Problem 96 - Chemistry: The Central Science

Question

Cadmium telluride, CdTe, adopts the zinc blende structure with a unit cell edge length of \(6.49 \AA\). There are four cadmium atoms and four tellurium atoms per unit cell. How many of each type of atom are there in a cubic crystal with an edge length of \(5.00 \mathrm{~nm}\) ?

Step-by-Step Solution

Answer

In the given cubic crystal, there are approximately 1832 cadmium (Cd) atoms and 1832 tellurium (Te) atoms.

Step 1: Calculate the volume of the cubic crystal

First, we need to find the volume of the crystal. Since the crystal is cubic-shaped, the volume can be calculated by cubing the edge length provided: \(V = a^3\), where \(V\) is the volume and \(a\) is the edge length. The edge length given is \(5.00\,\text{nm}\), which is equal to \(50.0\, \overset{\circ}A\). Therefore, the volume of the crystal: \(V_\text{crystal} = (50.0\, \overset{\circ}A)^3 = 125000\,\overset{\circ}A^3\).

Step 2: Determine the volume of the unit cell

Cadmium telluride (CdTe) has a zinc blende structure with a unit cell edge length of \(6.49\, \overset{\circ}A\). To find the volume of the unit cell, we cube the edge length: \(V_\text{unit cell} = (6.49\, \overset{\circ}A)^3 \approx 273\, \overset{\circ}A^3\).

Step 3: Calculate the number of unit cells in the crystal

Now, we can find the number of unit cells in the cubic crystal by dividing the volume of the crystal by the volume of one unit cell: \(\text{Number of unit cells} = \frac{V_\text{crystal}}{V_\text{unit cell}} = \frac{125000\,\overset{\circ}A^3}{273\, \overset{\circ}A^3} \approx 457.875\). However, we can't have a fraction of a unit cell, so we round up to the nearest whole number: \(\text{Number of unit cells} \approx 458\).

Step 4: Find the total number of atoms of each type

As there are four cadmium (Cd) atoms and four tellurium (Te) atoms per unit cell, we can now calculate the number of each type of atom in the whole crystal: \(\text{Cadmium atoms} = \text{Number of unit cells} \times 4 \approx 458 \times 4 = 1832\) \(\text{Tellurium atoms} = \text{Number of unit cells} \times 4 \approx 458 \times 4 = 1832\) So, in the given cubic crystal, there are approximately 1832 cadmium (Cd) atoms and 1832 tellurium (Te) atoms.

Key Concepts

Zinc Blende Structure

In the world of crystallography, the zinc blende structure is a significant concept. It's a face-centered cubic lattice, which is commonly seen in many binary compounds. The structure is named after the mineral zinc blende, also known as sphalerite, which has this type of lattice configuration.

The zinc blende structure is characterized by a repeating pattern where each atom is bonded to four others, creating a tetrahedral coordination. This setup can be visualized as two interpenetrating face-centered cubic (FCC) lattices, one for each type of atom, where the atoms can be of differing types, such as a metal and a nonmetal.

• This arrangement leads to notable properties, such as high symmetry and uniform bonding angles around each atom.
• It is the atomic arrangement of several important and industrially relevant materials.

This structure is crucial to understanding the properties and behaviors of many semiconducting and optoelectronic materials. The geometry and bonding create a stable structure conducive to these materials' desirable electrical and optical properties.

Cadmium Telluride (CdTe)

Cadmium telluride (CdTe) is a compound semiconductor that crystallizes in the zinc blende structure. This material is of significant interest, particularly in the field of photovoltaics. CdTe is widely used in solar panels, contributing to efficient energy conversion from sunlight into electrical power.

CdTe's zinc blende structure provides a favorable environment for its applications. Its lattice arrangement facilitates electron flow, which is essential for semiconductors. Here are a few reasons why CdTe is important:

• High absorption coefficient: CdTe efficiently absorbs sunlight, making it ideal for thin-film solar cells. It requires less material to absorb sunlight compared to other materials.
• Bandgap: CdTe has a direct bandgap of approximately 1.5 eV, which is considered near optimal for solar energy conversion.

By leveraging its zinc blende lattice, CdTe maximizes efficiency while minimizing resource use, thus playing a pivotal role in sustainable energy solutions.

Unit Cell Calculation

Understanding how to calculate the unit cell is crucial when analyzing crystal structures. The unit cell is the smallest repeating unit in a crystal, representing the entire lattice symmetry and structure within it. Calculations involving the unit cell are foundational to determining a crystal's overall atomic composition.

In the context of a zinc blende structured crystal like cadmium telluride (CdTe), precise volume assessments of the unit cell and the entire crystal are necessary to determine the number of atoms present in a given sample. Calculating the unit cell involves a few key steps:

• Unit cell volume: You cube the edge length of the unit cell to find its volume. For CdTe with an edge length of 6.49 Å, the calculation would be \(V_{\text{unit\,cell}} = (6.49\, \overset{\circ}{A})^3\).
• Crystal volume: Similarly, the total crystal volume is calculated by cubing its edge length.
• Number of unit cells: Dividing the total crystal volume by the unit cell volume gives the number of unit cells that can fit within the crystal.

This step-by-step calculation allows us to not only verify the structural makeup of the crystal but also provides a way to measure and predict the material properties based on atomic arrangements. This understanding is crucial in fields like materials science and electronics, where precise material characterization is key.