Problem 48 - Chemistry: The Central Science in SI Units, Global Edition

Question

An increase in temperature causes most metals to undergo thermal expansion, which means the volume of the metal increases upon heating. How does thermal expansion affect the unit cell length? What is the effect of an increase in temperature on the density of a metal?

Step-by-Step Solution

Answer

As the temperature of a metal increases, it undergoes thermal expansion causing an increase in the unit cell length, "a". This leads to an increase in the unit cell volume, \(V' = a^{3}\). As the volume increases, the density of the metal (ρ) decreases, given by the equation ρ' = \(\frac{m}{V'}\). Thus, an increase in temperature results in a decrease in the density of a metal.

Step 1: Understand thermal expansion of metals

As the temperature of a metal increases, its atoms gain kinetic energy which causes them to vibrate and move apart from each other. This process causes the metal to increase in volume and undergo thermal expansion.

Step 2: Define the unit cell of a metal

The unit cell is the smallest repeating structure of a crystal lattice that represents the entire crystal. The unit cell length, commonly represented as "a," is the distance between two vertices of the unit cell along the edge. In the simplest cubic unit cell, all edges have the same length (a).

Step 3: Relate temperature and unit cell length

As a metal undergoes thermal expansion due to an increase in temperature, the distance between the atoms in the metal increases. This causes the unit cell length, "a," to also increase. Therefore, the relationship between temperature and unit cell length is directly proportional: as the temperature increases, so does the unit cell length.

Step 4: Understand the effect of temperature on density

Density (ρ) of a metal can be defined as the ratio of its mass (m) to its volume (V). Mathematically, this can be expressed as: ρ = \(\frac{m}{V}\)

Step 5: Relate unit cell length to density

While the mass of the metal doesn't change when heated, the volume of the metal increases due to thermal expansion, as we've seen in step 1. This means the unit cell volume (V') also increases. For a simple cubic cell, the change in volume is directly related to the cube of "a" (unit cell length), \(V' = a^{3}\). As the unit cell length, "a," increases with temperature, \(a^{3}\) will also increase, leading to an increase in the volume (V'). Since density is inversely proportional to the volume, an increase in volume will result in a decrease in density: ρ' = \(\frac{m}{V'}\)

Step 6: Conclusion

In summary, as the temperature of a metal increases, it undergoes thermal expansion, which leads to an increase in the unit cell length. The increasing unit cell length causes the volume of the metal to increase as well, and as a result, its density decreases.

Key Concepts

Understanding Unit Cell

The concept of a unit cell is fundamental to understanding the structure of crystalline solids. A unit cell is the smallest structural unit or repeating unit that fully encompasses the symmetry and constitution of the entire crystal lattice. This means that, by repeating the unit cell in three-dimensional space, we can recreate the entire crystal.

For metals, the unit cell is often cubic, with the edges of this cube defined as "a." The length "a" forms the backbone for how we consider lattice dimensions and can directly impact various properties of a metal. When a metal is subject to a rise in temperature, the particles within vibrate more vigorously, causing an expansion. This expansion leads to a longer unit cell edge "a," hence it plays a significant part in the material property changes during thermal dynamics.

Exploring Crystal Lattice

A crystal lattice is the three-dimensional arrangement of atoms, molecules, or ions in a crystal. This ordered structure is vital because it determines many of the physical and mechanical properties of the material.

In a crystal, the repetition or periodicity of the unit cells forms the crystal lattice, setting the stage for structural integrity and properties. Lattice arrangements can vary, leading to classification into different types like face-centered cubic, body-centered cubic, or hexagonal close-packed, depending on the crystal system. When a temperature change occurs, the interatomic distances within the crystal lattice change, affecting the entire structure's stability and dimensions.

• The thermal expansion of a lattice means a broader spacing between atoms, impacting both flexibility and malleability of the metal.
• A change in the lattice structure due to thermal expansion is often reversible, usually returning to its original state upon cooling.

Understanding Metal Density

Metal density is a critical property, often dictating the strength, durability, and application of the metal. It is calculated as the mass of the metal divided by its volume. Mathematically, this is described by the expression: \[\rho = \frac{m}{V}\] where \(\rho\) is the density, \(m\) is the mass, and \(V\) is the volume.

When considering thermal expansion, the mass of the metal remains constant as it is primarily affected by its tightly bound atomic structure. However, the increase in temperature causes the volume to increase (as evident from the increased unit cell size), and thus, the density decreases. It is important because density change affects not just the material's weight, but also its buoyancy, electrical, and thermal conductivity.

• Thermal expansion affects volume, leading to lower density.
• Even slight changes in density can significantly alter a metal's application in design and engineering contexts.