Question

A mineral crystallizes in a cubic closest packed array of oxygen ions with aluminum ions in some of the octahedral holes and magnesium ions in some of the tetrahedral holes. Deduce the formula of this mineral, and predict the fraction of octahedral holes and tetrahedral holes that are filled by the various cations.

Short answer

With a cubic closest packed array of oxygen ions, we have 4 ${\mathrm{O}}^{2-}$ions per unit cell. We need to balance the total -8 charge of the anions with a + 8 charge from the overline${\mathrm{Al}}^{3+}$ and ${\mathrm{Mg}}^{2+}$cations. The only combination of ions that gives a + 8 charge is 2 Al³⁺ ions and 1 ${\mathrm{Mg}}^{2+}$. The formula is ${\mathrm{Al}}_2{\mathrm{MgO}}_4$.

There are an equal number of octahedral holes as anions (4) in a cubic closest packed array and twice the number of tetrahedral holes as anions in a cubic closest packed array. For the stoichiometry to work out, we need 2 Al³⁺and 1 ${\mathrm{Mg}}^{2+}$per unit cell. Hence one-half of the octahedral holes are filled with ${\mathrm{Al}}^{3+}$ions, and one-eighth of the tetrahedral holes are filled with${\mathrm{Mg}}^{2+}$ ions.

Step-by-step solution

Step 1:Cubic unit cell

The cubic unit cell is known as the smallest repeating unit, and it applies when all angles are 90^0,and all lengths are equal, with each axis being defined by a coordinate.

The simple cubic unit cell is denoted by eight atoms marked with the actual cube.

Body-centered cubic cell

Body-centered cubic (BCC) is the name given to a type of atom arrangement found in nature. A body-centered cubic unit cell structure consists of atoms arranged in a cube where each corner of the cube shares an atom and one atom positioned at the center.

Face-centered cubic cell

A face-centered cubic unit cell is a well-known structure that consists of atoms arranged in a cube, in which each corner of the cube has a fraction of an atom with six additional full atoms positioned at the center of each cube face.

 Step 4: Explanation

With a cubic closest packed array of oxygen ions, we have 4${\mathrm{O}}^{2-}$ ions per unit cell. We need to balance the total -8 charge of the anions with a + 8 charge from the overline ${\mathrm{Al}}^{3+}$and ${\mathrm{Mg}}^{2+}$cations. The only combination of ions that gives a + 8 charge is 2 Al³⁺ ions and 1 ${\mathrm{Mg}}^{2+}$. The formula is ${\mathrm{Al}}_2{\mathrm{MgO}}_4$

There are an equal number of octahedral holes as anions (4) in a cubic closest packed array and twice the number of tetrahedral holes as anions in a cubic closest packed array. For the stoichiometry to work out, we need 2 Al³⁺and 1${\mathrm{Mg}}^{2+}$ per unit cell. Hence one-half of the octahedral holes are filled with ${\mathrm{Al}}^{3+}$ions, and one-eighth of the tetrahedral holes are filled with ${\mathrm{Mg}}^{2+}$ions.