An ionic solid composed of Ba, Zr, and O ions has the cubic unit cell shown above. The unit cell has barium atoms at the corners, a zirconium atom in the cube center, and oxygen anions in the cube faces. What is the empirical formula of the compound?

Question

An ionic solid composed of Ba, Zr, and O ions has the cubic unit cell shown above.  The unit cell has barium atoms at the corners, a zirconium atom in the cube center, and oxygen anions in the cube faces.  What is the empirical formula of the compound?

BlackTom AI · Accepted Answer

The question asks for the empirical formula of an ionic solid with Ba at the corners of a cubic unit cell, Zr at the cube center, and O on the faces.
First, determine the contributions: Ba at corners contribute 1 Ba per formula unit (8 corners × 1/8 each = 1). Zr at the center contributes 1 Zr per formula unit (1 atom per cell). O at the faces contribute 3 O per formula unit (6 faces × 1/2 each = 3).
Thus the empirical formula that matches this arrangement is Ba1Zr1O3, i.e., BaZrO3.
Now evaluate each option:
- Option 1: 'Ba Zr O 6' (read as Ba1 Zr1 O6). This has six oxygens per formula unit, which is inconsistent with the 3 O per formula unit derived from the cubic arrangement; the stoichiometry is mismatched, indicating an incorrect empirical composition.
- Option 2: 'Ba Zr O 4' (Ba1Zr1O4). This has too many oxygens relative to the required 3; the ratio O:Ba is 4:1 instead of 3:1, so this option is not correct.
- Option 3: 'Ba 8 Zr O 3' (Ba8Zr1O3). Here the Ba count is eight, which conflicts with the 1 Ba per formula unit derived from corner sharing; the fractional contribution has been misrepresented, making this option invalid.
- Option 4: 'None of the others.' This is consistent with the fact that the correct empirical formula is BaZrO3, which is not represented correctly in any of the first three options.
- Option 5: 'Ba 8 Zr O 6' (Ba8Zr1O6). This also misstates the number of Ba and O compared to the derived 1 Ba and 3 O per formula unit, so it is incorrect.
In summary, each of the first three options misstates the stoichiometric ratios implied by the unit cell contents, while the fourth suggests none of the listed options match the correct BaZrO3 composition. The reasoning hinges on counting the contributions from corners, center, and faces to get Ba:Zr:O = 1:1:3 for the empirical formula, which is not reflected in options 1–3.

An ionic solid composed of Ba, Zr, and O ions has the cubic unit cell shown above. The unit cell has barium atoms at the corners, a zirconium atom in the cube center, and oxygen anions in the cube faces. What is the empirical formula of the compound?

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