12.77 P - Chemistry: Molecular Nature Of Matter And Change

Question

For structures consisting of identical atoms, how many atoms are contained in the simple, body-centered, and face-centered cubic unit cells? Explain how you obtained the values.

Step-by-Step Solution

Answer

If the structure consists of identical atoms, the simple cubic unit cell (SCC) consists of one atom, the body-centered cubic unit cell (BCC) will have two atoms, and the face-centered cubic unit cell (FCC) consists of four atoms.

Step 1: Different cubic unit cells

Considering the different cubic unit cells, the primary cubic unit cell consists of six atoms present in each corner of the cubic unit cell (Figure 77a).

A body-centered cubic unit cell (BCC) consists of those six atoms in each corner along with one atom present in the center of the body (Figure 77b).

A face-centered cubic unit cell (FCC) consists of the six corner atoms and atoms on the center of the six corresponding faces (Figure 77c).

Figure 77: (a) Simple cubic unit cell, (b) Body-centered cubic unit cell, (c) Face-centered cubic unit cell

Step 2: Calculation of the number of atoms in each cell

Now considering the contribution of atoms in the unit cell, the atoms in the corner of the cube contributes 1/8^TH to one cubic cell, the body-centered atom contributes the whole atom for that cell and the face-centered atom contributes half for the unit cell.

1. For SCC, 8 atoms are present in the corner of the cube, thus contributing one atom.
2. For BCC, 8 atoms are present in the corner of the cube and 1 in the center, thus contributing two atoms to the cell.

For FCC, 8 atoms are present in the corner of the cube and six on the corresponding faces of the cell, thus contributing a total of 4 atoms for the cell.