Animated Solution for Chemistry - States of Matter: Number of atoms in the unit cell of Na (bcc type crystal) and Mg (fcc type crystal) are, respectively

Imagine you are holding a tiny, microscopic building block of a crystal. This fundamental repeating unit is called a unit cell. By understanding the arrangement of atoms within this tiny box, we can unlock the macroscopic properties of the entire material. Today, we are going to calculate the effective number of atoms—denoted by $Z$—inside the unit cells of Sodium (Na) and Magnesium (Mg).

Sodium crystallizes in a Body-Centered Cubic (BCC) lattice. If you look at its unit cell, you will notice atoms positioned at all eight corners of the cube, and one single atom sitting right in the middle, the body center.

However, there is a catch. Those corner atoms do not belong entirely to this one unit cell. In a continuous crystal lattice, each corner is actually shared by eight neighboring cubes. Therefore, only $\frac{1}{8}$ of each corner atom is inside our specific unit cell.

Since we have eight corners, the total contribution from the corners is:

What about that atom in the middle? It is completely enclosed within the walls of our unit cell and is not shared with any other cube. So, its contribution is a full $1$.

Adding them up, the effective number of atoms in a BCC unit cell is:

Now let us shift our focus to Magnesium, which forms a Face-Centered Cubic (FCC) lattice. Here, we still have atoms at the eight corners, but instead of one in the middle, we have atoms sitting right on the center of each of the six faces of the cube.

The rule for the corners remains exactly the same. Eight corners, each contributing $\frac{1}{8}$, gives us $1$ effective atom.

Now, focus on the atoms on the faces. Imagine two rooms sharing a common wall. An atom on that wall is shared equally between the two rooms. So, each face-centered atom contributes exactly $\frac{1}{2}$ to our unit cell. Since a cube has six faces, the total contribution from the faces is:

Let us tally it up for Magnesium. One atom from the corners, plus three atoms from the faces, gives us:

We have our final answer! Sodium in BCC has $2$ atoms, and Magnesium in FCC has $4$ atoms. The correct option is (c) 2, 4. Always be careful with the order of the elements asked in the question to avoid silly mistakes!