common arrangement for many metallic crystals.

Tetrahedral Voids

The tetrahedral voids are the spaces between the atoms in a crystal lattice. In a fcc structure, there are four tetrahedral voids associated with each atom. These voids are located at the center of each face of the unit cell, forming a tetrahedron.

Occupation of Tetrahedral Voids

In this crystalline solid, atoms of element Y occupy all the tetrahedral voids. This means that for every atom of element X, there is one atom of element Y occupying a tetrahedral void.

Removal of Atoms along Body Diagonal

To determine the simplest formula of the crystalline lattice solid, we need to consider the arrangement of atoms after removing all the atoms along one body diagonal. The body diagonal of a unit cell connects opposite corners of the cube.

When all the atoms along one body diagonal are removed, the remaining atoms form a new unit cell. This new unit cell is smaller compared to the original unit cell, as it contains fewer atoms.

Simplest Formula of the Crystalline Lattice Solid

The simplest formula of the crystalline lattice solid can be determined by considering the ratio of the number of atoms of element X to the number of atoms of element Y in the new unit cell.

Since the atoms of element X form the fcc packing, there are 4 atoms of element X in the original unit cell. However, after removing all the atoms along one body diagonal, only 1/8th of each of the corner atoms and 1/2 of each of the face-centered atoms remain in the new unit cell.

On the other hand, each atom of element Y occupies a tetrahedral void, and there are 4 tetrahedral voids associated with each atom of element X in the original unit cell.

Therefore, the simplest formula of the crystalline lattice solid is X4Y4, indicating that there are 4 atoms of each element in the new unit cell.

Conclusion

In a crystalline solid with fcc packing, where atoms of element X form the fcc arrangement and atoms of element Y occupy the tetrahedral voids, the simplest formula of the crystalline lattice solid is X4Y4. This is determined by considering the ratio of the number of atoms of element X to the number of atoms of element Y after removing all the atoms along one body diagonal.