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.
To deduce the formula of this mineral, we need to determine the ratio of aluminum ions to oxygen ions and the ratio of magnesium ions to oxygen ions.
In a cubic closest packed (ccp) array, each oxygen ion is surrounded by four aluminum ions in octahedral holes and four magnesium ions in tetrahedral holes. The coordination number of oxygen ions in a ccp structure is 12, with six closest neighbors in the same layer and three close-packed layers above and below.
Let's assume we have "x" aluminum ions and "y" magnesium ions in the crystal structure.
The aluminum ions occupy octahedral holes, and since each aluminum ion is surrounded by six oxygen ions, the number of aluminum ions will be half the number of octahedral holes. So, the ratio of aluminum ions to oxygen ions will be 1:2.
Similarly, the magnesium ions occupy tetrahedral holes, and since each magnesium ion is surrounded by four oxygen ions, the number of magnesium ions will be one-fourth the number of tetrahedral holes. Therefore, the ratio of magnesium ions to oxygen ions will be 1:4.
Based on this information, we can deduce the formula of the mineral:
Ratio of aluminum ions to oxygen ions = 1:2
Ratio of magnesium ions to oxygen ions = 1:4
The formula will be Al2MgO4, indicating that two aluminum ions combine with one magnesium ion and four oxygen ions.
To predict the fraction of octahedral holes and tetrahedral holes filled by the various cations, we can use the concept of ion ratio and assume there are a total of "n" oxygen ions.
For each oxygen ion, there are two aluminum ions occupying the octahedral holes, which gives an octahedral hole filling fraction of (2/12) or (1/6). Similarly, for each oxygen ion, there are four magnesium ions occupying the tetrahedral holes, which gives a tetrahedral hole filling fraction of (4/12) or (1/3).
Therefore, in this mineral, the fraction of octahedral holes filled by aluminum ions is (1/6) and the fraction of tetrahedral holes filled by magnesium ions is (1/3).