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Ch.12 - Solids and Modern Materials
All textbooksBrown 15th EditionCh.12 - Solids and Modern MaterialsProblem 48
Chapter 12, Problem 48

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?

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Thermal expansion refers to the increase in volume of a material as its temperature increases. For metals, this means that the atoms vibrate more vigorously and tend to occupy more space, leading to an expansion of the material's dimensions.
In a crystalline solid, the smallest repeating unit is the unit cell. As the metal expands due to increased temperature, the dimensions of the unit cell, including its length, will also increase. This is because the atoms within the unit cell move slightly further apart.
Density is defined as mass per unit volume (\( \rho = \frac{m}{V} \)). As the temperature increases, the volume of the metal increases due to thermal expansion, while the mass remains constant.
Since the volume increases and the mass stays the same, the density of the metal will decrease. This is because the same amount of mass is now spread over a larger volume.
In summary, an increase in temperature causes the unit cell length of a metal to increase due to thermal expansion, and as a result, the density of the metal decreases because the volume increases while the mass remains constant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Thermal Expansion

Thermal expansion refers to the increase in volume of a material as its temperature rises. In metals, this occurs because the increased kinetic energy of atoms causes them to vibrate more vigorously, leading to greater distances between them. This phenomenon is typically linear for solids, meaning that as temperature increases, the dimensions of the material, including length, width, and height, expand proportionally.
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Unit Cell Length

The unit cell is the smallest repeating unit in a crystal lattice that defines the structure of a solid. As temperature increases and thermal expansion occurs, the dimensions of the unit cell also increase, leading to a longer unit cell length. This change can affect the overall properties of the material, including its mechanical and thermal characteristics, as the arrangement of atoms within the unit cell is altered.
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Density

Density is defined as mass per unit volume and is a critical property of materials. When a metal is heated and undergoes thermal expansion, its volume increases while its mass remains constant, leading to a decrease in density. This relationship is important in understanding how temperature changes can affect the buoyancy and structural integrity of metals in various applications.
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Related Practice
Textbook Question

Indicate whether each statement is true or false: (c) Nonmetallic elements are never found in alloys.

Textbook Question

State whether each sentence is true or false: (b) Metals have high electrical conductivities because they are denser than other solids.

Textbook Question

The molecular-orbital diagrams for two- and four-atom linear chains of lithium atoms are shown in Figure 12.21. Construct a molecular-orbital diagram for a chain containing six lithium atoms and use it to answer the following questions: a. How many molecular orbitals are there in the diagram?

Textbook Question

The molecular-orbital diagrams for two- and four-atom linear chains of lithium atoms are shown in Figure 12.21. Construct a molecular-orbital diagram for a chain containing six lithium atoms and use it to answer the following questions: c. How many nodes are in the highest-energy molecular orbital?