Question

{"given":"Aluminum oxide crystallizes with aluminum ions occupying two-thirds of the octahedral voids in a closest packed array of oxide ions. We need to determine the chemical formula based on this structural information.","key_observation":"In a closest packed structure, the number of octahedral voids equals the number of atoms in the structure. If aluminum ions occupy 2/3 of these octahedral voids, we can establish the stoichiometric ratio between aluminum and oxygen ions by considering their positions and charges.","option_analysis":[{"label":"(A)","text":"$\\mathrm{Al_2O_3}$","verdict":"correct","explanation":"For every 3 oxide ions in closest packing, there are 3 octahedral voids. Aluminum ions occupy 2/3 × 3 = 2 voids, giving Al:O ratio of 2:3, which corresponds to $\\mathrm{Al_2O_3}$."},{"label":"(B)","text":"$\\mathrm{AlO_2}$","verdict":"incorrect","explanation":"This formula suggests 1:2 ratio of Al:O, which would require aluminum to occupy all octahedral voids with additional oxygen, inconsistent with the given 2/3 occupancy."},{"label":"(C)","text":"$\\mathrm{Al_3O_4}$","verdict":"incorrect","explanation":"This 3:4 ratio cannot be achieved with 2/3 occupancy of octahedral voids in a closest packed oxide structure, as it would require a different packing arrangement."},{"label":"(D)","text":"$\\mathrm{Al_3O_2}$","verdict":"incorrect","explanation":"This formula has more aluminum than oxygen, which is impossible when aluminum occupies only 2/3 of the available octahedral voids in an oxide closest packed structure."}],"answer":"(A)","formula_steps":[]}