2.3 Empirical Formula
The number of equivalent atoms in a unit cell is directly related to the empirical formula for the substance. Consider the unit cell for cesium chloride (CsCl). Here, cesium is in purple and chlorine in green.
First, we recognize that this is a body-centered cubic. Therefore, there is exactly 1 equivalent Cs+ atom (located in the center of the cube) and 1 equivalent Cl– atom (8 chlorides located on the corners of the cube). Therefore, we write the formula
\[\mathrm{Cs_1Cl_1} \longrightarrow \mathrm{CsCl}\]
and reduce the subscripts down to the lowest common denominator. What remains is the empirical formula for cesium chloride.
Practice
Determine the empirical formula for substance represented with the following unit cell. The purple spheres represent sodium and the green spheres represent chlorine.
Solution
Identify atoms and positions
- 13 purple atoms (sodium): 12 on edges and 1 in the center
\[\begin{align*} \mathrm{eq.~Na~atoms} &= \dfrac{1}{4} ~\left ( \mathrm{12~edge~spheres} \right ) + 1 ~\left ( \mathrm{1~sphere~inside} \right ) \\ &= 4 \end{align*}\]
- 14 green atoms (chlorine): 8 on corners and 6 on edges
\[\begin{align*} \mathrm{eq.~Cl~atoms} &= \dfrac{1}{8} ~\left ( \mathrm{8~corner~spheres} \right ) + \dfrac{1}{2} ~\left ( \mathrm{6~faces~spheres} \right ) \\ &= 4 \end{align*}\]
Determine empirical formula
\[\mathrm{Na_4Cl_4} \longrightarrow \mathrm{NaCl}\]
Practice
Determine the empirical formula for substance represented with the following unit cell. The gray spheres represent calcium and the brown spheres represent fluorine.
Solution
Identify atoms and positions
- 14 gray atoms (calcium): 8 on corners and 6 on edges
\[\begin{align*} \mathrm{eq.~Ca~atoms} &= \dfrac{1}{8} ~\left ( \mathrm{8~corner~spheres} \right ) + \dfrac{1}{2} ~\left ( \mathrm{6~faces~spheres} \right ) \\ &= 4 \end{align*}\]
- 8 brown atoms (fluorine): 8 inside the cube (tetrahedral holes)
\[\begin{align*} \mathrm{eq.~F~atoms} &= 8 \end{align*}\]
Determine empirical formula
\[\mathrm{Ca_4F_8} \longrightarrow \mathrm{CaF_2}\]