Step-by-Step Solution

1. Analyze the Crystal Structure The problem states that oxygen atoms form a cubic close-packed (ccp) array. A ccp structure is equivalent to a face-centered cubic (FCC) lattice.

2. Determine the Number of Ions/Atoms and Voids per Unit Cell

   • In a ccp/fcc unit cell, the number of atoms (or ions) forming the lattice is 4. Since oxygen atoms form the lattice, the number of oxide ions ($O^{2-}$) per unit cell is $N = 4$.
   • The number of octahedral holes in a ccp lattice is equal to the number of lattice atoms, so there are $N = 4$ octahedral holes.
   • The number of tetrahedral holes in a ccp lattice is twice the number of lattice atoms, so there are $2N = 2 \times 4 = 8$ tetrahedral holes.

3. Calculate the Total Negative Charge per Unit Cell Each oxygen ion has a charge of -2. With 4 oxide ions per unit cell, the total negative charge is: $\text{Total Negative Charge} = 4 \times (-2) = -8$

4. Determine the Number of Cations and Total Positive Charge per Unit Cell

   • Aluminium ions ($Al^{3+}$) occupy a fraction 'm' of the octahedral holes. Number of $Al^{3+}$ ions = $m \times (\text{Number of octahedral holes}) = m \times 4 = 4m$.
   • Magnesium ions ($Mg^{2+}$) occupy a fraction 'n' of the tetrahedral holes. Number of $Mg^{2+}$ ions = $n \times (\text{Number of tetrahedral holes}) = n \times 8 = 8n$.

   The total positive charge is the sum of the charges from all the cations: $\text{Total Positive Charge} = (\text{Number of } Al^{3+} \text{ ions}) \times (+3) + (\text{Number of } Mg^{2+} \text{ ions}) \times (+2)$ $\text{Total Positive Charge} = (4m) \times 3 + (8n) \times 2 = 12m + 16n$

5. Apply the Principle of Electrical Neutrality For the crystal to be electrically neutral, the total positive charge must balance the total negative charge. $\text{Total Positive Charge} + \text{Total Negative Charge} = 0$ $(12m + 16n) + (-8) = 0$ $12m + 16n = 8$ Dividing the entire equation by 4 to simplify: $3m + 4n = 2$

6. Test the Given Options We now test each option to see which pair of (m, n) values satisfies the equation $3m + 4n = 2$.

   • A: m = 1/2, n = 1/8 $3\left(\frac{1}{2}\right) + 4\left(\frac{1}{8}\right) = \frac{3}{2} + \frac{4}{8} = \frac{3}{2} + \frac{1}{2} = \frac{4}{2} = 2$ This option satisfies the equation.

   • B: m = 1, n = 1/4 $3(1) + 4\left(\frac{1}{4}\right) = 3 + 1 = 4$ This does not equal 2.

   • C: m = 1/2, n = 1/2 $3\left(\frac{1}{2}\right) + 4\left(\frac{1}{2}\right) = \frac{3}{2} + 2 = 1.5 + 2 = 3.5$ This does not equal 2.

   • D: m = 1/4, n = 1/8 $3\left(\frac{1}{4}\right) + 4\left(\frac{1}{8}\right) = \frac{3}{4} + \frac{1}{2} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4} = 1.25$ This does not equal 2.

7. Conclusion Only the values in option A satisfy the condition for electrical neutrality. Therefore, m = 1/2 and n = 1/8. The resulting formula for the mineral is $Mg_{8n}Al_{4m}O_4 = Mg_{8(1/8)}Al_{4(1/2)}O_4 = Mg_1Al_2O_4$, which is the formula for the mineral spinel.