What is the empirical formula for a crystalline substance with atoms of element A at the corners and one at the center of each face, with eight atoms of element B inside?

To determine the empirical formula of the crystalline substance described, we first need to analyze the arrangement of the atoms. The description indicates that there are atoms of element A positioned at the corners of a crystal structure with additional atoms at the center of each face. This arrangement corresponds to a face-centered cubic (FCC) lattice.

In an FCC structure, each corner atom contributes one-eighth of an atom to the unit cell because it is shared among eight adjacent cells. There are 8 corners in a cube, so the total contribution from corner atoms is:

( 8 , \text{corners} \times \frac{1}{8} = 1 , \text{atom of A} )

In addition, each face-centered position contributes half an atom to the unit cell since it is shared between two unit cells. There are 6 faces on a cube, leading to a contribution of:

( 6 , \text{faces} \times \frac{1}{2} = 3 , \text{atoms of A} )

Summing these contributions from the corners and faces gives:

( 1 + 3 = 4 , \text{atoms of A} )

The problem states there