Problem 89 - Chemistry The Science in Context

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

Question: Determine the formula of the compound formed by elements X, Y, and Z inside a unit cell, where element X is at each corner, element Y is at the center, and element Z is at the center of each face. Answer: The formula of the compound is XY\(_3\).

Step 1: Determine the number of each element in the unit cell

In the given unit cell, there are different numbers of ions: - Element X is located at each of the 8 corners of the cube. - Element Y is located at the center of the cube. - Element Z is located at the center of each face, and there are 6 faces.

Step 2: Calculate the contribution of each ion to the unit cell

Each ion contributes a certain portion to the unit cell based on its position within the cell: - Each corner ion contributes \(\frac{1}{8}\) of itself to the unit cell (only one corner of the cube is inside the unit cell since the rest belongs to neighboring cells; so, in total, there is one full ion of element X in the unit cell). - The center ion contributes the whole of itself to the unit cell. - Each face-centered ion contributes \(\frac{1}{2}\) of itself to the unit cell (one half lies inside the unit cell and the other half outside), so in total, the 6 face-centered element Z contributes 3 Z ions to the unit cell.

Step 3: Determine the ratio of ions in the unit cell

Now that we know the contribution of each ion, we can find the ratio of ions: - There is 1 ion of element X. - There is 1 ion of element Y. - There are 3 ions of element Z. The ratio of ions in the unit cell is X:Y:Z = 1:1:3.

Step 4: Write the formula of the compound

Based on the ratio determined in Step 3, the formula of the compound is XY\(_3\).

Key Concepts

Unit Cell

The concept of a unit cell is fundamental in understanding crystal structures. In crystallography, a unit cell is the smallest repeating unit that makes up the crystal lattice, which extends in three-dimensional space. It acts as a building block, much like a single tile in a tiled floor.
Consider this analogy: if you imagine a crystal as being made from repeated stacking of unit cells, then each unit cell must pack together in a way that fills space efficiently and uniformly.
In the exercise, we explored a particular type of unit cell, which includes ions at different positions:

• Element X ions at each corner
• Element Y ion at the center
• Element Z ions at the center of each face

The arrangement of these ions in specific geometric positions within the unit cell gives rise to the overall symmetry and properties of the crystal. Understanding how individual atom positions contribute to the whole unit cell helps us deduce the compound's formula, as we see from how each ion contributes differently depending on its position within the cell.

Ionic Compounds

Ionic compounds are formed from ions - atoms that have gained or lost electrons, resulting in a charge. These compounds usually consist of a metal and a non-metal. Metals typically lose electrons to form positively charged cations, while non-metals gain electrons to form negatively charged anions. This electrostatic attraction between oppositely charged ions holds the compound together.
In our unit cell exercise, the ions X, Y, and Z are arranged in a specific order that reflects the ionic nature of the compound. These ions combine to achieve electrical neutrality, meaning the total positive charge should equal the total negative charge within the compound.
The crystal lattice in which these ions are arranged ensures that the attractive forces are maximized, leading to a stable structure. This lattice is what gives ionic compounds characteristics such as high melting and boiling points, as well as the ability to conduct electricity when dissolved in water or melted, because the ions are free to move.

Chemical Formulas

Chemical formulas provide a shorthand way of representing the types and numbers of atoms in a compound. They are derived from the ratio and arrangement of atoms or ions within a unit cell. The exercise specifically involves determining this ratio to write the compound's formula.
In the step-by-step solution, we calculated the contribution of each ion:

• X: Each corner ion's contribution results in a total of 1 full ion.
• Y: The center ion contributes entirely as it is fully enclosed within the cell.
• Z: With 6 face-centered positions, and each contributing \( \frac{1}{2} \), they sum to 3 Z ions.

This leads to a ratio of 1:1:3 for X:Y:Z. From here, the chemical formula of the compound is derived as XY\(_3\). This formula is significant because it precisely indicates the ratio of ions, reflecting the compound's stoichiometry and allowing for predictions about its properties and behavior in reactions.