Problem 83 - General Chemistry

Question

How many atoms are there in a simple cubic unit cell of an atomic crystal in which all atoms are at lattice points?

Step-by-Step Solution

Answer

There is 1 atom in a simple cubic unit cell.

Step 1: Visualizing the Unit Cell

A simple cubic unit cell is a cube where atoms are located at each of its eight corners. Each corner atom is shared by eight adjacent unit cells.

Step 2: Atom Contribution Calculation

Because each corner atom is shared by 8 unit cells, the contribution of each corner atom to one unit cell is \(\frac{1}{8}\).

Step 3: Sum of Atoms within Unit Cell

Since there are 8 corner atoms in the simple cubic unit cell, we multiply the number of corners by the atomic contribution per corner: \( 8 \times \frac{1}{8} = 1 \).

Step 4: Conclusion

The total number of atoms in a simple cubic unit cell is 1.

Key Concepts

Atomic Structure

In the world of chemistry and physics, atoms are the building blocks of all matter. An atom consists of a nucleus, made up of protons and neutrons, and electrons that orbit this nucleus. Each type of atom represents a different chemical element. In crystalline materials, atoms are arranged in a highly ordered and repeating pattern. This order is what distinguishes crystalline structures from amorphous ones like glass. Understanding the atomic structure is essential because it determines the material's properties such as strength, conductivity, and thermal expansion.

• The nucleus is tiny but contains almost all of the atomic mass.
• Electrons are negatively charged and influence atom interactions.
• Understanding the atomic structure helps us predict how elements will behave in different conditions.

This arrangement and properties also play a significant role in forming the basis of crystal lattices, which we will discuss next.

Crystal Lattice

A crystal lattice is a three-dimensional arrangement of atoms, molecules, or ions in a crystalline solid. This ordered structure extends in all directions and is defined by the lattice points where atoms are located. Each point can be visualized as the vertex of a geometric shape, such as a cube in a simple cubic lattice.
The concept of lattice helps to systematically categorize and describe crystalline structures according to their symmetry and bonding. Crystal lattices are fundamental to understanding the structure of solid materials. They affect how the material behaves chemically and physically.
Examples of properties influenced by the lattice structure include:

• Melting and boiling points
• Electrical conductivity
• Mechanical strength

In simple cubic lattices, the atoms are situated at the corners of the cube, directly providing a clear visual and mathematical framework for calculating the unit cell's atomic contribution.

Unit Cell

The unit cell is the smallest repeating component of a crystal lattice, acting as a building block from which the entire lattice is constructed. In a simple cubic unit cell, atoms are positioned only at the corners of the cube, with each corner atom being shared by eight neighboring cells.
Understanding the unit cell structure is crucial as it helps in determining the characteristics of the whole crystal structure, such as density, arrangement, and coordination number.

• Volume Calculation: The volume of the unit cell can be used to calculate material density.
• Coordination Number: This refers to the number of nearest atoms surrounding an atom in the lattice, which in the case of a simple cubic structure, is six.
• Lattice Parameters: The dimensions of the unit cell are essential for crystal characterization.

By learning about unit cells, one can better understand the macro-properties of materials based on their microscopic arrangement.