Problem 119 - Chemistry: An Atoms-Focused Approach

Question

A unit cell consists of a cube that has an ion of element X at each corner, an ion of element \(Y\) at the center of the cube, and an ion of element \(Z\) at the center of each face. What is the formula of the compound?

Step-by-Step Solution

Answer

Answer: The chemical formula of the compound is XYZ₃.

Step 1: Determine the number of X ions in the unit cell

There are 8 corners in a cube. Each corner has an X ion. But since each corner is shared by 8 adjacent unit cells, we need to divide the total number of X ions by 8 to get the number that belongs to a single unit cell. So, there's \((8 * 1/8)\) of an X ion in each unit cell, which equals to 1 X ion.

Step 2: Determine the number of Y ions in the unit cell

There's only 1 center in the cube, and it has a Y ion. Since the center is not shared with any other unit cells, we can say that there's 1 Y ion in each unit cell.

Step 3: Determine the number of Z ions in the unit cell’s faces

There are 6 faces in a cube, and each face has a Z ion at its center. Since each face is shared by 2 adjacent unit cells, we need to divide the total number of Z ions by 2. So, there's \((6 * 1/2)\) of a Z ion in each unit cell, which amounts to 3 Z ions.

Step 4: Find the formula

Now that we have the number of ions for each element in a unit cell, we can determine the formula of the compound. We have 1 X ion, 1 Y ion, and 3 Z ions within the unit cell. Therefore, the formula of the compound is \(XY_1Z_3\), which we can write more simply as \(XYZ_3\).