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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
Upon heating, metals undergo thermal expansion which increases their volume and unit cell length. With an increase in temperature, the unit cell length can be determined using the relationship: \(a_T = a_0 (1 + \alpha \Delta T) \), where \(\alpha\) is the coefficient of linear expansion, and \(\Delta T\) is the change in temperature. As the unit cell length and volume increase due to increased temperature, the density of the metal decreases since their mass remains constant: \(\rho_T = \frac{m}{a_T^3}\).

Step by step solution

01

Understand Thermal Expansion

Thermal expansion is a phenomenon where the volume of a material increases due to an increase in temperature. When a metal undergoes thermal expansion, it results in an increase in its unit cell length.
02

Relationship between Volume and Unit Cell Length

The volume of a unit cell in a metal can be calculated using the following formula: \[V = a^3\], where V is the volume and a is the unit cell length. As the temperature increases, the volume of the metal expands, thus leading to an increase in the unit cell length, a.
03

Calculate Unit Cell Length at Different Temperatures

To understand how the unit cell length increases with temperature, we can use the coefficient of linear expansion (\(\alpha\)), which is specific to each metal. The relationship between the unit cell length at a reference temperature (\(a_0\)), and at a higher temperature (\(a_T\)) is given by the following equation: \[a_T = a_0 (1 + \alpha \Delta T)\], where \(\Delta T\) is the change in temperature.
04

Effect of Temperature on Density

The density (\(\rho\)) of a metal is calculated by dividing its mass (m) by its volume (V). We can also express this density in terms of unit cell length, given that density and volume are related through the unit cell length: \[\rho = \frac{m}{V} = \frac{m}{a^3}\] If the unit cell length increases due to an increase in temperature, the volume of the unit cell (and, consequently, the entire metal) will increase as well. Since the mass of the metal remains constant, the increase in volume will cause the density of the metal to decrease: \[\rho_T = \frac{m}{a_T^3}\]
05

Conclusion

In conclusion, an increase in temperature leads to a corresponding increase in the unit cell length of a metal due to thermal expansion. As the unit cell length increases, the volume of the metal increases, resulting in a decrease in its density.

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Most popular questions from this chapter

State whether each sentence is true or false: $$ \begin{array}{l}{\text { (a) Metals have high electrical conductivities because the }} \\ {\text { electrons in the metal are delocalized. }} \\\ {\text { (b) Metals have high electrical conductivities because they }} \\\ {\text { are denser than other solids. }}\end{array} $$ $$ \begin{array}{l}{\text { (c) Metals have large thermal conductivities because they }} \\ {\text { expand when heated. }} \\ {\text { (d) Metals have small thermal conductivities because the }} \\ {\text { delocalized electrons cannot easily transfer the kinetic }} \\ {\quad \text { energy imparted to the metal from heat. }}\end{array} $$

You are given a gray substance that melts at \(700^{\circ} \mathrm{C}\) ; the solid is a conductor of electricity and is insoluble in water. Which type of solid (molecular, metallic, covalent-network, or ionic) might this substance be?

If you want to make a polymer for plastic wrap, should you strive to make a polymer that has a high or low degree of crystallinity?

What molecular structural features cause high-density polyethylene to be denser than low-density polyethylene?

The densities of the elements \(\mathrm{K}, \mathrm{Ca}, \mathrm{Sc},\) and Ti are \(0.86,1.5\) , \(3.2,\) and 4.5 \(\mathrm{g} / \mathrm{cm}^{3}\) , respectively. One of these elements crystallizes in a body-centered cubic structure; the other three crystallize in a face-centered cubic structure. Which one crystallizes in the body-centered cubic structure? Justify your answer.
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