Understanding Thermal Expansion of Copper and Glass

When materials are heated, they tend to expand. This phenomenon is called thermal expansion. The amount by which a material expands depends on several factors, including the material itself, the initial size of the material, and the change in temperature.

For solid materials, we often talk about linear expansion, which is the change in length. The change in length (\(\Delta L\)) of a material is directly proportional to its original length (\(L_0\)), the change in temperature (\(\Delta T\)), and a property of the material called the coefficient of linear expansion (\(\alpha\)). This relationship is expressed by the formula:

\(\Delta L = \alpha L_0 \Delta T\)

Coefficient of Linear Expansion (α)

The coefficient of linear expansion (\(\alpha\)) is a specific property of each material that indicates how much it expands per unit length for each degree Celsius or Kelvin rise in temperature. Materials with a higher coefficient of linear expansion expand more than materials with a lower coefficient for the same temperature change.

Comparing Copper and Glass Expansion

To compare how much copper expands relative to glass for the same rise in temperature and the same initial length, we need to compare their coefficients of linear expansion. If we assume the same initial length (\(L_0\)) and the same temperature change (\(\Delta T\)), the change in length (\(\Delta L\)) for each material is determined solely by its coefficient of linear expansion (\(\alpha\)).

Let \(\Delta L_{copper}\) be the expansion of copper and \(\Delta L_{glass}\) be the expansion of glass. We have:

\(\Delta L_{copper} = \alpha_{copper} L_0 \Delta T\)

\(\Delta L_{glass} = \alpha_{glass} L_0 \Delta T\)

To find how many times more copper expands than glass, we can take the ratio of their expansions:

\(\frac{\Delta L_{copper}}{\Delta L_{glass}} = \frac{\alpha_{copper} L_0 \Delta T}{\alpha_{glass} L_0 \Delta T}\)

Since \(L_0\) and \(\Delta T\) are the same for both materials (and not zero), they cancel out:

\(\frac{\Delta L_{copper}}{\Delta L_{glass}} = \frac{\alpha_{copper}}{\alpha_{glass}}\)

This means the ratio of their expansions is equal to the ratio of their coefficients of linear expansion.

Typical values for the coefficient of linear expansion (\(\alpha\)) are:

Using these typical values, let's calculate the ratio of copper's expansion coefficient to that of glass:

\(\frac{\alpha_{copper}}{\alpha_{glass}} = \frac{17 \times 10^{-6} \, \text{/°C}}{3.3 \times 10^{-6} \, \text{/°C}}\)

\(\frac{\alpha_{copper}}{\alpha_{glass}} \approx \frac{17}{3.3}\)

\(\frac{\alpha_{copper}}{\alpha_{glass}} \approx 5.15\)

This calculation shows that copper's coefficient of linear expansion is approximately 5.15 times larger than that of this type of glass. Therefore, for the same rise in temperature and the same initial length, copper will expand about 5.15 times more than this type of glass.

Looking at the options provided, 5.15 is closest to 'five'. This indicates that the question is likely comparing copper to a type of glass like borosilicate glass, which has a relatively low coefficient of thermal expansion.

Revision Table: Thermal Expansion Concepts

Additional Information on Thermal Expansion Types

Besides linear expansion, there are other types of thermal expansion:

• Area Expansion: When a material expands in two dimensions. The change in area (\(\Delta A\)) is given by \(\Delta A = \beta A_0 \Delta T\), where \(A_0\) is the original area and \(\beta\) is the coefficient of area expansion. For isotropic materials (materials that expand equally in all directions), \(\beta \approx 2\alpha\).
• Volume Expansion: When a material expands in three dimensions. The change in volume (\(\Delta V\)) is given by \(\Delta V = \gamma V_0 \Delta T\), where \(V_0\) is the original volume and \(\gamma\) is the coefficient of volume expansion. For isotropic materials, \(\gamma \approx 3\alpha\).

Different materials have different thermal expansion properties. This is why materials expand differently when heated, as shown by the comparison between copper and glass.