Calculations Involving Unit Cell Dimensions

From the dimensions of a unit cell we can calculate its volume, and if the density of the substance is known we can find the mass of the atoms in the cell. In fact, determining the mass of a single atom this way gives an accurate route to the Avogadro constant.

Suppose X-ray diffraction gives the edge length of a cubic unit cell as $a$, the density of the solid is $d$, and the molar mass is $M$. For a cubic crystal:

• Volume of the unit cell $= a^3$
• Mass of the unit cell $=$ (number of atoms in the cell) $\times$ (mass of one atom) $= z \cdot m$, where $z$ is the number of atoms per unit cell and $m$ the mass of a single atom.
• Mass of one atom: $m = \dfrac{M}{N_A}$, where $N_A$ is the Avogadro constant.

Combining these gives the density of the unit cell (which is the same as the density of the substance):

$\boxed{\,d = \dfrac{z\,M}{a^3\,N_A}\,}$ …