# How can I know the volume occupied by one element in a compound?

I struggle to find how to estimate the volume occupied by one atom in a compound. As an example let's take $\beta$-Ga$_2$O$_3$, since that's the material on which I work the most. The crystal structure is monoclinic (space group C2/m) with lattice constants $a=1.2214$nm, $b=0.30371$nm, $c=0.57981$nm and $\beta = 103.83^\circ$ (see attached picture). Additionally gallium oxide has a molar mass $M_{\text{Ga}_2\text{O}_3}=187.444$g$\cdot$mol$^{-1}$ and a density $\rho=5.88$g$\cdot$cm$^{-3}$.

I know how to calculate the volume occupied by each molecule of gallium oxide, but is there a way to know the mean volume occupied by each atom of gallium or of oxygen individually?

Here is my effort to try to answer the question: the unit cell has a volume $V=|\vec{a} \times \vec{b}| \cdot \vec{c}=0.20885$nm$^3$ and contains 12 atoms of oxygen and 8 atoms of gallium. The density of atoms/molecules can be calculated with:

$\rho = \frac{M}{N_\text{A} \cdot V}$

where $M$ is the molar mass of the atoms/molecules of interest and $N_\text{A}$ is the Avogadro constant. This formula seems to work approximately when I calculate the density for Ga$_2$O$_3$ molecules:

$\rho_{\text{Ga}_2\text{O}_3} = 5.9607$g$\cdot$cm$^{-3}$

If I do the same with the Ga and O atoms individually (that is, by inserting $M_\text{Ga}=69.723$g/mol and $M_\text{O}=15.999$g/mol) I get:

$\rho_\text{Ga} = 2.217196$g$\cdot$cm$^{-3}$

$\rho_\text{O} = 0.508769$g$\cdot$cm$^{-3}$

Does that make sense or should I proceed differently?

Thank you very much in advance.

• Electron orbitals in compounds tend to overlap so the concept of the volume of one element is rather fuzzy. – Lewis Miller May 20 '18 at 14:24
• Hi @LewisMiller and thank you for your contribution. The overlapping of electrons is an interesting point. Would you have another model to suggest other than the one that I presented above? Presented in another way, the question to which I am trying to provide a model for is "in average, how much volume do we lose when one atom of gallium/oxygen is removed?" Certainly the interactions between the remaining atoms would evolve as other atoms are removed, but I would already be satisfied to have an approximative model and to discuss its related uncertainties. – Jxx May 20 '18 at 15:50
• From crystallography data get the lattice position of each atom in a unit cell. Use your favorite software with a library to determine the Voronyi volumes. Enjoy. – Jon Custer May 20 '18 at 16:21