Thermal Expansion of Solids and Liquids
Thermal Expansion of Solids and Liquids
Thermal expansion joints like these in the Auckland Harbour Bridge in New Zealand allow bridges to change length without buckling. (credit: Ingolfson, Wikimedia Commons)
The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes.
What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.
What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory, an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.
Linear Thermal Expansion—Thermal Expansion in One Dimension
The change in length \(\text{Δ}L\) is proportional to length \(L\). The dependence of thermal expansion on temperature, substance, and length is summarized in the equation
\(\text{Δ}L=\mathrm{\alpha L}\text{Δ}T,\)
where \(\text{Δ}L\) is the change in length \(L\), \(\text{Δ}T\) is the change in temperature, and \(\alpha \) is the coefficient of linear expansion, which varies slightly with temperature.
The table below lists representative values of the coefficient of linear expansion, which may have units of \(1/\text{º}\text{C}\) or 1/K. Because the size of a kelvin and a degree Celsius are the same, both \(\alpha \) and \(\text{Δ}T\) can be expressed in units of kelvins or degrees Celsius. The equation \(\text{Δ}L=\mathrm{\alpha L}\text{Δ}T\) is accurate for small changes in temperature and can be used for large changes in temperature if an average value of \(\alpha \) is used.
Thermal Expansion Coefficients at \(\text{20}\text{º}\text{C}\)
| Material | Coefficient of linear expansion \(\alpha (1/\text{º}\text{C})\) | Coefficient of volume expansion \(\beta (1/\text{º}\text{C})\) |
|---|---|---|
| Solids | ||
| Aluminum | \(\text{25}×{\text{10}}^{–6}\) | \(\text{75}×{\text{10}}^{–6}\) |
| Brass | \(\text{19}×{\text{10}}^{–6}\) | \(\text{56}×{\text{10}}^{–6}\) |
| Copper | \(\text{17}×{\text{10}}^{–6}\) | \(\text{51}×{\text{10}}^{–6}\) |
| Gold | \(\text{14}×{\text{10}}^{–6}\) | \(\text{42}×{\text{10}}^{–6}\) |
| Iron or Steel | \(\text{12}×{\text{10}}^{–6}\) | \(\text{35}×{\text{10}}^{–6}\) |
| Invar (Nickel-iron alloy) | \(0\text{.}9×{\text{10}}^{–6}\) | \(2\text{.}7×{\text{10}}^{–6}\) |
| Lead | \(\text{29}×{\text{10}}^{–6}\) | \(\text{87}×{\text{10}}^{–6}\) |
| Silver | \(\text{18}×{\text{10}}^{–6}\) | \(\text{54}×{\text{10}}^{–6}\) |
| Glass (ordinary) | \(9×{\text{10}}^{–6}\) | \(\text{27}×{\text{10}}^{–6}\) |
| Glass (Pyrex®) | \(3×{\text{10}}^{–6}\) | \(9×{\text{10}}^{–6}\) |
| Quartz | \(0\text{.}4×{\text{10}}^{–6}\) | \(1×{\text{10}}^{–6}\) |
| Concrete, Brick | \(\text{~}\text{12}×{\text{10}}^{–6}\) | \(\text{~}\text{36}×{\text{10}}^{–6}\) |
| Marble (average) | \(7×{\text{10}}^{–6}\) | \(2\text{.}1×{\text{10}}^{–5}\) |
| Liquids | ||
| Ether | \(\text{1650}×{\text{10}}^{–6}\) | |
| Ethyl alcohol | \(\text{1100}×{\text{10}}^{–6}\) | |
| Petrol | \(\text{950}×{\text{10}}^{–6}\) | |
| Glycerin | \(\text{500}×{\text{10}}^{–6}\) | |
| Mercury | \(\text{180}×{\text{10}}^{–6}\) | |
| Water | \(\text{210}×{\text{10}}^{–6}\) | |
| Gases | ||
| Air and most other gases at atmospheric pressure | \(\text{3400}×{\text{10}}^{–6}\) | |
Example: Calculating Linear Thermal Expansion: The Golden Gate Bridge
The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge is exposed to temperatures ranging from \(–\text{15}\text{º}\text{C}\) to \(\text{40}\text{º}\text{C}\). What is its change in length between these temperatures? Assume that the bridge is made entirely of steel.
Strategy
Use the equation for linear thermal expansion \(\text{Δ}L=\mathrm{\alpha L}\text{Δ}T\) to calculate the change in length , \(\text{Δ}L\). Use the coefficient of linear expansion, \(\alpha \), for steel from the table above, and note that the change in temperature, \(\text{Δ}T\), is \(\text{55}\text{º}\text{C}\).
Solution
Plug all of the known values into the equation to solve for \(\text{Δ}L\).
\(\text{Δ}L=\mathrm{\alpha L}\text{Δ}T=(\cfrac{\text{12}×{\text{10}}^{-6}}{\text{º}\text{C}})(\text{1275 m})(\text{55}\text{º}\text{C})=0\text{.}\text{84 m.}\)
Discussion
Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small.
This lesson is part of:
Temperature, Kinetic Theory, and Gas Laws