Understanding Thermal Expansion Coefficients

Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. Different types of coefficients are used to quantify this change for different dimensions:

• Coefficient of Linear Expansion (\(\alpha\)): Describes how the length of a material changes per unit length per degree Celsius or Kelvin change in temperature.
• Coefficient of Areal (or Surface) Expansion (\(\beta\)): Describes how the area of a material changes per unit area per degree Celsius or Kelvin change in temperature.
• Coefficient of Volume Expansion (\(\gamma\)): Describes how the volume of a material changes per unit volume per degree Celsius or Kelvin change in temperature.

Relationship Between Coefficients for Isotropic Materials

For an isotropic material (a material that has the same properties in all directions), there are simple relationships between the coefficients of linear, areal, and volume expansion. These relationships are derived assuming the expansions are small:

• The coefficient of areal expansion is approximately twice the coefficient of linear expansion: \(\beta \approx 2\alpha\)
• The coefficient of volume expansion is approximately three times the coefficient of linear expansion: \(\gamma \approx 3\alpha\)

From these relationships, we can also find the relationship between the coefficient of areal expansion and the coefficient of volume expansion:

\(\gamma = 3\alpha\)

And \(\beta = 2\alpha \implies \alpha = \frac{\beta}{2}\)

Substituting the expression for \(\alpha\) into the equation for \(\gamma\):

\(\gamma = 3 \left(\frac{\beta}{2}\right)\)

\(\gamma = \frac{3}{2}\beta\)

Calculating the Coefficient of Volume Expansion

The question provides the value for the coefficient of areal expansion of the material, which is \(\beta = 1.6 \times 10^{-5} \text{ K}^{-1}\). We need to find the coefficient of volume expansion, \(\gamma\).

Using the relationship we derived for isotropic materials:

\(\gamma = \frac{3}{2}\beta\)

Substitute the given value of \(\beta\):

\(\gamma = \frac{3}{2} \times (1.6 \times 10^{-5} \text{ K}^{-1})\)

\(\gamma = 1.5 \times (1.6 \times 10^{-5} \text{ K}^{-1})\)

Now, perform the multiplication:

\(\gamma = (1.5 \times 1.6) \times 10^{-5} \text{ K}^{-1}\)

\(\gamma = 2.4 \times 10^{-5} \text{ K}^{-1}\)

Comparing with Options

Let's compare our calculated value of the coefficient of volume expansion with the given options:

• Option 1: \(0.8 \times 10^{-5} \text{ K}^{-1}\)
• Option 2: \(2.4 \times 10^{-5} \text{ K}^{-1}\)
• Option 3: \(3.2 \times 10^{-5} \text{ K}^{-1}\)
• Option 4: \(4.8 \times 10^{-5} \text{ K}^{-1}\)

Our calculated value, \(2.4 \times 10^{-5} \text{ K}^{-1}\), matches Option 2.

Revision Table: Thermal Expansion Coefficients

This table summarizes the key aspects of the different thermal expansion coefficients.

Additional Information: Thermal Expansion Concepts

• Isotropic vs. Anisotropic Materials: The relationships \(\beta = 2\alpha\) and \(\gamma = 3\alpha\) are strictly valid for isotropic materials, where the material expands equally in all directions. For anisotropic materials (like some crystals), the expansion is different in different directions, and the coefficients of expansion are tensors.
• Temperature Dependence: The coefficients of thermal expansion are generally not constant and can vary with temperature. However, for small temperature changes, they are often treated as constant.
• Units: The units \(\text{K}^{-1}\) (per Kelvin) and \(\text{°C}^{-1}\) (per degree Celsius) are equivalent for thermal expansion coefficients because a change of 1 Kelvin is equal to a change of 1 degree Celsius.
• Microscopic Origin: Thermal expansion is a result of the increased amplitude of atomic vibrations about their equilibrium positions as temperature increases. The average distance between atoms increases, leading to overall expansion.