Problem 88 - Chemistry: An Atoms-Focused Approach

Question

A mixture of gallium and arsenic is a widely used semiconductor. The arsenide ions are in a cubic close-packed arrangement and half the tetrahedral holes are occupied by the gallium ions. What is the formula of this compound?

Step-by-Step Solution

Answer

Answer: GaAs

Step 1: Calculate the number of arsenide ions in ccp structure

In a cubic close-packed structure, there are a total of 4 particles in the unit cell. Since arsenide ions form this ccp structure, there are 4 arsenide ions in one unit cell.

Step 2: Calculate the number of tetrahedral holes in one unit cell

In a ccp structure, the number of tetrahedral holes is twice the number of particles. So, in this case, there are 2 × 4 = 8 tetrahedral holes.

Step 3: Calculate the number of gallium ions occupying half the tetrahedral holes

As given, half the tetrahedral holes are occupied by gallium ions. So, the number of occupied tetrahedral holes is 8 × 0.5 = 4.

Step 4: Create the formula of the compound

Now that we have the number of arsenide ions (4) and gallium ions (4) in the unit cell, we can create the formula for the compound. Since both ions are present in equal numbers, the formula is GaAs.

Key Concepts

Gallium Arsenide

Gallium arsenide, often shortened to GaAs, is a compound that is widely used in the field of semiconductors. It combines gallium and arsenic to produce a material with unique electrical properties.
One of the main benefits of gallium arsenide is its ability to convert electrical energy into optical signals efficiently, making it an essential component in devices like LEDs and solar cells.
Additionally, GaAs offers several advantages over traditional silicon semiconductors, such as higher electron mobility and better efficiency at higher frequencies, which is particularly useful for high-speed electronics.

• Utilized in devices that require quick operation and high efficiency.
• Supports the conversion of electrical signals into light efficiently.
• Essential in technologies such as satellite communications and radar systems.

These features make GaAs a preferred material for applications that demand high performance and reliability under a variety of conditions.

Cubic Close-Packed Structure

The cubic close-packed (ccp) structure, also known as face-centered cubic (fcc), is a common arrangement of ions in a crystalline solid.
In this structure, each sphere (or atom) is surrounded by 12 other spheres, resulting in a highly efficient packing arrangement. This type of packing creates a unit cell that contains four atoms in total.
The ccp structure is not just theoretical—many real-world materials, including complex semiconductors like gallium arsenide, exhibit this crystalline configuration.

• Provides optimal use of space in the crystal lattice.
• Has a high packing efficiency of approximately 74%.
• Commonly found in metals such as copper, silver, and aluminum.

Understanding the ccp structure is crucial for determining other properties of materials, such as density and coordination number, which in turn influence material properties and their applications.

Tetrahedral Holes

In crystalline structures like the ccp, there exist empty spaces which are known as interstitial sites. Among these, tetrahedral holes are one of the most important types.
These holes are so named because they are surrounded by four spheres positioned in a way that resembles a tetrahedron. In a ccp lattice, each unit cell contains twice as many tetrahedral holes as there are particles.

• Essential for determining how ions like gallium can occupy these spaces.
• Help in the calculation of the stoichiometry of compounds.
• Accommodate smaller ions efficiently in a crystal lattice, influencing the overall stability of the structure.

By occupying these holes, specific ions can maintain the proper balance and composition of a compound, resulting in the unique properties associated with materials like gallium arsenide.