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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?

Short Answer

Expert verified
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 by step solution

01

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.
02

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).
03

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.
04

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 = \(\frac{m}{V}\)
05

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' = \(\frac{m}{V'}\)
06

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.

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Most popular questions from this chapter

Besides the cubic unit cell, which other unit cell(s) has edge lengths that are all equal to each other? (a) Orthorhombic, \((\mathbf{b})\) hexagonal, \((\mathbf{c})\) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

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