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?

Short answer

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

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.

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.

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.

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: \[\rho_T = \frac{m}{a_T^3}\]

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.