Problem 39 - Chemistry: An Atoms-Focused Approach

Question

The unit cell of an intermetallic compound consists of a face-centered cube that has an atom of element X at each corner and an atom of element \(Y\) at the center of each face. a. What is the formula of the compound? b. What would be the formula if the positions of the two elements were reversed in the unit cell?

Step-by-Step Solution

Answer

Question: Determine the formulas of the intermetallic compounds based on the positions of elements X and Y in the face-centered cubic unit cell. Short Answer: The formula for the compound with element X at the corners and Y at the center of each face is \(XY_3\). The formula for the compound with element Y at the corners and X at the center of each face is \(YX_3\).

Step 1: Determine the number of X atoms in a unit cell

In a face-centered cubic unit cell, there is one atom at each corner. Since there are 8 corners, there are 8 X atoms each contributing 1/8 (since each corner atom is shared by 8 unit cells). So the total number of X atoms in the unit cell is \((8 * 1/8) = 1\).

Step 2: Determine the number of Y atoms in a unit cell

There is one element \(Y\) atom at the center of each face. Since there are 6 faces, there are 6 Y atoms, each contributing 1/2 (since each face atom is shared by 2 unit cells). So the total number of Y atoms in the unit cell is \((6 * 1/2) = 3\).

Step 3: Find the formula of the compound

Since there is 1 X atom and 3 Y atoms in a unit cell, the formula of the compound is \(XY_3\). b. Formula of the compound with element Y at the corners and X at the center of each face:

Step 4: Determine the number of X and Y atoms in a unit cell with reversed positions

If the positions of the two elements were reversed in the unit cell, now the total number of Y atoms in the unit cell would be 1 (since each corner atom is shared by 8 unit cells), and the total number of X atoms in the unit cell would be 3 (since each face atom is shared by 2 unit cells).

Step 5: Find the formula of the compound with reversed positions

Now, since there is 1 Y atom and 3 X atoms in a unit cell, the formula of the compound with reversed positions is \(YX_3\).

Key Concepts

Face-Centered Cubic Unit Cell

In crystallography, a face-centered cubic (FCC) unit cell is a fundamental building block of crystal structures. It is a cubic unit cell with atoms positioned at each corner and the center of each face of the cube. This arrangement is known for its tightly packed structure, maximizing the use of available space.

For a better understanding:

• The FCC structure involves 8 corner atoms. However, each atom is shared by 8 adjacent unit cells. So, the contribution of each corner atom to one unit cell is just \(\frac{1}{8}\).
• Additionally, there are 6 face-centered atoms. Each of these atoms is shared with an adjacent unit cell, contributing \(\frac{1}{2}\) to each unit cell.

This arrangement results in a highly efficient packing and is common in metallic and intermetallic compounds, providing stability and uniformity. When solving problems related to FCC, understanding how atoms are shared between unit cells and calculating their contributions is essential.

Chemical Formula Determination

The determination of a chemical formula based on a unit cell involves counting the atoms of each element within the unit cell and understanding their contributions. This exercise can be simplified into a few key steps:

1. **Count the Corner Atoms:** These share space with multiple unit cells, meaning their contribution is less. For an FCC unit cell, an atom at each corner contributes \(\frac{1}{8}\) of an atom to the unit cell. Multiplied by the 8 corners, it results in 1 full atom per cell.
2. **Count the Face-Centered Atoms:** On the cube's faces, each atom is shared between 2 cells, contributing \(\frac{1}{2}\) of an atom. With 6 faces, this results in 3 atoms per unit cell.
3. **Combine Elements:** With these numbers, if element X is in the corners and element Y is at the faces, the formula results in \XY_3\. Reversing their positions produces \YX_3\.

Understanding the sharing of atoms and their proportions in the unit cell is vital in chemical formula determination, allowing precise depiction of the composition of intermetallic compounds.

Crystal Structure Analysis

Crystal structure analysis involves understanding how atoms are arranged in a crystalline solid. This impacts its chemical and physical properties. For intermetallic compounds, recognizing the spatial arrangement of atoms helps define their structural integrity and functional behavior.

**Key Aspects of Crystal Structure Analysis:**

• Symmetry and Arrangement: The distribution of atoms in the lattice contributes to the overall symmetry or asymmetry. In face-centered cubic structures, this uniform distribution offers balance and strength.
• Unit Cell Content: By examining exactly how many atoms of each type are present, it's possible to predict the stoichiometry of the compound. This is crucial for identifying the chemical formula accurately, as seen in the calculations of FCC unit cells.
• Material Properties: The arrangement can influence characteristics like hardness, ductility, and conductivity. Thus, a detailed crystal structure analysis is essential for material science applications.

Engaging deeply with these aspects aids in understanding not just the mathematical side of crystal structure problems but also the practical implications in real-world materials.